This is a revised edition of a classic and highly regarded...

This is a revised edition of a classic and highly regarded book, firstpublished in 1981, giving a comprehensive survey of the intensive researchand testing of general relativity that has been conducted over the last threedecades. As a foundation for this survey, the book first introduces theimportant principles of gravitation theory, developing the mathematicalformalism that is necessary to carry out specific computations so thattheoretical predictions can be compared with experimental findings. Acompletely up-to-date survey of experimental results is included, not onlydiscussing Einstein's "classical" tests, such as the deflection of light andthe perihelion shift of Mercury, but also new solar system tests, neverenvisioned by Einstein, that make use of the high precision space andlaboratory technologies of today. The book goes on to explore new arenasfor testing gravitation theory in black holes, neutron stars, gravitationalwaves and cosmology. Included is a systematic account of the remarkable"binary pulsar" PSR 1913+16, which has yielded precise confirmation ofthe existence of gravitational waves.

The volume is designed to be both a working tool for the researcher ingravitation theory and experiment, as well as an introduction to the subjectfor the scientist interested in the empirical underpinnings of one of thegreatest theories of the twentieth century.

Comments on the previous edition:

"consolidates much of the literature on experimental gravity and should beinvaluable to researchers in gravitation" Science

"a c»ncise and meaty book . . . and a most useful reference work . . .researchers and serious students of gravitation should be pleased withit" Nature

Theory and Experiment in Gravitational Physics

Revised Edition

THEORY ANDEXPERIMENT INGRAVITATIONALPHYSICS

CLIFFORD M.WILLMcDonnell Center for the Space Sciences, Department of PhysicsWashington University, St Louis

Revised Edition

[CAMBRIDGEUNIVERSITY PRESS

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© Cambridge University Press 1981, 1993

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University Press.

First published 1981First paperback edition 1985

Revised edition 1993

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication Data

Will, Clifford M.Theory and experiment in gravitational physics / Clifford M. Will.Rev. ed.

p. cm.Includes bibliographical references and index.ISBN 0 521 43973 61. Gravitation. I. Title.QC178.W47 199353i'.4�dc20 92-29555 CIP

ISBN 978-0-521-43973-2 Paperback

Cambridge University Press has no responsibility for the persistence oraccuracy of URLs for external or third-party internet websites referred to inthis publication, and does not guarantee that any content on such websites is,or will remain, accurate or appropriate. Information regarding prices, traveltimetables, and other factual information given in this work is correct atthe time of first printing but Cambridge University Press does not guaranteethe accuracy of such information thereafter.

To Leslie

Contents

Preface to Revised Edition page xiiiPreface to First Edition xv

1 Introduction 12 The Einstein Equivalence Principle and the

Foundations of Gravitation Theory 132.1 The Dicke Framework 162.2 Basic Criteria for the Viability of a Gravitation Theory 182.3 The Einstein Equivalence Principle 222.4 Experimental Tests of the Einstein Equivalence Principle 242.5 Schiff 's Conjecture 382.6 The THsu Formalism 45

3 Gravitation as a Geometric Phenomenon 673.1 Universal Coupling 673.2 Nongravitational Physics in Curved Spacetime 683.3 Long-Range Gravitational Fields and the Strong

Equivalence Principle 794 The Parametrized Post-Newtonian Formalism 86

4.1 The Post-Newtonian Limit 874.2 The Standard Post-Newtonian Gauge 964.3 Lorentz Transformations and the PPN Metric 994.4 Conservation Laws in the PPN Formalism 105

5 Post-Newtonian Limits of Alternative MetricTheories of Gravity 116

5.1 Method of Calculation 1165.2 General Relativity 1215.3 Scalar-Tensor Theories 1235.4 Vector-Tensor Theories 1265.5 Bimetric Theories with Prior Geometry 1305.6 Stratified Theories 1355.7 Nonviable Theories 138

ix

Contents x

6 Equations of Motion in the PPN Formalism 1426.1 Equations of Motion for Photons 1436.2 Equations of Motion for Massive Bodies 1446.3 The Locally Measured Gravitational Constant 1536.4 N-Body Lagrangians, Energy Conservation, and the Strong

Equivalence Principle 1586.5 Equations of Motion for Spinning Bodies 163

7 The Classical Tests 1667.1 The Deflection of Light 1677.2 The Time-Delay of Light 1737.3 The Perihelion Shift of Mercury 176

8 Tests of the Strong Equivalence Principle 1848.1 The Nordtvedt Effect and the Lunar Eotvos Experiment 1858.2 Preferred-Frame and Preferred-Location Effects:

Geophysical Tests 1908.3 Preferred-Frame and Preferred-Location Effects: Orbital Tests 2008.4 Constancy of the Newtonian Gravitational Constant 2028.5 Experimental Limits on the PPN Parameters 204

9 Other Tests of Post-Newtonian Gravity 2079.1 The Gyroscope Experiment 2089.2 Laboratory Tests of Post-Newtonian Gravity 2139.3 Tests of Post-Newtonian Conservation Laws 21510 Gravitational Radiation as a Tool for Testing Relativistic Gravity 221

10.1 Speed of Gravitational Waves 22310.2 Polarization of Gravitational Waves 22710.3 Multipole Generation of Gravitational Waves and Gravitational

Radiation Damping 23811 Structure and Motion of Compact Objects in

Alternative Theories of Gravity 25511.1 Structure of Neutron Stars 25711.2 Structure and Existence of Black Holes 26411.3 The Motion of Compact Objects: A Modified EIH Formalism 266

12 The Binary Pulsar 28312.1 Arrival-Time Analysis for the Binary Pulsar 28712.2 The Binary Pulsar According to General Relativity 30312.3 The Binary Pulsar in Other Theories of Gravity 306

13 Cosmological Tests 31013.1 Cosmological Models in Alternative Theories of Gravity 31213.2 Cosmological Tests of Alternative Theories 316

Contents xi

1414.114.2

14.314.414.514.614.7

An UpdateThe Einstein Equivalence PrincipleThe PPN Framework and Alternative Metric Theories ofGravityTests of Post-Newtonian GravityExperimental Gravitation: Is there a Future?The Rise and Fall of the Fifth ForceStellar-System Tests of Gravitational TheoryConclusionsReferencesReferences to Chapter 14Index

320320

331332338341343352353371375

Preface to the Revised Edition

Since the publication of the first edition of this book in 1981, experimentalgravitation has continued to be an active and challenging field. However,in some sense, the field has entered what might be termed an Era ofOpportunism. Many of the remaining interesting predictions of generalrelativity are extremely small effects and difficult to check, in some casesrequiring further technological development to bring them into detectablerange. The sense of a systematic assault on the predictions of generalrelativity that characterized the "decades for testing relativity" has beensupplanted to some extent by an opportunistic approach in which noveland unexpected (and sometimes inexpensive) tests of gravity have arisenfrom new theoretical ideas or experimental techniques, often from unlikelysources. Examples include the use of laser-cooled atom and ion traps toperform ultra-precise tests of special relativity, and the startling proposalof a "fifth" force, which led to a host of new tests of gravity at short ranges.Several major ongoing efforts continued nonetheless, including theStanford Gyroscope experiment, analysis of data from the Binary Pulsar,and the program to develop sensitive detectors for gravitational radiationobservatories.

For this edition I have added chapter 14, which presents a brief updateof the past decade of testing relativity. This work was supported in part bythe National Science Foundation (PHY 89-22140).

Clifford M. Will1992

xm

Preface to First Edition

For over half a century, the general theory of relativity has stood as amonument to the genius of Albert Einstein. It has altered forever our viewof the nature of space and time, and has forced us to grapple with thequestion of the birth and fate of the universe. Yet, despite its subsequentlygreat influence on scientific thought, general relativity was supportedinitially by very meager observational evidence. It has only been in the lasttwo decades that a technological revolution has brought about a con-frontation between general relativity and experiment at unprecedentedlevels of accuracy. It is not unusual to attain precise measurements withina fraction of a percent (and better) of the minuscule effects predicted bygeneral relativity for the solar system.

To keep pace with these technological advances, gravitation theoristshave developed a variety of mathematical tools to analyze the new high-precision results, and to develop new suggestions for future experimentsmade possible by further technological advances. The same tools are usedto compare and contrast general relativity with its many competingtheories of gravitation, to classify gravitational theories, and to under-stand the physical and observable consequences of such theories.

The first such mathematical tool to be thoroughly developed was a"theory of metric theories of gravity" known as the Parametrized Post-Newtonian (PPN) formalism, which was suited ideally to analyzing solarsystem tests of gravitational theories. In a series of lectures delivered in1972 at the International School of Physics "Enrico Fermi" (Will, 1974,referred to as TTEG), I gave a detailed exposition of the PPN formalism.However, since 1972, significant progress has been made, on both theexperimental and theoretical sides. The PPN formalism has been refined,and new formalisms have been developed to deal with other aspects of

xv

Preface to First Edition xvi

gravity, such as nonmetric theories of gravity, gravitational radiation,and the motion of condensed objects. A irecent review article (Will, 1979)1

summarizes the principal results of these new developments, but givesnone of the physical or mathematical details. Since 1972, there has beena need for a complete treatment of techniques for analyzing gravitationtheory and experiment.

To fill this need I have designed this study. It analyzes in detail gravita-tional theories, the theoretical formalisms developed to study them, andthe contact between these theories and experiments. I have made noattempt to analyze every theory of gravity or calculate every possibleeffect; instead I have tried to present systematically the methods for per-forming such calculations together with relevant examples. I hope such apresentation will make this book useful as a working tool for researchersboth in general relativity and in experimental gravitation. It is written at alevel suitable for use as either a reference text in a standard graduate-levelcourse on general relativity or, possibly, as a main text in a more special-ized course. Not the least of my motivations for writing such a book is thefact that it was my "centennial project" for 1979 - the 100th anniversaryof Einstein's birth.

It is a pleasure to thank Bob Wagoner, Martin Walker, Mark Haugan,and Francis Everitt for helpful discussions and critical readings of portionsof the manuscript. Ultimate responsibility for errors or omissions rests, ofcourse, with the author. For his constant support and encouragement, Iam grateful to Kip Thorne. Victoria LaBrie performed her usual feats ofspeedy and accurate typing of the manuscript. Thanks also go to RoseAleman for help with the typing.

Preparation of this book took place while the author was in the PhysicsDepartment at Stanford University, and was supported in part by theNational Aeronautics and Space Administration (NSG 7204), the Na-tional Science Foundation (PHY 76-21454, PHY 79-20123), the AlfredP. Sloan Foundation (BR 1700), and by a grant from the Mellon Foun-dation.

1 See also Will (1984).

Introduction

On September 14,1959,12 days after passing through her point of closestapproach to the Earth, the planet Venus was bombarded by pulses ofradio waves sent from Earth. Anxious scientists at Lincoln Laboratoriesin Massachusetts waited to detect the echo of the reflected waves. Totheir initial disappointment, neither the data from this day, nor from anyof the days during that month-long observation, showed any detectableecho near inferior conjunction of Venus. However, a later, improved re-analysis of the data showed a bona fide echo in the data from one day:September 14. Thus occurred the first recorded radar echo from a planet.

On March 9, 1960, the editorial office of Physical Review Letters re-ceived a paper by R. V. Pound and G. A. Rebka, Jr., entitled "ApparentWeight of Photons." The paper reported the first successful laboratorymeasurement of the gravitational red shift of light. The paper was ac-cepted and published in the April 1 issue.

In June, 1960, there appeared in volume 10 of the Annals of Physics apaper on "A Spinor Approach to General Relativity" by Roger Penrose. Itoutlined a streamlined calculus for general relativity based upon "spinors"rather than upon tensors.

Later that summer, Carl H. Brans, a young Princeton graduate studentworking with Robert H. Dicke, began putting the finishing touches onhis Ph.D. thesis, entitled "Mach's Principle and a Varying GravitationalConstant." Part of that thesis was devoted to the development of a "scalar-tensor" alternative to the general theory of relativity. Although its authorsnever referred to it this way, it came to be known as the Brans-Dicketheory.

On September 26,1960, just over a year after the recorded Venus radarecho, astronomers Thomas Matthews and Allan Sandage and co-workersat Mount Palomar used the 200-in. telescope to make a photographic

Theory and Experiment in Gravitational Physics 2

plate of the star field around the location of the radio source 3C48. Al-though they expected to find a cluster of galaxies, what they saw at theprecise location of the radio source was an object that had a decidedlystellar appearance, an unusual spectrum, and a luminosity that varied ona timescale as short as 15 min. The name quasistellar radio source or"quasar" was soon applied to this object and to others like it.

These disparate and seemingly unrelated events of the academic year1959-60, in fields ranging from experimental physics to abstract theoryto astronomy, signaled a new era for general relativity. This era was to beone in which general relativity not only would become an importanttheoretical tool of the astrophysicist, but would have its validity challengedas never before. Yet it was also to be a time in which experimental toolswould become available to test the theory in unheard-of ways and tounheard-of levels of precision.

The optical identification of 3C48 (Matthews and Sandage, 1963) andthe subsequent discovery of the large red shifts in its spectral lines and inthose of 3C273 (Schmidt, 1963; Greenstein and Matthews, 1963),presentedtheorists with the problem of understanding the enormous outpouringsof energy (1047 erg s"1) from a region of space compact enough to per-mit the luminosity to vary systematically over timescales as short as daysor hours. Many theorists turned to general relativity and to the strongrelativistic gravitational fields it predicts, to provide the mechanism under-lying such violent events. This was the first use of the theory's strong-fieldaspect (outside of cosmology), in an attempt to interpret and understandobservations. The subsequent discovery of pulsars and the possible iden-tification of black holes showed that it would not be the last. However,the use of relativistic gravitation in astrophysical model building forcedtheorists and experimentalists to address the question: Is general rela-tivity the correct relativistic theory of gravitation? It would be difficultto place much confidence in models for such phenomena as quasars andpulsars if there were serious doubt about one of the basic underlyingphysical theories. Thus, the growth of "relativistic astrophysics" inten-sified the need to strengthen the empirical evidence for or against generalrelativity.

The publication of Penrose's spinor approach to general relativity(Penrose, 1960) was one of the products of a new school of relativitytheorists that came to the fore in the late 1950s. These relativists appliedthe elegant, abstract techniques of pure mathematics to physical problemsin general relativity, and demonstrated that these techniques could alsoaid in the work of their more astrophysically oriented colleagues. The

Introduction 3

bridging of the gaps between mathematics and physics and mathematicsand astrophysics by such workers as Bondi, Dicke, Sciama, Pirani, Pen-rose, Sachs, Ehlers, Misner, and others changed the way that research(and teaching) in relativity was carried out, and helped make it an activeand exciting field of physics. Yet again the question had to be addressed:Is general relativity the correct basis for this research?

The other three events of 1959-60 contributed to the rebirth of a pro-gram to answer that question, a program of experimental gravitation thathad been semidormant for 40 years.

The Pound-Rebka (1960) experiment, besides verifying the principle ofequivalence and the gravitational red shift, demonstrated the powerfuluse of quantum technology in gravitational experiments of high preci-sion. The next two decades would see further uses of quantum technologyin such high-precision tools as atomic clocks, laser ranging, supercon-ducting gravimeters, and gravitational-wave detectors, to name only a few.

Recording radar echos from Venus (Smith, 1963) opened up the solarsystem as a laboratory for testing relativistic gravity. The rapid develop-ment during the early 1960s of the interplanetary space program maderadar ranging to both planets and artificial satellites a vital new tool forprobing relativistic gravitational effects. Coupled with the theoretical dis-covery in 1964 of the relativistic time-delay effect (Shapiro, 1964), it pro-vided new and accurate tests of general relativity. For the next decadeand a half, until the summer of 1974, the solar system would be the solearena for high-precision tests of general relativity.

Finally, the development of the Brans-Dicke (1961) theory provided aviable alternative to general relativity. Its very existence and agreementwith experimental results demonstrated that general relativity was not aunique theory of gravity. Many even preferred it over general relativity onaesthetic and" theoretical grounds. At the very least, it showed that dis-cussions of experimental tests of relativistic gravitational effects shouldbe carried on using a broader theoretical framework than that providedby general relativity alone. It also heightened the need for high-precisionexperiments because it showed that the mere detection of a small generalrelativistic effect was not enough. What was now required was measure-ments of these effects to accuracy within 10%, 1%, or fractions of a per-cent and better, to distinguish between competing theories of gravitation.

To appreciate more fully the regenerative effect that these events hadon gravitational theory and its experimental tests, it is useful to reviewbriefly the history of experimental gravitation in the 45 years followingthe publication of the general theory of relativity.


In deriving general relativity, Einstein was not particularly motivatedby a desire to account for unexplained experimental or observationalresults. Instead, he was driven by theoretical criteria of elegance and sim-plicity. His primary goal was to produce a gravitation theory that incor-porated the principle of equivalence and special relativity in a naturalway. In the end, however, he had to confront the theory with experiment.This confrontation was based on what came to be known as the "threeclassical tests."

One of these tests was an immediate success - the ability of the theoryto account for the anomalous perihelion shift of Mercury. This had beenan unsolved problem in celestial mechanics for over half a century, sincethe discovery by Leverrier in 1845 that, after the perturbing effects of theplanets on Mercury's orbit had been accounted for, and after the effectof the precession of the equinoxes on the astronomical coordinate systemhad been subtracted, there remained in the data an unexplained advancein the perihelion of Mercury. The modern value for this discrepancy is43 arc seconds per century (Table 1.1). A number of ad hoc proposalswere made in an attempt to account for this excess, including, amongothers, the existence of a new planet, Vulcan, near the Sun; a ring ofplanetoids; a solar quadrupole moment; and a deviation from the inverse-square law of gravitation (for a review, see Chazy, 1928). Although theseproposals could account for the perihelion advance of Mercury, they eitherinvolved objects that were detectable by direct optical observation, orpredicted perturbations on the other planets (for example, regressions ofnodes, changes in orbital inclinations) that were inconsistent with obser-vations. Thus, they were doomed to failure. General relativity accounted

Table 1.1. Perihelion advance of Mercury

Cause of advance

General precession (epoch 1900)VenusEarthMarsJupiterSaturnOthers

SumObserved AdvanceDiscrepancy

Rate (arc s/century)

5025'.'6211".%9070

27515376

773072

555770559977

4277

Introduction 5

for the anomalous shift in a natural way without disturbing the agreementwith other planetary observations. This result would go unchallengeduntil 1967.

The next classical test, the deflection of light by the Sun, was not onlya success, it was a sensation. Shortly after the end of World War I, twoexpeditions set out from England: one for Sobral, in Brazil; and one forthe island of Principe off the coast of Africa. Their goal was to measurethe deflection of light as predicted by general relativity -1.75 arc secondsfor a ray that grazes the Sun. The observations had to be made in thepath of totality of a solar eclipse, during which the Moon would blockthe light from the Sun and reveal the field of stars behind it. Photographicplates taken of the star field during the eclipse were compared with platesof the same field taken when the Sun was not present, and the angulardisplacement of each star was determined. The results were 1.13 + 0.07times the Einstein prediction for the Sobral expedition, and 0.92 ±0.17for the Principe expedition (Dyson et al., 1920). The announcement ofthese results confirming the theory caught the attention of a war-wearypublic and helped make Einstein a celebrity. But Einstein was so con-vinced of the "correctness" of the theory because of its elegance and inter-nal consistency that he is said to have remarked that he would have feltsorry for the Almighty if the results had disagreed with the theory (seeBernstein, 1973). Nevertheless, the experiments were plagued by possiblesystematic errors, and subsequent independent analyses of the Sobralplates yielded values ranging from 1.0 to 1.3 times the general relativityvalue. Later eclipse expeditions made very little improvement (Table 1.2).The main sources of error in such optical deflection experiments are un-known scale changes between eclipse and comparison photographicplates, and the precarious conditions, primarily associated with badweather and exotic locales, under which such expeditions are carried out.By 1960, the best that could be said about the deflection of light wasthat it was definitely more than 0'.'83, or half the Einstein value. Thiswas the amount predicted from a simple Newtonian argument, by Soldnerin 1801 (Lenard, 1921),1 or from an extension of the principle of equiv-alence, by Einstein (1911). Beyond that, "the subject [was] still a liveone" (Bertotti et al., 1962).

The third classical test was actually the first proposed by Einstein (1907):the gravitational red shift of light. But by contrast with the other two

1 In 1921, the physicist Philipp Lenard, an avowed Nazi, reprinted Soldner'spaper in the Annalen der Physik in an effort to discredit Einstein's "Jewish" scienceby showing the precedence of Soldner's "Aryan" work.


tests, there was no reliable confirmation of it until the 1960 Pound-Rebkaexperiment. One possible test was a measurement of the red shift of spec-tral lines from the Sun. However, 30 years of such measurements revealedthat the observed shifts in solar spectral lines are affected strongly byDoppler shifts due to radial mass motions in the solar photosphere. Forexample, the frequency shift was observed to vary between the center ofthe Sun and the limb, and to depend on the line strength. For the gravita-tional red shift the results were inconclusive, and it would be 1962 beforea reliable solar red-shift measurement would be made. Similarly incon-clusive were attempts to measure the gravitational red shift of spectrallines from white dwarfs, primarily from Sirius B and 40 Eridani B, bothmembers of binary systems. Because of uncertainties in the determinationof the masses and radii of these stars, and because of possible complica-tions in their spectra due to scattered light from their companions, reli-able, precise measurements were not possible [see Bertotti et al. (1962)for a review].

Furthermore, by the late 1950s, it was being suggested that the gravita-tional red shift was not a true test of general relativity after all. Accordingto Leonard I. Schiff and Robert H. Dicke, the gravitational red shift wasa consequence purely of the principle of equivalence, and did not test thefield equations of gravitational theory. Schiff took the argument one step

Table 1.2. Optical measurements of light deflection by the Suri*

Eclipse

191919191922192219221922192919361936194719521973"

Approximatenumberof stars

75

92145141817258

511039

Minimum distance fromcenter of Sunin solar radii

222.12.1221.5243.32.12

Result inunits ofEinsteinprediction

1.13 + 0.070.92 + 0.170.98 ± 0.061.04 + 0.090.7 to 1.30.8 to 1.21.28 ±0.061.55 + 0.150.7 to 1.21.15 ±0.150.97 + 0.060.95 + 0.11

Results fromdifferentanalyses

1.0 to 1.3

1.3 to 0.91.2

0.9 to 1.21.55 ± 0.2

1.0 to 1.40.82 ± 0.09

a See Bertotti et al. (1962) for details.b Texas Mauritanian Eclipse Team (1976), Jones (1976).

Introduction 7

further and suggested that the gravitational red-shift experiment wassuperseded in importance by the more accurate Eotvos experiment, whichverified that bodies of different composition fall with the same accelera-tion (Schiff, 1960a; Dicke, 1960).

Other potential tests of general relativity were proposed, such as theLense-Thirring effect, an orbital perturbation due to the rotation of abody, and the de Sitter effect, a secular motion of the perigee and nodeof the lunar orbit (Lense and Thirring, 1918; de Sitter, 1916), but theprospects for ever detecting them were dim.

Cosmology was the other area where general relativity could be con-fronted with observation. Initially the theory met with success in itsability to account for the observed expansion of the universe, yet by the1940s there was considerable doubt about its applicability. According topure general relativity, the expansion of the universe originated in a denseprimordial explosion called the "big bang." The age of the universe sincethe big bang could be determined by extrapolating the expansion of theuniverse backward in time using the field equations of general relativity.However, the observed values of the present expansion rate were so highthat the inferred age of the universe was shorter than that of the Earth.One result of this doubt was the rise in popularity during the 1950s of thesteady-state cosmology of Herman Bondi, Thomas Gold, and Fred Hoyle.This model avoided the big bang altogether, and allowed for the ex-pansion of the universe by the continuous creation of matter. By thismeans, the universe would present the same appearance to all observersfor all time.

But by the late 1950s, revisions in the cosmic distance scale had reducedthe expansion rate by a factor of five, and had thereby increased the ageof the universe in the big bang model to a more acceptable level. Never-theless, cosmological observations were still in no position to distinguishamong different theories of gravitation or of cosmology [for a detailedtechnical and historical review, see Weinberg (1972), Chapter 14].

Meanwhile, a small "cottage industry" had sprung up, devoted to theconstruction of alternative theories of gravitation. Some of these theorieswere produced by such luminaries as Poincare, Whitehead, Milne, Birk-hoff, and Belinfante. Many of these authors expressed an uneasiness withthe notions of general covariance and curved spacetime, which were builtinto general relativity, and responded by producing "special relativistic"theories of gravitation. These theories considered spacetime to be "specialrelativistic" at least at a background level, and treated gravitation as aLorentz-invariant field on that background. As of 1960, it was possible


to enumerate at least 25 such alternative theories, as found in the primaryresearch literature between 1905 and 1960 [for a partial list, see Whitrowand Morduch (1965)].

Thus, by 1960, it could be argued that the validity of general relativityrested on the following empirical foundation: one test of moderate preci-sion (the perihelion shift, approximately 1%), one test of low precision(the deflection of light, approximately 50%), one inconclusive test thatwas not a real test anyway (the gravitational red shift), and cosmologicalobservations that could not distinguish between general relativity andthe steady-state theory. Furthermore, a variety of alternative theorieslaid claim to viability.

In addition, the attitude toward the theory seemed to be that, whereasit was undoubtedly of importance as a fundamental theory of nature, itsobservational contacts were limited to the classical tests and cosmology.This view was present for example in the standard textbooks on generalrelativity of this period, such as those by Mcller (1952), Synge (1960), andLandau and Lifshitz (1962). As a consequence, general relativity was cutoff from the mainstream of physics. It was during this period that oneyoung, beginning graduate student was advised not to enter this field,because general relativity "had so little connection with the rest of physicsand astronomy" (his name: Kip S. Thorne).

However, the events of 1959-60 changed all that. The pace of researchin general relativity and relativistic astrophysics began to quicken and,associated with this renewed effort, the systematic high-precision testingof gravitational theory became an active and challenging field, with manynew experimental and theoretical possibilities. These included new ver-sions of old tests, such as the gravitational red shift and deflection of light,with accuracies that were unthinkable before 1960. They also includedbrand new tests of gravitational theory, such as the gyroscope precession,the time delay of light, and the "Nordtvedt effect" in lunar motion, thatwere discovered theoretically after 1959. Table 1.3 presents a chronologyof some of the significant theoretical and experimental events that oc-curred in the two decades following 1959. In many ways, the years 1960-1980 were the decades for testing relativity.

Because many of the experiments involved the resources of programsfor interplanetary space exploration and observational astronomy, theircost in terms of money and manpower was high and their dependenceupon increasingly constrained government funding agencies was strong.Thus, it became crucial to have as good a theoretical framework as possiblefor comparing the relative merits of various experiments, and for pro-

Introduction

Table 1.3. A chronology: 1960-80

Time Experimental or observational events Theoretical events

1960

1962

1964

1966

1968

1970

19721974

1976

1978

1980

Hughes-Drever mass-anisotropyexperiments

Pound-Rebka gravitational red-shiftexperiment

Discovery of nonsolar x-ray sourcesDiscovery of quasar red shiftsPrinceton Eotvos experiment

Pound-Snider red-shift experimentDiscovery of 3K microwave

background

Reported detection of solaroblateness

Discovery of pulsars

Planetary radar measurement of timedelay

Launch of Mariners 6 and 7Acquisition of lunar laser echoFirst radio deflection measurements

CygXl: a black hole candidateMariners 6 and 7 time-delay

measurements

Moscow Eotvos experiment

Discovery of binary pulsar

Rocket gravitational red-shiftexperiment

Lunar test of Nordtvedt effectTime delay results from Mariner 9

and VikingMeasurement of orbit period

decrease in binary pulsarSS433

Discovery of gravitational lens

Penrose paper on spinorsGyroscope precession (Schiff)

Brans-Dicke theoryBondi mass-loss formulaKerr metric discovery

Time-delay of light (Shapiro)

Singularity theorems ingeneral relativity

Element production in thebig bang

Nordtvedt effect and earlyPPN framework

Preferred-frame effectsRefined PPN frameworkArea increase of black holes in

general relativity

Quantum evaporation ofblack holes

Dipole gravitational radiationin alternative theories


posing new ones that might have been overlooked. Another reason thatsuch a theoretical framework was necessary was to make some sense ofthe large (and still growing) number of alternative theories of gravitation.Such a framework could be used to classify theories, elucidate their sim-ilarities and differences, and compare their predictions with the results ofexperiments in a systematic way. It would have to be powerful enough tobe used to design and assess experimental tests in detail, yet generalenough not to be biased in favor of general relativity.

A leading exponent of this viewpoint was Robert Dicke (1964a). It ledhim and others to perform several high-precision null experiments whichgreatly strengthened our faith in the foundations of gravitation theory.Within this viewpoint one asks general questions about the nature ofgravity and devises experiments to test them. The most important div-idend of the Dicke framework is the understanding that gravitationalexperiments can be divided into two classes. The first consists of experi-ments that test the foundations of gravitation theory, one of these founda-tions being the principle of equivalence. These experiments (Eotvosexperiment, Hughes-Drever experiment, gravitational red-shift experi-ment, and others, many performed by Dicke and his students) accuratelyverify that gravitation is a phenomenon of curved spacetime, that is, itmust be described by a "metric theory" of gravity. General relativity andBrans-Dicke theory are examples of metric theories of gravity.

The second class of experiments consists of those that test metric theo-ries of gravity. Here another theoretical framework was developed thattakes up where the Dicke framework leaves off. Known as the "Param-etrized Post-Newtonian" or PPN formalism, it was pioneered by Ken-neth Nordtvedt, Jr. (1968b), and later extended and improved by Will(1971a), Will and Nordtvedt (1972), and Will (1973). The PPN frameworktakes the slow motion, weak field, or post-Newtonian limit of metrictheories of gravity, and characterizes that limit by a set of 10 real-valuedparameters. Each metric theory of gravity has particular values for thePPN parameters. The PPN framework was ideally suited to the analysisof solar system gravitational experiments, whose task then became oneof measuring the values of the PPN parameters and thereby delineatingwhich theory of gravity is correct. A second powerful use of the PPNframework was in the discovery and analysis of new tests of gravitationtheory, examples being the Nordtvedt effect (Nordtvedt 1968a), preferred-frame effects (Will, 1971b) and preferred-location effects (Will, 1971b,1973). The Nordtvedt effect, for instance, is a violation of the equalityof acceleration of massive bodies, such as the Earth and Moon, in an

Introduction 11

external field; the effect is absent in general relativity but present in manyalternative theories, including the Brans-Dicke theory. The third use ofthe PPN formalism was in the analysis and classification of alternativemetric theories of gravitation. After 1960, the invention of alternativegravitation theories did not abate, but changed character. The crude at-tempts to derive Lorentz-invariant field theories described previouslywere mostly abandoned in favor of metric theories of gravity, whosedevelopment and motivation were often patterned after that of the Brans-Dicke theory. A "theory of gravitation theories" was developed aroundthe PPN formalism to aid in their systematic study.

The PPN formalism thus became the standard theoretical tool foranalyzing solar system experiments, looking for new tests, and studyingalternative metric theories of gravity. One of the central conclusions ofthe two decades of testing relativistic gravity in the solar system is thatgeneral relativity passes every experimental test with flying colors.

But by the middle 1970s it became apparent that the solar systemcould no longer be the sole testing ground for gravitation theories. Onereason was that many alternative theories of gravity agreed with generalrelativity in their post-Newtonian limits, and thereby also agreed with allsolar system experiments. But they did not necessarily agree in other pre-dictions, such as cosmology, gravitational radiation, neutron stars, orblack holes. The second reason was the possibility that experimental tools,such as gravitational radiation detectors, would ultimately be availableto perform such extra-solar system tests.

This suspicion was confirmed in the summer of 1974 with the discoveryby Joseph Taylor and Russell Hulse of the binary pulsar (Hulse and Tay-lor, 1975). Here was a system that combined large post-Newtonian grav-itational effects, highly relativistic gravitational fields associated with thepulsar, and the possibility of the emission of gravitational radiation bythe binary system, with ultrahigh precision data obtained by radio-telescope monitoring of the extremely stable pulsar clock. It was also asystem where relativistic gravity and astrophysics became even more in-tertwined than in the case, say, of quasars. In the binary pulsar, relativisticgravitational effects provided a means for accurate measurement of astro-physical parameters, such as the mass of a neutron star. The role of thebinary pulsar as a new arena for testing relativistic gravity was cementedin the winter of 1978 with the announcement (Taylor et al., 1979) thatthe rate of change of the orbital period of the system had been measured.The result agreed with the prediction of general relativity for the rate oforbital energy loss due to the emission of gravitational radiation. But it


disagreed violently with the predictions of most alternative theories, eventhose with post-Newtonian limits identical to general relativity.

As a young student of 17 at the Poly technical Institute of Zurich,Einstein studied closely the work of Helmholtz, Maxwell, and Hertz, andultimately used his deep understanding of electromagnetic theory as afoundation for special and general relativity. He appears to have beenespecially impressed by Hertz's confirmation that light and electromag-netic waves are one and the same (Schilpp, 1949). The electromagneticwaves that Hertz studied were in the radio part of the spectrum, at 30MHz. It is amusing to note that, 60 years later, the decades for testingrelativistic gravity began with radio waves, the 440 MHz waves reflectedfrom Venus, and ended with radio waves, the signals from the binarypulsar, observed at 430 MHz.

During these two decades, that closed on the centenary of Einstein'sbirth, the empirical foundations of general relativity were strengthened asnever before. But this does not end the story. The confrontation betweengeneral relativity and experiment will proceed, using new tools, in newarenas. Whether or not general relativity will continue to survive is amatter of speculation for some, pious hope for another group, and supremeconfidence for others. Regardless of one's theoretical prejudices, it cancertainly be agreed that gravitation, the oldest known, and in many waysmost fundamental interaction, deserves an empirical foundation secondto none.

Throughout this book, we shall adopt the units and conventions ofMisner, Thorne, and Wheeler, 1973 (hereafter referred to as MTW).Although we have attempted to produce a reasonably self-contained ac-count of gravitation theory and gravitational experiments, the reader'spath will be greatly smoothed by a familiarity with at least the equivalentof "track 1" of MTW. A portion of the present book (Chapters 4-9) ispatterned after the author's 1972 Varenna lectures "The Theoretical Toolsof Experimental Gravitation" (Will, 1974a, hereafter referred to as TTEG),with suitable modification and updating. An overview of this book withoutthe mathematical details is provided by the author's "The Confrontationbetween Gravitation Theory and Experiment" (Will, 1979). Other usefulreviews of this subject are of three types: (i) semipopular: Nordtvedt (1972),Will (1972, 1974b); (ii) technical: Richard (1975), Brill (1973), Rudenko(1978); (iii) "early": Dicke (1964a,b), Bertotti et al. (1962). The reader isreferred to these works for background or for different points of view.

The Einstein Equivalence Principle and theFoundations of Gravitation Theory

The Principle of Equivalence has played an important role in the develop-ment of gravitation theory. Newton regarded this principle as such acornerstone of mechanics that he devoted the opening paragraphs of thePrincipia to a detailed discussion of it (Figure 2.1). He also reported therethe results of pendulum experiments he performed to verify the principle.To Newton, the Principle of Equivalence demanded that the "mass" ofany body, namely that property of a body (inertia) that regulates itsresponse to an applied force, be equal to its "weight," that property thatregulates its response to gravitation. Bondi (1957) coined the terms "iner-tial mass" mb and "passive gravitational mass" mP, to refer to these quan-tities, so that Newton's second law and the law of gravitation take the forms

F = m,a, F = mPg

where g is the gravitational field. The Principle of Equivalence can thenbe stated succinctly: for any body

mP = m1

An alternative statement of this principle is that all bodies fall in a gravi-tational field with the same acceleration regardless of their mass or in-ternal structure. Newton's equivalence principle is now generally referredto as the "Weak Equivalence Principle" (WEP).

It was Einstein who added the key element to WEP that revealed thepath to general relativity. If all bodies fall with the same accelerationin an external gravitational field, then to an observer in a freely fallingelevator in the same gravitational field, the bodies should be unaccelerated(except for possible tidal effects due to inhomogeneities in the gravita-tional field, which can be made as small as one pleases by working in asufficiently small elevator). Thus insofar as their mechanical motions are

Figure 2.1. Title page and first page of Newton's Principia.

PHILOSOPHISENATURALIS

P R I N C I P I AMATHEMATICA

Autore JS. UEfFTON, Trin. CM. Cantab. Soc. MathefeosProfeflbre Lucafuoto, & Sodetatis Regalis Sodali.

IMPRIMATURS. P E P Y S, Reg. Soc. P R R S E S.

Jutii 5. 1686.

L 0 N D I N /,

Juflii Societatis Regia ac Typis Jofepbi Streater. Proftat apudplures Bibliopolas. Anno MDCLXXXVIl.

14

Figure 2.1 (continued)

MATHEMATICAL PRINCIPLESOF

NATURAL PHILOSOPHY1

D eft nitions

DEFINITION I

The quantity of matter is the measure of the same, arising from its densityand bulk, conjointly.2

THUS AIR of a double density, in a double space, is quadruple in quan-tity; in a triple space, sextuple in quantity. The same thing is to beunderstood of snow, and fine dust or powders, that are condensed

by compression or liquefaction, and of all bodies that are by any causeswhatever differently condensed. I have no regard in this place to a medium,if any such there is, that freely pervades the interstices between the partsof bodies. It is this quantity that I mean hereafter everywhere under thename of body or mass. And the same is known by the weight of each body,for it is proportional to the weight, as I have found by experiments on pen-dulums, very accurately made, which shall be shown hereafter.

D E F I N I T I O N II s

The quantity of motion is the measure of the same, arising from thevelocity and quantity of matter conjointly.

The motion of the whole is the sum of the motions of all the parts; andtherefore in a body double in quantity, with equal velocity, the motion isdouble; with twice the velocity, it is quadruple.

t l Appendix, Note 10.] [2 Appendix, Note 11.] [3 Appendix, Note 12.]

CO

15


concerned, the bodies will behave as if gravity were absent. Einstein wentone step further. He proposed that not only should mechanical lawsbehave in such an elevator as if gravity were absent but so should all thelaws of physics, including, for example, the laws of electrodynamics. Thisnew principle led Einstein to general relativity. It is now called the"Einstein Equivalence Principle" (EEP).

Yet, it is only relatively recently that we have gained a deeper under-standing of the significance of these principles of equivalence for gravi-tation and experiment. Largely through the work of Robert H. Dicke,we have come to view principles of equivalence, along with experimentssuch as the Eotvos experiment, the gravitational red-shift experiment, andso on, as probes more of the foundations of gravitation theory, than ofgeneral relativity itself. This viewpoint is part of what has come to beknown as the Dicke Framework described in Section 2.1, allowing one todiscuss at a very fundamental level the nature of space-time and gravity.Within it one asks questions such as: Do all bodies respond to gravitywith the same acceleration? Does energy conservation imply anythingabout gravitational effects? What types of fields, if any, are associatedwith gravitation-scalar fields, vector fields, tensor fields... ? As oneproduct of this viewpoint, we present in Section 2.2 a set of fundamentalcriteria that any potentially viable theory should satisfy, and as another,we show in Section 2.3 that the Einstein Equivalence Principle is thefoundation for all gravitation theories that describe gravity as a mani-festation of curved spacetime, the so-called metric theories of gravity. InSection 2.4 we describe the empirical support for EEP from a varietyof experiments.

Einstein's generalization of the Weak Equivalence Principle may nothave been a generalization at all, according to a conjecture based on thework of Leonard Schiff. In Section 2.5, we discuss Schiif 's conjecture,which states that any complete and self-consistent theory of gravity thatsatisfies WEP necessarily satisfies EEP. Schiff's conjecture and the DickeFramework have spawned a number of concrete theoretical formalisms,one of which is known as the THsu formalism, presented in Section 2.6,for comparing and contrasting metric theories of gravity with nonmetrictheories, analyzing experiments that test EEP and WEP, and provingSchiff's conjecture.

2.1 The Dicke FrameworkThe Dicke Framework for analyzing experimental tests of gravi-

tation was spelled out in Appendix 4 of Dicke's Les Houches lectures

Einstein Equivalence Principle and Gravitation Theory 17

(1964a). It makes two main assumptions about the type of mathematicalformalism to be used in discussing gravity:

(i) Spacetime is a four-dimensional differentiable manifold, with eachpoint in the manifold corresponding to a physical event. The manifoldneed not a priori have either a metric or an affine connection. The hopeis that experiment will force us to conclude that it has both.

(ii) The equations of gravity and the mathematical entities in them areto be expressed in a form that is independent of the particular coordinatesused, i.e., in covariant form.

Notice that even if there is some physically preferred coordinate systemin spacetime, the theory can still be put into covariant form. For example,if a theory has a preferred cosmic time coordinate, one can introducea scalar field T{0>) whose numerical values are equal to the values of thepreferred time t:

T(0>) = t{0>), 0> a point in spacetime

If spacetime is endowed with a metric, one might also demand that VTbe a timelike vector field and be consistently oriented toward the future(or the past) throughout spacetime by imposing the covariant constraints

VT-VT<0, V<g>VT = 0

where V is a covariant derivative with respect to the metric. Other typesof theories have "flat background metrics" IJ; these can also be writtencovariantly by defining i; to be a second-rank tensor field whose Riemanntensor vanishes everywhere, i.e.,

Riem(>r) = 0

and by defining covariant derivatives and contractions with respect to i\.In most cases, this covariance is achieved at the price of the introductioninto the theory of "absolute" or "prior geometric" elements (T, i/), thatare not determined by the dynamical equations of the theory. Someauthors regard the introduction of absolute elements as a failure of generalcovariance (Einstein would be one example), however we shall adopt theweaker assumption of coordinate invariance alone. (For further discussionof prior geometry, see Section 3.3.)

Having laid down this mathematical viewpoint [statements (i) and (ii)above] Dicke then imposes two constraints on all acceptable theories ofgravity. They are:

(1) Gravity must be associated with one or more fields of tensorialcharacter (scalars, vectors, and tensors of various ranks).


(2) The dynamical equations that govern gravity must be derivablefrom an invariant action principle.

These constraints strongly confine acceptable theories. For this reasonwe should accept them only if they are fundamental to our subsequentarguments. For most applications of the Dicke Framework only the firstconstraint is often needed. It is a fact, however, that the most successfulgravitation theories are those that satisfy both constraints.

The Dicke Framework is particularly useful for designing and inter-preting experiments that ask what types of fields are associated withgravity. For example, there is strong evidence from elementary particlephysics for at least one symmetric second-rank tensor field that is approxi-mated by the Minkowski metric i\ when gravitational effects can beignored. The Hughes-Drever experiment rules out the existence of morethan one second-rank tensor field, each coupling directly to matter, andvarious ether-drift experiments rule out a long-range vector field couplingdirectly to matter. No experiment has been able to rule out or reveal theexistence of a scalar field, although several experiments have placedlimits on specific scalar-tensor theories (Chapters 7 and 8). However, thisis not the only powerful use of the Dicke Framework.

2.2 Basic Criteria for theViability of a Gravitation TheoryThe general unbiased viewpoint embodied in the Dicke Frame-

work has allowed theorists to formulate a set of fundamental criteria thatany gravitation theory should satisfy if it is to be viable [we do not imposeconstraints (1) and (2) above]. Two of these criteria are purely theoretical,whereas two are based on experimental evidence.

(i) It must be complete, i.e., it must be capable of analyzing from"first principles" the outcome of any experiment of interest. It is notenough for the theory to postulate that bodies made of different materialfall with the same acceleration. The theory must incorporate a completeset of electrodynamic and quantum mechanical laws, which can be usedto calculate the detailed behavior of bodies in gravitational fields. Thisdemand should not be extended too far, however. In areas such as weakand strong interaction theory, quantum gravity, unified field theories,spacetime singularities, and cosmic initial conditions, even special andgeneral relativity are not regarded as being complete or fully developed.We also do not regard the presence of "absolute elements" and arbitraryparameters in gravitational theories as a sign of incompleteness, eventhough they are generally not derivable from "first principles," rather we


view them as part of the class of cosmic boundary conditions. Fortunately,so simple a demand as one that the theory contain a set of gravitationallymodified Maxwell equations is sufficiently telling that many theories failthis test. Examples are given in Table 2.1.

(ii) It must be self-consistent, i.e., its prediction for the outcome ofevery experiment must be unique, i.e., when one calculates the predictionsby two different, though equivalent methods, one always gets the same

Table 2.1. Basically nonviable theories of gravitation - a partial list

Theory and references Comments"

Newtonian gravitation theory

Milne's kinematical relativity(Milne, 1948)

Kustaanheimo's various vectortheories (Kustaanheimo andNuotio, 1967; Whitrow andMorduch, 1965)

Poincare's theory (asgeneralized by Whitrow andMorduch, 1965)

Whitrow-Morduch (1965)vector theory

Birkhoff's (1943) theory

Yilmaz's (1971,1973) theory

Is not relativistic

Was devised originally to handle certaincosmological problems. Is incomplete: makesno gravitational red-shift predictionContain a vector gravitational field in flatspacetime. Are incomplete: do not mesh withthe other nongravitational laws of physics(viz. Maxwell's equations) except by imposingthem on the flat background spacetime. Arethen inconsistent: give different results for lightpropagation for light viewed as particles andlight viewed as waves.

Action-at-a-distance theory in flat spacetime.Is incomplete or inconsistent in the samemanner as Kustaanheimo's theories

Contains a vector gravitational field in flatspacetime. Is incomplete or inconsistent in thesame manner as Kustaanheimo's theories.Contains a tensor gravitational field used toconstruct a metric. Violates the Newtonianlimit by demanding that p = pc2, i.e.''sound = "light-

Contains a tensor gravitational field used toconstruct a metric. Is mathematicallyinconsistent: functional dependence of metricon tensor field is not well defined.

° These theories are nonviable in their present form. Future modifications or special-izations might make some of them viable. If I have misinterpreted any theory hereI apologize to its proponents, and urge them to demonstrate explicitly its com-pleteness, self-consistency, and compatibility with special relativity and Newtoniangravitation theory.


results. An example is the bending of light computed either in the geo-metrical optics limit of Maxwell's equations or in the zero-rest-mass limitof the motion of test particles. Furthermore, the system of mathematicalequations it proposes should be well posed and self-consistent. Table 2.1shows some theories that fail this criterion.

(iii) It must be relativistic, i.e., in the limit as gravity is "turned off"compared to other physical interactions, the nongravitational laws ofphysics must reduce to the laws of special relativity. The evidence for thiscomes largely from high-energy physics and from a variety of opticalether-drift experiments. Since these experiments are performed at highenergies and velocities and over very small regions of space and time, theeffects of gravity on their outcome are negligible. Thus we may treat suchexperiments as if they were being performed far from all gravitatingmatter. The evidence provided by these experiments is of two types. Firstare experiments that measure space and time intervals directly, e.g.,measurements of the time dilation of systems ranging from atomic clocksto unstable elementary particles, experiments that verify the velocity oflight is independent of the velocity of the source for sources ranging frompions at 99.98% of the speed of light to pulsating binary x-ray sources at10"3 of the speed of light [for a thorough review and reference list, seeNewman et al. (1978)] and Michelson-Morley-type experiments [for re-cent high-precision results, see Trimmer et al. (1973) and Brillet and Hall(1979); see also Mansouri and Sexl (1977a,b,c) for theoretical discussion].Second are experiments which reveal the fundamental role played bythe Lorentz group in particle physics, including verifications of four-momentum conservation and of the relativistic laws of kinematics, elec-tron and muon "g-2" experiments, and tests of esoteric predictions ofLorentz-in variant quantum field theories [Lichtenberg (1965), Blokhintsev(1966), Newman et al. (1978), Combley et al. (1979), and Cooper et al.(1979)].

The fundamental theoretical object that enters these laws is the Min-kowski metric i\, with a signature of + 2, which has orthonormal tetradsrelated by Lorentz transformations, and which determines the tickingrates of atomic clocks and the lengths of laboratory rods. If we view q asa field [Dicke statement (ii)], then we conclude that there must exist atleast one second-rank tensor field in the Universe, a symmetric tensor ^,which reduces to r\ when gravitational effects can be ignored.

Let us examine what particle physics experiments do and do not tellus about the tensor field V- First, they do not guarantee the existence ofglobal Lorentz frames, i.e., coordinate systems extending throughout


spacetime in which

(-1,1,1,1)

Nor do they demand that at each event 2P, there exist local frames relatedby Lorentz transformations, in which the laws of elementary-particlephysics take on their special form. They only demand that, in the limit asgravity is "turned off," the nongravitational laws of physics reduce tothe laws of special relativity.

Second, elementary-particle experiments do tell us that the times mea-sured by atomic clocks in the limit as gravity is turned off dependonly on velocity, not upon acceleration. The measured squared interval,ds2 = i^dx^dx", is independent of acceleration. Equivalently, but morephysically, the time interval measured by a clock moving with velocity vJ

relative to a coordinate system in the absence of gravity is

ds = (-q^tordx*)112 = dt(l - |v|2)1/2

independent of the clock's acceleration d2xi/dt2. (For a review of experi-mental tests, see Newman et al., 1978.)

We shall henceforth assume the existence of the tensor field $.(iv) It must have the correct Newtonian limit, i.e., in the limit of weak

gravitational fields and slow motions, it must reproduce Newton's laws.Massive amounts of empirical data support the validity of Newtoniangravitation theory (NGT), at least as an approximation to the "true"relativistic theory of gravity. Observations of the motions of planets andspacecraft agree with NGT down to the level (parts in 108) at whichpost-Newtonian effects can be observed. Observations of planetary, solar,and stellar structure support NGT as applied to bulk matter. LaboratoryCavendish experiments provide support for NGT for small separationsbetween gravitating bodies. One feature of NGT that has recently comeunder experimental scrutiny is the inverse-square force law. Despite oneclaim to the contrary (Long, 1976), there seems to be no hard evidence fora deviation from this law (other than those produced by post-Newtonianeffects) over distances ranging from a few centimeters to several astro-nomical units (see Mikkelson and Newman, 1977; Spero et al., 1979; Paik,1979; Yu et al., 1979; Panov and Frontov, 1979; and, Hirakawa et al.,1980).

Thus, to at least be viable, a gravitation theory must be complete,self-consistent, relativistic, and compatible with NGT. Table 2.1 showsexamples of theories that violate one or more of these criteria.


2.3 The Einstein Equivalence PrincipleThe Einstein Equivalence Principle is the foundation of all curved

spacetime or "metric" theories of gravity, including general relativity. Itis a powerful tool for dividing gravitational theories into two distinctclasses: metric theories, those that embody EEP, and nonmetric theories,those that do not embody EEP. For this reason, we shall discuss it insome detail and devote the next section (Section 2.4) to the supportingexperimental evidence.

We begin by stating the Weak Equivalence Principle in more preciseterms than those used before. WEP states that if an uncharged test bodyis placed at an initial event in spacetime and given an initial velocity there,then its subsequent trajectory will be independent of its internal structureand composition. By "uncharged test body" we mean an electrically neutralbody that has negligible self-gravitational energy (as estimated usingNewtonian theory) and that is small enough in size so that its couplingto inhomogeneities in external fields can be ignored. In the same spirit,it is also useful to define "local nongravitational test experiment" to beany experiment: (i) performed in a freely falling laboratory that is shieldedand is sufficiently small that inhomogeneities in the external fields can beignored throughout its volume, and (ii) in which self-gravitational effectsare negligible. For example, a measurement of the fine structure constantis a local nongravitational test experiment; a Cavendish experiment isnot.

The Einstein Equivalence Principle then states: (i) WEP is valid, (ii) theoutcome of any local nongravitational test experiment is independent of thevelocity of the (freely falling) apparatus, and (iii) the outcome of any localnongravitational test experiment is independent of where and when in theuniverse it is performed.

This principle is at the heart of gravitation theory, for it is possible toargue convincingly that if EEP is valid, then gravitation must be a curved-spacetime phenomenon, i.e., must satisfy the postulates of Metric Theoriesof Gravity. These postulates state: (i) spacetime is endowed with a metricg, (ii) the world lines of test bodies are geodesies of that metric, and (iii) inlocal freely falling frames, called local Lorentz frames, the nongravitationallaws of physics are those of special relativity. General relativity, Brans-Dicke theory, and the Rosen bimetric theory are metric theories ofgravity (Chapter 5); the Belinfante-Swihart theory (Section 2.6) is not.

The argument proceeds as follows. The validity of WEP endows space-time with a family of preferred trajectories, the world lines of freelyfalling test bodies. In a local frame that follows one of these trajectories,

test bodies have unaccelerated motions. Furthermore, the results of localnongravitational test experiments are independent of the velocity of theframe. In two such frames located at the same event, 9, in spacetime butmoving relative to each other, all the nongravitational laws of physicsmust make the same predictions for identical experiments, that is, theymust be Lorentz invariant. We call this aspect of EEP Local LorentzInvariance (LLI). Therefore, there must exist in the universe one or moresecond-rank tensor fields i/t(1), ij/(2\ . . . , that reduce in a local freelyfalling frame to fields that are proportional to the Minkowski metric,(j)(1\^)tl, 0(2)(^)«J,..., where 4>(A\0>) are scalar fields that can vary fromevent to event. Different members of this set of fields may couple todifferent nongravitational fields, such as boson fields, fermion fields, elec-tromagnetic fields, etc. However, the results of local nongravitationaltest experiments must also be independent of the spacetime location ofthe frame. We call this Local Position Invariance (LPI). There are thentwo possibilities, (i) The local versions of ijf{A) must have constant co-efficients, that is, the scalar fields 4>(A\^) must be constants. It is thereforepossible by a simple universal rescaling of coordinates and couplingconstants (such as the unit of electric charge) to set each scalar field equalto unity in every local frame, (ii) The scalar fields <f>iA)(<P) must be constantmultiples of a single scalar field ${&), i.e., 4>{A\0>) = cA4>(0>). If this istrue, then physically measurable quantities, being dimensionless ratios,will be location independent (essentially, the scalar field will cancel out).One example is a measurement of the fine structure constant; anotheris a measurement of the length of a rigid rod in centimeters, since such ameasurement is a ratio between the length of the rod and that of a standardrod whose length is defined to be one centimeter. Thus, a combinationof a rescaling of coupling constants to set the cA's equal to unity (re-definition of units), together with a "conformal" transformation to a newfield ij/ = cj>~ V. guarantees that the local version of if/ will be ij.

In either case, we conclude that there exist fields that reduce to r\ inevery local freely falling frame. Elementary differential geometry thenshows that these fields are one and the same: a unique, symmetric second-rank tensor field that we now denote g. This g has the property that itpossesses a family of preferred worldlines called geodesies, and that ateach event $* there exist local frames, called local Lorentz frames, thatfollow these geodesies, in which

<W^) = 1** + 0(Y |X« - x\0>)\\ dgjdx* = 0, at 0>


However, geodesies are straight lines in local Lorentz frames, as are thetrajectories of test bodies in local freely falling frames, hence the testbodies move on geodesies of g and the Local Lorentz frames coincidewith the freely falling frames.

We shall discuss the implications of the postulates of metric theoriesof gravity in more detail in Chapter 3. Because EEP is so crucial to thisconclusion about the nature of gravity, we turn now to the supportingexperimental evidence.

2.4 Experimental Tests of the Einstein Equivalence Principle(a) Tests of the Weak Equivalence PrincipleA direct test of WEP is the Eotvos experiment, the comparison

of the acceleration from rest of two laboratory-sized bodies of differentcomposition in an external gravitational field. If WEP were invalid, thenthe accelerations of different bodies would differ. The simplest way toquantify such possible violations of WEP in a form suitable for com-parison with experiment is to suppose that for a body of inertial massm,, the passive mass mP is no longer equal to mv Now the inertial mass ofa typical laboratory body is made up of several types of mass energy:rest energy, electromagnetic energy, weak-interaction energy, and so on.If one of these forms of energy contributes to mP differently than it doesto m,, a violation of WEP would result. One could then write

�p = m, + I r]AEA/c2 (2.1)A

where EA is the internal energy of the body generated by interaction A,and nA is a dimensionless parameter that measures the strength of theviolation of WEP induced by that interaction, and c is the speed of light.1

For two bodies, the acceleration is then given by

^ ( + S r,AEA/m2Ag (2.2)

where we have dropped the subscript I on mj and m2.

1 Throughout this chapter we shall avoid units in which c = 1. The reasonfor this is that if EEP is not valid then the speed of light may depend on thenature of the devices used to measure it. Thus, to be precise we should denotec as the speed of light as measured by some standard experiment. Once we acceptthe validity of EEP in Chapter 3 and beyond, then c has the same value in everylocal Lorentz frame, independently of the method used to measure it, and thuscan be set equal to unity by appropriate choice of units.

A measurement or limit on the relative difference in acceleration thenyields a quantity called the "Eotvos ratio" given by

K + a\ t \mC2 m2c

2j v '

Thus, experimental limits on r\ place limits on the WEP-violation param-eters rjA.

Many high-precision Eotvos-type experiments have been performed,from the pendulum experiments of Newton, Bessel, and Potter to theclassic torsion-balance measurements of Eotvos, Dicke, and Braginskyand their collaborators. The latter experiments can be described heuris-tically. Two objects of different composition are connected by a rod oflength r, and suspended in a horizontal orientation by a fine wire ("torsionbalance"). If the gravitational acceleration of the bodies differs, therewill be a torque N induced on the suspension wire, given by

N = tjr(g x ew) � er

where g is the gravitational acceleration, and ew and er are unit vectorsalong the wire and rod, respectively (see Figure 2.2). If the entire apparatusis rotated about a direction <o with angular velocity |co|, the torque willbe modulated with period 2JT/CO. In the experiments of Baron Roland vonEotvos, g was the acceleration of the Earth (note g and ew were not quiteparallel because of the centripetal acceleration on the apparatus due tothe Earth's rotation), and the apparatus was rotated about the directionof the wire. In the Princeton (Roll, et al., 1964) and Moscow (Braginskiand Panov, 1972) experiments, g was that of the Sun, and the rotationof the Earth provided the modulation of N at a period of 24 hr. Themodulated torque was determined either by measuring the torsional mo-tion of the rod (Moscow) or by measuring the force required to counteractthe torque and keep the rod in place (Princeton). The resulting upperlimits on measurable torques |N| yielded limits on r\ given by

\\ x 10"J1 [Princeton]II x 10"12 [Moscow] ( '

where the limits are \a formal standard deviations. For further discussionof the experiments, see Dicke (1964a) and Braginsky (1974). The primarysources of error in these experiments are seismic noise and coupling ofthe torsion balance to gradients in the external gravitational field (pro-duced, for example, by the experimenters). Attempts to improve theseresults have centered on different forms of suspension of the masses,


Figure 2.2. Schematic arrangement of a torsion-balance Eotvos experi-ment; g is the external gravitational acceleration, and to is the angularvelocity vector about which the apparatus is rotated. The unit vectors ew

and er are parallel to the wire and rod, respectively. In the Eotvos experi-ments, g was the acceleration toward the Earth, and to was parallel to ew;in the Princeton and Moscow experiments g was that of the Sun, and cowas parallel to the Earth's rotation axis.

\\W\\\\\

including magnetic levitation (Worden and Everitt, 1974), flotation onliquids (Keiser and Faller, 1979), and free fall in orbit.

Experiments to test WEP for individual atoms and elementary particleshave been inconclusive or inaccurate, with the exception of neutrons(Fairbank et al, 1974 and Koester, 1976).

Table 2.2 discusses various experiments and quotes the limits they seton Y) for different pairs of materials. Future improved tests of WEP mustreduce noise due to thermal, seismic, and gravity-gradient effects, andmay have to be performed in space using cryogenic techniques. Antici-pated limits on r\ in such experiments range between 10""15 and 10"18

(Worden, 1978).To determine the limits placed on individual parameters r\K by, say,

the best of the torsion-balance experiments, we must estimate the co-

Table 2.2. Tests of the weak equivalence principle

Experiment

NewtonBesselEotvosPotterRennerPrincetonMoscowMunichStanfordBoulderOrbital-

Reference

Newton (1686)Bessel (1832)Eotvos, Pekar, and Fekete (1922)Potter (1923)Renner (1935)Roll, Krotkov, and Dicke (1964)Braginsky and Panov (1972)Koester (1976)Worden (1978)Keiser and Faller (1979)Worden (1978)

Method

PendulaPendulaTorsion balancePendulaTorsion balanceTorsion balanceTorsion balanceFree fallMagnetic suspensionFlotation on waterFree fall in orbit

Substances tested

VariousVariousVariousVariousVariousAluminum and goldAluminum and platinumNeutronsNiobium, EarthCopper, tungstenVarious

Limit on \tj\

10"3

2 x 10"5

5 x 10"9

2 x 10"5

2 x lO"9

io-»lO"1 2

3 x 10"4

io-4

4 x 10" u

10~15 - lO"18"

" Experiments yet to be performed.

efficients EA/m for the different interactions and for different materials.For laboratory-sized bodies, the dominant contribution to £A comesfrom the atomic nucleus.

We begin with the strong interactions. The semiempirical mass formula(see, for example, Leighton, 1959) gives

Es = -15.74 + 17.SA213 + 23.604 - 2Z)2A~l + 132/1"lS MeV (2.5)

where Z and A are the atomic number and mass number, respectively,of the nucleus, and where S = 1 if {Z,A} = {odd, even}, 5 � � 1 if{Z,A} = {even, even} and 8 = 0 if A is odd. Then,

Es/mc2 = -1.7 x 1(T2 + 1.9 x 10"2,4"1/3

+ 2.5 x 10~2(l - 2Z/A)2 + 1.41 x 10"M-2<5 (2.6)

For platinum (Z = 78, A = 195), and aluminum (13,27) the difference inEs/mc2 is approximately 2 x 10"3, so from the limit \n\ < 10"12, weobtain the limit \ns\ < 5 x 10"10.

In the case of electromagnetic interactions, we can distinguish amonga number of different internal energy contributions, each potentiallyhaving its own n* parameter. For the electrostatic nuclear energy, thesemiempirical mass formula yields the estimate

£ES = 0.71Z(Z - X)A ~1/3 MeV (2.7)

Thus,

EES/mc2 = 7.6 x 10"4Z(Z - \)A"4/3 (2.8)

with the difference for platinum and aluminum being 2.5 x 10"3. Theresulting limit on nES is |T/ES| < 4 X 10"10. Another form of electromag-netic energy is magnetostatic, resulting from the nuclear magnetic fieldsgenerated by the proton currents. To estimate the nuclear magnetostaticenergy requires a detailed shell model computation. For example, the netproton current in any closed angular momentum shell vanishes, hencethere is no energy associated with the magnetostatic interaction betweensuch a closed shell and any particle outside the shell. For aluminum andplatinum, Haugan and Will (1977) have shown

(£MS/mc2)A1 = 4.1 x 10~7, (£M7mc2)pt = 2.4 x 10"7 (2.9)

thus \r\us\ < 6 x 10 "6. A third form of electromagnetic energy that hasbeen studied is hyperfine, the energy of interaction between the spins ofthe nucleons and the magnetic fields generated by the proton and neutronmagnetic moments. Computations by Haugan (1978) have yielded the

estimate

£HF = (2n/V)ntig2pZ

2 + g2(A - Z)2] (2.10)

where V is the nuclear volume, (iN is the nuclear magneton, and gp = 2.79and ga = �1.91 are gyromagnetic ratios for the proton and neutron,respectively. Then,

Em/mc2 = 2.1 x 10"5[>2Z2 + g2(A - Zf^A'2 (2.11)

with the difference between aluminum and platinum being 4 x 10~6;thus|>7HF|<2 x 10" 7.

For some time, it was believed that the contribution of the weakinteractions to nuclear energy was of the order of a part in 1012, and thatthe Eotvos experiment was not yet sufficiently accurate to test WEP forweak interactions (see for example, Chiu and Hoffman, 1964; Dicke,1964a). However, these estimates took into account only the parity non-conserving parts of the weak interactions, which make no contribu-tion to the energy of a nucleus in its ground state, to first order in theweak-interaction coupling constant Gw. On the other hand, the parity-conserving parts of the weak interactions do contribute at first order in Gw

and yield a value E�/me2 ~ 10"8 (Haugan and Will, 1976). Specifically,in the Weinberg-Salam model for weak and electromagnetic interactions,the result is

E^/mc2 = 2.2 x 10"8(iVZM2)[l + g(N,Z)],

g(N,Z) = 0.295[i(iV - Z)2/ATZ + 4 sin2 0W

+ (Z/N) sin2 0W(2 sin2 0W - 1)] (2.12)

where N = (A � Z) is the neutron number, and where 0W ~ 20° is the"Weinberg" angle. For aluminum and platinum, the difference is 2 x10~10, yielding |>?w| < 10~2.

Gravitational interactions are specifically excluded from WEP andEEP. In Chapter 3, we shall extend these two principles to incorporatelocal gravitational effects, thereby defining the Gravitational Weak Equiv-alence Principle (GWEP) and the Strong Equivalence Principle (SEP).These two principles will be useful in classifying alternative metric theoriesof gravity. In any case, for laboratory Eotvos experiments, gravitationalinteractions are totally irrelevant, since for an atomic nucleus

Ea/mc2 ~ Gmp/Raucleusc2 ~ 10"3 9

To test for gravitational effects in GWEP, it will be necessary to employplanetary objects and planetary Eotvos experiments (Section 8.1).

(b) Tests of Local Lorentz InvarianceAny experiment that purports to test special relativity (Section 2.2)

also tests some aspect of Local Lorentz Invariance, since every Earth-bound laboratory resides in a gravitational field (although it is only par-tially in free fall). However, very few of these experiments have been usedto make quantitative tests of LLI in the same way that Eotvos experimentshave been used to test WEP. For example, although elementary-particleexperimental results are consistent with the validity of Lorentz invariancein the description of high-energy phenomena, they are not "clean" testsbecause in many cases it is unlikely that a violation of Lorentz invariancecould be distinguished from effects due to the complicated strong andweak interactions. For instance, the observed violation of conservationof four momentum in beta decay was found to be due not to a violationof LLI, but to the emission of a hitherto unknown particle, the neutrino.

However, there is one experiment that can be interpreted as a "clean"test of Local Lorentz invariance, and an ultrahigh precision one at that.This is the Hughes-Drever experiment, performed in 1959-60 indepen-dently by Hughes and collaborators at Yale University and by Dreverat Glasgow University (Hughes et al., 1960; Drever, 1961). In the Glasgowversion, the experiment examined the J = § ground state of the 7Li nu-cleus in an external magnetic field. The state is split into four levels bythe magnetic field, with equal spacing in the absence of external perturba-tions, so the transition line is a singlet. Any external perturbation asso-ciated with a preferred direction in space (the velocity of the Earth relativeto the mean rest frame of the universe, for example) that has a quadrupole(/ = 2) component will destroy the equality of the energy spacing andsplit the transition lines. Using NMR techniques, the experiment set alimit of 0.04 Hz (1.7 x 10"16 eV) on the separation in frequency (energy)of the lines. One interpretation of this result is that it sets a limit on apossible anisotropy 3m\j in the inertial mass of the 7Li nucleus: |5mjJc2| ;$1.7 x 10"16 eV. If any of the forms of internal energy of the 7Li nucleussuffered a breakdown of Local Lorentz Invariance, one would expect acontribution to 5m{J of the form

dmij ~ X 5AEx/c2 (2.13)A

where <5A is a dimensionless parameter that measures the strength ofanisotropy induced by interaction A. Using formulae from Section 2.4(a),we can then make estimates of EA for 7Li and obtain the following limits

on<5\

|<5S| < 1(T23, |<5HF| < 5 x 1(T22,|£ES| < 1 0 - 2 2 ) |gW| < 5 x 1Q-18 (2.14)

Notice that the magnetostatic energy for 7Li is zero, since the protonshell structure is ls1/2lp3/2 and there is no magnetostatic interactioneither within the closed s-shell (/ = 0) or between that shell and the valenceproton. Because of the remarkably small size of these limits, the Hughes-Drever experiment has been called the most precise null experiment everperformed.

If Local Lorentz Invariance is violated, then there must be a preferredrest frame, presumably that of the mean rest frame of the universe, or,equivalently of the cosmic microwave background, in which the locallaws of physics take on their special form. Deviations from this formwould then depend on the velocity of the laboratory relative to the pre-ferred frame. Since the anisotropy is a quadrupole effect, one would expectit to be proportional to the square of the velocity w of the laboratory.If <>o is a parameter that measures the "bare" strength of LLI violation,then one would expect

For the motion of the Earth relative to the universe rest frame, w2 ~ 10 ~6.Limits on the <5Q can then be inferred from Equation (2.14). As a specialcase of this general argument, the Hughes-Drever experiment has alsobeen interpreted as a test of the existence of additional long-range tensorfields that couple directly to matter (Peebles and Dicke, 1962; Peebles,1962).

Other experiments that can be interpreted as tests of LLI include var-ious ether-drift experiments, such as the Turner-Hill experiment (Dicke,1964a; Haugan, 1979).

(c) Tests of Local Position InvarianceThe two principal tests of Local Position Invariance are gravita-

tional red-shift experiments that test the existence of spatial dependenceon the outcomes of local experiments, and measurements of the constancyof the fundamental nongravitational constants that test for temporaldependence.

Gravitational Red-Shift Experiments A typical gravitational red-shiftexperiment measures the frequency or wavelength shift Z = Av/v =� AA/A between two identical frequency standards (clocks) placed atrest at different heights in a static gravitational field. To illustrate howsuch an experiment tests LPI, we shall assume that the remaining partsof EEP, namely WEP and Local Lorentz Invariance, are valid. (In Sec-tions 2.5 and 2.6, we shall discuss this question under somewhat differentassumptions.) WEP guarantees that there exist local freely falling frameswhose acceleration g relative to the static gravitational field is the sameas that of test bodies. Local Lorentz Invariance guarantees that in theseframes, the proper time measured by an atomic clock is related to theMinkowski metric by

c2dx2 oc - r\^dx% dx\ oc c2dt\ - dx\ - dy\ - dz% (2.15)

where x% are coordinates attached to the freely falling frame. However, ina local freely falling frame that is momentarily at rest with respect to theatomic clock, we permit its rate to depend on its location (violation ofLocal Position Invariance), that is, relative to an arbitrarily chosen atomictime standard based on a clock whose fundamental structure is differentthan the one being analyzed, the proper time between ticks is given by

T = T(O) (2.16)

where O is a gravitational potential whose gradient is related to the test-body acceleration by g = £V.

Now the emitter, receiver, and gravitational field are assumed to bestatic, therefore in a static coordinate system (ts,xs), the trajectories ofsuccessive wave crests of emitted signal are identical except for a timetranslation Ats from one crest to the next. Thus, the interval of time Afsbetween ticks (passage of wave crests) of the emitter and of the receivermust be equal (otherwise there would be a build up or depletion of wavecrests between the two clocks, in violation of our assumption that thesituation is static). The static coordinates are not freely falling coordi-nates, but are accelerated upward (in the +z direction) relative to thefreely falling frame, with acceleration g. Thus, for \gts/c\ ~ |#zs/c

2| « 1(i.e., for g uniform over the distance between the clocks), a sequence ofLorentz transformations yields (MTW, Section 6.6)

ctF = (zs + c2/g)sinh{gts/c),

zF = (zs + c2/g)cosh(gts/c),

x F =x s , yF = ys (217)

Thus, the time measured by the atomic clocks (relative to the standardclock) is given by

c2dx2 = T2(<D)(C2 dtl - dxl - dyl - dz2)

= T2(<J>)[(1 + gzs/c2)2c2 dti - dx2 - dy2 - dz2} (2.18)

Since the emission and reception rates are the same (1/Afj) when mea-sured in static coordinate time, and since dxs = dys = dzs = 0 for bothclocks, the measured rates (v = AT" ') are related by

rJc2)~\

Wc2) J�7 Vrec Vem_ , [T(g>rec)(l + gZrJc)\

~^T~ U<"U(i + flWc2) J ( }

For small separations, Az = zrec � zem, we can expand T(<D) in the form

T(4>rec) = T0 + ^ r ^ A z (2.20)

where T0 = T(<Dem), x'o = Bx/d<b\tm. Then

Z = (1 + a)At//c2 (2.21)

where a =�C2C~1TJ)/T0 and where AC/ = g � Az = � g(zrec� zem). If thereis no location dependence in the clock rate, then a = 0, and the red shiftis the standard prediction, i.e.,

Z = AU/c2 (2.22)

An alternative version of this argument assumes the validity of bothLLI and LPI and shows that, if the red shift is given by Equation (2.22),then the acceleration of the local frames in which Lorentz and PositionInvariance hold is the same as that of test bodies, i.e., the local framesare freely falling frames (Thorne and Will, 1971).

Although there were several attempts following the publication of thegeneral theory of relativity to measure the gravitational red shift of spec-tral lines from white dwarf stars, the results were inconclusive (see Bertottiet al., 1962 for a review). The first successful, high-precision red-shift mea-surement was the series of Pound-Rebka-Snider experiments of 1960-65,which measured the frequency shift of y-ray photons from Fe57 as theyascended or descended the Jefferson Physical Laboratory tower at Har-vard University. The high accuracy achieved (1%) was obtained bymaking use of the Mossbauer effect to produce a narrow resonance linewhose shift could be accurately determined. Other experiments since 1960measured the shift of spectral lines in the Sun's gravitational field andthe change in rate of atomic clocks transported aloft on aircraft, rockets,

Table 2.3. Gravitational red-shift experiments

Experiment

Pound-Rebka-Snider

BraultJenkins

SniderJet-Lagged Clocks (A)

Jet-Lagged Clocks (B)

Vessot-Levine RocketRed-shift Experiment

Null Red-shift ExperimentClose Solar Probe"

Reference

Pound and Rebka (1960)Pound and Snider (1965)Brault (1962)Jenkins (1969)

Snider (1972,1974)Hafele and Keating (1972a,b)

Alley (1979)

Vessot and Levine (1979)Vessot et al. (1980)Turneaure et al. (1983)Nordtvedt (1977)

Method

Fall of photons fromMossbauer emitters

Solar spectral linesCrystal oscillator clocks

on GEOS-1 satelliteSolar spectral linesCesium beam clocks on

jet aircraftRubidium clocks on

jet aircraftHydrogen maser on rocket

Hydrogen maser vs. SCSOHydrogen maser or SCSO

on satellite

Limit on |a|

io-2

5 x 10"2

9x 10"2

6 x 10"2

10"1

2 x 10"2

2 x 10"4

io-2

10-6«

' Experiments yet to be performed.

and satellites. Table 2.3 summarizes the important red-shift experimentsthat have been performed since 1960.

Recently, however, a new era in red-shift experiments has been usheredin with the development of frequency standards of ultrahigh stability -parts in 1015 to 1016 over averaging times of 10 to 100 s and longer. Ex-amples are hydrogen-maser clocks (Vessot, 1974), superconducting-cavitystabilized oscillator (SCSO) clocks (Stein, 1974; Stein and Turneaure,1975), and cryogenically cooled monocrystals of dielectric materials suchas silicon and sapphire (McGuigan et al., 1978). The first such experi-ment was the Vessot-Levine Rocket Red-shift Experiment that tookplace in June, 1976. A hydrogen-maser clock was flown on a rocket toan altitude of about 10,000 km and its frequency compared to a similarclock on the ground. The experiment took advantage of the high frequencystability of hydrogen-maser clocks (parts in 1015 over 100 s averagingtimes) by monitoring the frequency shift as a function of altitude. A so-phisticated data acquisition scheme accurately eliminated all effects ofthe first-order Doppler shift due to the rocket's motion, while trackingdata were used to determine the payload's location and velocity (to eval-uate the potential difference AU, and the second-order Doppler shift).Analysis of the data yielded a limit (Vessot and Levine, 1979; Vessot et al.,1980)

|a| < 2 x 10"4 (2.23)

Coincidentally, the Scout rocket that carried the maser aloft stood 22.6 min its gantry, almost exactly the height of the Harvard Tower. In an inter-planetary version of this experiment, a stable clock (H-maser or SCSOclock) would be flown on a spacecraft in a very eccentric solar orbit(closest approach ~4 solar radii); such an experiment could test a to apart in 106 (Nordtvedt, 1977) and could conceivably look for "second-order" red-shift effects of O(AC/)2 (Jaffe and Vessot, 1976).

Advances in stable clocks have also made possible a new type of red-shift experiment that is a direct test of Local Position Invariance (LPI):a "null" gravitational red-shift experiment that compares two differenttypes of clocks, side by side, in the same laboratory. If LPI is violated,then not only can the proper ticking rate of an atomic clock vary withposition, but the variation must depend on the structure and compositionof the clock, otherwise all clocks would vary with position in a universalway and there would be no operational way to detect the effect (since oneclock must be selected as a standard and ratios taken relative to that clock).

Thus, we must write for a given clock type A,

T = t\U) = TA(1 - aA AU/c2) (2.24)

Then a comparison of two different clocks at the same location wouldmeasure

TA/TB = (TA/tB)o [ 1 _ (aA _ aB) A [//C2] £.25)

where (TA/TB)0 is t n e constant ratio between the two clock times observed

at a chosen initial location.A null red-shift experiment of this type was performed in April, 1978

at Stanford University. The rates of two hydrogen maser clocks and ofan ensemble of three SCSO clocks were compared over a 10 day period(Turneaure et al., 1983). During this time, the solar potential U changedsinusoidally with a 24 hour period by 3 x 10~13 because of the Earth'srotation, and changed linearly at 3 x 10""12 per day because the Earthis 90° from perihelion in April. However, analysis of the data set an upperlimit on both effects, leading to a limit on the LPI violation parameter

| a H_ a scso | < 1 0 - 2 (2.26)

The art of atomic timekeeping has advanced to such a state that it maysoon be necessary to take red-shift and Doppler-shift corrections intoaccount in making comparisons between timekeeping installations atdifferent altitudes and latitudes.

Constancy of the constants The other key test of Local PositionInvariance is the constancy of the nongravitational constants over cos-mological timescales (we delay discussion of the gravitational "constant"until Section 8.5). We shall not review here the various theories andproposals, originating with Dirac, that permit variable fundamental con-stants [for detailed review and references, see Dyson (1972)], rather weshall cite the most recent observational evidence (Table 2.4). The observa-tions range from comparisons of spectral lines in distant galaxies andquasars, to measurements of isotopic abundances of elements in the solarsystem, to laboratory comparisons of atomic clocks. Recently, Shlyakhter(1976a,b) has made significant improvements in the limits on variationsin the electromagnetic, weak, and strong coupling constants by studyingisotopic abundances in the "Oklo Natural Reactor," a sustained U235

fission reactor that evidently occurred in Gabon, Africa nearly two billionyears ago (Maurette, 1976). Measurements of ore samples yielded anabnormally low value for the ratio of two isotopes of samarium (Sm149/

Table 2.4. Limits on cosmological variation of nongravitational constants

Constant k

Fine structure Constant:a = e2/hc

Weak Interaction Constant:P = g(mlc/h3

Electron-Proton Mass Ratio:mjmp

Proton Gyromagnetic Factor:g me/mp

Strong Interactions:

gl

Limit on kjkper Hubble time2 x 1010 yr(H0 = 55kms" 1 Mpc" 1 )

4 x 10"4

8 x 10"2

8 x 10"2

2

1

10"1

8 x 10"2

Method

Re187 ft decay rate overgeological time

Mgll fine structure and21 cm line in radiosource at Z = 0.5

SCSO clock vs. cesiumbeam clock

Re187, K40 decay rates

Mass shift in quasarspectral lines (Z ~ 2)

Mgll, 21 cm line

Nuclear stability

Reference

Dyson (1972)

Wolfe, Brown, and Roberts(1976)

Turneaure and Stein (1976)

Dyson (1972)

Pagel (1977)

Wolfe, Brown, and Roberts(1976)

Davies (1972)

Limit fromOklo reactor(Shlyakhter, 1976a,b)

10"7

2 x 10~2

8 x 10"9


Sm147). Neither of these isotopes is a fission product, but Sm149 can bedepleted by a dose of neutrons. Estimates of the neutron fluence (inte-grated dose) during the reactor's "on" phase, combined with the measuredabundance anomaly yielded a value for the neutron capture cross sectionfor Sm149 two billion years ago which agrees with the modern value.However, the capture cross section is extremely sensitive to the energyof a low-lying level (E ~ 0.1 eV) of Sm149, so that a variation of only20 x 10 ~3 eV in this energy over 109 years would change the capturecross section from its present value by more than the observed amount.By estimating the contributions of strong, electromagnetic, and weakinteractions to this energy, Shlyakhter obtained the limits on the rate ofvariation of the corresponding coupling constants shown in Table 2.4,column 5 (see also Dyson, 1978).

2.5 Schiff 's ConjectureBecause the three parts of the Einstein Equivalence Principle dis-

cussed above are so very different in their empirical consequences, it istempting to regard them as independent theoretical principles. However,any complete and self-consistent gravitation theory must possess suffi-cient mathematical machinery to make predictions for the outcomes ofexperiments that test each principle, and because there are limits to thenumber of ways that gravitation can be meshed with the special relativisticlaws of physics, one might not be surprised if there were theoretical con-nections between the three subprinciples. For instance, the same mathe-matical formalism that produces equations describing the free fall of ahydrogen atom must also produce equations that determine the energylevels of hydrogen in a gravitational field, and thereby determine theticking rate of a hydrogen maser clock. Hence a violation of EEP in thefundamental machinery of a theory that manifests itself as a violation ofWEP might also be expected to emerge as a violation of Local PositionInvariance. Around 1960, Leonard I. Schiff conjectured that this kind ofconnection was a necessary feature of any self-consistent theory of gravity.More precisely, Schiff's conjecture states that any complete, self-consistenttheory of gravity that embodies WEP necessarily embodies EEP. In otherwords, the validity of WEP alone guarantees the validity of Local Lorentzand Position Invariance, and thereby of EEP. This form of Schiff's con-jecture is an embellished classical version of his original 1960 quantummechanical conjecture (Schiff, 1960a). His interest in this conjecture wasrekindled in November, 1970 by a vigorous argument with Kip S. Thorneat a conference on experimental gravitation held at the California Insti-


tute of Technology. Unfortunately, his untimely death in January, 1971cut short his renewed effort.

If Schiff's conjecture is correct, then the Eotvos experiments may beseen as the direct empirical foundation for EEP, and for the interpretationof gravity as a curved-spacetime phenomenon. Some authors, notablySchiff, have gone further to argue that if the conjecture is correct, thengravitational red-shift experiments are weak tests of gravitation theory,compared to the more accurate Eotvos experiment. For these reasons,much effort has gone into "proving" Schiff's conjecture. Of course, arigorous proof of such a conjecture is impossible, yet a number ofpowerful "plausibility" arguments using a variety of assumptions can beformulated.

The most general and elegant of these arguments is based upon theassumption of energy conservation. This assumption allows one to per-form very simple cyclic gedanken experiments in which the energy at theend of the cycle must equal that at the beginning of the cycle. This ap-proach was pioneered by Dicke (1964a), and subsequently generalizedby Nordtvedt (1975) and Haugan (1979).

Specifically, we restrict attention to theories of gravity in which thereis a conservation law of energy for nongravitating "test" systems thatreside in given static and external gravitational fields. To guarantee theexistence of such a law, it is sufficient for the theory to be based on aninvariant action principle [cf. Dicke's constraint (2)], but it is not necessary.We consider an idealized composite body made up of structureless testparticles that interact by some nongravitational force to form a boundsystem. For a system that moves sufficiently slowly in a weak, staticgravitational field, the laws governing its motion can be put into a quasi-Newtonian form (we assume the theory has a Newtonian limit); in par-ticular, the conserved energy function Ec associated with the conservationlaw can be assumed to have the general form

Ec = MKc2 - MRU(X) + |M R F 2 + O(MRU2, MRV\MRUV2) (2.27)

where X and V are quasi-Newtonian coordinates of the center of mass ofthe body, MR is the "rest" energy of the body, U is the external gravita-tional potential, and c is a fundamental speed used to convert units ofmass into units of energy. If EEP is violated, we must allow for thepossibility that the speed of light and the limiting speed of materialparticles may differ in the presence of gravity; to maintain this possibilitywe do not set c = 1 automatically in Equation (2.27) (see also footnote,p. 24). Note that V is the velocity relative to some preferred frame. In

problems involving external, static gravitational potentials, the preferredframe is generally the rest frame of the external potential, while in problemsinvolving cosmological gravitational effects where localized potentialscan be ignored, the preferred frame is that of the universe rest frame.[In problems involving both kinds of effects, the simple form of Equa-tion (2.27) no longer holds.] The possible occurrence of EEP violationsarises when we write the rest energy MRc2 in the form

MRc2 = Moc2 - £B(X, V) (2.28)

where Mo is the sum of the rest masses of the structureless constituentparticles, and £B is the binding energy of the body. It is the position andvelocity dependence of £B, a dependence that in general is a functionof the structure of the system, which signals the breakdown of EEP.Roughly speaking, an observer in a freely falling frame can monitor thebinding energy of the system, thereby detecting the effects of his locationand velocity in local nongravitational experiments. Haugan (1979) hasmade this more precise by showing that in fact it is the possible functionaldifference in £B(X, V) between the system under study and a "standard"system arbitrarily chosen as the basis for the units of measurement thatleads to measurable effects. Because the location and velocity dependencein £B is a result of the external gravitational environment, it is usefulto expand it in powers of U and V2. To an order consistent with thequasi-Newtonian approximation in Equation (2.27), we write

£B(X,V) = El + 8myUiJ(X) - tfntfV'V1 + O(E%U2,...) (2.29)

where UiJ is the external gravitational potential tensor [cf. Equa-tion (4.28)]; it is of the same order as U and satisfies U" = U. The quan-tities <5w# and bm\' are called the anomalous passive gravitational andinertial mass tensors, respectively. They are expected to be of order t\E%,where r\ is a dimensionless parameter that characterizes the strength ofEEP-violating effects; they depend upon the detailed internal structureof the composite body. Summation over repeated spatial indices i, j isassumed. The conserved energy can thus be written, to quasi-Newtonianorder,

Ec = (M0c2 - £g) - [(Mo - E°c-2)S»

^ ' J + O(M0U2,...) (2.30)

We first give examples of violations of Local Position and LorentzInvariance generated by £B(X, V). Consider two different systems at restin the gravitational potential. Each system makes a transition from one

quantum energy level to another, and emits a quantum of frequencyv = AEc/h. The ratio of the two frequencies is given from Equation (2.30) by

L (AEE)x (AEg) 2 Jc 2

In the case where <5mj/ oc diJ, the quantity in square brackets can beidentified as the coefficient a. 1 � a2 in Equation (2.25). Thus the anomalouspassive gravitational mass tensor dmtf produces preferred-location effectsin a null gravitational red-shift experiment. Consider the same two systemsfar from gravitating matter, but moving relative to the universe rest framewith velocity V. Then the ratio of the two frequencies is given by

jAQSmiV A(<5m{V2l V'VJ

(A£g)211 \ (A£°)x (A£°)2 J c2

Thus, the anomalous inertial mass tensor produces preferred-frame effectsin an experiment such as the Hughes-Drever experiment, where the twosystems in question are two excited states of 7Li nuclei of different azi-muthal quantum numbers in an external magnetic field. In this case,because the Zeeman splitting is the same for all levels in the absence ofa preferred-frame effect, (A£B)X = (A£B)2, however because of the possibleanisotropy in dm?, one would expect A(dm\J) to differ for transitionsbetween different pairs of levels [for further details, see Section 2.6]. Thusdmy is responsible for violations of Local Position Invariance and <5m{J

is responsible for violations of Local Lorentz Invariance.

In order to verify Schiff s conjecture, it remains only to show that 5mj,J

and dm\3 also produce violations of WEP. To do this we make use of acyclic gedanken experiment first used by Dicke (1964a). We begin with aset of n free particles of mass m0 at rest at X = h. From Equation (2.27),the conserved energy is simply nm0c2[l � t/(h)/c2]. We then form acomposite body and release the binding energy £B(h,0), in the form offree particles of rest mass m0, stored in a massless reservoir. The conservedenergy of the composite body is [nmoc

2 � £B(h,0)][l - [7(h)/c2] andthat of the reservoir is £B(h,0)[l � t/(h)/c2]. The composite body fallsfreely to X = 0 with an acceleration assumed to be A = g 4- <5A whilethe stored test particles fall with acceleration g = Vt/ (by definition). AtX = 0 we bring both systems to rest, and place the energies thus gained,

- [nm0 - £B(0, V)/c2]A � h - dniJgihi, and - EB(h, 0)g � h/c2

into the reservoir (we have assumed g, h, and V are parallel). Droppingterms of order (g � h)2, we see that the reservoir now contains conserved

energy

£B(h,O)[l - C/(0)/c2] - £jjg � h/c2 - (nmo - E°/c2)A � h - &nitfh>

From this we extract enough energy £B(0,0)[l � t/(0)/c2] to disassemblethe composite system into its n constituents, and enough energy � nmog � hto give the particles sufficient kinetic energy to return to their initialstate of rest at X = h. The cycle is now closed, and if energy is to be con-served, the reservoir must be empty. To quasi-Newtonian order, thisimplies

£B(h, 0) - £B(0,0) - (nm0 - E$/c2)SA � h - <5m{W = 0 (2.33)

Since

£B(h, 0) - £B(0,0) = dmy\UlJ � h (2.34)

we obtain

A' = g> + (5mik/MR)U{f - (<5mp/MRV (2.35)

where MR s nm0 � £B/c2. The first term is the universal gravitationalacceleration that would be expected in a theory satisfying WEP. Theremaining terms depend upon the body's structure through the anomalousmass tensors in £B(X, V). Hence a violation of Local Lorentz or PositionInvariance implies a violation of WEP. Equivalently, WEP {dm^ =dm{k = 0) implies Local Lorentz and Position Invariance. Equivalently,WEP implies EEP.

The gravitational red-shift experiment can also be studied within thisframework, using a cyclic gedanken experiment suggested by Nordtvedt(1975). The cycle begins as before with a set of n free particles of mass m0

at rest at X = h. We form a composite body and release the bindingenergy £B(h,0)[l � L/(h)/c2] in the form of a massless quantum whichpropagates to X = 0. Its energy there, compared to the energy £B(0,0)[l �l/(0)/c2] of a quantum emitted from an identical system at X = 0, isassumed to be given by (1 - Z)£B(0,0)[l - t/(0)/c2]. This energy isstored in a reservoir. Our goal is to evaluate the red shift Z. The body isthen allowed to fall freely to X = 0, where it is brought to rest, with thekinetic energy of motion,

-{nm0 - £B(0,V)/c2]A � h - <5mJW

added to the reservoir. If we substitute for A from Equation (2.35), wesee that the reservoir now contains energy (to quasi-Newtonian order)

(1 - Z)£B(0,0)[l - t/(0)/c2] - [rnno - £B(0,0)/c2]g � h -


We extract from the reservoir enough energy £B(0,0)[l - t/(O)/c2] todisassemble the system, and enough energy � nmog � h to return the nfree particles to the starting point. Again, conservation of energy requiresthat the reservoir be empty and therefore that Z must satisfy (to firstorder in g � h)

- ZE% + Elg � h/c2 - dmyVUiJ � h = 0 (2.36)

or

Z = [At/ - (8r4Jc2/E$) AUir\/c2 (2.37)

where A C / s g h and AUiJ = Vt/° � h. By a similar analysis one canshow that the second-order Doppler shift between an emitter moving atvelocity V and a receiver at rest, relative to a preferred universe restframe, is given by (Haugan, 1979)

ZD = - i V2/c2 + ttSnf/E&VV' (2.38)

Thus, the simple assumption of energy conservation has allowed us to"prove" Schiff's conjecture, as well as elucidate the empirical consequencesof possible violations of the three aspects of the Einstein EquivalencePrinciple.

Thorne, Lee, and Lightman (1973) have proposed a more qualitative"proof" of Schiff's conjecture for that class of gravitation theories thatare based on an invariant action principle, so-called Lagrangian-basedtheories of gravity. They begin by defining the concept of "universalcoupling": a generally covariant Lagrangian-based theory is universallycoupled if it can be put into a mathematical form (representation) inwhich the action for matter and nongravitational fields /NG containsprecisely one gravitational field: a symmetric, second-rank tensor # withsignature + 2 that reduces to J/ when gravity is turned off; and when ^ isreplaced by if, 7NG becomes the action of special relativity. Clearly, amongall Lagrangian-based theories, one is universally coupled if and only if itis a metric theory (for details see Thorne, Lee, and Lightman, 1973).

Let us illustrate this point with a simple example. Consider aLagrangian-based theory of gravity that possesses a globally flat back-ground metric i\ and a symmetric, second-rank tensor gravitational fieldh. The nongravitational action for charged point particles of rest massm0 and charge e, and for electromagnetic fields has the form

/NG = h + /in, + hm (2-39)


where

Io= -m0 jdr, dx2 = -(rj^ + h^)dx"dx\

^ ^ d4x (2.40)

where F^ = AVfll � A ^ , and where

IMNIWI1 ' (2.41)We work in a coordinate system in which if = diag( �1,1,1,1). To seewhether this theory is universally coupled, the obvious step is to assumethat the single gravitational field i/ v is given by

"/V = V + V (2.42)

This would make Jo and Jinl appear universally coupled. However, in theelectromagnetic Lagrangian, we obtain, for example,

tf* _ W = yj,** _ }f% + O(h3) (2.43)

where ||^""|| = | | ^ | | " 1 . Thus, there is no way to combine */�� and h^ intoa single gravitational field in ING, hence the theory is not universallycoupled. To see that the theory is also not a metric theory, we transformto a frame in which at an event 2P,

Note that in this frame, h^ ^ 0 in general, thus the action can be putinto the form

/NG = SRT + A/ (2.44)

where

JSRT = -m0 Jdx + e JAndx* - (len)'1 j^F^F^-fj)1!2d*x,

AI =

+ O(h3F2) (2.45)

where fcflC = h*9]? ¥= 0. So in a local Lorentz frame, the laws of physicsare not those of special relativity, so the theory is not a metric theory.Notice that in this particular case, for weak gravitational fields (|fy,v| « 1),the theory is metric to first order in h^, while the deviations from metricform occur at second order in h^. In the next section, we shall present a

mathematical framework for examining a class of theories with non-universal coupling and for making quantitative computations of itsempirical consequences.

Consider now all Lagrangian-based theories of gravity, and assumethat WEP is valid. WEP forces 7NG to involve one and only one gravita-tional field (which must be a second-rank tensor ty which reduces to r\far from gravitating matter). If /NG were to involve some other gravita-tional fields <j>, Kp, h^,... they would all have to conspire to produceexactly the same acceleration for a body made largely of electromagneticenergy as for one made largely of nuclear energy, etc. This is unlikelyunless i/ v and the other fields appear everywhere in JNG in the same form,for example, /((p)^^ if a scalar field is present, i//^ + ah^ if a tensor fieldis present, and so on. In this case, one can absorb these fields into a newfield g^ and end up with only one gravitational field in JNG. This meansthat the theory must be universally coupled, and therefore a metrictheory, and must satisfy EEP.

One possible counterexample to Schiff 's conjecture has been proposedby Ni (1977): a pseudoscalar field <j> that couples to electromagnetism ina Lagrangian term of the form ^""F^F^, where s*�11 is the completelyantisymmetric Levi-Civita symbol. Ni has argued that such a term,while violating EEP, does not violate WEP, although it does have theobservable effect of producing an anomalous torque on systems of elec-tromagnetically bound charged particles. Whether this torque then canlead to observable WEP violations is an open question at present.

2.6 The THs/i FormalismThe discussion of Schiff's conjecture presented in the previous

section was very general, and perhaps gives compelling evidence for thevalidity of the conjecture. However, because of the generality of thosearguments, there was little quantitative information. For example, nomeans was presented to compute explicitly the anomalous mass tensors(5mj/ and 8m\J for various systems. In order to make these ideas moreconcrete, we need a model theory of the nongravitational laws of physicsin the presence of gravity that incorporates the possibility of both non-metric (nonuniversal) and metric coupling. This theory should be simple,yet capable of making quantitative predictions for the outcomes of ex-periments. One such "model" theory is the THe/x formalism, devised byLightman and Lee (1973a). It restricts attention to the motions and elec-tromagnetic interactions of charged structureless test particles in an ex-ternal, static, and spherically symmetric (SSS) gravitational field. It


assumes that the nongravitational laws of physics can be derived froman action /NG given by

f NG = Jo + hat + Iem>

\{T -

E2 - li-^d'x (2.46)

(we use units here in which x and t both have units of length) where mOa,ea, and x£(t) are the rest mass, charge, and world line of particle a, x° = t,v»a = dxljdt, E = \A0 - A o, B = (V x A), and where scalar productsbetween 3-vectors are taken with respect to the Cartesian metric 8ij.The functions T, H, e, and n are assumed to be functions of a single ex-ternal gravitational potential 4>, but are otherwise arbitrary. For an SSSfield in a given theory, T, H, e, and /x will be particular functions of O. Itturns out that, for SSS fields, equations (2.46) are general enough to en-compass all metric theories of gravitation and a wide class of nonmetrictheories, such as the Belinfante-Swihart (1957) theory and the nonmetrictheory discussed in Section 2.5. In many cases, the form of /N G in equation(2.46) is valid only in special coordinate systems ("isotropic" coordinatesin the case of metric theories of gravity). An example of a theory thatdoes not fit the THsfi form of /NG is the Naida-Capella nonmetric theory(see Lightman and Lee, 1973a for discussion). Cases such as this mustthen be analyzed on an individual basis. For an "en" formalism, seeDicke (1962).

(a) Einstein Equivalence Principle in the THe/x formalismWe begin by exploring in some detail the properties of the form-

alism as presented in equations (2.46). Later, we shall discuss the physicalrestrictions built into it, and shall apply it to the interpretation of experi-ment.

In order to examine the Einstein Equivalence Principle in this formalismwe must work in a local freely falling frame. But we do not yet knowwhether WEP is satisfied by the THsn theory (and suspect that it is not,in general), so we do not know to which freely falling trajectories localframes should be attached. We must therefore arbitrarily choose a setof trajectories: the most convenient choice is the set of trajectories ofneutral test particles, i.e., particles governed only by the action l0, since

their trajectories are universal and independent of the mass mOa. We makea transformation to a coordinate system x" = (?, x) chosen according tothe following criteria: (i) the origins of both coordinate systems coincide,that is, for a selected event 3P, xx{@) - x\0>) = 0, (ii) at 0>, a neutral testbody has zero acceleration in the new coordinates, i.e., d2xJ'/dt2^ = 0,and in the neighborhood of 9 the deviations from zero acceleration arequadratic in the quantities Ax* = x* � x%0>), and (iii) the motion of theneutral-test body is derivable from an action Jo. The required transforma-tion, correct to first order in the quantities g0? and gj, � x, assumed small, is

x = Hy\x + | t f 0-»Togof2 + ±Ho ' H 2xg0 � x - gox

2)] (2.47)

where the subscript (0) and superscript (') on the functions T, H, E, andfi denote

To = T(x* = xs = 0), r 0 = ^r/a<D|x.=xa=0 (2.48)

and where

go = V* (2.49)

The action Io in the new coordinates then has the form

'o = - I > o a fd - v2a)

il2 dt{\ + O[(xa)2]} (2.50)

where va = dxjdf. Note that our choice of the multiplicative factors Tj / 2

and HQ12 resulted in unit coefficients in /0, making it look exactly likethat of special relativity. Similarly the actions /int and Iem can be rewrittenin the new coordinates, with the result

/in, = 2 X UfV? dt{\ + O[(XS)2]}, (2.51)

Im = (Snr1eoTlo'

2Ho1

- To 'HOEE V O ' ^ ( l - JT'OTO lHE 1/2Aogo � x)

+ H0TE 1 / 2 r 0 fg 0 -(E x S)(l - To 'HoEE V O 1 ) } ^

+ [corrections of order (x*)2] (2.52)

where

A% = (dx*/dxii)Aa

E = *At - A o, 6 = V x A (2.53)


and

Ao = (2ro /T'o)(^o * + iT'oTo ' - iH'oHo *) (2.54)

Let us now examine the consequences for EEP of physics governed byING. Focus on the form of JNG at the event 0>(xj = t = 0), since local testexperiments are assumed to take place in vanishingly small regions sur-rounding 3". Because such experiments are designed to be electricallyneutral overall, we can assume that the E and B fields do not extendoutside this region. Then at 9

/ N G = - X > o a f ( l -v2a)

+ (8TT)- hoT^Ho 1/2 J [ £ 2 - (To 'Hoeo Vo X)B2] d*x (2.55)

We first see that, in general, /NG violates Local Lorentz Invariance. Asimple Lorentz transformation of particle coordinates and fields in 7NG

shows that JNG is a Lorentz invariant if and only if

To1/fo£oVo1 = l or eofio=TolHo (2.56)

Since we have not specified the event 0>, this condition must hold through-out the SSS spacetime. Notice that the quantity (TQ lHtfo Vo x)1/2 playsthe role of the speed of light in the local frame, or more precisely, of theratio of the speed of light clight to the limiting speed c0 of neutral testparticles, i.e.,

To 'Hoeo Vo l = (clight/c0)2 (2.57)

Our units were chosen in such a way that, in the local freely falling frame,c0 = 1; equivalently, in the original THsfi coordinate system [cf. Equa-tion (2.46)]

c0 = (To/Ho)1'2, clight = (EoAio)-1/2 (2.58)

These speeds will be the same only if Equation (2.56) is satisfied. If not,then the rest frame of the SSS field is a preferred frame in which /NG takesits THe/j. form, and one can expect observable effects in experiments thatmove relative to this frame. Thus, the quantity 1 � ToHo lfioMo plays therole of a preferred-frame parameter: if it is zero everywhere, the formalismis locally Lorentz invariant; if it is nonzero anywhere, there will be pre-ferred-frame effects there. As we shall see, the Hughes-Drever experimentprovides the most stringent limits on this preferred-frame parameter.


Next, we observe that /NG is locally position invariant if and only if

o 1/2 = [constant, independent of 9\o 1/2 = [constant, independent of >] (2.59)

Even if the theory is locally Lorentz invariant (TQ 1H0EQ VO * = 1>independent of &) there may still be location-dependent effects if thequantities in Equation (2.59) are not constant. This would correspond,for example, to the situation discussed in Section 2.3, in which differentparts of the local physical laws in a freely falling frame couple to differentmultiples of the Minkowski metric; in this case, free particle motioncoupling to 7 itself, electrodynamics coupling to the position-dependenttensor i\* =. eTi/2H'i/2ti in the manner given by the field Lagrangianri*'"'rivfFllvFltf. The nonuniversality of this coupling violates EEP andleads to position-dependent effects, for example, in gravitational red-shift experiments (also see Section 2.4). An alternative way to characterizethese effects in the case where Local Lorentz Invariance is satisfied is torenormalize the unit of charge and the vector potential at each event& according to

e*a = ettso 1/2To 1/4H S/4, Af = A^T^H^ (2.60)

then the action, (2.55), takes the form

+ (8TT)- * j(E*2 - B*2) d4x (2.61)

This action has the special relativistic form, except that the physicallymeasured charge e* now depends on location via Equation (2.60), unlessE0TII2HQXI2 is independent of 9. In the latter case, the units of chargecan be effectively chosen so that everywhere in spacetime,

soTlol

2Ho112 = 1 (2.62)

Note, however, that if LPI alone is satisfied, one can renormalize thecharge and vector potential to make either £0TQI2HQ

1/2 = 1 orfioTll2Ho 1/2 = 1, but not both, thus in general LLI need not be satisfied.

Combining Equations (2.56), (2.59), and (2.62), we see that a necessaryand sufficient condition for both Local Lorentz and Position Invarianceto be valid is

e0 = n0 = (Ho/T0)112, for all events 9 (2.63)

Consider now the terms in /NG, in Equations (2.50)-(2.52) that dependon the first-order displacements x, t from the event 9. These occur onlyin /em, and presumably produce polarizations of the electromagnetic fieldsof charged bodies proportional to the external "acceleration" g0 = V4>.One would expect these polarizations to result in accelerations of com-posite bodies made up of charged test particles relative to the local freelyfalling frame (i.e., relative to neutral test particles), in other words, toresult in violations of WEP. These terms are absent if Fo = Ao = 0, and

U i.e.,

eoTlo

l2Ho 1/2 = const, noTl'2Ho m = const,

EoHo^HoTo1 (2.64)

Again, the units of charge can be normalized so that

e0 = Ho = (H0/T0)1'2, for all 9 (2.65)

But this condition also guarantees Local Lorentz and Position Invariance.Thus, within the THe/j. formalism, for SSS fields

[Equation (2.65)] => WEP,

[Equation (2.65)] => EEP (2.66)

However, the above discussion suggests that WEP alone may guaranteeEquation (2.65) and thereby EEP. We can demonstrate this directly bycarrying out an explicit calculation of the acceleration of a compositetest body within the THsfi framework. The resulting restricted proof ofSchiff's conjecture was first formulated by Lightman and Lee (1973a).

(b) Proof of Schiff's conjectureWe work in the global THsfi coordinate system in which JNG

has the form Equation (2.46). Variation of /NG gives a complete set ofparticle equations of motion and "gravitationally modified" Maxwell(GMM) equations, given by

(d/dt)(HW~ \ ) + {W- XV(T - Hvl) = aL(xa),

aL(x0) = (ea/mOa){VAo(xa) + V[va � A(xfl)] - dA{xa)/dt},

V � (EE) = 4np,

V x ( ^ » B ) = 4TTJ + d{eE)/dt (2.67)

where W = {T- Hv2a)

112,p = Y.aea<53(x - xfl), J = £aeaya53(x - xa), andaL is the Lorentz acceleration of particle a. These equations are used to

calculate the acceleration from rest of a bound test body consisting ofcharged point particles. A number of approximations are necessary tomake the computation tractable. First, the functions T, H, e, and fi,considered to be functions of <D are expanded about the instantaneouscenter of mass location X = 0, in the form

T(O) = To + T'ogo � x + O(g0 � x)2 (2.68)

where To = T(x = 0), T'o = dT/d<S>\x=0. As long as the body is smallcompared to the scale over which d> varies, we can assume that g0 � x « 1,and work to first order in g0 � x. Second, we assume that the internalparticle velocities and electromagnetic fields are sufficiently small so wecan expand the equations of motion and GMM equations in terms ofthe small quantities

v1 ~ e2/mor « 1

where r is a typical interparticle distance. By analogy with the post-Newtonian expansion to be described in Chapter 4, we call this a post-Coulombian expansion; for the purpose of the present discussion weshall work to first post-Coulombian order. We expect the single particleacceleration to contain terms that are O(g0) (bare gravitational accelera-tion), O(v2) (Coulomb interparticle acceleration), O(gf0t;

2) (post-Cou-lombian gravitational acceleration), O(v4) (post-Coulombian interparticleacceleration), O(g0v*) (post-post-Coulombian gravitational acceleration),and so on. To O(g0v

2), we obtain

o + itf'oHo'goi;2

+ (T'oTo 1 - H'0H0- l)g0 � vavfl + Ty2H» X W (2-69)

To write the Lorentz acceleration aL(xa) directly in terms of particlecoordinates, we must obtain the vector potential A^ in this form to anappropriate order. In a gauge in which

£Mo,o - V � A = 0 (2.70)

the GMM equations take the form

V240 - ej^o.oo = 4ns~1p-e~1\e- (\A0 - A,o),

V2A - e/xA.oo = ~^nJ + (sp)-xV(sp)V � A - ^ V / z x (V x A) (2.71)

These equations can be solved iteratively by writing

Ao = A^ + A%\ A = A(0) + A(1) (2.72)

where A(v/A(°y ~ O(g0), and solving for each term to an appropriateorder in v2. The result is

A o = -4>A = A(0) + O(0O) (2.73)

where

a

The resulting single-particle acceleration is inserted into a definition ofcenter of mass. It turns out that to post-Coulombian order, it sufficesto use the simple center-of-mass definition

Y - - i v - v (2-75)A = m 2 J wOaXa> m = ZJ m0a

a a

We then compute d2X/dt2, substituting the single-particle equations ofmotion to the necessary order, and using the fact that, at t = 0, X = 0,dX/dt = 0. The resulting expression is simplified by the use of virialtheorems that relate internal structure-dependent quantities to eachother via total time derivatives of other internal quantities. As long aswe restrict attention to bodies in equilibrium, these time derivatives canbe assumed to vanish when averaged over intervals of time long comparedto internal timescales. Errors generated by our choice of center-of-massdefinitions similarly vanish. To post-Coulombian order, the requiredvirial relation is

= o (2.76)

where angular brackets denote a time average, and where

(2.77)a ab

where xab = xa � xb, rab = |xa(,|, and the double sum over a and b excludesthe case a = b. The final result is

d2Xl/dt2 = g{- ^[TE1'%1rojea/m)

+ 0J'[ro1/2eo1(l-ToHo1eo/Xo)]

x (JEjf/m + <5yEE» (2.78)

where

Vo (2-79)

where F o is given by Equation (2.54), and where

Ef? = <Q">, EES = ( £ Q " ) (2-8°)

The first term gl is the universal acceleration of a neutral test body (gov-erned by Io alone); the other two terms depend on the body's electro-magnetic self-energy and self-energy tensor. These terms vanish for allbodies (i.e., WEP is satisfied) if and only if

(2.81)

at any event 0>, which is equivalent to Equation (2.63). Hence

WEP => EEP

and Schiff's conjecture is verified, at least within the confines of theTHEH formalism.

It is useful to define the gravitational potential U whose gradientyields the test-body acceleration g; modulo a constant

U(x)= ~^T'0Ho xg0-x (2.82)

If the functions T, H, e, and fi are now considered as functions of Uinstead of <D, then because of Equations (2.48), (2.54), (2.58), and (2.82),

T o = - c 20 (

] | x = 0 (2.83)

(c) Energy conservation and anomalous mass tensorsBecause the THefi formalism is based on an action principle, it

possesses conservation laws, in particular a conservation law for energy,and so is amenable to analysis using the conserved-energy frameworkdescribed in Section 2.5. The main products of that framework are theanomalous inertial mass tensor Sm[J and passive gravitational mass tensorSm'j obtained from the conserved energy. These two quantities then yieldexpressions for violations of WEP, Local Lorentz Invariance, and LocalPosition Invariance.

As a concrete example (Haugan, 1979), we consider a classical boundsystem of two charged particles. As in the above "proof" of Schiff's con-jecture we work to post-Coulombian order and to first order in g0 � x. Wefirst formulate the equations of motion in terms of a truncated action

?NG = h + An, (2-84)


where 7im is rewritten entirely in terms of particle coordinates by substitut-ing the post-Coulombian solutions for A^, Equation (2.73), into Iint.Variation of 7NG with respect to particle coordinates then yields the com-plete particle equations of motion. We identify a Lagrangian L using thedefinition

7NG = $Ldt (2.85)

We next make a change of variables in L from xu x2, vl5 and v2, to thecenter of mass and relative variables

X = ( m ^ + m2x2)/m, x = x t � x2,

V = dX/dt, v = dx/dt (2.86)

where m = mx + m2, n = mlm2/m. A Hamiltonian H is constructed fromL using the standard technique

Pj s dL/dV{ pi = BL/dv3,

H s PJVj + pV - L (2.87)

The result is

H = Tj'2m(l + iTiTo »g0 � X) + n'2HE lP2/2m

eie2/r)go � X - Tj/2Ho 2[(p � P)2

o \e,e2lr)\P2 + (n � P)2]/2m2}

+ hT0li0Ho \m2 - m1)(e1e2/r)(p P + ii pfi

+ O(p4) + O(P4) (2.88)

where n = x/r. The post-Coulombian terms O(p4) and O(P)4 neglected inEquation (2.88) do not couple the internal motion and the center-of-massmotion and thus do not lead to violations of EEP. We now average H overseveral timescales for the internal motions of the bound two-body system,assumed short compared to the timescale for the center-of-mass motion.The average is simplified using virial theorems obtained from Hamilton'sequations for the internal variables derived from H. The relevant expres-sions are

+ post-Coulombian terms), (2.89)

H nj(n � p)]> (2.90)

Notice that although the post-Coulombian corrections in Equation (2.89)may depend on the center of mass variables P or X, this dependence doesnot affect the form of if; it is only the explicit dependence on P and X inEquation (2.88) that generates the center-of-mass motion. The resultingaverage Hamiltonian is then rewritten in terms of V using VJ = d<H>/dPJ.The conserved energy function Ec used in Section 2.5 is then defined to beEc = Tll2Ho\H}, so that at lowest order, Ec = mc\ = m(ToHo *). Theresult is

Ec = M(T0Ho ') + i

x (£«% )] | [ ^^o}

xT'oHo'go-X (2.91)

where

M = m + <tfo V / t y + To-1/2£0- W ) (2-92)

By defining the "binding" energy and energy tensor by

Ef = -co2{i/o-y/2/i + To 1 / 2 e o ' e ^ / r )

1EES + [post-Coulombian corrections],

& lEf? (2.93)

and using Equation (2.82), we cast Equation (2.91) into the form

£c = MRC2, - MRt/(X) + \MKV2 (2.94)

where

MRC2. = mcl - Ef + tymiWVi - dm\!Ui} (2.95)

with

dm? = 2(1 - Toffo'eoMoX^w + £^)/cS.

j/ isi)<5ij' (2.96)

Substitution of these formulae into Equation (2.35) for the center-of-massacceleration of the system yields precisely Equation (2.78).

One advantage of the Hamiltonian approach is that it can also beapplied to quantum systems (Will, 1974c). This is especially useful indiscussing gravitational red-shift experiments since it is transitions be-tween quantized energy levels that produce the photons whose red shiftsare measured. For the idealized gravitational red shift experiments dis-cussed in Section 2.5, only the anomalous passive mass tensor 5m^ isneeded. The simplest quantum system of interest is that of a charged

particle (electron) moving in a given external electromagnetic potential ofa charged particle (proton) at rest in the SSS field, i.e., a hydrogen atom.For such a system the truncated Lagrangian [Equation (2.85)] has the form

L = - me(T0 - Hov2)112 - eA^if (2.97)

where m0 = me and e� + \e\ for the electron. We shall ignore the spatialvariation of T, H, e, and \i across the atom, hence we evaluate each at x = 0.The Hamiltonian obtained from L is given by

H = n /2[me2 + Ho > + eA|2]1/2 + eA0 (2.98)

where Pj = dL/dvK Introducing the Dirac matrices

where / is the two-dimensional unit matrix and <rt are the constant Paulispin matrices, we perform the "square root" in H and obtain

H = Ti^lmJ + Ho 1/2a � (p + <?A)] + eAol (2.100)

The gravitationally modified Dirac (GMD) equation is then

H\\l>y = ih(8/dt)\il/y (2.101)

For most applications it is more convenient to use the semirelativisticapproximation to if obtained by means of a Foldy-Wouthuysen trans-formation, yielding

H = Ty\me + Ho J |p + eA\2/2me - Ho 2p*/Sm2 + HQ leho � B/2m,]

+ eA0-HQ l{eh/4m2)a - (Exp-%ih\ xE)

(2.102)

where we have made the usual identification p -» � ih\ and have ignoredthe effects of spatial variations in T0,H0,s0, or fi0 on the atomic structure.For a charged particle with magnetic moment Mp at rest at the origin, thevector potential as obtained from the GMM equations is given (to thenecessary accuracy) by

Ao = -e/sor, A = iu0Mp x x/r3 (2.103)

The Hamiltonian then takes the form

H = Hr + Hs + H( + HM + O(p6) (2.104)

where

Hr = Tl'2me,

H(=-Tlo'

2Ho2p4/^m2 - Hvh^{e2hl4m2er

3)o � L,

Hu = T^2[Ho l(ehl2me)o � B] (2.105)

where L = r x p is the angular momentum of the electron. The four piecesof H are the usual rest mass (Hr), Schrodinger (Hs), fine-structure (H{) andhyperfine-structure (Hhf) contributions. We have ignored the Darwin term(oc V � E). The magnetic field produced by the proton is given by

B = V x A = - i^o{[M p - 3n(fi � Mp)]/r3 - (87t/3)Mp<53(x)} (2.106)

We must first identify the proton magnetic moment. From the hyperfineterm Hhl, it is clear that the magnetic moment of the electron is given by

Me = T£/2#o H - eh/2me)o (2.107)

It is then reasonable to assume that the magnetic moment of the proton hasthe same dependence on To and Ho,

Mp = T^Ho l(gpeh/2mjap (2.108)

where gp is the gyromagnetic ratio of the proton and mp is its mass. Then

Hbf = -yoToHo2(gpe2h2/4memp)

x ae � {|>p - 3fi(il � ffp)]/r3 - (8rt/3)«Tp^

3(x)} (2.109)

Solving for the eigenstates of the Hamiltonian using perturbation theoryyields

£ = Ty\me + £p(HoTolEo2) + *AH0TZ h^ 2)2

l (2.H0)

where ip, Su and SM are the usual expressions for the principal, fine-structure, and hyperfine-structure energy levels in terms of atomic con-stants me, e, mp, gp, h, and quantum numbers. In order to calculate theanomalous mass tensors <5mj/, we must determine the manner in whichE varies as the location of the atom is changed. Expanding E to first orderin g � X, substituting Equation (2.82), and converting to the conservedenergy function Ee - E(Tk'2/H0), we obtain Equation (2.30) (with V = 0),with

E% = Ef + El-¥EW (2.111)

where

£ B � �e0 <5p, £ B � �ttoio e0 0(,

£gF=-Wo3'hf (2-112)

and

H * 2ro(£|s/cg)^ii (2.113)

= 4ro{El/cl)Sij (2.114)

= (3r0 - A0)(E^/c2)8^ (2.115)

Compare Equation (2.113) with Equation (2.96).A useful fact that emerges from the solution for the energy eigenstates

is that the Bohr radius is given by

a = (e0Ti'2/H0){h2/mee2) (2.116)

This will be important in analyzing the gravitational red shift of micro-wave cavities.

(d) Limitations of the THefi formalismThe THefi formalism is a very strong - perhaps overly strong -

idealization of the coupling of electromagnetism to gravity. The questionnaturally arises, can the formalism be applied to realistic physical situa-tions where there are no SSS fields and where strong and weak interactionsmay be present? We shall discuss each of these points in turn.

(i) SSS Fields In practical experimental situations, say in an Earth-bound laboratory, there are, strictly speaking, no SSS fields: orbital androtational motions of the planets cause the gravitational potentials tochange with time, and the superposition of fields from the Sun and planetsleads to aspherical fields. However, the evolution of the gravitational fieldsoccurs on a much longer timescale than the internal (atomic) timescalesof typical laboratory experiments, and so the fields can be treated quasi-statically. Furthermore, most experiments of interest single out one static,nearly spherical gravitational field by exploiting a symmetry, by modula-tion, or by some other technique. (For example, singling out of the solarfield by searching for a torque with a 24 h period in the Dicke�Braginskyversions of the Eotvds experiment.) A potentially more serious criticismof the SSS restriction is the possibility of relativistic, nonisotropic effectsdue for example to the orbital motion of the planets, or to the motion ofthe solar system relative to the mean rest frame of the universe. These ef-


fects would produce off-diagonal terms in the action /NG, such as F � v in Jo

or G � (E x B) in /em, where F and G are vector gravitational functions.In the case of the overall motion of the solar system, one can see that theframe in which the solar potential is spherical is in motion relative tothe frame in which the cosmic background field is spherical (isotropic),therefore there must be two limiting actions of the THe/i form, oneapplicable to each situation. These limiting cases can be handled by asingle action of the THefi form only if the theory is Lorentz invariant,i.e., only if TH~1e/i = 1. Nevertheless, if either of these off-diagonal effectsoccurs, they will be smaller than the dominant SSS effects by factors oforder |v| ~ [orbital velocity of planets] ~ 10 ~4 or |w| ~ [solar systemvelocity] ~ 10"3. The simplest way to summarize is as follows: the re-striction to SSS fields is an approximation that may overlook observableeffects, however, the experimental consequences that emerge from thepure SSS version are sufficiently interesting and, we believe, sufficientlygeneric to a broad class of gravitational theories, that powerful conclu-sions about the nature of gravity can be made within the standard THs/iframework. With this caveat in mind, for most of the remainder of thischapter we will assume that every experiment discussed takes place in aSSS field.

(ii) Weak and strong interactions The coupling of classical electromag-netic fields to gravitation is well understood within metric theories ofgravity (see Section 3.2) and has been formulated in many nonmetrictheories. By contrast, the laws of weak and strong interactions have onlyrecently been given an adequate mathematical representation even in theabsence of gravity, and the problem of their coupling to gravity is madeeven more complicated by the fact that the theories of these interactionsfundamentally involve quantum field theory. Thus, at present, electro-magnetism is the only interaction amenable to a detailed analysis of EEPusing something like the THe/j. formalism. Nevertheless, a violation ofEEP by electrodynamics alone can lead to many observable effects, barringfortuitous cancellations, and to several important experimental tests. Con-sequently, for the remainder of this discussion we shall simply ignore thestrong and weak interactions, or if necessary assume that they obey EEP.

(e) Application to tests of EEPWe now turn to the experiments that test EEP and study the

constraints they place on the coupling of electromagnetism to gravity inSSS gravitational fields.

Tests of WEP Equation (2.78) gives the acceleration of a com-posite body through post-Coulombian order in an external SSS field.However, for the purpose of comparing the predicted acceleration withthe results of Eotvos experiments, that expression is not accurate enough.The WEP-violating terms in Equation (2.78) are of order EBS/m ~ 10" 3

for atomic nuclei; therefore, WEP-violating terms of order (EES/m)2 ~{EES/m)v2 ~ 10 ~6 would also be strongly tested by Eotvos experimentsaccurate to a part in 1012. To obtain these terms, Haugan and Will (1977)extended the Lightman-Lee computation to post-post-Coulombian order(the Hamiltonian method could also have been used). When specializedto composite bodies that are spherical on average (a good approximationfor experimental situations), the resulting acceleration is given by

d2X/dt2 = g{l + (Efs/Mc2)[2r0 - f(l - ToH

(2.117)

where [cf. Equations (2.77), (2.80), and (2.93)]

Ef = -ab

o Vo ( l eaebr;b\va � yb + (vfl �ab

VVo ( I WaV[v a � yb- (ya � *ab)(yb � x j r i 2 ] ) (2.118)\ Iab

Because we shall shortly obtain a very tight upper limit on the coefficient1 � T0HQ

1e0^i0 from the Hughes-Drever experiment, we shall simply setit equal to zero in Equation (2.117). Then the results for the Eotvos ratiosdefined in Equation (2.2) are

^ES = |2T0|, r,� = |2A0| (2.119)

The quantities E|S an<l £ B S given by Equation (2.118) were estimated forvarious substances in Section 2.4 [Equations (2.8) and (2.9)] and providedexperimental limits on nES and nm that are equivalent to

|ro | < 2 x 10-10, |A0| < 3 x 10-6 (2.120)

Recall that if EEP is satisfied, r 0 = Ao = 0.

Tests of LLI The Hughes-Drever experiment can now beanalyzed in detail using the TH&n formalism (Haugan, 1978). Equations(2.95) and (2.96) demonstrate the possibility of an inertial mass anisotropy3m\j that leads to a contribution to the binding energy given by

SEB = -$8m\'ViVj (2.121)

where V is the velocity of the body relative to the THefi coordinatesystem. This term could lead to energy shifts of states having differentvalues of 5m\j and thus to observable effects in a quantum mechanicaltransition between these states. In the case of the Hughes-Drever experi-ment, the system, a 7Li nucleus, can be approximated as a two-bodysystem consisting of a J = 0 core (two protons and four neutrons) of charge+ 2, and a valence proton in a ground state with angular momentum of1. The spin of the proton couples to its angular momentum to yield atotal angular momentum J = f. In an applied magnetic field, the fourmagnetic substates Af, = ±j, +§ are split equally in energy, giving asinglet emission line for transitions between the three pairs of states. Howdoes SEB alter the energies of these four states? The isotropic part ofdm\J oc EB

sd'J simply shifts all four levels equally, since < JMi\e1e2r~ l\ JM3}is independent of Mj. However, the other contribution to 8m\J oc £{[? doesshift the levels unequally. We first decompose V into a component V^parallel to the applied magnetic field and a component V± perpendicularto it. Then

0] (2.122)

where 0, (j> are polar coordinates appropriate to the orbital wave function�Aim, =/('")5inii(0>0)- By combining the orbital wave function and spinstates into states of total J, Ms, we then calculate the expectation valueof (x'xJ/r3)ViVJ in states of different M,. Inserting these results into theformula, (2.121), for 8EB, and taking the difference in the energy shiftsbetween adjacent Af, states, we find that the singlet line splits into a trip-let with relative energies

Mi - -i) = o,M-i--!)=-<5 (2-123)

where

S = &(£f/cg)(l - ToHo lHH0){Vl - 2VD (2.124)

In the notation of Section 2.4, Equation (2.13), we have

<5ES = £ ( 1 - ToHo 'eolhKVl - 2F(j) (2.125)

The limit set by the experiment was |<5ES| < 10~ 22. If we treat the laboratoryas being in motion in the SSS field of the Sun, then V^ ~ VL ~ 10"4;hence, as evaluated at the Earth,

|1 - ToHo'Bo/ioU = I1 - (Co/clighl)2U < 10"13 (2.126)

We can also assume the laboratory to be moving in the quasistatic, spher-ically symmetric background field of the universe, with velocity V^ ~VL ~ 10"3, then for that portion of the THEH fields associated with theasymptotic cosmological model (labeled by the subscript oo), we obtain

Il-TVO^^HT15 (2.127)

Although there may be observable effects due to the possible nonmeshingof these two SSS fields into a single THefi field, they are unlikely to cancelthe effects we have derived and negate the limits obtained above.

The central conclusion is that to within at least a part in 1013, LocalLorentz Invariance is valid.

Tests of Local Position Invariance Consider gravitationalred-shift experiments. Suppose, for example, one measures the gravita-tional red shift of photons emitted from various transitions of hydrogen,such as principal transitions, fine-structure transitions within a principallevel, or a hyperfine transition in, say, the ground state (21 cm line, basisfor hydrogen maser clocks). Then, substituting Equations (2.113)�(2.115)into Equation (2.37), we obtain (Will, 1974c)

Z p = [ l - 2 r 0 ] A C / / c 2 ,

Z f = [ l - 4 r 0 ] A l / / c &

Zhf = [1 - (3r0 - Ao)] AU/c2. (2.128)

Notice that the three shifts are different in general. Thus the gravitationalred shift depends on the nature of the clock whose frequency shift isbeing measured unless Fo = Ao = 0, i.e., unless LPI is satisfied [Equation(2.59)]. The red-shift parameters a discussed in Section 2.4(c) can thus beread off from Equations (2.128). The Vessot-Levine Rocket red-shift ex-periment thus sets the limit

|(3r0 - Ao)| < 2 x KT4 (2.129)

To analyze the Stanford null gravitational red-shift experiment, we mustcalculate the energy of a microwave cavity. The energy in question is thatof an electromagnetic mode whose wavelength is determined by thelength of the cavity. The vector potential for the mode can be written, insecond quantized notation,

A = N(ateexp[i(k � x - cot)] + h.c.) (2.130)

where a+ is a creation operator, e a polarization vector, N a normaliza-tion constant, and h.c. denotes Hermitian conjugate. We have suppressedthe sum over k and e. The GMM equations (2.67) yield the dispersionrelation

|k|2 - E0H0CO2 = 0 (2.131)

The energy of the electromagnetic field obtained from the canonicalHamiltonian is

E = %(aa< + a^a)hco (2.132)

However, for a stationary mode, the wave number k must satisfy

k L = nn (2.133)

where |L| is the length of the cavity and n is an integer. But it is clear that|L| is proportional to an integer (number of atoms in a line along thelength of the cavity) times the Bohr radius a (which determines the inter-atomic spacing). But from Equation (2.116) we find L cc(eoTo/2Ho l) x(atomic constants, integers), hence, |k| cc H0TQ 1/2SQ

1. Combining Equa-tions (2.131) and (2.132), we finally obtain

E = c s o H o V / V o ^ e o 3 / 2 (2-134)

where < SGSO depends only on atomic constants and integers. Expandingin terms of g0 � x, and calculating the conserved energy function Ec, weobtain Equation (2.30) with

pSCSO _ _ v -1/2.-3/2C B � 0scsoA to fco >

6n#= i (3r 0 + Ao)(EF°/co)*y (2.135)

Thus for a superconducting-cavity stabilized oscillator clock

Zscso = [1 - l (3r 0 + Ao)] AL//c2 (2.136)

or, in the comparison between a cavity clock and a hydrogen maserclock [see Equation (2.31)]

+ f (r0 - A0)t//c2] (2.137)

The experimental limit is thus

|ro-Ao|<l(T2 (2.138)

(f) The Belinfante-Swihart nonmetric theoryAs a specific example of the application of the THz\i formalism

to the analysis of gravitational theory and experiment we consider theBelinfante-Swihart (1957a,b,c) theory. This theory treats gravity as asymmetric second rank tensor field B on a Riemann-fiat backgroundmetric (prior geometry), t\. We first define a "particle metric" g^ accordingto

H ~ &,) = $ (2-139)where K is an arbitrary constant, and where indices on B^ and A v areraised and lowered using n^. In a coordinate system in which t\ = diag( � 1,1,1,1), the nongravitational action can be put into the form (Lee andLightman, 1973)

r ( dx11 dxv\1/2 c/NO = - 1 «o. J ( - 0,v -£-ft) dt + ^ea JA.ixl) dx" -

(2.140)

where, through second order in B, H^ is related to the Maxwell field

H,v = FMV(1 + iB + i^2) + 2FAUBJ,(1 + B)

- 2Fx(MBt}Bl - 2Fi.B£,B;, + O(Ffi3) (2.141)

It turns out that, to first order in B, the electromagnetic part of the actioncan be put into metric form (see Section 3.2 for discussion of this form),but not to second and higher orders. The particle and interaction partsof /NG are already in metric form. The action for the gravitational field

IG = -(167T)-1 §(aB»JB& + fB<aB'*)(-rj)ll2d*x (2.142)

where a and / are arbitrary constants.In the weak field, post-Newtonian limit appropriate for application to

solar system experiments (see Chapter 5), the theory can be made to agreewith all experiments performed to date. Thus, the theory was thought

to be a completely viable alternative to general relativity. However, be-cause of the deviations from metric form in the electromagnetic action,the theory violates EEP. We therefore expect it to violate WEP, althoughat second order in B^. To demonstrate that this is indeed the case, wefirst compute B^ for a SSS field, then recast /NG into THefi form. Thesolution of the gravitational field equations (Section 5.5) yields the form

Boo = b0, Bij & b&j (2.143)

where b0 and b t are functions of a gravitational potential U. Then, fromEquations (2.143) and (2.139), we find to O(b2),

0Oo = -(l-bo-2Kb + fb2 + 2Kbb0 + K2b2),

g..= ^.(l + bt- 2Kb + Jfef - 2Kbb1 + K2b2) (2.144)

where b = � b0 + 3b We have assumed for simplicity that far from thegravitating source, b0 and b^ vanish (see Section 5.5 for discussion).Substituting Equations (2.144) and (2.141) into Equation (2.140) puts/NG into THe/j. form to O(b2), with

T = 1 - b0 - 2Kb + Ibl + 2Kbb0 + K2b2,

+ K2b2,

/ * = [ ! + i(*o + *»i)] (2-145)

In the weak-field limit, it turns out that the SSS solutions for b0 andfcx have the form (see Section 5.5)

bo = 2CoU, b1 = 2C1U (2.146)

where U is the Newtonian gravitational potential and Co and Ct arearbitrary constants. Then

T = 1 - 21/ + 2t/2[i + Co] + O(l/3),

H = 1 + 2U[C0 + d - 1] + C/2[(C! + C0)(3C1 + Co)

- 4CX - 2C0 + 1] + O(t/3),

£ = 1 + U(C0 + d ) + U2(C0 + Cx)2 + O(t/3),

H = 1 + U(C0 + Ct) + O(t/3) (2.147)

where we have chosen the values of Co, d , an<i ^ s u c n t n a t

Co + 2K(3Cl - Co) = 1 (2.148)

in order to ensure that T = 1 � 1U + .... This will guarantee that theparticle Lagrangian will yield the correct Newtonian limit. Notice that,

to first order in U, these functions satisfy the EEP constraint in Equa-tion (2.63), but to second order they do not in general. Now in solarsystem tests of post-Newtonian effects, where the consequences of electro-magnetic violations of EEP are negligible, the coefficients ( + Co) and(Co + Ct � 1) in T and H are simply the PPN parameters /? and y (seeChapter 4). Solar system measurements of light deflection, radar-timedelay, and the perihelion shift of Mercury (see Chapter 7) constrain theseparameters by

|2C0 + 4Ct - 7| < 0.1,

\C0 + d - 2| < 0.002 (2.149)

Equations (2.83) and (2.147) then yield

r0 = -2co(co + cx)u + o(t/2),Ao = 2C1(C0 -I- d)U + O(U2) (2.150)

Using the above constraints on Co and Ct along with the value U =[/Q s 10"8, the relevant local potential for the Princeton-Moscow Eotvosexperiments, we obtain

|ro | ^ 1.7 x 10-8 (2.151)

which violates the experimental limit, Equation (2.120), by a factor 80.Thus, the Belinfante-Swihart theory is unviable.

Gravitation as a Geometric Phenomenon

The overwhelming empirical evidence supporting the Einstein Equiva-lence Principle, discussed in the previous chapter, has convinced manytheorists that only metric theories of gravity have a hope of being com-pletely viable. Even the most carefully formulated nonmetric theory - theBelinfante-Swihart theory - was found to be in conflict with the MoscowEotvos experiment. Therefore, here, and for the remainder of this book,we shall turn our attention exclusively to metric theories of gravity.

In Section 3.1, we review the concept of universal coupling, first definedin Section 2.5. Armed with EEP and universal coupling, we then develop,in Section 3.2, the mathematical equations that describe the behavior ofmatter and nongravitational fields in curved spacetime. Every metrictheory of gravity possesses these equations.

Metric theories of gravity differ from each other in the number andtype of additional gravitational fields they introduce and in the fieldequations that determine their structure and evolution; nevertheless, theonly field that couples directly to matter is the metric itself. In Section 3.3,we discuss general features of metric theories of gravity, and present anadditional principle, the Strong Equivalence Principle that is useful forclassifying theories and for analyzing experiments.

3.1 Universal CouplingThe validity of the Einstein Equivalence Principle requires that

every nongravitational field or particle should couple to the same sym-metric, second rank tensor field of signature � 2. In Section 2.3, we denotedthis field g, and saw that it was the central element in the postulates ofmetric theories of gravity: (i) there exists a metric g, (ii) test bodies followgeodesies of g, and (iii) in local Lorentz frames, the nongravitational lawsof physics are those of special relativity.


The property that all nongravitational fields should couple in the samemanner to a single gravitational field is sometimes called "universalcoupling" (see Section 2.5). Because of it, one can discuss the metric gas a property of spacetime itself rather than as a field over spacetime.This is because its properties may be measured and studied using a varietyof different experimental devices, composed of different nongravitationalfields and particles, and, because of universal coupling, the results willbe independent of the device. Consider, as a simple example, the propertime between two events as measured by two different clocks. To bespecific, imagine a Hydrogen maser clock and a SCSO clock at rest in astatic spherically symmetric gravitational field. If each clock is governedby a Hamiltonian H, then the proper time (number of clock "ticks")between two events separated by coordinate time dt is given by

where E is the eigenstate energy of the Hamiltonian (or energy difference,for a transition). The results of Section 2.6 show that if, for instance, theTHefi formalism is applicable, and if EEP is satisfied, e0 = /x0 = (Ho/To)

112

everywhere, thus using Equations (2.110) and (2.134) we obtain for eachclock

JVH oc dt(H0To

oc dt (H0TZ m Mo 1/2 6o 3/2) = TV2 dt

where the proportionality constants are fixed by calibrating each clockagainst a standard clock far from gravitating matter. Thus, each clockmeasures the same quantity To (in metric theories of gravity, in SSSfields, To � �gOo) a n d the proper time between two events is a charac-teristic of spacetime and of the location of the events, not of the clocksused to measure them.

Consequently, if EEP is valid, the nongravitational laws of physics maybe formulated by taking their special relativistic forms in terms of theMinkowski metric r\ and simply "going over" to new forms in terms ofthe curved spacetime metric g, using the mathematics of differentialgeometry. The details of this "going over" are the subject of the nextsection.

3.2 Nongravitational Physics in Curved SpacetimeIn local Lorentz frames, the nongravitational laws of physics are

those of special relativity. For point, charged, test particles coupled toelectromagnetic fields, for example, these laws may be derived from the

Gravitation as a Geometric Phenomenon 69

action

a

-(167T)-1 UAV^FAv-F^(-f?)1/2d4x (3.1)

where

F . = A �A

fj = det| |^, | | (3.2)

Here, n^ is the Minkowski metric, which in Cartesian coordinates hasthe form

In the local Lorentz frame, t}^ is assumed to have this form only up tocorrections of order [xs � x s(^)]2 , where xs(^) is the coordinate of achosen fiducial event in the local frame, in other words, r\^ is describedmore precisely as

-1,1,1,1),

According to the discussion in Section 3.1, the general form of these lawsin any frame is obtained by a simple coordinate transformation fromthe freely falling frame to the chosen frame. This transformation is given by

(3.4)

Then, the vectors and tensors that appear in JNG transform according to

n-^ = (dx«/dx»)(dxll/dxi)rlxlh

dx* = {dx*/dx*)dx*,

where J is the Jacobian of the transformation. Partial derivatives offields, as for example in the formula for F^, transform according to

a'" + dx* dx* * ( }

However, in the local frame, n^^ = 0. Thus,

_dx*_dif_dx>_ d2x* dx» 82xp dx*- ^ * - ^ ? a ? &? n*"-y + a* d& ex1 n<* + dx* dx& dx*n" { '

Using the fact that

(dxydx^idx^/dx*) = <5f (3.8)

we obtain

dx^dx* d2xs dx* dx° d2xd

n^'y ~ ~ fa* ~W dx^dx^ n»~fa7~fa7 "dx^dx1* n<* { '

If we now define

9*e = 1ae, (3-10)

M»IUl--gg-r. (in)

then Equation (3.9) can be written

or, using Equations (3.11), (3.12), and (3.13), the Christoffel symbolsF^y (also known as connection coefficients) take the form

% P,y y,D - gyfi,s) (3.14)

Then Equation (3.6) becomes

dx" dx0

^^faJdx1^11'1^^ ( 1 1 5 )

We define the covariant derivative ";" by

Ax;P = A^ - n,A, (3.16)

and notice that it transforms as a tensor; it can be shown that

Atf^A'f + VnA1 (3.17)

where A* = gafiAp. Taking the determinant of Equation (3.5) yields

n = [det(Sx7dx*)]2g (3.18)

where g = det g^, then

1 / 2 y i 2 (3.19)


Substituting these results into /NG gives

_ r( dx" dxv\112 . r, f i J uJNG = ~ L mOa J I -g^ � � I A + 2, eB J A^dx"

$ ( 3 . 2 0 )

where

We notice that the transformation to an arbitrary frame has resultedsimply in the replacements

%v by #��

"comma" by "semicolon"

(-f7)1/2d4x by (-gY'Wx (3.22)

This is the mathematical manifestation of EEP. We must point out thatthe specific mathematical forms given above for the Christoffel symbols,transformation laws, and so on are valid only in coordinate bases (seeMTW, Chapter 10 for further discussion). However, in this book we shallwork exclusively in coordinate bases.

Generally speaking, then, the procedure for implementing EEP is: putthe local special relativistic laws into a frame-invariant form using Lorentz-invariant scalars, vectors, tensors, etc., then make the above replacements.

It is simple to show that the same rules apply to the field equations andequations of motion derived from the Lagrangian. In the local frame theyare

(3.23)

where

dx = ( - f,p. dx" dx")112, if = dx*ld%,

rfr (3.24)

However, these are not in frame-invariant form. We must write

= M v U / i i p,

(3.25)

where we have used the fact that for the four-dimensional delta function(�fj)~ll2SA is invariant (since J<S4d4x = 1 or 0 regardless of the frame).Then in the general frame the equations are

mOaDuJD% = ej^ul, (3.26)

Ffvv = 47r./" (3.27)

where

fa = (-9^ dx" dxv)112, u" = dx"/dt,

DuJDx = u"umv,

�/" = I ea(-g)-V25\x - xa)dx»/dt (3.28)a

However, here there is a potential ambiguity in the application of EEPto electrodynamics if one writes Maxwell's equations, (3.27), in terms ofthe vector potential A^. In the local Lorentz frame, Maxwell's equationshave the special relativistic form

Ay~A»-»,v= -47tJ" (3.29)

It is always possible to choose a gauge (Lorentz gauge) in which A" � = 0,thus, since AV'")V = A"^11, we have

= A»'\v = -47tJ* (3.30)

It is tempting then to apply the rules of EEP to this equation to obtain

ngA" s A":v.v = <TM?vA = -4nJ»,A% = 0 (3.31)

However, there is another alternative. The curved-spacetime Maxwellequation, Equation (3.27), yields

(3.32)

But covariant derivatives of vectors and tensors do not commute incurved spacetime, in fact in general

A?* = Ke + *UAV (3-33)

where R%ap is the Riemann curvature tensor, given by

*U = rf^. - r^ + r ^ r j , - rj^rj. (3.34)

Then

Ar-».v = A " , * + R^A", (3.35)

where R% is the Ricci tensor given by

Ri = g**Ry» Ryf = R-U (3.36)

This version of Maxwell's equations in Lorentz gauge becomes

n0A»-R$Al>= -4nJ",

A], = 0 (3.37)

It is generally agreed that this second version is correct (although thereis no experimental evidence one way or the other). To resolve suchambiguities, the following rule of thumb should be applied: the simplereplacements (i; -*� g, comma -* semicolon) should be used without cur-vature terms in equations involving physically measurable quantities(F"v is physically measurable, A* is not); and coupling to curvature shouldoccur only with good physical reason (as in tidal coupling). (For a fullerdiscussion, see MTW, box 16.1.)

An uncharged test body follows a trajectory given by Equation (3.26)with e = 0, namely Du^/Dx = 0. This equation can be written usingEquations (3.17) and (3.28) in the form

d V/dr 2 + r^(dxx/dr)(dxp/dz) = 0 (3.38)

This is the geodesic equation.The mathematics of measurements made by atomic clocks and rigid

measuring rods follow the same rules since the structure of such measuringdevices is governed by solutions of the nongravitational laws of physics.In special relativity, the proper time between two events separated by aninfinitesimal coordinate displacement dx", as measured by any atomicclock moving on a trajectory that connects the events, is given by

dT = (_^vdx"dxv)1/2 (3.39)

if the separation is timelike, i.e., »/�� dx" dxv < 0. The proper distancebetween two events as measured by a rigid rod joining them is given by

ds = (r,liydx»dxv)112 (3.40)

if the separation is spacelike, i.e., n^ dx* dxv > 0. These results are inde-pendent of the coordinates used. Then in curved spacetime we have

[timelike] «> g^dx"dxv < 0,

[spacelike] «» Sllv dx" dx" > 0 (3.41)

There is a third class of separation dx* between events, those for which

rjllvdx"dxv = 0 (3.42)

These are called null or lightlike separations, and pairs of events thatsatisfy this condition are connectible by light rays. It is a tenet of specialrelativity that light rays move along straight, null trajectories, i.e., ifk" = dx"/da is a tangent vector to a light-ray trajectory, then

dW/d<r = 0, if fc'ifc, = 0 (3.43)

where a is a parameter labeling points along the trajectory. It should notbe forgotten, however, that this is at bottom a consequence of Maxwell'sequations, valid only in the "geometrical optics" limit, in which thecharacteristic wavelength X [a^fc0)"1] is small compared to the scale £Pover which the amplitude of the wave changes. (For example, if mightbe the radius of curvature of a spherical wavefront.) Since the first ofequations (3.43) can be written, in flat spacetime

dkf/do = {dx*/da)k*y = kvk% = 0 (3.44)

then EEP yields the equations

/cv/c?v = 0, fe"/cv^v = 0 (3.45)

i.e., the trajectories of light rays in the geometrical optics limit are nullgeodesies.

It is useful to derive this result directly from the curved-spacetime formof Maxwell's equations, in order to illustrate the role and the limits ofvalidity of the geometrical-optics assumption. In curved spacetime, thegeometrical-optics limit requires that X be small compared both to ££and to ffl, the scale over which the background geometry changes {01 isrelated to the Riemann curvature tensor), i.e.,

A/(min{&, Si}) = 1/L « 1 (3.46)

In this limit, the electromagnetic vector potential can be written in termsof a rapidly varying real phase and a slowly varying complex amplitudein the form (see MTW, Section 22.5 for details)

K = (a, + <*� + �� y / £ (3.47)

where 6 is the real phase, a,,, b^,... are complex, and e is a formal ex-pansion parameter that keeps track of the powers oiXjL. Ultimately, onetakes only the real part of A^ in any physical calculations. We define the


wave vector

K = e,v> k" = / V 0 v (3-48)

Then Maxwell's equations in Lorentz gauge [Equations (3.37)], yield

0 = A% = [(i/s)kv(av + sbv) + a]v + 0(e)]e

i9'E,

0 = DgA" - R$A'

= [ - s - 2Jfc,fcV + fib") + 2(i/e)fc'aJ + (i/e)fef a" + 0(e °j]emie (3.49)

Setting the coefficients of each power of e equal to zero, we obtain for theleading terms in each equation

fa, = 0, (3.50)

k% = 0, (3.51)

in other words, the amplitude is orthogonal to the wave vector, and thewave vector is null. Taking the gradient of Equation (3.51) and notingthat &�.� = kv;il since /cM itself is a gradient, we get

fcyc" = 0 (3.52)

which is the geodesic equation for k". The trajectory x^a) of the ray canthen be shown to be related to k" by the differential equation

dx"(a)/da = fe"(xv) (3.53)

where a is an affine parameter along the ray. For further discussion of thehigher-order terms in Equation (3.49), see MTW, Section 22.5.

Another useful and important form of the equations of motion formatter and nongravitational fields can be derived in the case where theequations are obtained from a covariant action principle. This will es-sentially always be the case, for the following reason: in special relativity,all modern viable theories of nongravitational fields and their interactionstake an action principle as their starting point, leading to an action /NG.The use of EEP does not alter the fact that the equations of motion arederivable from an action. Consequently, one is led in curved spacetimeto an action of the general form

= J(3-54)

where qA and qAifl are the nongravitational fields under consideration andtheir first partial derivatives (e.g., M", A^, A^,...) and g^ and g^^ are

the metric and its first derivative. (The extension to second and higherderivatives is straightforward). The action principle <5/NG = 0 is covariant,thus, under a coordinate transformation, i?N G must be unchanged in func-tional form, modulo a divergence [see Trautman (1962) for discussion].Consider the infinitesimal coordinate transformation

x* -> x" + <5x", <5x" = £" (3.55)

Then the metric changes according to [cf. Equation (3.5)],

a = - g^% - gvx^ - g^J" (3.56)

Assume the matter and nongravitational field variables change accordingto

<5<7A = « . - «A.*£V (3-57)

where d^v are functions of x". Under this transformation, JSf NG changes by

S 9 ^ ( 3-5 8 )

Substituting Equations (3.56) and (3.57), integrating by parts, droppingdivergence terms, and demanding that JS?NG be unchanged for arbitraryfunctions £*, yields the "Bianchi identities"

w J l = 0 (3.59)where S£CNO/3qx is the "variational" derivative of i ? N G defined, for anyvariable \j/, by

dx" V # /and T*" is the "stress-energy tensor," defined by

T^^2(-g)-^8^NG/dg,v (3.61)

Using the fact that

(-ff).V2 = ( - 0 ) 1 / 2 n . (3.62)

we can rewrite Equation (3.59) in the form


However, the nongravitational field equations and equations of motionare obtained by setting the variational derivative of J£NG with respect toeach field variable qA equal to zero, i.e.,

. = 0 (3.64)

which by Equation (3.63) is equivalent to

Tl,y = 0 (3.65)

Thus, the vanishing of the divergence of the stress-energy tensor T"v is aconsequence of the nongravitational equations of motion. This resultcould also have been derived, first working with i ? N G in flat spacetime,to obtain the equation T)j>v = 0 by the above method, then using EEPto obtain Equation (3.65). Notice that Equation (3.65) is a consequencepurely of universal coupling (EEP) and of the invariance of the non-gravitational action, and is valid independent of the field equations forthe gravitational fields.

The stress-energy tensor T*"1 for charged particles and electromagneticfields may be obtained from the action 7NG, Equation (3.20), by firstrewriting it in the form

. _ ^ , , dx'^W*-�NG �

(3.66)

Since only g^ (and not its derivatives) appears in 7NG, we obtain

= £ mOau"u\uorl{-g)-ll28\x - xa(x)]a

+ (4*)- \F^F\ - kg"vF^Fa0) (3.67)

where we have used the fact that

(3-68)

Throughout most of this book we shall use the perfect fluid as ourmodel for matter. This model is an average of the properties of matterover scales that are large compared to atomic scales, but small comparedto the scales over which the bulk properties of the fluid vary. Thus, onecan speak of density, pressure, velocity of fluid elements at a point within

the fluid. A perfect fluid is one that has negligible viscosity, heat transport,and shear stresses. It is then possible to show that the stress-energy tensorfor the fluid has the following property: in a local Lorentz frame, momen-tarily comoving with a chosen element of the fluid, the stress-energytensor for that element has the form

T"v = diag[p(l + n),p,p,p-] (3.69)

where p is the rest-mass-energy density of atoms in the fluid element, IIis the specific density of internal kinetic and thermal energy in the fluidelement, and p is the isotropic pressure. This can also be written in thecovariant form

+ J/"V) (3.70)

where u" = dx^jdi is the four-velocity of the fluid element (=<58 m thecomoving frame). Then in curved spacetime, T"v has the form

T"v = (p + PU + p)u"Mv + pgT (3.71)

This can also be derived from Equation (3.67) using suitable techniquesin relativistic kinetic theory (see Ehlers, 1971).

To obtain a complete metric theory of gravity one must now specifyfield equations for the metric and for the other possible gravitationalfields in the theory. There are two alternatives. The first is to assume thatthese equations, like the nongravitational equations can be derived froman invariant action /G which will be a function of the gravitational fields(£A (which could include #��):

IG = IG(A,<1>AJ (3.72)

The complete action is thus

1 = IG(4>A,\J + /NGOZA^A.^V.^V./S) (3.73)

Variation with respect to <f>K yields the gravitational field equations

or, using Equation (3.61),

dSeol64>K= -U-g)ll2T^dgJdcl>A (3.75)

Theories of this type are called Lagrangian-based covariant metric theo-ries of gravity. Many important general properties of such theories aredescribed by Lee, Lightman, and Ni (1974). The other alternative is tospecify gravitational field equations that are not derivable from an action.


These are called non-Lagrangian-based theories. Although many suchtheories have been devised, they have not met with great success inagreeing with experiment. All the metric theories to be described inChapter 5 that agree with solar system experiments are Lagrangian based.

3.3 Long-Range Gravitational Fields and theStrong Equivalence PrincipleIn any metric theory of gravity, matter and nongravitational fields

respond only to the spacetime metric g. In principle, however, there couldexist other gravitational fields besides the metric, such as scalar fields,vector fields, and so on. If matter does not couple to these fields what cantheir role in gravitation theory be? Their role must be that of mediating themethod by which matter and nongravitational fields generate gravitationalfields and produce the metric. Once determined, however, the metric aloneinteracts with the matter as prescribed by EEP.

What distinguishes one metric theory from another, therefore, is thenumber and kind of gravitational fields it contains in addition to themetric, and the equations that determine the structure and evolution ofthese fields. From this viewpoint, one can divide all metric theories ofgravity into two fundamental classes: "purely dynamical" and "prior geo-metric." (This division is independent of whether or not the theory isLagrangian based.)

By "purely dynamical metric theory" we mean any metric theory whosegravitational fields have their structure and evolution determined by cou-pled partial differential field equations. In other words, the behavior ofeach field is influenced to some extent by a coupling to at least one of theother fields in the theory. By "prior geometric" theory, we mean any metrictheory that contains "absolute elements," fields or equations whose struc-ture and evolution are given a priori and are independent of the structureand evolution of the other fields of the theory. These "absolute elements"could include flat background metrics IJ, cosmic time coordinates T, andalgebraic relationships among otherwise dynamical fields, such as

where h^ and k^ may be dynamical fields. Note that a field may be absoluteeven if it is determined by partial differential equations, as long as theequation does not involve any dynamical fields. For instance, a flat back-ground metric is specified by the field equation

Riemfa) = 0 (3.76)


or a cosmic time function is specified by the field equations

vwvvr = 0, VT � VT = - 1

where the gradient and inner product are taken with respect to a non-dynamical background metric, such as i\.

General relativity is a purely dynamical theory since it contains only onegravitational field, the metric itself, and its structure and evolution isgoverned by a partial differential equation (Einstein's equations). Brans-Dicke theory is a purely dynamical theory; the field equation for the metricinvolves the scalar field (as well as the matter as source), and that for thescalar field involves the metric. Rosen's bimetric theory is a prior-geometric theory: it has a flat background metric of a type described inEquation (3.76), and the field equations for the physical metric g involve t\.In Chapter 5, we will discuss these and other theories in more detail.

By discussing metric theories of gravity from this broad, "Dicke" pointof view, it is possible to draw some general conclusions about the nature ofgravity in different metric theories, conclusions that are reminiscent of theEinstein Equivalence Principle, but that will be given a new name: theStrong Equivalence Principle.

Consider a local, freely falling frame in any metric theory of gravity. Letthis frame be small enough that inhomogeneities in the external gravita-tional fields can be neglected throughout its volume. However, let theframe be large enough to encompass a system of gravitating matter and itsassociated gravitational fields. The system could be a star, a black hole, thesolar system, or a Cavendish experiment. Call this frame a "quasilocalLorentz frame". To determine the behavior of the system we must calcu-late the metric. The computation proceeds in two stages. First, we deter-mine the external behavior of the metric and gravitational fields, therebyestablishing boundary values for the fields generated by the local system,at a boundary of the quasilocal frame "far" from the local system. Second,we solve for the fields generated by the local system. But because the metricis coupled directly or indirectly to the other fields of the theory, its structureand evolution will be influenced by those fields, particularly by the bound-ary values taken on by those fields far from the local system. This will betrue even if we work in a coordinate system in which the asymptotic formof g^ in the boundary region between the local system and the externalworld is that of the Minkowski metric. Thus, the gravitational environmentin which the local gravitating system resides can influence the metric gen-erated by the local system via the boundary values of the auxiliary fields.Consequently, the results of local gravitational experiments may depend

on the location and velocity of the frame relative to the external environ-ment. Of course, local nongravitational experiments are unaffected sincethe gravitational fields they generate are assumed to be negligible, andsince those experiments couple only to the metric whose form can alwaysbe made locally Minkowskian. Local gravitational experiments mightinclude Cavendish experiments, measurements of the acceleration of mas-sive bodies, studies of the structure of stars and planets, and so on. We cannow make several statements about different kinds of metric theories (Willand Nordtvedt, 1972).

(a) A theory that contains only the metric g yields local gravitationalphysics that is independent of the location and velocity of the localsystem. This follows from the fact that the only field coupling the localsystem to the environment is g, and it is always possible to find a coordinatesystem in which g takes the Minkowski form at the boundary between thelocal system and the external environment. Thus, the asymptotic values ofg^ are constants independent of location, and are asymptotically Lorentzinvariant, thus independent of velocity. General relativity is an example ofsuch a theory.

(b) A theory that contains the metric g and dynamical scalar fields</>A yields local gravitational physics that may depend on the location ofthe frame but which is independent of the velocity of the frame. This followsfrom the asymptotic Lorentz invariance of the Minkowski metric and ofthe scalar fields, except now the asymptotic values of the scalar fields maydepend on the location of the frame. An example is Brans-Dicke theory,where the asymptotic scalar field determines the value of the gravitationalconstant, which can thus vary as <j> varies.

(c) A theory that contains the metric g and additional dynamicalvector or tensor fields or prior-geometric fields yields local gravitationalphysics that may have both location- and velocity-dependent effects. Thiswill be true, for example, even if the auxiliary field is a flat backgroundmetric IJ. The background solutions for g and t\ will in general be different,and therefore in a frame in which g^ takes the asymptotic form diag(� 1,1,1,1), r\^ will in general have a form that depends on location andthat is not Lorentz invariant (although it will still have vanishing curva-ture). The resulting location and velocity dependence in q will act back onthe local gravitational problem. (For a clear example of this, see Rosen'stheory in Chapter 5.) Be reminded that these effects are a consequence ofthe coupling of auxiliary gravitational fields to the metric and to eachother, not to the matter and nongravitational fields. For metric theoriesof gravity, only g^ couples to the latter.

These ideas can be summarized in the form of a principle called theStrong Equivalence Principle that states that (i) WEP is valid for self-gravitating bodies as well as for test bodies (GWEP), (ii) the outcome of anylocal test experiment is independent of the velocity of the (freely falling)apparatus, and (iii) the outcome of any local test experiment is independentof where and when in the universe it is performed. The distinction betweenSEP and EEP is the inclusion of bodies with self-gravitational interactions(planets, stars) and of experiments involving gravitational forces (Caven-dish experiments, gravimeter measurements). Note that SEP contains EEPas the special case in which gravitational forces are ignored.

It is tempting to ask whether the parallel between SEP and EEP extendsas far as a Schiff-type conjecture; e.g., "any theory that embodies GWEPalso embodies SEP." As in Section 2.5, we can give a plausibility argumentin support of this, for the special case of metric theories of gravity with aconservation law for total energy (Haugan, 1979). Generally speaking, thismeans Lagrangian-based theories. Consider a local gravitating systemmoving slowly in a weak, static, and external gravitational field. We assumethat the laws governing its motion can be put into a quasi-Newtonianform, with the conserved energy Ec given by

(3.77)

where

MR = M 0 -E B (X ,V) ,

£B(X, V) = £g + 8my UiJ(X) - $Sm[j VlV> + O(Egt / 2 , . . . ) (3.78)

(see Section 2.5 for detailed definitions). Here, we use units in which thespeed of light as measured far from the local system is unity. The positionand velocity dependence in £B can manifest itself, for example, as positionand velocity dependence in the locally measured gravitational constant.For two bodies in a local Cavendish experiment, the gravitational constantis given by

Gcavendish = r2Fr/mim2 = r\dE^dr)lmxm1 (3.79)

and thus the anomalous mass tensors will contribute to GCavendUh (seeSection 6.4). However, a cyclic gedanken experiment identical to thatpresented in Section 2.5 shows that the anomalous mass tensors bml4 and8m\J also generate violations of GWEP

A1 = g' + (Smf/MJU* - (<5mjJ/AW (3.80)


where g = \U. Hence, GWEP (dm$ = Sm{k = 0) implies no preferred-location or preferred-frame effects, thence SEP. In Chapters 4, 5, and 6we shall see specific examples of GWEP and SEP in action in the post-Newtonian limits of arbitrary metric theories of gravity, and in Chapter 8,shall study experimental tests of SEP.

The above discussion of the coupling of auxiliary fields to local gravita-ting systems indicates that if SEP is valid, there must be one and only onegravitational field in the universe, the metric g. Those arguments were onlysuggestive however, and no rigorous proof of this statement is availableat present.

The assumption that there is only one gravitational field is the founda-tion of many so-called derivations of general relativity. One class ofderivations uses a quantum-field-theoretic approach. One begins withthe assumption that, in perturbation theory, the gravitational field isassociated with the exchange of a single massless particle of spin 2 (corre-sponding to a single second-rank tensor dynamical field), and by makingcertain reasonable assumptions that the S-matrix be Lorentz invariantor that the theory be derivable from an action, one can generate the fullclassical Einstein field equations (Weinberg, 1965; Deser, 1970). Anotherclass of derivations attempts to build the most general field equation forg out of tensors constructed only from g, subject to certain constraints(no higher than second derivatives, for instance). By demanding that thefield equations should imply the matter equations of motion Tfv

v = 0,one is led (except for the possible cosmological term) to Einstein's equa-tions. For a review of these and other derivations of general relativity thereader is referred to MTW, box 17.2. However, the implicit use of SEP inall these derivations cannot be emphasized enough. Empirically, it hasbeen found that every metric theory other than general relativity intro-duces auxiliary gravitational fields, either dynamical or prior geometric,and thus predicts violations of SEP at some level. General relativity seemsto be the only metric theory that embodies SEP completely. Thus, thewide variety of derivations of general relativity assuming SEP, plus evi-dence from alternative theories lends some credence to the conjecture

SEP => [General Relativity] (3.81)

In Chapters 8 and 12, we shall discuss experimental evidence for thevalidity of SEP.

This qualitative discussion of alternative metric theories of gravity hasneglected two subjects, each of which could generate a monograph of its


own. The first is "torsion." In applying EEP to the nongravitational lawsof physics we assumed the rule "comma goes to semicolon," where semi-colon denoted covariant derivative with respect to the metric g [Equations(3.14), (3.16), and (3.17)]. However, it is possible that the correct covariantderivative is given by

A% = A'j, + §y}A> (3.82)

where

{?,} = T% + S$y (3.83)

with Sjiy antisymmetric on ft and y, i.e.,

Sfryi-Si, (3.84)

In general, S^y is a tensor called the "torsion" tensor, and thus does notvanish in the local Lorentz frame. Torsion has been introduced into gravi-tation theory either as a means to incorporate quantum mechanical spinin a consistent way, as a byproduct of attempts to construct gauge theoriesof gravitation, and as a possible route to a unified theory of gravity andelectromagnetism. However, in almost all experiments discussed in thisbook, the observable effects of torsion are negligible [see, however, Ni(1979)]. Instead, torsion has an effect primarily in the realm of elementaryparticle physics or in the very early universe. Thus, we shall neglect torsioncompletely for the rest of this book, and shall refer the interested readerto the review by Hehl et al. (1976).

The second topic to be neglected falls under the heading "general rela-tivity with R2 terms." Although this is an old subject (Weyl, 1919;Eddington, 1922), it has recently attracted some interest. The standardgravitational action of classical general relativity (Section 5.2) has the form

(3.85)

where R is the Ricci scalar given by

R = g^R,, (3.86)

However, some attempts to make a renormalizable quantum theory ofgravity based on general relativity lead to the introduction of "counterterms" into the action, to eliminate the nonrenormalizable infinities. Thesecounter terms are quadratic and higher in the Riemann tensor, Riccitensor, and Ricci scalar, leading to a gravitational action of the form

1G = (16TT)-l J(R + aR2 + bR^R*"* + cR^R^i-g)1'2dAx (3.87)


Since the theory has only one gravitational field #��, one suspects that itsatisfies SEP, and so represents a possible counter example to our con-jecture that SEP => general relativity. However, in most theories of thistype, the constants a, b, and c [units of (length)2] have sizes ranging fromthe Planck length, 10~33 cm, to nuclear dimensions, 10"13 cm, so the ob-servable effects of these terms will be confined to elementary particleinteractions or to the very early universe. Thus the issue of "R2 terms,"too, will be ignored throughout this book (see Havas, 1977).

The Parametrized Post-Newtonian Formalism

We have seen that, despite the possible existence of long-range gravita-tional fields in addition to the metric in various metric theories of gravity,the postulates of metric theories demand that matter and nongravitationalfields be completely oblivious to them. The only gravitational field thatenters the equations of motion is the metric g. The role of the other fieldsthat a theory may contain can only be that of helping to generate the space-time curvature associated with the metric. Matter may create these fields,and they, plus the matter, may generate the metric, but they cannotinteract directly with the matter. Matter responds only to the metric.

Consequently, the metric and the equations of motion for matter becomethe primary theoretical entities, and all that distinguishes one metrictheory from another is the particular way in which matter and possiblyother gravitational fields generate the metric.

The comparison of metric theories of gravity with each other and withexperiment becomes particularly simple when one takes the slow-motion,weak-field limit. This approximation, known as the post-Newtonianlimit, is sufficiently accurate to encompass all solar system tests that canbe performed in the foreseeable future. The post-Newtonian limit is notadequate, however, to discuss gravitational radiation, where the slow-motion assumption no longer holds, or systems with compact objectssuch as the binary pulsar, where the weak-field assumption is not valid,or cosmology, where completely different assumptions must be made.These issues will be dealt with in later chapters.

In Section 4.1, we discuss the post-Newtonian limit of metric theoriesof gravity, and devise a general form for the post-Newtonian metric for asystem of perfect fluid. This form should be obeyed by most metric theories,with the differences from one theory to the next occurring only in the

The Parametrized Post-Newtonian Formalism 87

numerical coefficients that appear in the metric. When the coordinatesystem is appropriately specialized (standard gauge), and arbitrary param-eters used in place of the numerical coefficients, the result, described inSection 4.2, is known as the Parametrized post-Newtonian (PPN) for-malism, and the parameters are called PPN parameters. In Section 4.3,we discuss the effect of Lorentz transformations on the PPN coordinatesystem, and show that some theories of gravity may predict gravitationaleffects that depend on the velocity of the gravitating system relative to therest frame of the universe (perferred-frame effects). In Section 4.4, weanalyze the existence of post-Newtonian integral conservation laws forenergy, momentum, angular momentum, and center-of-mass motionwithin the PPN formalism and show that metric theories possess suchlaws only if their PPN parameters obey certain constraints.

This formalism then provides the framework for a discussion of specificalternative metric theories of gravity (Chapter 5) and for the analysis ofsolar system tests of relativistic gravitational effects (Chapters 7-9). Mostof this chapter is an updated version of Chapter 4 of TTEG (Will, 1974a).

4.1 The Post-Newtonian Limit(a) Newtonian gravitation theory and the Newtonian limitIn the solar system, gravitation is weak enough for Newton's

theory of gravity to adequately explain all but the most minute effects. Toan accuracy of about one part in 105, light rays travel on straight lines atconstant speed, and test bodies move according to

a = \U (4.1)

where a is the body's acceleration, and U is the Newtonian gravitationalpotential produced by rest-mass density p according to1

= -4np, U(x, t) = [-~Ar d2x' (4.2)|x � x |

A perfect, nonviscous fluid obeys the usual Eulerian equations of hydro-dynamics

dp/dt + V � (pv) = 0,

pdv/dt^ pVU - Vp,

d/dt = d/dt + v � V (4.3)

1 We use "geometrized" units in which the speed of light is unity and in whichthe gravitational constant as measured far from the solar system is unity.

where v is the velocity of an element of the fluid, p is the rest-mass densityof matter in the element, p is the total pressure (matter plus radiation) onthe element, and d/dt is the time derivative following the fluid.

From the standpoint of a metric theory of gravity, Newtonian physicsmay be viewed as a first-order approximation. Consider a test body mo-mentarily at rest in a static external gravitational field. From the geodesicEquation (3.38), the body's acceleration a* = d2xk/dt2 in a static (t,x) co-ordinate system is given by

«*=-n<> = ie*W, (4-4)Far from the Newtonian system, we know that in an appropriately chosencoordinate system, the metric must reduce to the Minkowski metric (seesubsection (c))

ff,,,-»if,, = diag(-1,1,1,1) (4.5)

In the presence of a very weak gravitational field, Equation (4.4) can yieldNewtonian gravitation, Equation (4.1) only if

gfi-S*, goo^-l+2U (4.6)

It can be straightforwardly shown that with this approximation and astress-energy tensor for perfect fluids given by

T00 = p, TOj = pvJ, TJk = pvV + p5Jk (4.7)

the Eulerian equations of motion, (4.3), are equivalent to

T?; ~ r?; + rgoT00 = o (4.8)

where we retain only terms of lowest order in v2 ~ U ~ p/p.But the Newtonian limit no longer suffices when we begin to demand

accuracies greater than a part in 105. For example, it cannot accountfor Mercury's additional perihelion shift o f~5 x 10 ~7 radians per orbit.Thus we need a more accurate approximation to the spacetime metricthat goes beyond or "post" Newtonian theory, hence the name post-Newtonian limit.

(b) Post-Newtonian bookkeepingThe key features of the post-Newtonian limit can be better

understood if we first develop a "bookkeeping" system for keeping trackof "small quantities." In the solar system, the Newtonian gravitationalpotential U is nowhere larger than 10"5 (in geometrized units, U isdimensionless). Planetary velocities are related to U by virial relations


which yield

v2 Z U (4.9)

The matter making up the Sun and planets is under pressure p, but thispressure is generally smaller than the matter's gravitational energy densitypU; in other words

P/P £ U, (4.10)

{p/p is ~10~5 in the Sun, ~10~1 0 in the Earth). Other forms of energyin the solar system (compressional energy, radiation, thermal energy,etc.) are small: the specific energy density II (ratio of energy density torest-mass density) is related to U by

nzu (4.ii)

(II is ~ 10"5 in the Sun, ~ 10~9 in the Earth). These four small quantitiesare assigned a bookkeeping label that denotes their "order of small-ness":

U ~ v2 ~ p/p ~ n ~ O(2). (4.12)

Then single powers of velocity i; are O(l), U2 is O(4), Uv is O(3), UHis O(4), and so on. Also, since the time evolution of the solar system isgoverned by the motions of its constituents, we have

d/dt ~ v � V

and thus,

\s/et\\d/e> O(l) (4.13)

We can now analyze the "post-Newtonian" metric using this book-keeping system. The action, Equation (3.20), from which one can derivethe geodesic Equation (3.38) for a single neutral particle, may be re-written

-i - Cf dx»dx*yi2

- ' o - -m0 Jl ~a^~Jf~^fJ dt

(- 000 - 2gop' - gjkvV)112 dt (4.14)

The integrand in Equation (4.14) may thus be viewed as a Lagrangian Lfor a single particle in a metric gravitational field. From Equation (4.6),we see that the Newtonian limit corresponds to

L = (1 - 2(7 - v2)112 (4.15)

as can be verified using the Euler-Lagrange equations. In other words,Newtonian physics is given by an approximation for L correct to O(2).Post-Newtonian physics must therefore involve those terms in L of nexthighest order, O(4).

What happened to odd-order terms, O(l) or O(3)? Odd-order termsmust contain an odd number of factors of velocity v or of time derivativesd/dt. Since these factors change sign under time reversal, odd-orderterms must represent energy dissipation or absorption by the system.But conservation of rest mass prevents terms of O(l) from appearing inL, and conservation of energy in the Newtonian limit prevents terms ofO(3). Beyond O(4), different theories may make different predictions. Ingeneral relativity, for example, the conservation of post-Newtonian energyprohibits terms of O(5). However, terms of O(7) can appear; they rep-resent energy lost from the system by gravitational radiation.

In order to express L to O(4), we must know the various metric com-ponents to an appropriate order:

L = {1 - 2V - v2 - 0oo[O(4)] - 2gOJ[p0)y

V<}1/2 (4.16)

Thus the post-Newtonian limit of any metric theory of gravity requiresa knowledge of

0OO to O(4),

g0J to O(3),

gJk to O(2) (4.17)

The post-Newtonian propagation of light rays may also be obtainedusing the above approximations to the metric. Since light moves alongnull trajectories (dx � 0), the Lagrangian L must be formally identical tozero. In the first order Newtonian limit this implies that light must moveon straight lines at speed 1, i.e.,

0 = L = (1 - v2)112, v2 = 1 (4.18)

In the next, post-Newtonian order, we must have

0 = L={l-2U -v2- gjk[O(2)~]vJvk}112 (4.19)

Thus to obtain post-Newtonian corrections to the propagation of lightrays, we need to know

goo to O(2),

gjk to O(2) (4.20)


In a similar manner, one can verify that if one takes the perfect-fluidstress-energy tensor

T"v = (p + pU + p)u"u" + pg»v (4.21)

expanded through the following orders of accuracy:

T00 to pO(2),

T0J to pO(3),

TJk to pO(4) (4.22)

and combined with the post-Newtonian metric, then the equation ofmotion 7?v

v = 0 will yield consistent "post-Eulerian" equations of hydro-dynamics.

(c) Post-Newtonian coordinate systemTo discuss the post-Newtonian limit properly, we must specify

the coordinate system. We imagine a homogeneous isotropic universe inwhich an isolated post-Newtonian system resides. We choose a coordinatesystem whose outer regions far from the isolated system are in free fallwith respect to the surrounding cosmological model, and are at rest withrespect to a frame in which the universe appears isotropic (universe restframe). In these outer regions, one expects the physical metric to varyaccording to

ds2 = -dt\+ [a(t)/ao]2(l + kr2IAal)'2dijdxidxi + h^dx^dx1 (4.23)

where the first two terms comprise the Robertson-Walker line elementappropriate to a homogeneous isotropic cosmological model and thethird term represents the perturbation due to the local system. Here,r is the distance from the local system to the field point, a = a{t)[aQ =a(toj] is the cosmological scale factor, and k is the curvature parameter(k = 0, + 1). At a given radius r0 and at a particular moment t0, we cantransform to a coordinate system

t' == t, xy = x\l - krl/4al)-1 (4.24)

in which

ds2 = (ifc, + fcJJ dx»' dxv' (4.25)

This must be done at a value of r0 large enough that we can then regardn^ as the asymptotic form of g^, i.e., that h^ ~ M/r0 « 1, where M isthe mass of the isolated system, yet small enough that the deviation ofthe cosmological metric from n^ for r « r0 is small, in fact smaller than

the post-Newtonian terms in h^ of order (M/r)2. The value of r0

that optimizes these constraints is given by (M/r0)2 > {ro/ao)

2, or M «r0 <, (Mao)

1/2. Since a0 ~ 1010 light yr, we have, for the solar systemr0 <: 1011 km ^ 103 a.u., with maximum deviations from n^ of order(ro/ao)

2 ~ 10"24. These are much smaller than the expected post-Newtonian deviations (M/r)2 > 10"16 that influence solar system experi-ments. Thus, to a precision of about one part in 1022, we can regard thespace time metric of the solar system as being asymptotically Minkowskianin its outer regions, out to 103 a.u., with deviations of order M/r and{M/r)2 in its interior. The above discussion ignores the variation of thecosmological scale factor a(t) with time. However, because this variationtakes place over a timescale (1010 yr) long compared to a dynamicaltimescale (1 yr) for the solar system, we can treat the effects of the variationadiabatically.

The coordinate system thus constructed we shall call "local quasi-Cartesian coordinates." In this coordinate system it is useful to definethe following conventions and quantities:

(i) Unless otherwise noted, spatial vectors are treated as Cartesianvectors, with x* = xk.

(ii) Repeated spatial indices or the symbol |x| denotes a Cartesian innerproduct, for example

xkxk = Xkxk = xkxk = |x|2 s x2 + y2 + z2 (4.26)

(iii) The volume element d3x = dxdydz.

(d) Post-Newtonian potentialsWe assume throughout that the matter composing the solar

system can be idealized as perfect fluid. For the purposes of most solarsystem experiments in the coming decades, this is an adequate assumption(see, however, Section 9.2). As we shall see in more detail in Chapter 5,the post-Newtonian limit for a system of perfect fluid in any metric theoryof gravity is best calculated by solving the field equations formally,expressing the metric as a sequence of post-Newtonian functionals of thematter variables, with possible coefficients that may depend on thematching conditions between the local system and the surroundingcosmological model and on other constants of the theory. The evolutionof the matter variables, and thence of the metric functionals, is deter-mined by means of the equations of motion Tfv

v = 0 using the matterstress-energy tensor and the post-Newtonian metric all evaluated to an


order consistent with the post-Newtonian approximation. The evolutionof the cosmological matching coefficients is determined by a solution ofthe appropriate cosmological model. Thus, the most general post-Newtonian metric can be found by simply writing down metric termscomposed of all possible post-Newtonian functionals of matter variables,each multiplied by an arbitrary coefficient that may depend on thecosmological matching conditions and on other constants, and addingthese terms to the Minkowski metric to obtain the physical metric.Unfortunately, there is an infinite number of such functionals, so that inorder to obtain a formalism that is both useful and manageable, we mustimpose some restrictions on the possible terms to be considered, guidedin part by a subjective notion of "reasonableness" and in part by evidenceobtained from known gravitation theories. Some of these restrictions areobvious:

(i) The metric terms should be of Newtonian or post-Newtonian order;no post-post-Newtonian or higher terms are included.

(ii) The terms should tend to zero as the distance |x � x'| between thefield point x and a typical point x' inside the matter becomes large. Thiswill guarantee that the metric becomes asymptotically Minkowskian inour quasi-Cartesian coordinate system.

(iii) The coordinates are chosen so that the metric is dimensionless.(iv) In our chosen quasi-Cartesian coordinate system, the spatial origin

and initial moment of time are completely arbitrary, so the metric shouldcontain no explicit reference to these quantities. This is guaranteed byusing functionals in which the field point x always occurs in the com-bination x � x', where x' is a point associated with the matter distribution,and by making all time dependence in the metric terms implicit via theevolution of the matter variables and of the possible cosmologicalmatching parameters.

(v) The metric corrections h00, h0J, and hJk should transform underspatial rotations as a scalar, vector, and tensor, respectively, and thusshould be constructed out of the appropriate quantities. For variablesassociated with the matter distribution, examples are: scalar, p, |x � x'|,v'2, v' � (x � x') etc.; vector, v), (x � x')f, and tensor, (x � x')/* � x')k,VjVk, etc. For variables associated with the structure of the field equationsof the theory or with the cosmological matching conditions, there areonly two available quantities in the rest frame of the universe: scalarcosmological matching parameters or numerical coefficients; and a tensor,Sjk. In the rest frame of an isotropic universe, no vectors or anisotropic


tensors can be constructed. [If the universe is assumed to be slightlyanisotropic, other terms may be possible (Nordtvedt, 1976).]

(vi) The metric functionals should be generated by rest mass, energy,pressure, and velocity, not by their gradients. This restriction is purelysubjective, and can be relaxed quite easily if there is ever any reason todo so. No reason has yet arisen.

A final constraint is extremely subjective:(vii) The functionals should be "simple."With those restrictions in mind, we can now write down possible terms

that may appear in the post-Newtonian metric.(1) gJk to O(2): From condition (v), gjk must behave as a three-

dimensional tensor under rotations, thus the only terms that can appearare

gjk[O(2)-]:U3jk,Ujk (4.27)

where Ujk is given by

Ujk s f PV,t)(x-x')M-x')k ^ (4 2g)X � X

The term Ujk can be expressed more conveniently in terms of the "super-potential" %(x, t), given by

X(x,t)=-jp(x',t)\x-x'\d3x',

Xjk=-SjkU+UJk, V2x=-2t/ (4.29)

Thus, the only terms that we shall consider are

gjk[O(2)l. U5jk,x,Jk (4.30)

(2) gOj to O(3): These metric components must transform as three-vectors under rotations, and thus contain only the terms

: VJtWj (4.31)

where

y CPWMpJ \x � x'\

The functionals V} and Wj are also related to the superpotential % by

X.oj =VJ-WJ (4.33)

(3) goo t° O(4): This component should be a scalar under rotations.The only terms we shall consider are

0oo[O(4)]: U\<bw,<bu<b2,<bi,<bt,s/,a (4.34)

where

Y , , , 2 f , , , ,x-x 2 J x-x'

(4.35)

| | |x-x'| dt

Restriction (vii) has been used liberally to eliminate otherwise possiblemetric functionals, for example

VJVJU-\

'\'Yd3x\...

Should one of these terms ever appear in the post-Newtonian metric ofa gravitational theory, the formalism could be modified accordingly.

There are a number of simple and useful relationships satisfied by thefunctionals that we have included in the metric:

! = -4npv2, V2O2 = -AnpU,

V2O4 = -4np,

= rf + ® - <E>X (4.36)

To derive many of these relationships one makes repeated use of theformula, obtained using the continuity Equation (4.3),

8 r rn(x' t)f(x x')d3x' = o(x' tW � V'/Yx x'1ii3jc'ri + O<2)1 (4 371

at J �>


4.2 The Standard Post-Newtonian GaugeWe can restrict the form of the post-Newtonian metric somewhat

by making use of the arbitrariness of coordinates embodied in statement(ii) of the Dicke framework. An infinitesimal coordinate or "gauge" trans-formation [see Equations (3.5), (3.13), and (3.17)]

xu = x" + £"(xv) (4.38)

changes the metric to

U *«» - £*,� (439)

We wish to retain the post-Newtonian character of g^ and the quasi-Cartesian character of the coordinate system, and to remain in the uni-verse rest frame, thus the functions £� must satisfy: (i) £mv + £v;/, arepost-Newtonian functions; (ii) £�.� + £v;A, -*� 0, far from the system; and(iii) |^"|/|x"| -> 0, far from the system. The only "simple" functional thathas this property is the gradient of the superpotential x,»- Thus, we choose

and obtain, to post-Newtonian order

9~jk = 9jk ~ 2A2Xjk>

9oo = 9oo ~ 2AiX,oo + 2X2HoX,j (4.41)

To the necessary order, the Christoffel symbol rJ00 is equal to � Uj [see

Equation (3.14)]. We must also transform the functional integrals overxk' that appear in g^ into integrals over xF. The only place where thischanges anything is in g00 ^ � 1 + 2U(x,l), where

- f P(X'J

Now the quantity p(x',T) is an invariant; it is the rest-mass density asmeasured in a comoving local Lorentz frame. Furthermore, the quantity(�g)ll2u°d3x is an invariant proper volume element, where u° is the four-velocity of the matter. Thus,

d3x' = d3x'[(-g)ll2i/i/(-g)ll2u0'] (4.42)

Using Equation (4.41) plus the relation u° = dt/dt, we get to the requiredorder,

p 'dV = p'd3x'[l + 212[/(x',l)] (4.43)

We also have

k

|x � x'l ~ |x-x'| 2 |x-xf

Thus,l/(x,T) = U{x,7) + 212O2 - k2 J P ^ _ ~ _ ^ , | 3

? T d3x (4.45)

Using Equations (4.33), (4.36), and (4.41), we obtain, finally

9jk = 9jk ~ ^iXjk,

055 = 0oo - 2A2(C/2 +OW- $2) - 2X^ +<%-*>!) (4.46)

By an appropriate choice of kt and X2 w e c a n eliminate certain terms

from the post-Newtonian metric. We will thus adopt a standard post-Newtonian gauge - that gauge in which the spatial part of the metric isdiagonal and isotropic (i.e., x,jk eliminated) and in which g00 contains noterm Si. There is no physical significance in this gauge choice; it is purelya matter of convenience.

We now have a very general form for the post-Newtonian perfect-fluidmetric in any metric theory of gravity, expressed in a local, quasi-Cartesiancoordinate system at rest with respect to the universe rest frame, and in astandard gauge. The only way that the metric of any one theory candiffer from that of any other theory is in the coefficients that multiplyeach term in the metric. By replacing each coefficient by an arbitraryparameter we obtain a "super metric theory of gravity" whose specialcases (particular values of the parameters) are the post-Newtonian metricsof particular theories of gravity. This "super metric" is called the param-etrized post-Newtonian (PPN) metric, and the parameters are calledPPN parameters.

This use of parameters to describe the post-Newtonian limit of metrictheories of gravity is called the Parametrized Post-Newtonian (PPN) For-malism. A primitive version of such a formalism was devised and studied

by Eddington (1922), Robertson (1962), and Schiff (1967). This Eddington-Robertson-Schiff formalism treated the solar system metric as that of aspherical nonrotating Sun, and idealized the planets as test bodies movingon geodesies of this metric. The metric in this version of the formalismreads

9oj = 0.

9jk = (1 + 2yM/r)6jk (4.47)

where M is the mass of the Sun, and )5 and y are PPN parameters.These two parameters may be given a physical interpretation in this

formalism. The parameter y measures the amount of curvature of spaceproduced by a body of mass M at radius r, in the sense that the spatialcomponents of the Riemann curvature tensor are given to post-Newtonianorder by [see Equations (3.14) and (3.34)]

Riju = (3yM/r3)(«j«^a + n^S^ - nfi^j, - n/i,<5tt - f 8jkSa + ^Sikdjt)

where

n = x/r

independent of the choice of post-Newtonian gauge. The parameter ftis said tb measure the amount of nonlinearity (M/r)2 that a given theoryputs into the g00 component of the metric. However, this statementis valid only in the standard post-Newtonian gauge. The coefficient ofU2 = (M/r)2 depends upon the choice of gauge, as can be seen fromEquation (4.46). In general relativity, for example (/? = y = 1), the (M/r)2

term can be completely eliminated from g00 by a gauge transformationthat is the post-Newtonian limit of the exact coordinate transforma-tion from isotropic coordinates to Schwarzschild coordinates for theSchwarzschild geometry. Thus, this identification of fi should be viewedonly as a heuristic one.

Schiff (1960b) generalized the metric [Equation (4.47)] to incorporaterotation (Lense-Thirring effect, Section 9.1), and Baierlein (1967) de-veloped a primitive perfect-fluid PPN metric. But the pioneering develop-ment of the full PPN formalism was initiated by Kenneth Nordtvedt, Jr.(1968b), who studied the post-Newtonian metric of a system of gravitatingpoint masses. Will (1971a) generalized the formalism to incorporate matterdescribed by a perfect fluid. A unified version of the PPN formalism wasthen presented by Will and Nordtvedt (1972) and summarized in TTEG.The Whitehead term Ow was added by Will (1973). Henceforth, we shall

adopt the Will-Nordtvedt version (as augmented with the Whiteheadterm), altered to conform with MTW signature and index conventions, andwith minor notational modifications (see Table 4.1). As in the Eddington-Robertson-Schiff version of the PPN formalism, we introduce an arbi-trary PPN parameter in front of each post-Newtonian term in the metric.Ten parameters are needed; they are denoted y, j8, & alt a2, a3, d, d> (3.and C4. In terms of them, the PPN metric reads

0Oo = - 1 + 217 - 2pU2 - 2&bw + (2y + 2 + a3 + Ci -

+ 2(3y - 2/J + 1 + £2 + £)<D2 + 2(1 + {3)«>3

9oj = -i(4y + 3 + ax - a2 + d - 2 $ ^ - | ( 1 + a2 - Ci^ = (1 + 2yU)5jk (4.48)

Although we have used linear combinations of PPN parameters inEquation (4.48), it can be seen quite easily that a given set of numericalcoefficients for the post-Newtonian terms will yield a unique set of valuesfor the parameters. The linear combinations were chosen in such a waythat the parameters at, a2, a3, £l5 f2, £3, and £4 will have special physicalsignificance.

Other versions of the formalism have been developed to deal with pointmasses with charge (Section 9.2), fluid with anisotropic stresses (MTWSection 39), and isolated systems in an anisotropic universe (Nordtvedt,1976).

4.3 Lorentz Transformations and the PPN MetricIn Section 4.1, the PPN metric was devised in a coordinate system

whose outer regions are at rest with respect to the universe rest frame.For some purposes - for example, the computation of the post-Newtonianmetric in a given theory of gravity - this is a useful coordinate system.But for other purposes, such as the computation of observable post-Newtonian effects in systems, such as the solar system, that are in motionrelative to the universe rest frame, it is not a convenient coordinate system.In such cases, a better coordinate system might be one in which the centerof mass of the system under study is approximately at rest. Again, this isa matter of convenience; the results of experiments cannot be affected byour choice of coordinate system. Because many of our computations willbe carried out for such moving systems, it is useful to reexpress the PPNmetric in a moving coordinate system. This will also yield some insightinto the significance of the PPN parameters au oe2, and <x3 (Will, 1971c).


To do this we make a Lorentz transformation from the original PPNframe to a new frame which moves at velocity w relative to the old frame.In order to preserve the post-Newtonian character of the metric, weassume that |w| is small, i.e., of O(l). This transformation from rest co-ordinates (t,\) to moving coordinates (T,£) can be expanded in powers ofw to the required order: this approximate form of the Lorentz transforma-tion is sometimes called a post-Galilean transformation (Chandrasekharand Contopoulos, 1967), and has the form

x = | + (1+ » T + ±({ � w)w + O(4) x {,

t = T(1 + W + fw4) + (1 + W)S � w + O(5) x T (4.49)

where wr is assumed to be O(0).We use the standard transformation law,

and express the functional that appear in gaf(x, t) in terms of the newcoordinates. Since p, n , and p are all measured in comoving local Lorentzframes, they are unchanged by the transformation: for any given elementof fluid,

p(x,t) = p(Z,i),

p(x,t) = p«,t) (4.51)

If v(x, t) and v(£, T) are the matter velocities in the two frames, they arerelated by

v = v + w + O(3) (4.52)

The elements of volume d3x' and d3£' in the two frames are related bythe transformation law [Equation (4.42)]

- v' � w - W + O(4)] (4.53)

The quantity x(t) � x'(t) that appears in the post-Newtonian potentialstransforms according to

- T') + K«T) - S'(T')] � WW + O(4),0 = (t - T')(1 + W) + [«t) - £'(*')] � w + O(3) (4.54)


But in the (§,T) coordinates, the quantity £ � §' must be evaluated at thesame time x, hence we must use the fact that

£'(T') = %{x) + v'(t' - t) + O(T ' - T)2 (4.55)

Combining Equations (4.54) and (4.55), we obtain

1 1 .,, {1 + i(w � fi')2 + (w � fi')(v' � n') + O(4)} (4.56)S I

, ,, = .« .,,|X - X | |S � S I

where

S'| (4.57)

We then find, using Equations (4.51)-(4.53), and (4.56), along with thedefinitions of the metric functions, Equations (4.2), (4.32), and (4.35), that

U(x,t) = (1 - WWlr) - wtVj&z) +

T) + 2WiVj($, T) + W2 U(Z, T) + O(6),

.4)($,t) + O(6),

) + 2 ^ W J « , T ) + w V ^ ( { , t ) + O(6),

Vj(x, t) = Vj{l t) + wy!/(«, T) + O(5),

W5(x, t) = Wj(Z, x) + wkUjk($, T) + O(5) (4.58)

Applying the transformation Equations (4.49) and (4.50) to the PPNmetric Equation (4.48) and making use of Equations (4.58) we obtain, forthe metric in the moving (£, T) system, to post-Newtonian order,

- 2)5 + 1 + C2 + �)«2«,T) + 2(1

2(3y + 3(a, - a2

(2a3 - a ^ F / t r ) - (1 - a2 -3 + «t - a2

,T) - | ( 1 - a2 - Ci J

T)],5;t (4.59)

Because we now have available an additional post-Newtonian variable,w, we have an additional gauge freedom that can be employed withoutaltering the standard PPN gauge, which is valid in the frame in whichw = 0 (and which, incidently, was not affected by the post-Galilean trans-formation). By making the gauge transformation

T = T + J(l - <x2 - d + 2Z)w%j, V = V (4.60)

we can eliminate the terms

-(1 - a 2 - Ci + 2£)wJx>Oj from g00,

-Ul - « 2 ~ Ci + 2f)w*JU from g0J

This then becomes part of the standard PPN gauge in a coordinate systemmoving at velocity w relative to the universe rest frame: that gauge inwhich gjk is diagonal and isotropic, and in which the terms 36 and wJx,Ojare absent from g00. It is then possible to show that a further post-Galileantransformation (plus a possible gauge transformation to maintain thestandard gauge) does not alter the form of the PPN metric, it merelychanges the value of the coordinate system velocity w that appears there.

At first glance, one might be disturbed by the presence of metric termsthat depend on the coordinate system's velocity w relative to the universerest frame. These terms do not violate the principles of special relativitysince they are purely gravitational terms, while special relativity is validonly when the effects of gravitation can be ignored; but they do suggestthat the gravitation generated by matter may be affected by motion rela-tive to the universe (violation of the Strong Equivalence Principle). Never-theless, the results of physical measurements must not depend on thevelocity w (this is a consequence of general covariance). For a systemsuch as the Sun and planets, the only physically measureable velocities arethe velocities of elements of matter relative to each other and to the centerof mass of the system, and the velocity, w0, of the center of mass relativeto the universe rest frame (as measured for example by studying Dopplershifts in the cosmic microwave radiation). Thus, the PPN prediction forany physical effect can depend only on these relative velocities and onw0, never on w. Therefore, the terms in the PPN metric that depend onw must signal the presence of effects that depend on w0. This can be seenmost simply by working in a coordinate system in which the system understudy is at rest, i.e., where w = w0. Then, if any one of the set of parameters{ai,a2,a3} is nonzero, there may be observable effects which depend onw0; if at = a2 = a3 = 0, there is no reference to w or w0 in the metric in

any coordinate system, and no such effects pan occur. Thus, we see that theparameters au a2, and <x3 measure the extent and manner in which motionrelative to the universe rest frame affects the post-Newtonian metric andproduces observable effects. These parameters are called "preferred-frameparameters" since they measure the size of post-Newtonian effects pro-duced by motion relative to the "preferred" rest frame of the universe. Ifall three are zero, no such effects are present, and there is no preferredframe (to post-Newtonian order).

Notice that even if one works in the universe rest frame, where w = 0,physical effects will be unchanged, for even though the explicit preferred-frame terms are absent, the velocities of elements of matter vJ that appearin the PPN metric and in the equations of motion must be decomposedaccording to

v = w0 + v"

where v is the velocity of each element relative to the center of mass, and,unless alt <x2, and <x3 are all zero, the same effects dependent upon w0

will result.At this point the PPN metric has taken on its standard form. Table 4.1

summarizes the basic definitions and formulae that enter the PPN for-malism and compares the present version with previous versions.

Table 4.1. The parametrized post-Newtonian formalism

A. Coordinate system: the framework uses a nearly globally Lorentz coordinatesystem [Section 4.1(c)] in which the coordinates are (t,xl,x2,x3). Three-dimen-sional, Euclidean vector notation is used throughout. All coordinate arbitrariness("gauge freedom") has been removed by specialization of the coordinates to thestandard PPN gauge (Section 4.2).

B. Matter variables:1. p = density of rest mass as measured in a local freely falling frame momentarily

comoving with the gravitating matter.2. v' = (dx'/dt) = coordinate velocity of the matter.3. w' = coordinate velocity of PPN coordinate system relative to the mean rest

frame of the universe.4. p = pressure as measured in a local freely falling frame momentarily comoving

with the matter.5. n = internal energy per unit rest mass. It includes all forms of nonrest mass,

nongravitational energy - e.g., energy of compression and thermal energy.

C. PPN parameters:v, P, L «i, &2, «3, Ci, £2, C3> C4

Table 4.1. (continued)

D. Metric:

goo= -1 + 2U- 2PU1 - 2^w + (2y + 2 + a3 + Ci - 2{)®t+ 2(3y - 20 + 1 + C2 + £)*2 + 2(1 + C3)O3 + 2(3y + 3- (£, - 2§st - (at - «2 - a3)w2l/ - a2wWl7y + (2a3

0oi = - i (4y + 3 + a, - <x2 + Ct - 2{W - ftl + a2 -- i ( B l - 2*2)w

tU - >gtj = (1 + 2yU)Sij

E. Metric potentials:

| - x I J |x - x |3

/� p'p"(X - X') ( X' - X" X- X" \®W = - j ^ j � � -j 77s n 7T ) d X d X

J |x � x | J \ |x � x I |x � x I /' [ v ' - (x -xQP f p V ^

|x - x'|3 J |x - x I

J |x - x'|

| I |x - x I

J |x � x |

F. Stress-energy tensor (perfect fluid)

T00 = p(l + U + v2 + 217)

T0i = p(l + Tl + v2 + 2U + p/p)v'

TiJ = pt>V(l + n + v2 + 217 + p/p) + p5iJ(l - 2yU)

G. Equations of motion1. Stressed Matter,

7?vv = 0

2. Test Bodies

d2x"/dX2 + TUdxVd).)(dxx/dX) = 0

3. Maxwell's Equations

H. Differences between this version and the TTEG version1. Adoption of MTW signature (� 1,1,1,1) and index convention (Greek indices

run 0,1,2,3; Latin run 1,2,3)2. New symbol for Whitehead parameter: ^ instead of Cn- as in Will (1973)3. Modified conservation-law parameters incorporating effects of Whitehead

term (see Lee et al., 1974)


4.4 Conservation Laws in the PPN FormalismConservation laws in Newtonian gravitation theory are familiar:

for isolated gravitating systems, mass is conserved, energy is conserved,linear and angular momenta are conserved, and the center of mass ofthe system moves uniformly. This does not apply to every metric theoryof gravity, however. Some theories violate some of these conservationlaws at the post-Newtonian level, and it is the purpose of this section toexplore such violations using the PPN formalism.

One can distinguish two kinds of conservation laws: local and global.Local conservation laws are laws that are valid in any local Lorentzframe, and are independent of the metric theory of gravity. They dependrather, upon the structure of matter that one assumes. Global conserva-tion laws, however, are statements about gravitating systems in asymp-totically flat spacetime. Because they incorporate the structure of boththe matter and the gravitational fields, they depend on the metric theoryin question.

(a) Local conservation lawsConservation of baryon number is one of the most fundamental

laws of physics, and should certainly be valid in the presence of gravity.This law can be expressed as a continuity equation for the baryon numberdensity n: in a local Lorentz frame momentarily comoving with the matter,the equation expressing conservation of baryon number 5A

0 = d(SA)/dt = dind V)/dt (4.61)

is equivalent to

dn/dt + V � (nv) = 0 (4.62)

where v is the baryon velocity in the comoving frame (v = 0 but V � v =SV~1d(SV)/dt # 0). The Lorentz-invariant version of this continuityequation, valid in any local Lorentz frame is

0 = ^(n«°) + ^ ( n ^ ) = (nu")>, (4.63)

where w" is the baryon four-velocity given by u" = dx^jdx. Equation (4.63)can then be generalized to any frame in curved spacetime using the stan-dard "comma-goes-to-semicolon" rule

0 = (nu").^ (4.64)

This is the law of baryon conservation in covariant form. If the matteris assumed to have a chemical composition that is homogeneous and


static, then there is a direct proportionality between the baryon numberdensity (we assume negligible numbers of antibaryons) and the rest massdensity p of the atoms in the element of fluid, namely

p = \m (4.65)

where \i is the mean rest mass per baryon in the element and a constant.Proceeding by a similar argument to the one presented above, one obtainsthe law of rest-mass conservation,

(pu% = 0. (4.66)

By combining this equation with the equations of motion for stressedmatter Tfv

v = 0 along with the assumption that matter is a perfect fluid,we obtain a third local law, the law of local energy conservation or thelaw of isentropic flow. The equation

ujt: = 0 (4.67)

may be evaluated, using Equation (3.71). We work in a local Lorentzframe, momentarily comoving with the element 8V of fluid. From Equa-tion (4.67),

(d/dt)(p + pU) + V � (p + pll + p)\ = 0 (4.68)

This can be rewritten

(d/dt){p + pU) + (p + pU) V � v + p V � v = 0 (4.69)

or,

{d/dt)[(p + pU)8V\ + pd(5V)/dt = 0 (4.70)

So, in a local comoving inertial frame, the change in the total energy (rest-mass plus internal) of an element of fluid is balanced by the work done[pd(8V)2: this simply expresses Local Conservation of Energy or IsentropicFlow, since from the First Law of Thermodynamics, and from Equation(4.70)

d(energy) + pdV = 2f(heat) = TdS = 0 (4.71)

Actually, the absence of heat flow was built into the stress-energy tensorfrom the start by assuming the perfect-fluid form. Had we permitted heattransport, we would have added an additional piece to T"v,

1 heat Z M H

where qv is a "heat-flux four-vector." For further discussion of nonperfectfluids see MTW, Section 22.3 and Ehlers (1971).


Because of the conservation of rest mass, pSV is constant, and Equation(4.70) can be written in the form

pdTl/dt - (p/p)dp/dt = 0 (4.72)

Then in frame-invariant language, Equation (4.72) has the form

«*[n � + p(l/p)J = 0 (4.73)

We can obtain a useful form of the law of conservation of rest mass(or baryon number) by noticing that for any four-vector field, A11,

A^ = (-grll2i{-g)mA"lll (4.74)

hence

(pu% = (-gr1/2l(-g)ll2pu"l, = 0 (4.75)

In a coordinate system (t, x), Equation (4.75) can thus be written

0 = ip(-g)ll2u0l0 + ip(-g)ll2u°v% (4.76)

since uJ = u°vJ.By defining the "conserved density" p*

P* = p(-g)ll2u° (4.77)

we can cast Equation (4.75) in the form of an "Eulerian" continuity equa-tion, valid in our (t,x) coordinate system:

dp*/dt + V � p*v = 0 (4.78)

This "conserved" density is useful because for any function /(x,t)defined in a volume V whose boundary is outside the matter

(d/dt) j y p*f d3x = JK p*(df/dt)d3x (4.79)

Notice that Equation (4.79) implies

dm/dt^Q, m=[ p * d 3 x (4.80)

where m is the total rest mass of the particles in the volume V; from Equa-tion (4.77), we get,

m � \ [pu°(�g)ll2~\d3x = Jpd(proper volume)

= total rest mass of particles (4.81)

(b) Global conservation lawsThe conservation laws discussed above are purely local conserva-

tion laws; they depend only on properties of matter as measured in local,

comoving Lorentz frames, where relativistic and gravitational effects arenegligible (hence they are theory independent). Equation (4.80) representsour first "global" or "integral" conservation law; it is really nothing morethan conservation of baryons coupled with our specific model for matter.

However, when we attempt to devise more general integral conserva-tion laws, such as for total energy (as opposed to exclusively rest mass),total momentum, or total angular momentum, we run into difficulties. Itis well known that integral conservation laws cannot be obtained directlyfrom the equation of motion for stressed matter Tfv

v = 0 because of thepresence of the Christoffel symbols in the covariant derivative. Rather,one searches for a quantity 0"v which reduces to T"v in flat spacetimeand whose ordinary divergence in a coordinate basis vanishes, i.e.,

0?vv = 0 (4.82)

Then, provided 0*" is symmetric, one finds that the quantities

P" = £ ©"v p ^ J"v = 2 £ xl»®vU <*% (4.83)

are conserved, i.e., the integrals in Equation (4.83) vanish when taken overa closed three-dimensional hypersurface E. If one chooses a coordinatesystem (t, x) in which £ is a constant-time hypersurface that extends in-finitely far in all spatial directions, then, provided 0"v vanishes suffi-ciently rapidly with spatial distance from the matter, P" and J1" are inde-pendent of time and are given by

p* = J©*° d3x, J"v = 2 J x ^ 0V>° d3x (4.84)

An appropriate choice of 0"v allows one to interpret the components ofP" and J*� in the usual way: as measured in the asymptotically flat space-time far from the matter, P° is the total energy, PJ is the total momentum,JiJ is the total angular momentum, and J0J determines the motion of thecenter of mass of the matter. If 0"v exists but is not symmetric, then P"is conserved but J"v varies according to

dJ"v/dt= -2 J© M d 3 x (4.85)

The quantity 0"v, normally called the stress-energy complex, has beenfound for the exact versions of general relativity (Landau and Lifshitz,1962), Brans-Dicke theory (Nutku, 1969b), and others (Lee et al, 1974).A wide variety of nonsymmetric stress-energy complexes have been de-vised and discussed within general relativity, but only the symmetricversion guarantees conservation of angular momentum.


There is a close connection between integral conservation laws andcovariant Lagrangian formulations of metric theories. It has been shown(Lee et al., 1974) that every Lagrangian based, generally covariant metrictheory of gravity that either (i) is purely dynamical (possesses no absolutevariables), or (ii) contains prior geometry, with a simple constraint on thesymmetry group of its absolute variables (a constraint satisfied by allspecific metric theories known), possesses conservation laws of the form

0?vv = 0

where 0"v is a function of certain variational derivatives of the Lagrangianof the theory that reduces to T** in the absence of gravity. When thereare no absolute variables, the conservation laws are the result of invari-ance under coordinate transformations, and the stress-energy complexes0"v are not tensors (or tensor densities); moreover, there may be infinitelymany of them. When absolute variables are present, their symmetry groupproduces the conservation laws and 0"v typically are tensors (or tensordensities). Although &lv is guaranteed to exist for any Lagrangian-basedmetric theory, there is no guarantee that it will be symmetric, and nogeneral argument is known to determine the conditions under which itwill be symmetric.

In the post-Newtonian limit, the existence of conservation laws of theform of Equation (4.82) can be translated into a condition on values ofsome of the PPN parameters. The form of 0"v that we shall attempt toconstruct is given by

0*v = (1 - aU)(T"v + t"v) (4.86)

where a is a constant, and t"v is a quantity (which may be interpretedunder some circumstances as "gravitational stress energy") which vanishesin flat spacetime, and which is a function of the fields U, UJk, ®w, Vp

Wp ..., their derivatives, and w (and may also contain the matter vari-ables p, II, p, and v). We reject terms in 0"v of the form

w2T»"

since such terms do not vanish in general in regions of negligible gravita-tional field.

By combining Equations (3.65), (4.82), and (4.86), we find that, to post-Newtonian order, t"v must satisfy

"v (4.87)

In our attempt to integrate Equation (4.87) we will make use of Table 4.1and Equations (4.36) along with the following identity, which is valid forany function/:

+ 17,,V2/ (4.88)

where

riJ(f)=UAif,j)-iSijVU-\f (4.89)

Another useful identity is

-2rij(^w + 3/4172 - VX � Vl/)

^U^ - <5,/l/,0)2] + (2n)-l(d/dt)(U,iU,0)

+ U,£(4ny*V2^ + pv2 + 2p- (87t)-^VL/j2] (4.90)

where \j/j is the solution of the equation

VV ;= -4npUj (4.91)

Then, Equation (4.87) can be put into the form

4nt°; = 47i(t°0° + t°>)

2a- 5)|Vt/|2]

+ a - 3 ) l / j ^ , n + (3y + a - 2)UtiU<0\ (4.92)

5/5t[(4y + 4 + OLXWJVM + i(4y + 2 + a, - 2a2 + 2C1)C/,iC/,0- (5y + a - l)UV2Vi + ia1wI-[/V

2t/ + a2Uti(<n �

+ a/3x^{[l - (f 2 + 4£ - a)C7 + i(a3 - a i)w2]

+ 2ri7(0») + (2a3 -

- (1 + a2 - Ci- 2(4y + 4 + «I)(T^,

+ {Ay + 4 + aiMl/

2 + a i - 2a2 + 2tl)5ij(U,0)2

Vl/)2 - 17.ow � Vl/]

(5y + a- l)U(pv'vJ + pdij) + ziJ} + 4nQ' (4.93)

where

<£ = i(2y + 2 + «3 + Ci - 2{)®i + (3y - 2/3 + 1 + C2 +

+ (1 + £3)<J>3 + (3y + 3C4 - 20*4, (4-94)

x'J = iaiWit /V 2^- + ajWj-t/^-t/.o - OL2WJU,,{W � VC7), (4.95)

Q' = t/j[i(«3 + CI)P»2 + (8n)-K2\vu\2 + c3pn

+ 3Up + (SnrKiV2** + «3pv � w] (4.96)

It has been found to be impossible to write Q, as a combination of gradi-ents and time derivatives of gravitational fields and matter variables.Thus, integrability of Equations (4.92) and (4.93) requires that each of theterms in Q( vanish identically, i.e.,

«3 = fl = Ci = t 3 = C4 3E 0 (4.97)

These constraints must be satisfied by any metric theory in order thatthere be conservation laws of the form of Equation (4.82).

If these conditions hold, then expressions for the conserved energy andmomentum can be obtained using Equations (4.84), (4.86), (4.92), and(4.93). The results are (after integrations by parts):

, (4.98)

n + p/p] - | ( i + 0L2)Wi

- fawjUtj} d3x (4.99)

where we have used the PPN version of the conserved density [Equation(4.77)]

p* = p[i + %V2 + 3yU + O(4)] (4.100)

In the expression for P°, the first term is the total conserved rest mass ofparticles in the fluid. The other terms are the total kinetic, gravitational,and internal energies in the fluid, whose sum is conserved according toNewtonian theory (which can be used in any post-Newtonian terms).Thus, P° is simply the total mass energy of the fluid, accurate to O(2)beyond the rest mass, and is conserved irrespective of the validity of theconditions in Equation (4.97). However, if those conditions were violated,one would expect violations of the conservation of P° at O(4).

An alternative derivation of the conserved momentum uses Chandra-sekhar's (1965) technique of integrating the hydrodynamic equations ofmotion T� = 0 over all space, and searching for a quantity P' whose timederivative vanishes. This procedure is blocked by a term ^Qtd

3x where Qt

is given by Equation (4.96). This integral can be written as a total time

derivative only if (?, can be written as a combination of time derivativesand spatial divergences (which lead to surface integrals at infinity thatvanish). But according to the reasoning given above, this can be true onlyif the five parameter constraints of Equation (4.97) are satisfied. ThenQi 25 0 and the conserved P' derived by this method agrees with Equation(4.99).

We now see the physical significance of the parameters <x3, £u £2, C3,and £4: they measure the extent and manner in which a given metric theoryof gravity predicts violations of conservation of total energy and momen-tum. If all five are zero in any given theory, then energy and momentum areconserved; if some are nonzero, then energy and momentum may not beconserved. According to the theorem of Lee, et al., 1974, every Lagrangian-based metric theory of gravity has all five conservation law parameterszero. Notice that the parameter a3 plays a dual role in the PPN formal-ism, both as a conservation-law parameter and as a preferred-frameparameter.

In order to guarantee conservation of the angular momentum tensorJ^, t"v must be symmetric. Equations (4.92) and (4.93) show that there arenonsymmetric terms, xiJ [Equation (4.95)], in tiJ, and that tOi # t'°. How-ever, in integrating Equations (4.92) and (4.93), we have the freedom toadd to the nominal solutions for t"v any quantity S"v that satisfies

Sfvv = 0 (4.101)

However, we have been utterly unable to find an SiJ that will eliminate orsymmetrize the offending terms tiJ in t'J. As for the toi and ti0 components,the best we can do is to make use of the identity

d/dt(UV2U + |VC/|2) + d/dxJ(UW2Vj- Ui0Uj - 2U_kVlkJ s 0 (4.102)

to eliminate or symmetrize one of the offending terms. A convenientchoice is to match the term involving f/V2 Vt in ti0 with an identical termin toi. With this choice, all dependence on the constant a is eliminatedfrom ®"v. The result is

2a2)UtOUti -

-&lWiUV2U - a2l/>;w � VU, (4.103)

2ti-n = 2 TWI

= aiUWliV2Vn - 2a2l/>ow[|.l/,jl + 2a2w[il/,J.jW � Vl/ (4.104)

Symmetry of t"v requires that each of the terms in Equations (4.103) and(4.104) vanish identically, i.e.,

a!=«2E0 (4.105)

We apply the name Fully Conservative Theory to any theory of gravitythat possesses a full complement of post-Newtonian conservation laws:energy, momentum, angular momentum, and center-of-mass motion, i.e.,whose PPN parameters satisfy

ttl = a2 s a3 = d = £2 = C3 = U = 0 (4.106)

A fully conservative theory cannot be a preferred frame theory to post-Newtonian order since at = a2 = a3 = 0. For such theories, only threePPN parameters, y, /?, and f may vary from theory to theory, and &"v

and t"v have the form

0"v = [1 + (5y - 1)[/](T"V + t"v),

too= �(8w)-1(4y + 3) |Vt/|2,tot = t.o = (47t)-

1[(2y + lyUjUjo + 4(y

tu = [i _ (5y + 4 |

- 8(T

-i(2y + 1)<50([/,0)2,

O == i(2y + 2 � 2^)Oj + (3y � 2/? + 1 -

+ (3y - 2{)0>4 (4.107)

and the conserved quantities are

P° = Jp*(l + ^ 2 - \\J + Jl)d3x

F = Jp*[V(l + iu2 - {7 + n + p/p) - i^£] d3x,

J" = 2 Jp*x[I>J1[l + *»2 + (2y +1)C/ + n + p/p]

JOi = Jp*x'(l + if2 - i l / + n)<i3x - PH (4.108)

By defining a center of mass X1 given by

fp*x'(l+ ?v2 - i[7 + U)d3xX' = ^ (4.109)


we find from Equations (4.108) and the constancy of J0 ' that

(4.110)i.e., the center of mass moves uniformly with velocity P'/P°.

Some theories of gravity may possess only energy and momentumconservation laws, i.e., their parameters may satisfy

a3 = d = C2 = C3 = U s 0, one of {ai,a2} # 0 (4.111)

We call such theories Semiconservative Theories. Their conserved P"may be obtained from Equations (4.98) and (4.99); their nonconservedJ"v may be obtained from Equations (4.84), (4.92), and (4.93). A peculiarfeature of the semiconservative case is that in a coordinate system at restwith respect to the universe, w = 0, and the spatial components t'J areautomatically symmetric, irrespective of the values of at and a2 (sincerij = 0 if w = 0). Thus, spatial angular momentum J'j is a conservedquantity in this frame, whereas it is not in a moving frame. The center-of-mass component J°\ however, is not conserved in any frame, since%oi _£ T>o for a n v w j j j j s discrepancy Can be understood by noting that

the distinction between JiJ and J0J is not a Lorentz-invariant distinction.Because the PPN metric is post-Galilean invariant, the quantities P"and J"v should transform as a vector and antisymmetric tensor respec-tively under post-Galilean transformations. This can be verified explicitlyby applying the transformation Equation (4.49) to the integrals thatcomprise P" and J"v, with the result, valid to post-Newtonian order

P0' = P°(l + |u2) - u P,

P' = P - (1 + i«2)uP° + |u(u � P),

fr = fJ _ j * W + 2(1 + %u2)JoliuJ\

Ji0' = J'°(l + lu2) - uJJiJ - yu}JJ0 (4.112)

where u = uJe, is the velocity of the boost. Thus a boost from the universerest frame where (d/di)JiJ = 0 to a frame moving with velocity w yields

jtr = 2J°(V1[1 + O(w2)] (4.113)

thus, the violation of angular momentum conservation is intimatelyconnected with.the violation of uniform center-of-mass motion. This isour reason for stating that semiconservative theories of gravity possessonly energy and momentum conservation laws. Equation (4.113) may beverified explicitly using Equations (4.103), (4.104), and the fact that

JiJ = 2 J tmd3x, joi = 2 J tli0]d3x (4.114)

Every Lagrangian-based theory of gravity is at least semiconservative.


Nonconservative Theories possess no conservation laws (other thanthe trivial one for P°); their parameters satisfy

oneoffo.fc.Ca.CW^O (4.115)Table 4.2 summarizes these conservation law properties of metric theoriesof gravity, and Table 4.3 summarizes the significance of the various PPNparameters.

Table 4.2. Post-Newtonian integral conservation laws

PPN parameter values

{C1.C2.C3.C4,1*3}

all zeroall zeromay be nonzero

{«i.«2}

all zeromay be nonzeroany values

Type of theory

Fully conservativeSemiconservativeNonconservative

Conservedquantities

P", J"v

P"pOa

" In nonconservative theories, P° is only conserved through lowest Newtonianorder, i.e., to O(2) beyond the conserved rest mass.

Table 4.3. The PPN parameters and their significance

Value in Value in Value inWhat it measures, relative general semiconservative fully conservative

Parameter to general relativity" relativity theories theories

y How much space-curvature 1 y yis produced by unitrest mass?

/} How much "nonlinearity" 1 /? j?is there in the superpositionlaw for gravity?

£ Are there preferred-location O f (effects?

0 at 0Are there preferred-frame 0 a 0

effects? 0 02 00 0 0

Is there violation of conservation 0 0 0of total momentum? 0 0 0

0 0 0

" These descriptions are valid only in the standard PPN gauge, and should not be construed ascovariant statements. For examples of the misunderstandings that can arise if this caution is notheeded, especially in the case of P, see Deser and Laurent (1973), and Duff (1974).

Post-Newtonian Limits of AlternativeMetric Theories of Gravity

We now breathe some life into the PPN formalism by presenting a chapterfull of metric theories of gravity and their post-Newtonian limits. Thischapter will illustrate an important application of the PPN formalism,that of comparing and classifying theories of gravity. We begin in Section5.1 with a discussion of the general method of calculating post-Newtonianlimits of metric theories of gravity. The theories to be discussed in thischapter are divided into three classes. The first class is that of purelydynamical theories (see Section 3.3). These include general relativity inSection 5.2; scalar-tensor theories, of which the Brans-Dicke theory is aspecial case in Section 5.3; and vector-tensor theories in Section 5.4.The second class is that of theories with prior geometry. These includebimetric theories in Section 5.5; and "stratified" theories in Section 5.6.The theories described in detail in these five sections are those of whichwe are aware that have a reasonable chance of agreeing with presentsolar system experiments, to be described in Chapters 7, 8, and 9. Table5.1 presents the PPN parameter values for the theories described in thesefive sections. The third class of theories includes those that, while perhapsthought once to have been viable, are in serious violation of one or moresolar system tests. These will be described briefly in Section 5.7.

5.1 Method of CalculationDespite the large differences in structure between different metric

theories of gravity, the calculation of the post-Newtonian limit possessesa number of universal features that are worth summarizing. It is just thesecommon features that cause the post-Newtonian limit to have a nearlyuniversal form, except for the values of the PPN parameters. Thus, thecomputation of the post-Newtonian limits of various theories tends to

Table 5.1. Metric theories of gravity and their PPN parameter values

Theory and itsgravitational fields

Arbitrary functionsor constants

Cosmological PPN parameters"ma idling

parameters

none

00

thWo

ih<Po

KKK

K l

co,cua,b,c,dco,cua,b,c,d

y

l

1 +0)2 + 0)

1 +0)2 + 0)

1+0)2 + o)

yi

ii

aco/cl

P

1

1 + A

1 + A

1

1

11P

bc0

0

0

0

0

000

000

0

«1

0

0

0

0

0

00

%

a2

0

0

0

0

a'2

( c o / c i ) - l

a'2

a'2«2

(«3,0

0

0

0

0

000

000

00

(a) Purely dynamical theories(i) General relativity (g)(ii) Scalar-tensor (g, <j>)

BWN

Bekenstein's VMT

Brans-Dicke

(iii) Vector-tensor (g, K)GeneralHellings-NordtvedtWill-Nordtvedt

none

o)(0),r,

CO

0)none

(b) Theories with prior geometry(iv) Bimetric theories

Rosen (g, 9) noneRastall(g,ir,K) noneBSLL(g,V)B) a,f,k

(v) Stratified theories

" Prime over a PPN parameter (e.g., / ) denotes a complicated function of arbitrary constants and cosmological matching parameters. See textfor explicit formulae.

have a repetitive character, the major variable usually being the amountof algebraic complexity involved. In order to streamline the presentationof specific theories in the following sections, and to establish a uniformnotation, we present a "cookbook" for calculating post-Newtonian limitsof any metric theory of gravity.

Step 1: Identify the variables: (a) dynamical gravitational variablessuch as the metric g^, scalar field <f>, vector field K", tensor field J5^v,and so on; (b) prior-geometrical variables such as a flat backgroundmetric n^, cosmic time function t, and so on; and (c) matter and non-gravitational field variables.

Step 2: Set the cosmological boundary conditions. Assume a homo-geneous isotropic cosmology, and at a chosen moment of time andasymptotic coordinate system define the values of the variables far fromthe post-Newtonian system. With isotropic coordinates in the rest frameof the universe, a convenient choice that is compatible with the symmetryof the situation is, for the dynamical variables,

9^ -> gfv = diag{-co,ci,cuci), < / > 0 ,

*�->(*:, o,o,o),B^ -* B$ = diag(coo, <ou cou coj (5.1)

and for the prior-geometric variables (these values are valid everywhere,since these variables are independent of the local system),

t = t, with Vt = (l,0) (5.2)

The relationships among and the evolution of these asymptotic values willbe set by a solution of the cosmological problem. Because these asymptoticvalues may affect the values of the PPN parameters, a complete deter-mination of the post-Newtonian limit may in fact require a completecosmological solution. This can be very complicated in some theories.For the present, we shall avoid these complications by simply assumingthat the cosmological matching constants are arbitrary constants (ormore precisely, arbitrary slowly varying functions of time). In Chapter 13,we shall turn to the cosmological question and discuss the relationshipbetween cosmological models and observations that may fix the asymp-totic values of the fields and post-Newtonian gravity. Notice that if a flatbackground metric q is present, it is almost always most convenient towork in a coordinate system in which it has the Minkowski form, for in

Post-Newtonian Limits 119

many theories the resulting field equations involve flat-spacetime waveequations, which are easy to solve. Then the asymptotic form of g shownis determined by the cosmological solution. If r\ is present it is not generallypossible (unless in a special cosmology or at a special cosmological epoch)to make both it and g have the asymptotic Minkowski form simulta-neously. Of course, once the post-Newtonian metric g has been deter-mined, one can always choose a local quasi-Cartesian coordinate system[see Section 4.1(c)] in which it takes the asymptotic Minkowski form. Theform that IJ now takes is irrelevant since, unlike g, it does not couple tomatter. In theories without if, it is usually convenient to choose asymp-totically Minkowski coordinates right away.

Step 3: Expand in a post-Newtonian series about the asymptoticvalues:

Guv Gfiv ' 'Vv)

<f> = 4>0 + <p,

K^iK + ko, fcl5 fe2, /c3),

B,v = &°> + &�� (5.3)

Generally, the post-Newtonian orders of these perturbations are given by

htj ~ 0(2)," 0 0

9k0

boo

~ O(2) H~ O(2) H

~ O(2) H

~ O(2) H

h 0(4), /- 0(4),H 0(4),

- 0(4), Z

»<y ~ 0(3),

/c,- ~ 0(3),

»oi ~ 0(3), fty ~ O(2) (5.4)

Step 4: Substitute these forms into the field equations, keeping onlysuch terms as are necessary to obtain a final, consistent post-Newtoniansolution for h^. Make use of all the bookkeeping tools of the post-Newtonian limit (Section 4.1), including the relation (d/dt)/(d/dx) ~ O(l).For the matter sources, substitute the perfect-fluid stress-energy tensorT"v and associated fluid variables.

Step 5: Solve for h00 to O(2). Only the lowest post-Newtonian orderequations are needed. Assuming that h00 -»0 far from the system, oneobtains the form

/loo = 2aU (5.5)

where U is the Newtonian gravitational potential [Equation (4.2)], andwhere a. may be a complicated function of cosmological matching param-eters and of other coupling constants that may appear in the theory'sfield equations (such as a "gravitational constant"). To Newtonian order,


the metric thus has the form

0 o o = -co + 2ccU, gOJ = 0, giJ = Sif1 (5.6)

To put the metric into standard Newtonian and post-Newtonian form inlocal quasi-Cartesian coordinates, we must make the coordinate trans-formation

x5 = (co)1 / 2x0 , x1 = (Cl)ll2xJ (5.7)

then

065 = Co o o , 06J = (c oci)" il2g0j, 9iJ = cf 1gii,

£7 = cxt7 (5.8)

and

goo = - 1 + 2(cc/coCl)U, 0sj = 0, gij = dtj (5.9)

Because we work in units in which the gravitational constant measuredtoday far from gravitating matter is unity, we must set

Gtoday = a/coC! = 1 (5.10)

The constraint provided by this equation often simplifies other calcula-tions, however there is no physical constraint implied; it is merely adefinition of units.

Step 6: Solve for hu to O(2) and h0J to O(3). These solutions canbe obtained from the linearized versions of the field equations. The fieldequations of some theories have a gauge freedom, and a certain choiceof gauge often simplifies solution of the equations. However, the gaugeso chosen need not be the standard PPN gauge (Section 4.2), and a gauge(coordinate) transformation into the standard gauge [Equations (4.40)and (4.46)] may be necessary once the complete solution has been obtained.

Step 7: Solve for h00 to O(4). This is the messiest step, involving allthe nonlinearities in the field equations, and many of the lower-ordersolutions for the gravitational variables. The stress-energy tensor T"v

must also be expanded to post-Newtonian order. Using Equations (3.71),(5.6), and (5.10), we obtain

T00 = Co V [ l + n + 2cxU + Co'cy + O(4)],

T'J= Co'pv'vJ + c^'pS1' + pO(4) (5.11)

Step 8: Convert to local quasi-Cartesian coordinates [Equation(5.7)] and to the standard PPN gauge (Section 4.2).


Step 9: By comparing the result for g^ with Equation (4.48), or withTable 4.1 (with w = 0), read off the PPN parameter values.

In obtaining these post-Newtonian solutions, the following formulaeare useful

u = - iv2x ,

\\U\2 = V2(iU2 - O2) (5.12)

along with Equations (4.29), (4.33), (4.36), and (4.37).

5.2 General Relativity(a) Principal references: Standard textbooks such as MTW and

Weinberg (1972).(b) Gravitational fields present: the metric g.(c) Arbitrary parameters and functions: None (we shall ignore the

cosmological constant, which is too small to be measured in the solarsystem).

(d) Cosmological matching parameters: None.(e) Field equations: The field equations are derivable from an invariant

action principle 51 = 0, where

+ W « A , 0 U (5-13)

where R is the Ricci scalar [Equation (3.86)] and JNG is the universallycoupled nongravitational action, and G is the gravitational couplingconstant. By varying the action with respect to g^, we obtain the fieldequations

(5.14)

(f) Post-Newtonian limit: Because g is the only gravitational field present,we can choose it to be asymptotically Minkowskian without affecting anyother fields. Thus we have initially c0 = cl = 1. It is convenient to rewritethe field Equation (5.14) in the equivalent form

R^ = 8TTG(T,V - i ^ T ) (5.15)

where T = T^^. To the required order in the perturbation h^, R^v hasthe form

j � nk0,jk + nkk,Oj ~

fcy - fcoo.« + Kk.ii ~ hki,kj ~ hkJ,k,i)

(i) h00 to O(2): To the required order,

Roo = -iV2fc00, TOo = - T * p, 0oo = - 1 (5-17)

thus

V2h00 = -8nGp, h00 = 2GU (5.18)

We now choose units in which G = 1, hence

Ko = 21/ (5.19)

(ii) hy to O(2): If we impose the three gauge conditions (i = 1,2,3)

K, - \Ki = o, K = yf*hH (5.20)

Equation (5.16) for Rtj becomes

V 2 fcy=-8«piw , hiJ = 2UStJ (5.21)

(iii) h0J to O(3): If we impose the further gauge condition

^ - R o = -ifcoco (5-22)

Equation (5.15) becomes

V2h0j + U.oj = 16npvj (5.23)

or, using Equations (4.29), (4.32), and (4.33),

h0J = -4Vj + ix.o; =-ty-iWj (5.24)

It is useful to check that the solutions for h00, hOj, and htJ do satisfy thegauge conditions, Equations (5.20) and (5.22), to the necessary order.

(iv) h00 to O(4): In the chosen gauge, Roo evaluated correctly to O(4)using the known lower-order solutions for h^v where possible, has the form

Roo = -iV2( / j0 0 + 2U2 - 8<D2) (5.25)

To the necessary order, we also have

Too - k o o T = M l + 2(v2 -U + ±I1 + fp/p)] (5.26)

Then the solution to Equation (5.15) is

h00 = 21/ - 2U2 + 4 $ t + 4<D2 + 2<D3 + 6O4 (5.27)

(v) g^ and the PPN parameters: The final form for the metric is

<?oo = - 1 + 21/ - 2[/2 + 4 ^ + 4<D2 + 2«D3 + 64>4,

gtJ = (1 + 2l/)5y (5.28)

Since the metric is already in the standard PPN gauge, the PPN param-eters can be read off immediately

y = p = 1, £ = 0,

a i = a2 = a3 = Ci = C2 = C3 = C4 = 0 (5.29)

(g) Discussion: Notice that general relativity is a fully conservativetheory of gravity (af = £; = 0) and predicts no preferred-frame effects(a* = 0).

5.3 Scalar-Tensor TheoriesA variety of metric theories of gravity have been devised which

postulate in addition to the metric, a dynamical scalar gravitational field<t>. The most general such theory was examined by Bergmann (1968) andWagoner (1970), and special cases have been studied by Jordan (1955),Thiry (1948), Brans and Dicke (1961), Nordtvedt (1970b), and Bekenstein(1977). We shall examine the Bergmann-Wagoner theory in detail, thenshall discuss the various special cases.

(a) Principle references: Bergmann (1968), Wagoner (1970).(b) Gravitational fields present: the metric g, a dynamical scalar field <j>.(c) Arbitrary parameters and functions: Two arbitrary functions of (j),

the coupling function a>{4>) and the cosmological function A(#).(d) Cosmological matching parameters: <f>0.(e) Field equations: The field equations are derived from the action

^ (5.30)

The resulting field equations are

(5.31)

( 5 3 2 )

The field equation for (j> can be rewritten by substituting the contractionof Equation (5.31) into Equation (5.32), with the result

dco^ 4

The cosmological function l(<f>) causes two effects in this theory. First,in the field equation for g, it plays the same role as the cosmological

constant in general relativity. Second, in the field equation for <j>, it givesthe scalar field <l> a range I related to X, co and their derivatives, in thesense that the solutions for cf> for an isolated system contain Yukawa-liketerms exp(�r/l). The result in g00 (Wagoner, 1970) is a "Newtoniangravitational potential" U of the form

x,0 = J^j^^^V (5.34)

where the effective gravitational "constant" is given by

G(x - x') = a + bexp(- |x - x'\/l) (5.35)

Experiments that test the inverse square law for gravitation (see Sec-tion 2.2) could thus set limits on the cosmological function X. However,henceforth we shall assume X = 0.

(f) Post-Newtonian limit: We choose coordinates (local quasi-Cartesian)in which g is asymptotically Minkowskian; <j) takes the asymptoticvalue (f>0 (which presumably varies on a Hubble timescale as the universeevolves). We define

co = co(<p0), co =A == <u'(3 + 2co)- 2(4 + 2co)-1 (5.36)

Following the method of Section 5.1, we obtain for the post-Newtonianmetric

goo= -1 +2U -2 (1 +A)U2

+ 4 ( , . � - A )<E>2 + 2«D3 + 6 ( � - |<D4

In going to geometrized units, we have set

1 <5J8)

Notice that if c/>0 changes as a result of the evolution of the universe,then Gtoday may change from its present value of unity (see Section 8.4).

The PPN parameters may now be read off:

a t = a2 = a3 = d = C2 = £3 = U = 0 (5.39)

For details of the derivation see Nutku (1969a), Nordtvedt (1970b).(g) Other theories and special cases: (i) Nordtvedt's (1970b) scalar-tensor

theory is equivalent to the Bergmann-Wagoner theory in the specialcase of zero cosmological function X = 0. Its PPN parameters are thesame as in the Bergmann-Wagoner theory. We shall denote these generalversions the BWN scalar-tensor theories.

(ii) Brans-Dicke theory is the special case a> = constant, 1 = 0. ItsPPN parameters may be obtained from the BWN PPN parametersby setting a>' = 0 s A. In the limit a> -* oo, the Brans-Dicke theoryreduces to general relativity.

(iii) Bekenstein's (1977) Variable Mass Theory (VMT) is a special caseof the BWN theory with a restricted form for the coupling functionoi((j)). Beginning with a theory in which the rest masses of elementaryparticles are allowed to vary in spacetime via a scalar field <f>, the variationbeing determined by a field equation with two arbitrary parameters rand q, Bekenstein has shown that, when transformed to a metric re-presentation, the theory is a BWN scalar-tensor theory with

l ] [ r + (1 - r)qf(<f>)y2,

0 = [1 - qf(4>)~\f(<t>rr (5-40)

Note that for chosen values for r and q, the present values of a> and Aare determined by the asymptotic value </>0, which in turn is found througha cosmological solution using the theory. For further details, see Beken-stein and Meisels (1978,1980) and Bekenstein (1979).

(iv) Barker's Constant G Theory (1978) is the special case in which

0 * 0 = (4 - 30)/(20 - 2) (5.41)

thus

G,oday = 1 = [constant] (5.42)

and

A = (1 - </>o)/2<Ao = - ( 8 + 4co)-x (5.43)

(h) Discussion: We note that scalar-tensor theories are all fully con-servative theories (a, = £, = 0), with no preferred-frame effects (<x; = 0).In the limit a -*� oo, they reduce to general relativity, both in the post-Newtonian limit and in the exact, strong-field theory, for all except aset of measure zero of pathological coupling functions co(<t>). In particular,this is true for Brans-Dicke theory, Bekenstein's VMT, and Barker'stheory. Generally speaking, for large values of a>($0), these theories makepredictions at the current epoch for all gravitational situations - post-Newtonian limit, neutron stars, black holes, gravitational radiation,cosmology - that differ from general relativity at most by corrections ofO(l/co). However, in theories in which co is a function of <t>, there couldbe significant differences with general relativity in the early universe,even if the present value of co(0o) is large (Chapter 13). In Brans-Dicketheory (constant co) all predictions are within O(l/co) of those of generalrelativity [see Ni (1972) for an extensive list of references for Brans-Dicketheory].

5.4 Vector-Tensor TheoriesWithin the class of purely dynamical metric theories of gravity,

one simple way to devise a theory that is different from the scalar-tensortheories is to postulate a dynamical four-vector gravitational field K"in addition to the metric, thus obtaining a vector-tensor theory of gravity.A broad class of such theories can be analyzed if we restrict attention toLagrangian-based theories, and to theories whose differential equationsfor the vector field are linear and at most of second order. The mostgeneral gravitational action for such theories is given by

iG = (167CG)-1 §[atR + a2KliK"R + aJPlTR^ + a4Kp.vK":v

2 (5.44)

(we have ignored the possible term K^ICg^, since it presumably playsthe same role as the cosmological function A in scalar-tensor theories).In fact this action is too general; it can be simplified by an integrationby parts, dropping divergence terms which do not contribute to the varia-tion of /. Thus the sixth term in JG can be eliminated. Furthermore, theconstant a! can be absorbed into G, resulting in a four-parameter set ofvector-tensor theories.

(a) Principal references: Will and Nordtvedt (1972), Hellings andNordtvedt (1973).


(b) Gravitational fields present: The metric g, a dynamical vector fieldK (assumed timelike).

(c) Arbitrary parameters and functions: Four arbitrary parameters co, rj,e, T.

(d) Cosmological matching parameters: K.(e) Field equations: The field equations are derived from the action

+ xK^K^J - gf'2 d4x + ING(qA, gj (5.45)

where

F,v = *,.� - KK, (5.46)

The resulting field equations are

= SnGT^, (5.47)

eFfvv + \tK%v - ^coK'R - iriKvR$ = 0 (5.48)

where

0£> - K^R + K ^ - \g^K2R - (K%v

= 2K"K(I1RV)X - faj

+ (K*K(fl;V) � K*(flKv) � K ( / JK^).a (5.49)

where K2 = K^K". Throughout, we assume that one of {e, T} is nonzeroin order to have a well-defined free dynamical vector field. An importantproperty of these equations is worth examining here. If one takes a co vari-ant divergence of the left-hand side of Equation (5.47), one finds explicitlythat it vanishes, in agreement with the law Tfv

v = 0, in other words, noadditional constraint on the fields is imposed by the vanishing divergenceof T"v. This is a result of the fact that the action / is generally covariantand contains no prior-geometric variables [see Lee, Lightman, and Ni(1974) for discussion]. However, a divergence of the left-hand side ofEquation (5.48) yields the constraint

ll - (a)K»R + tilPR/i).,, = 0 (5.50)


This is a result of the fact that the action is not fully gauge invariant,i.e., invariant under the transformation

K, -> K^ + A � (5.51)

where A is a scalar function. Only the term involving Fpv is gauge invariant.A Lagrangian that admits such a partial gauge group is called "singular,"and can be shown to satisfy a "Bianchi identity," which, in the case ofthe partial gauge group of a vector field, has the form

. .,. = 0 (5.52)

This is equivalent to Equation (5.50). This means that, in general, thesolution for K^ will be constrained. It is useful to examine the form thatthis constraint takes in the linearized approximation, in which we write

g^ = n^ + Ky K? = K3° + K (5-53)

If we adopt a coordinate system (coordinate "gauge" as opposed to vectorgauge) in which

>,*; - ±h-° = 0 (5.54)

where indices on h^ and kv are raised and lowered using if, and whereh =�= hi, Equation (5.50), to first order in h^ and fcp, takes the form

{3v{?k*v � jK(a> + ^n � \x)h 0} = 0 (5.55)

Since this equation must be satisfied for arbitrary sources, then to firstorder in h^ and k^ we must have

+ %n- %i)ht0 = 0 (5.56)

In the weak field limit, in the chosen gauge, h^v must have the form

h00 = 2(7, hu = 2yUStJ - (y - l)x,,j (5.57)

where y is the PPN parameter. Then Equation (5.56) becomes

T/cyv - 2K{co + \n - ±r)(2y - 1)17.0 = 0 (5.58)

In the case x =£ 0, this represents a constraint on the gauge of the vectorfield k^ imposed by the lack of full gauge invariance of the action /. Inthe case x = 0, no constraint is placed on the vector field; however, inorder to obtain consistent solutions of the equations, with a hope ofagreeing with experiment, we must have co + \r\ = 0, since K # 0 andt / 0 ^ 0 in general, and since experiments (Chapter 7) place the value of

y close to unity, so that 2y � 1 ^ 0. These constraints will be importantin our discussion of the post-Newtonian limit.

(f) Post-Newtonian limit: We choose local quasi-Cartesian coordinatesin the universe rest frame, with K^ taking the asymptotic form K8®, whereK may vary on a Hubble timescale. Following the method of Section 5.1,we compute the post-Newtonian limit, and obtain for the PPN parameters

_ 1 + K2[co - 2co{2co + n~ -Q/(2e - T)]77~ 1 - K2[co + 8«2/(2e - T)]

/*= i(3 + y) + M i + y(v - 2)/G],

a i = 4(1 - y)[l - (2e - T)A] + 4coK2 Aa,

a2 = 3(1 - y)[l - |(28 - T)A] + 2coK2 Aa - \bK2IG,

<*3 = Ci = Ci = Cs = U - 0 (5.59)

The quantities a, A, a, and b are given by

(1 - coK2)(2co -n + 2£) _ (1 2

- T) - 8o>2K2

A = {(2e - t)[l - K2(co + n - T)] + i (^ - T ) 2 K 2 } - \

a = (2B - r)(3y - 1) - 2(n - T)(27 - 1),

f(2o> + n - x)[(2y - l)(r + 1) + <r(y-2)]

= \ -(2y - l)2(2co + i/)[l - T - 1(2co + »,)], % # 0

0, T = 0 (5.60)

and G is related to the other parameters by our choice of geometricalunits, namely

Gtoday s G[i(y + 1) + f coK2(y - 1) - i(ij - T)K 2 (1 + <r)] "» = 1 (5.61)

(g) (Mer theories and special cases: (i) The Will-Nordtvedt (1972)theory is the special case a> = n = s = 0, x = 1. Its PPN parameters aregiven by

y = fi = 1, f = a3 = d = C2 = C3 = C4 = 0,ox = 0, a2 = K2/(l + | K 2 ) (5.62)

with

2 ) = l (5.63)


(ii) The Hellings-Nordtvedt (1973) theory is the special case T = 0,e = 1, r\ = � 2ft). Its PPN parameters are given by

l + coK2 o . . / 1 + coy

= «3 = d = C2 = Cs = C* = 0,

4coK\2(l + co)y + co(y - 1)]

1 + a)X2(l + co)

_2c»K[y + c o ( y l ) ]0(2 ~ 1 + coKHl + co) ( 5 > 6 4 )

with

G,oday = G[cDK\y + 1)] - 1 = 1 (5.65)

We point out that the original computations of Hellings and Nordtvedt(1973) were in error, since their method failed to take into account theconstraint Equation (5.58).

(h) Discussion: These vector-tensor theories are semiconservative(a3 = C, = 0) with possible post-Newtonian preferred-frame effects (oneof{ax,a2} ¥" 0). In the limit {co, r\, E, T} -» 0, they reduce to general relativityboth in the post-Newtonian limit, and in the exact, strong field theory.However, there are other possible limiting cases in which the theoriesmay agree with general relativity (and thus with experiment) in the post-Newtonian limit. For instance, in the limit K -»0, the PPN parameterscoalesce with those of general relativity. However, the present value ofK depends upon a solution of the cosmological problem, and in theearly universe K could be sufficiently large to produce significant dif-ferences.

5.5 Bimetric Theories with Prior GeometryTheories in this class contain dynamical scalar, vector, or tensor

gravitational fields, and a nondynamical metric ij of signature + 2. Intypical theories, t\ is chosen to be Riemann flat everywhere in spacetime,that is

Rlem(i/) = 0 (5.66)

(in some versions, IJ is chosen to correspond to a spacetime of constantcurvature). Because of the above constraint, we can always choose globalcoordinates in which t]^ = diag( � 1,1,1,1); this is usually the most con-venient choice for the computation of the post-Newtonian metric.


Rosen's bimetric theory(a) Principal references: Rosen (1973,1974,1977,1978), Rosen and Rosen

(1975), Lee et al. (1976).(b) Gravitational fields present: the metric g, a flat, nondynamical metric

fl-ic) Arbitrary parameters and functions: None.(d) Cosmological matching parameters: co,cy.(e) Field equations: The field equations are derived from the action

= (647TG)-

x (-V)ll2d*x + ING(qA,g,v) (5.67)

where the vertical line " |" denotes covariant derivative with respect toThe field equations may be written in the form

Riemfo) = 0 (5.68)where � , is the d'Alembertian with respect to q, and T s T^g1�.

(f) Post-Newtonian limit: We choose coordinates in which if has theform diag(� 1,1,1,1) everywhere. In the universe rest frame, g then hasthe asymptotic form diag(�c^c^c^c^) [see Equation (5.1)], where c0

and c t may vary on a Hubble timescale. Following the method of Section5.1, we obtain for the PPN parameters (Lee et al., 1976)

y = p = 1, £ = «3 = Ci = C2 = C3 = U = 0,

Kl = 0, a2 = (co/Cl) - 1 (5.69)with

Gtoday = G{coCl)112 = 1 (5.70)

(g) Discussion: The PPN parameters are identical to those of generalrelativity except for a2, which may be nonzero if c0 # cx. Notice thatthe ratio cjco is equal to the square of the velocity of weak gravitationalwaves, in units in which the speed of light is unity. This can be seen asfollows. In a quasi-Cartesian coordinate system, in which gffl = diag(� 1,1,1,1 ),»;�� must have the form

n^ = diag( - Co \ c^ \ erf *, c r ' )

and the vacuum, linearized field equations for g^v (wave equations forweak gravitational waves) take the form

(co/cite^oo - V V = 0 (5.71)

whose solution is a wave propagating with speed vg = (cjco)112. Thus,

in Rosen's theory, the PPN parameter a2 measures the relative difference

in speed (as measured by an observer at rest in the universe rest frame)between electromagnetic and gravitational waves. The values of c0 andCj are determined by a solution of the cosmological problem. They canalso be related to the covariant expressions

c0 + 3cj = n^gtg!, CQ i + 3cf* = /7JIV0<O)"V

Rastall's theory(a) Principal references: Rastall (1976, 1977a,b,c, 1979).(b) Gravitational fields present: the metric g, a dynamical timelike vector

field K, a nondynamical flat metric r\.(c) Arbitrary parameters and functions: None.(d) Cosmological matching parameters: K.(e) Field equations: The physical metric g is an algebraic function of

the fields if and K, given by

g = (1 + rfKJS.,)-^2^ + K ® K) (5.72)

where \\rfp\\ = H^H"1 . The field equations are derivable from the action

: + W « A » M (5-73)

where indices on K^ are raised using g, and where

F(N) = - N(2 + N)~ \ N = g^K^ (5.74)

We also have Riem(if) = 0. The resulting field equations are

0)- 1/2(0"v - k

= 87tG(l + n^KxK^yll2{T"v - ^VT)KV (5.75)

where F'(N) = dF/dN,

+ F'{N)KX'IIKO[.I)K'1KV (5.76)

and & = ©''"^v, T= T"v^v . In varying the action /, with respect toKp, we have taken account of the fact that the dependence on K^ is bothexplicit and implicit via gMV, thus for example, although the action formatter and nongravitational fields 7NG contains only g^, we have

(5.77)

(f) Post-Newtonian limit: We choose coordinates in which n^ =diag( �1,1.1.1), then from Equation (5.72) g^ takes the form, to post-Newtonian order

0oo = - c o ( l - Kco2k0 - |CQ 4feg),

gOJ = Kc^kj,

gij = ColSJk(l + Kco2k0) (5.78)

where c0 = (1 � K2)112, \K\ < 1. Solving the field equations for k^ to therequired order, substituting into Equation (5.78), and transforming tolocal quasi-Cartesian coordinates in the standard PPN gauge yields thePPN parameters

y = /? = 1, Z. = a3 = d = C2 = C3 = U = 0,

In choosing geometrical units, we set

Gtoday = G = 1 (5.80)

(g) Discussion: RastalFs theory is semiconservative (a3 = Ci � 0), withpreferred-frame effects (a2 # 0). Its PPN parameters are identical to thoseof general relativity, except for a2, which maybe nonzero. The value of<x2 depends upon K, whose value is determined by a solution of the cos-mological problem.

The BSLL bimetric theoryThis theory is a variant of the Belinfante-Swihart nonmetric

theory of gravity, discussed in Section 2.6. Instead of the nongravitationalaction /NG shown in Equations (2.140) and (2.141), one chooses a uni-versally coupled action, thereby obtaining a metric theory of gravity(Lightman and Lee, 1973b). Otherwise the equations of the theory arethe same as those presented in Section 2.6.

(a) Principal references: Belinfante and Swihart (1957a,b,c), Lightmanand Lee (1973b).

(b) Gravitational fields present: the metric g, a dynamical second ranktensor field B, a nondynamical flat metric i\.

(c) Arbitrary parameters and functions: three arbitrary parameters a,f,K.

(d) Cosmological matching parameters: a>0, (o^.

(e) Field equations: The metric is constructed algebraically from t\ andB according to the equations

; - ± 3 D = <5v (5-81)where indices on Apv and B^ only are raised and lowered using n^;indices on all other tensors are raised and lowered using g^; B = B^n*".The field equation for i\ is Riem(//) = 0. The field equations for B arederived from the action

/ = -(167c)"1 j(aB"^B^x + / B ^ X - i / ) 1 ' 2 * * * * + W « A , 0 , « ) (5-82)

where vertical line denotes a covariant derivative with respect to r\. Theresulting field equations are

, , (5.83)

which may be rewritten in the form

D ^ = -(4w/fl)to/f7)1/27^[0|5 - f(a + 4f)-ieil>r,»%d] (5.84)where

Kl = 8gxl,/dB,v (5.85)

(f) Post-Newtonian limit: We work in the universe rest frame, choosecoordinates in which n^ = diag( � 1,1,1,1), and assume that

o.co^co^co!). We further assume that |coo| « 1, \a>^ « 1, assump-tions that turn out to be consistent with experimental limits. Then to thenecessary order, g^ has the form

fifoo = -Do + E0b00 - Fob - K2b2 - 2Kbb00 - |fcg0.

g0J = HbOj,

SiJ = Ddtj + EbtJ + FdiP (5.86)where

Do = 1 - 2Kco -coo + K2co2 + 2Kcoco0 + |coo + O(co3),

Eo = 1 - 2Kco - f a»0 + O(co2),

Fo = -2K + 2K2co + 2Kco0 + O(co2),

H = 1 - 2Kco - |(coo - co^ + O(a>2),

D = 1 - 2Kw + w1+ K2co2 - 2Kcoco1 + |cof + O(co3),

E = 1 - 2Kco + Icoj + O(co2),

F= -2K + 2K2co - 2Kco1 + O(co2),

co = 3(0! � coo (5.87)

Solving the field equations for fcJlv, substituting into Equation (5.86), thentransforming to quasi-Cartesian coordinates and to the standard PPNgauge yields the PPN parameters

p = i [ l + l a " 1 - ia-^Sa2 - 3a)1/2(a + 4/)-1 / 2] + O(co),

i = a3 = Ci = £2 = C3 = U = 0,

a t = (2a)~1[ojo + » ! - (8X - 2)w] + O(co2),

a 2 = � (<o0 + coi) + O(co2) (5.88)

In using geometrized units, we set

_a + 3/-4Xa-16*2a"today � /) j . jn T- vj^ti); � 1. (J.O?,)

2a(a + 4/)

(g) Discussion: The BSLL Theory is semiconservative (a3 = Ct = 0),with potential preferred-frame effects if a>0 or col are nonzero. However,solar system experiments (Chapter 8) demand that la^ and |a2| be small,in keeping with our original assumption that jcoo| « 1, leo^ « 1. Whethera>0 and co1 in fact satisfy this constraint depends upon a solution of thecosmological problem. Notice that if m^ ~ a>0 � 0, the PPN parameterscan be made identical to those of general relativity if

0 = (i -A,*} (5-90)

5.6 Stratified TheoriesThese theories are characterized by the presence, in addition to a

flat background metric t\, of a nondynamical scalar field t whose gradientis covariantly constant and timelike with respect to tj, i.e.,

This scalar field selects out preferred spatial sections or "strata" in theuniverse that are orthogonal to \t. In a frame in which V / = <5°, the equa-tions of a stratified theory take on some special form.

(a) Principal references: Lee, Lightman, and Ni (1974), Ni (1973).(b) Gravitational fields present: the metric g; dynamical scalar, vector,

and symmetric tensor fields <p, K, B; nondynamical flat metric i\ and scalarfield t.

(c) Arbitrary parameters and functions: functions /1 (</>), fii^parameters e, KU K2.

(d) Cosmological matching parameters: co,cud,a,b, c.

(e) Field Equations: The field equations for the prior-geometric vari-ables are

Riem(i;) = 0,

*l,* = 0, t / ^ = - l ,

= 0,

= 0 (5.91)

The last two equations constrain the vector and tensor fields to havecomponents only in the strata orthogonal to \t. The metric g is constructedalgebraically from t/, <j>, t, B, and K according to

9 = / 2 ( # / - E/iW>) - /2(0)]dt ® d t + K ® d t + d t ® K + B (5.92)

The field equations for the dynamical variables are derived from theaction

- <£>* - \_Mcj>)

+ /NG(qA,^v) (5.93)

where all indices on the variables <f>, t, B, and K are raised and loweredusing i\. The result is

\_M<t>)

(5.94)

The constraints on the prior geometric variables allow one to choosea global coordinate system in which t]^ = diag( � 1,1,1,1), ttll = d°,Ko = B^o = 0, and in which the field equations simplify to

'1((/») - Tilf'2(4>)l(5.95)

In this preferred frame (presumably the universe rest frame), g^v has theform

9oj = Kj,

9u = 8M4) + Bij (5-96)

(f) Post-Newtonian limit: In the preferred frame we expand <j>, Kit andBij about cosmological boundary values:

(j> = 4>0 + <P, Kt = k,, By = fi)15y+fty (5.97)

We then define the cosmological matching parameters c0, cu a, b, c, daccording to

M<t>) = c0 - 2c) = (ci - cOi) + 2ac2),

M4) = d + O(<p) (5.98)

Then to post-Newtonian order, g^ has the form

0oo = - c 0 + 2ccp - 2bczcp2,

9oj = kj,

Qii = ci3u + 2acq>5tj + bu (5.99)

Solving the field Equation (5.95) for $, kj, and bip and transforming toquasi-Cartesian coordinates and the standard PPN gauge, we obtainthe PPN parameters

y = aco/ci, P = bc0 + (K2 /8K1C)(C0 /C1),

£, = (K2/8K1C)(C0/C1), a3 = Ci = t2 = C3 = U = 0,

a t = 2e/(coc1)1'2 - 4a(co/Cl) - 4,

a2 = - 1 - ( c o / c J t a ^ c + (d + K22/4Kl)(l + K2/4Kl)-

x] (5.100)

In choosing geometrical units, we set

Gtoday = c2c\'2co3/2(l + K1/4K!)-1 = 1 (5.101)

(g) Other theories and special cases: (i) Ni's (1973) stratified theory isthe special case K^ * = K2 = 0 (no tensor field). Its PPN parameters canbe obtained from Equation (5.100) by setting K2 = 0, with the result

y = acQ/cu P = bc0,

i = a3 = Ci = C2 = C3 = U = 0,

ax = 2e/(coci)1'2 - 4a(co/Cl) - 4,

a2=-l-d(c0/Cl) (5.102)

(ii) Ni's (1972) stratified theory is the special case e = K^ l = K2 = 0(no vector or tensor field). However, as we shall see in the next section,this theory is not viable because its PPN parameter a1 satisfies

a i = -(4y + 4) (5.103)

which is in serious violation of experiment.

5.7 Nonviable TheoriesAll the metric theories of gravity previously discussed have the

property that by making an appropriate choice of values for arbitraryconstants and for cosmological matching parameters, one can producePPN parameter values in agreement with present-day solar system experi-ments, to be described in Chapters 7,8, and 9. In some theories, a particularchoice of these quantities can yield PPN parameters that are identicalwith general relativity at the current epoch. Therefore, in order to testand possibly rule out some of these competing theories, we will have toexplore new arenas for testing relativistic gravity outside the solar system,such as gravitational radiation, the binary pulsar, and cosmology.

However, there is a sizable set of metric theories that, while perhapsonce thought to have been viable, are now known to be in serious violationof solar system experiments. Some of these theories agree with the "clas-sical" tests: deflection of light, time delay, perihelion shift of Mercury (seeChapter 7 for discussion). But this is not enough. There are now manyfurther solar system tests, discovered through the use of the PPN for-malism, that place tight limits on the preferred-frame parameters a1? anda2, on conservation-law parameters such as a3, and on the parameter £,.Many theories violate these limits. The lesson to be learned is that it isno longer sufficient for the inventor of an alternative gravitation theoryto compare the predictions of the theory with experiment by simplyderiving the static spherically symmetric solution (analogue of theSchwarzschild solution in general relativity), obtaining the PPN param-eters ft and y. He or she must determine the full post-Newtonian metricfor a dynamical system of bodies or fluid, possibly moving relative to theuniverse rest frame, including cosmological matching parameters. Onlywith a complete set of values for the PPN parameters can the theory becompared with the results of solar system experiments.

Many of the nonviable theories that we shall describe were discussedin more detail in TTEG. We shall touch upon them here only briefly,referring the interested reader to TTEG and the original references fordetails.

(a) Quasilinear theoriesQuasilinear theories of gravity are theories whose post-

Newtonian metric, in a particular post-Newtonian gauge, contains onlylinear potentials, in particular lacks the potentials U2 and <SW. This is aproperty of many theories that attempt to describe gravity by means ofa linear field theory on a flat spacetime background. If the gauge in which

this occurs is not the standard PPN gauge, then a gauge transformation,as in Equations (4.38) and (4.40) yields

06o = 0oo ~ 2X2(U2 + ®w- <D2) - 21^,00 (5.104)

Since g00 did not contain U2 or <bw, we see immediately that

{ = P (5.105)

We shall see that this is in severe violation of Earth-tide measurements(Chapters 8 and 9).

The most famous example of a quasilinear theory is Whitehead's(1922) theory. The theory has a nondynamical flat background metric IJ,and a physical metric constructed algebraically from IJ and the mattervariables according to

= n,,v � 2( w-)3iA »/ r>

( / ) - = X" - (*�)-, (yT(y»)~ = o,w - = (/)-(uM)-, u" = dx^/da,

da2 = rifl,dx"dxy (5.106)

where the superscript (�) indicates quantities to be evaluated along thepast i/-light cone of the field point x*. The post-Newtonian metric has

y = P = £, = 1, oti = <x2 = a3 = 0,{Ci,C2,C3,C*}^0 (5.107)

Although the theory was thought for a long time to have been viable,the value £ = 1 is now known to be in violation of Earth-tide measure-ments.

Another group of theories in this class is known as Linear Fixed-Gauge (LFG) theories. The standard field theoretic approach to theconstruction of a tensor gravitation theory on a flat spacetime backgroundis to use the gauge invariant action for a spin-two tensor field h^, com-bined with the universally coupled nongravitational action to yield

- h*v[ V l J ( - f ) 1 / 2 d 4 x + /NG(<2A,0/1V) (5.108)

where g^ = */�� + h^. However, the Lagrangian is singular: the gravita-tional part is invariant under the gauge transformation

hpv -> h ^ � £(�!�) (5.109)

while /N G is not. The Bianchi identity associated with this partial gaugeinvariance is

= 0 (5.110)

which is in conflict with the equation of motion that results from thegeneral coordinate invariance of / [Equation (3.63)],

T? v =0 (5.111)

LFG theories seek to remedy this by breaking the gauge invariance ofthe gravitational action through the introduction of auxiliary gravita-tional fields that couple to h in such a way as to fix the gauge of h. Never-theless these theories, devised by Deser and Laurent (1968) and Bolliniet al. (1970), turn out to be quasilinear in the sense defined above, andpredict <* = P in violation of experiment (see Will, 1973).

(b) Stratified theories with time-orthogonal space slicesThese theories are special cases of the stratified theories dis-

cussed in Section 5.6, in which there is no vector field K^, i.e., e = 0.

Table 5.2. Nonviable metric theories of gravity

Theory" Description Reasons for nonviability

(a) Quasilinear theoriesWhiteheadDeser-LaurentBollini-Giambiagi-Tiomno

For some gauge, U2

and Q>w are absentfrom 0oo; thus £ = fi

(b) Stratified theories with time-orthogonal space slicesEinstein (1912) Metric is given byWhitrow-Morduch g = / idt ® it + f2t\;Rosen thus a : = � 4(y + 1)PapapetrouNi (2 versions)YilmazPage-TupperColeman

(c) Conformally flat theoriesNordstromEinstein-FokkerNi (2 versions)Whitrow-MorduchLittlewood-Bergmann

Metric is given byg = fii; thus y = � 1

Predict galaxy inducedperihelion shifts andEarth tides, in violation ofobservation

Predict preferred-frameeffects on Earth's rotationrate and on perihelionshifts, in violation ofobservation

Predict no deflection ortime delay of light, inviolation of observation

1 For discussion and references, see TTEG, Ni (1972), and Will (1973).

They therefore have the property that

f "0T + W)g^ = - K" = 0 (5.112)

independently of the nature of the source. In the preferred frame, thismeans goj = 0. However, under a possible coordinate transformation toput the post-Newtonian limit of the theory into the standard PPN gauge,g0J becomes

goj = sx,oj = sVj-eWj (5.113)

By comparing this with the PPN metric [Equation (4.48)], it is possibleto obtain in a straightforward manner, independently of £,

B l = -(4y + 4) (5.114)

This is a gross violation of geophysical experiments that demand loc « 1,while time-delay measurements demand y « 1 (see Chapters 7 and 8).Prior to the placing of the limit on ax, theories of this type were popularalternatives to general relativity, largely because of their mathematicalsimplicity. Table 5.2 lists nine theories of this type, all nonviable.

(c) Conformally flat theoriesThese theories typically possess a flat background metric IJ and

a scalar field <f>. The metric g is constructed from r\ and </> according to

g = /(#/ (5.H5)

where / is some function of <f>. However, in order to obtain the correctNewtonian limit, /($) must have the form (in a suitable coordinate system)

/ = 1-217 + 0(4) (5.116)

Thus,

flfy = [1 - 21/+ O(4)]5y (5.117)

hence y = � 1. We shall see in Chapter 7 that this implies zero bendingof light and zero time delay, in violation of experiment. This result canalso be deduced from the conformal invariance of Maxwell's equations(i.e., invariance under the transformation g^ -» </>#��): propagation of lightrays in the metric f(4>)ti is identical to propagation in the flat spacetimemetric i/, namely straight-line propagation at constant speed. Table 5.2lists six conformally flat theories, all nonviable.

Equations of Motion in the PPN Formalism

One of the consequences of the fundamental postulates of metric theoriesof gravity is that matter and nongravitational fields couple only to themetric, in a manner dictated by EEP. The resulting equations of motioninclude

Tfvv = 0, [stressed matter and nongravitational fields] (6.1)

wvwfv = 0, [neutral test body: geodesies] (6.2)

F?vv = 4nJ", [Maxwell's equations] (6.3)

/cv/cfv = 0, [light rays: geodesies] (6.4)

(see Section 3.2 for discussion). In Chapter 4, we developed the generalspacetime metric through post-Newtonian order as a functional of mattervariables and as a function of ten PPN parameters. If this metric issubstituted into these equations of motion, we obtain coupled sets ofequations of motion for matter and nongravitational field variables interms of other matter and nongravitational field variables. For specificproblems, these equations can be solved using standard techniques toobtain predictions for the behavior of matter in terms of the PPN param-eters. These predictions can then be compared with experiment. It is thepurpose of this chapter to cast the above equations of motion into a formthat can be simply applied to specific situations and experiments. Thatapplication will be made in Chapters 7, 8, and 9. In Section 6.1, we carryout this procedure for light rays. Section 6.2 deals with massive, self-gravitating bodies and presents appropriate n-body equations of motion.In Section 6.3, we derive the relative acceleration between two bodies,including the effects of nearby gravitating bodies and of motion withrespect to the universe rest frame, and put it into a form from which one

Equations of Motion in the PPN Formalism 143

can identify a "locally measured" Newtonian gravitational constant.Section 6.4 specializes to semiconservative theories and presents an n-bodyaction from which the semiconservative n-body equations of motion canbe derived. We also develop in Section 6.4 a conserved-energy formalismof the type discussed in Section 2.5, and discuss the Strong EquivalencePrinciple from this viewpoint. In Section 6.5, we analyze equations ofmotion for spinning bodies.

6.1 Equations of Motion for PhotonsWe begin with the geodesic equation obtained from Maxwell's

equations in the geometrical-optics limit [Equation (6.4)]:

fev/cfv = 0 (6.5)

where k11 is the wave vector tangent to the "photon" trajectory, with

*"*� = 0 (6.6)Substituting k" = dx*/da where a is an "affine" parameter measured alongthe trajectory, we obtain

We can rewrite Equation (6.7) using PPN coordinate time t = x° ratherthan a as affine parameter by noticing that

Then the spatial components of Equation (6.7) can be rewritten

~dW +

Equation (6.6) can be written

^ v ^ ! ^ L = 0 (6.10)

To post-Newtonian accuracy, Equations (6.9) and (6.10) take the form(see Table 6.1 for expressions for the ChristorTel symbols T*k):

dx2S

dt2 � = [ / , . 1 + ydt

0 = 1 - 2C7 - \dx/dt\2(l + 2yU) (6.11)

The Newtonian, or zeroth order solution of these equations is

x£ - n\t - t0), \n\ = 1 (6.12)

Table 6.1. Christoffel symbols for the PPN metric

3 + a, - a2 + C, - 2 © ^ + i ( l + a2 -t/,J-) + a2

+ y)U2

3 + B l - a, + Ci - 2 « ^ +1(1 + a2 - d(at - 2a2)w'U + ajvWl/y],C/>0 - i(4y + 4 + a j ^ ^ - i

where

2 + <x3 + {, - 2^a>x + (3y - 2)3

(3y + 3C4

in other words, straight-line propagation at constant speed |dxN/di| = 1.By writing

xj = n\t - to) + xJp (6.13)

and substituting into Equation (6.11) we obtain post-Newtonian equationsfor the deviation xJ

p of the photon's path from uniform, straight linemotion:

^ £ = (l + y)[yu - 2n(n � VC7)], (6.14)

dxi'-£=-(l+y)U (6.15)

In Chapter 7 we shall use these equations to derive expressions for thedeflection and the time delay of photons passing near the Sun.

6.2 Equations of Motion for Massive BodiesOne method of obtaining equations of motion for massive bodies

is to assume that each body moves on a test-body geodesic in a space-time whose PPN metric is produced by the other bodies in the system aswell as by the body itself (with proper care taken of infinite self-fieldterms). However, the resulting equations of motion cannot be applied tomassive self-gravitating bodies, such as planets, stars, or the Sun (exceptin general relativity, as it turns out), because such bodies do not necessarilyfollow geodesies of any PPN metric. Rather, their motion may depend


upon internal structure (a violation of GWEP). This was first demon-strated by Nordtvedt (1968b).

Therefore, one must treat each body realistically, as a finite, self-gravitating "clump" of matter and solve the stressed-matter equations ofmotion [Equation (6.1)] to obtain equations of motion for a suitablychosen center of mass of each body. For the purposes of solar systemexperiments, it is adequate to treat the matter composing each body asperfect fluid (see Will, 1971a for discussion).

In Newtonian gravitation theory, this program is straightforward. Bydefining an inertial mass and a center of mass for each body according to

ma = I pd3x,Joth body

xa = m-1 f pxd3x (6.16)Ja

one can show, using the Newtonian equation of continuity [Equation (4.3)]that

dmjdt = 0,

va = dxjdt = m~1 ja p\d3x,

K = dyjdt = m;l I p{dv/dt)d3x (6.17)

By using the Newtonian perfect-fluid equations of motion [Equation (4.3)]we obtain the following expression for aa

S \ \ Ql # (^)] (6-18)b*a \Jab L rab J

where mb is the inertial mass of the bth body, Q'J is its quadrupole momentdefined by

J i | | 2 ) 3 (6.19)

and \ab and rah are given by

xai, = xa ~ xb, r^ = IxJ (6.20)

We now wish to generalize these equations to the post-Newtonianapproximation, using the PPN formalism. Because there are many dif-ferent "mass densities" in the post-Newtonian limit - rest-mass of baryonsp, mass-energy density p{\ + U), "conserved" density p*, and so on -there is a variety of possible definitions for inertial mass and center ofmass. The definition we shall adopt is chosen in order to yield the simplestclosed-form result for the equations of motion. It turns out that as long

as we average the equations of motion over several internal dynamicaltimescales of each body (assumed short compared to the orbital dynamicaltimescale), the final equation of motion is insensitive to the precise formof the definition. We define the inertial mass of the ath body to be

ma = f p*(l + iF2 - \V + n)d3x (6.21)Ja

where p* is the conserved density [Equation (4.77)], v = v � va(0), wherev = f n*\d3x (6 77\Yo(0) � I r " " �*� \v.£^.)

andU = £ p(x',t)\x -x'l'1 d3x' (6.23)

Note that, roughly speaking, ma is the total mass energy of the body -rest mass of particles plus kinetic, gravitational, and internal energies - asmeasured in a local, comoving, nearly inertial frame surrounding thebody. As long as we ignore tidal forces on the ath body, then accordingto our discussion of conservation laws in the PPN formalism [see Equa-tion (4.108)], ma is conserved to post-Newtonian accuracy, i.e.,

dmjdt = 0 (6.24)

This can also be shown by explicit calculation using Equations (6.21),(6.22), and (6.23). We now define the center of inertial mass

xa s m~1 f p*(l + iv2 - \V + U)xd3x (6.25)Ja

By making use of the equation of continuity for p* [Equation (4.78)]and by using Newtonian equations of motion in any post-Newtonianterms, we obtain

vfl = dxjdt = m~1 f [p*(l + iu2 - iU + IT)v + pv - |p*W] d3x (6.26)

where

The acceleration aa is thus given byao = d\Jdt

= m'1 < Ja p*(l + it;2 - if/ + U)(d\/dt)d3x

+ v{ £ pjf d3x + £ |>>ov - (p/p*)Vp] d3x

P*Wd3x + \g-a - \«r*a + g>>\ (6.28)


where &~a, 3~*, and 0>a are determined purely by the internal structureof the ath body. Formulae for these and other "internal" terms are givenin Table 6.2. Notice that the acceleration of our chosen center of mass ismore than just the weighted average of the accelerations of individualfluid elements, as it is in Newtonian theory.

We now evaluate the first integral in Equation (6.28) using the PPNperfect-fluid equations of motion. We substitute the post Newtonianexpressions for T"v (Table 4.1) and Y*x (Table 6.1) into the equation ofmotion, (6.1), and rewrite it in terms of the conserved density p*. The

Table 6.2. Integrals for massive bodies in the PPN equations of motion.

Vector integrals

' "II '13X � X I [X � X I

o*p*'p*"(x' - x") � (x - x')(x - x'Y**v

r H y u, ^ ~>d3Xd3X',

|x - x'|3

y*i - I P*P*'lr " (x ~ x')]2(* ~ x')3

X - X' 5

Tensor and scalar integrals:

P*v'vJd3x

n« = - i j»�prz^p�r f xd x> "-= -*J.-prr7j-J xd

I'J = £ p*(x - xj'(x - xa)^3x, / . = ja p*\x - xa\2 d3x

result is

p*dvJ/dt = p*Utj - |>(1 + 3yU)lj + Pj&2 + U + pip*)

v\p*Ui0 - p,0) - i ( l + <x2 -

iP*[(4y + 4 + atf + (ax -

p*(d/dxJ)[® - £«V - i ( d -

- (2jS - 2)U + 3yp/p*] (6.29)

where O is given in Table 6.1.We now substitute this expression for p*d\/dt into Equation (6.28) and

perform the integration, using Newtonian equations where necessary tosimplify post-Newtonian terms. Considerable simplification of the equa-tions results if we average over several internal dynamical timescales ofeach body. Then we can set equal to zero any total time derivatives ofinternal quantities. This is a reasonable approximation for the solarsystem, since any secular changes in the structure of the sun or planetsthat would prevent the vanishing of such averaged time derivatives occursover timescales much longer than an orbital timescale. This allows us touse several Newtonian virial relations to simplify post-Newtonian expres-sions. These relations, easily derived using the Newtonian equations ofmotion have the form for each massive body

H"= -<Q> = 0,

ST*> + 33T**J - Q*i - 0>J = (~ [p*WWx) = 0,\dt J /

=(j Jp*VJ d3x)=0 (6.30)

The final form of the equation of motion is

K = (aAelf + (aa)Newt + (aJnbody (6-31)

where

{ai)seK = -m^iu** + CM + Ci(ri - ir**J)vttfH

kJ, (6.32)

(6.33)

(27 + 2f})rr± ^r

I ^ {(7 f) (y p fa C2)b*a rab (. rab rab

+(2j8-l-2«-C2) E ^+(2y + 2/?-2£) ^ �c*a* r6c c*ab rac

.,2> + 2 + a2+<x3)»i

a2)(v6 � Kb)2 + | « 2 (w � fla!,)2 + 3a2(w � hab)(vb � nab)

| ( 7 + ^ + 1 2 + C1)E I #*#a rab c*ab rbc

- t Z ^ ( ^ - 3 « 4 ^ ) E "C^

+ I 5 xa6 � [(2y + 2)va - (2y

1 *w~ 5 I ^xa6-[(47 + 4+a1)vfl

z b*a "ab

~ * I ^ r x«fc' [«iv« - (ai ~ 2«2)T6 + 2a2w]w ' (6.34)z i>#n "aft

where nab = \ab/rab

The first six terms in (aa)self, Equation (6.32), involving terms such ast{, 3~'a, and so on, depend only on the internal structure of the ath massivebody, and thus represent "self-accelerations" of the body's center of mass.Such self-accelerations are associated with breakdowns in conservationof total momentum, since they depend on the PPN conservation-lawparameters a3, £i> 2> C3, and £4- In any semiconservative theory of

gravity,

«3 EE d = C2 = C3 s C4 = 0 (6.35)

and these self-accelerations are absent. Also note that spherically sym-metric bodies suffer no acceleration regardless of the theory of gravity,since for them the terms t{, PJ

a, ^~**j, Q.{, &{, and &[ are identically zero.The same is true for a composite massive body made up of two bodies ina nearly circular orbit, when the self-acceleration is averaged over anorbital period. Thus, there is little hope of testing the existence of theseterms in the solar system. However, in the binary pulsar, for instance,where the orbit eccentricity is large, there may be a potential test. Weshall discuss this possibility in Section 9.3.

The next term in Equation (6.32), �m~1a.3(w + vafHk.j, is a self-

acceleration which involves the massive body's motion relative to theuniverse rest frame. It depends on the conservation-law/preferred-frameparameter <x3, which is zero in any semiconservative theory of gravity.For any static body, v = 0, thus HkJ is zero, but for a body that rotatesuniformly with angular velocity o>,

v = <o x (x - xa) (6.36)

and

\X � X I

= e/llmco\Sla)Jm (6.37)

For a nearly spherical body, the isotropic part of QJm makes the dominantcontribution to Equation (6.37), i.e.,

(Qaym * &jmna, HkJ =s ±e*WQ. (6.38)

Then the acceleration term in Equation (6.32) becomes

-!<x3(Qa/ma)(w + ya) x to (6.39)

In Chapter 8, we shall see that this term may produce strikingly largeobservable effects in the solar system, if a3 is different from zero.

The next term, (ao)Newt in Equation (6.31) is the quasi-Newtonian ac-celeration of the massive body. Here (mP)a* is the "passive gravitationalmass tensor" given by

(mP)ik=ma{<5*[l + (4/J - y - 3. - 3{ - a, + a2 - Ci W.M, - 3£nafcnam^i7ma]

C2)fiJ*MJ (6.40)

and U(xa) is the quasi-Newtonian potential, given by

U(xa) = £ & ^ ) irab

where [mA(habf\b is the "active gravitational mass" of the bth body, given by

(6-42)

Note that the active and passive gravitational mass tensors may be func-tions of direction n^ relative to the other bodies. It is useful to rewritethe quasi-Newtonian acceleration in a form involving inertial, active andpassive mass tensors that are independent of position, and a gravitationalpotential U'm, as follows

W'm= £ frJTK&r* (6.43)

where

(ax - a2 + CiRM] + (a2 - d(4/J - y - 3 - 3fl«./<| - ZflT/(4/? - y - 3 - 3{ - i«3 - Ki -

+ Ca^M, - (fa3 + Ci - X&M - « - KiJOfM} (6-44)In Newtonian theory, the active gravitational mass, the passive gravita-tional mass, and the inertial mass are the same, hence each massive body'sacceleration is independent of its mass or structure ("Equivalence princi-ple"). However, according to Equation (6.44), passive gravitational massneed not be equal to inertial mass in a given metric theory of gravity (andin fact both may be anisotropic); their difference depends on several PPNparameters, and on the gravitational self energy (Q and Qik) of the body.This is a breakdown in the gravitational Weak Equivalence Principle(GWEP) (see Section 3.3), also called the "Nordtvedt effect" after its dis-coverer (Nordtvedt, 1968a, b). The possibility of such an effect was firstnoticed by Dicke [1964b; see also Dicke (1969), Will (1971a)]. The observ-able consequences of the Nordtvedt effect will be discussed in Chapter8. Its existence does not violate EEP or the Eotvos experiment (Chapter 2),because the laboratory-sized bodies considered in those situations havenegligible self gravity, i.e., (il/m)^^ bodies < 10~39. In Section 6.3, we shallsee that there is a close connection between violations of GWEP andthe existence of preferred-location and preferred-frame effects in post-Newtonian gravitational experiments.


According to Equation (6.44), active gravitational mass for massivebodies may also differ from inertial mass and from passive gravitationalmass. In Newtonian gravitation theory, the uniform center-of-mass mo-tion of an isolated system is a result of the law "action equals reaction,"i.e., of the law "active gravitational mass equals passive gravitationalmass." In the PPN formalism, one can still use such Newtonian languageto describe the quasi-Newtonian acceleration (ao)Newt. From Section 4.4,we know that uniform center-of-mass motion is a property of fully conser-vative theories of gravity, whose parameters satisfy

a, = a2 = oc3 = Ci = C2 = C3 = U = 0 (6.45)

By substituting these values into Equation (6.44), we find that for fullyconservative theories, the inertial mass is equal to ma, and the active andpassive mass tensors are indeed equal, and are given by

(«ptf=(«A)^ = { ^ [ l + ( 4 / » - 7 - 3 - 3 W . / m J - { n f / m . } (6.46)

equivalently, (a£)Newt can be written to post-Newtonian order in the form

(ae^/TS*^ sxjjj^Xj {6A7)

\ma mj\ r%b r^ ))

The term in braces is manifestly antisymmetric under interchange of aand b, hence action equals reaction, and £ , ma(a{)Newt = 0. Note that ingeneral relativity, the mass tensors of Equation (6.44) are isotropic andequal to the inertial mass, i.e., (dropping the Kronecker deltas)

fhl = fhp = mA = ma [general relativity] (6.48)

There is no Nordtvedt effect in general relativity. However, in scalar-tensor theories, there is in general a Nordtvedt effect, since

mP = mA = ma{\ + [(2 + co)"1 + 4A]fta/ma} (6.49)

For most practical situations, we may assume that the bodies in questionare spherically symmetric, then using the equation ClJ

ak m %SJkQa to sim-

plify the mass tensors, we may write

« = I (MAV^ (6.50)b*


where (we combine (m{k)~1 and ml� into one quantity mP)

K)fl/ma = 1 + (40 - y - 3 - Aft - Kl + | a 2 - f £

K ) » M = 1 + (4/J - y - 3 - ^ - i«3 - Ki+ Cs^/m - (|a3 + d - SCJPJm, (6.51)

The remaining term (ajnbody in Equation (6.31) is called the n-bodyterm. It contains the post-Newtonian corrections to the Newtonianequations of motion which would result from treating each body as a"point mass" moving along a geodesic of the PPN metric produced byall the other bodies, assumed to be point masses, taking account ofcertain post-Newtonian terms generated by the gravitational field of thebody itself. It is the n-body acceleration which produces the "classical"perihelion shift of the planets, as well as a host of other effects, to beexamined in Chapters 7 and 8. For the case of general relativity, the n-bodyterms in Equation (6.34) are in agreement with the equations obtained byde Sitter (1916) [once a crucial error in de Sitter's work has been corrected],Einstein, Infeld, and Hoffmann (1938), Levi-Civita (1964), and Fock (1964).

6.3 The Locally Measured Gravitational ConstantHere, we derive an equation which is not really an equation of

motion, but is nevertheless a fundamental result in the PPN formalism.In the previous section, we found that some metric theories of gravitycould predict a violation of GWEP (Nordtvedt effect). Such effects wouldrepresent violations of the Strong Equivalence Principle (SEP). As dis-cussed in Section 3.3, the existence of preferred-frame and preferred-location effects in local gravitational experiments would also representviolations of SEP. One such local gravitational experiment is the Caven-dish experiment. In an idealized version of such a Cavendish experimentone measures the relative acceleration of two bodies as a function of theirmasses and of the distance between them. Distances and times are mea-sured by means of physical rods and atomic clocks at rest in the labora-tory. The gravitational constant G is then identified as that number withdimensions cm3 g"1 s"2 which appears in Newton's law of gravitationfor the two bodies. This quantity is called the locally measured gravita-tional constant GL.

The analysis of this experiment proceeds as follows: a body of mass mt

("source") falls freely through spacetime. A test body with negligible massmoves through spacetime, maintained at a constant proper distance rp

from the source by a four-acceleration A. The line joining the pair ofmasses is nonrotating relative to asymptotically flat inertial space. An in-variant "radial" unit vector Er points from the test mass toward the source.


Then according to Newton's law of gravitation the radial component ofthe four acceleration of the test mass is given by

/KEr=-GlmJr2p (6.52)

for rp small compared to the scale of inhomogeneities in the external grav-itational fields. Since the quantity A � Er is invariant, we can calculate it ina suitably chosen PPN coordinate system, then use Equation (6.52) toread off the locally measured GL.

Before carrying out the computation, however, it is instructive to askwhat might be expected for the form of A � Er to post-Newtonian order.We imagine that the source and the test body are moving with respect tothe universe with velocity w1 and are in the presence of some externalsources, idealized as point masses of mass ma at location xa. It is simplestto do the calculation in a PPN coordinate system in which the sourceis momentarily at rest. Then we would expect A � Er to contain post-

rlm1ml mt ma m^ ma mY 2

Newtonian corrections to the equation A � E, = m1frl of the form

mY 2A E r ; - 2 � ' 72�> -rzr> 72 -K) (6-53)

YV rP rP ria 'P rl" P

where rla = |xx � xa|. In obtaining this form, we have neglected the varia-tion of the external gravitational potentials across the separation rp. Thisvariation will produce the standard Newtonian tidal gravitational force,which is of the form

( A E ) - m a rrla

and post-Newtonian corrections to this force. The latter we shall neglectthroughout. The first term in Equation (6.53) represents post-Newtonianmodifications in the two-body motion of the test body about the source,which can be understood and analyzed separately from a discussion of GL.The third term represents effects due to the gradients of the external fields;however, if we fit A � Er to an r~2 curve in order to determine GL, theseterms will have no effect [in most practical situations, they are negligiblysmall anyway (Will, 1971d)]. Both of these types of terms will be droppedthroughout the analysis. Thus, we retain only terms of the form (m,/rl)(mjrla) or (m1/^)(wj).

The form of the PPN metric that we shall use is given by the expressionin Table 4.1, where now the velocity w is the source's velocity relative tothe mean rest frame of the universe, denoted wt. We label the test body bya = 0, the source by a � 1, and the remaining bodies by a = 2, 3 , . . . Ini-tially, both the source and test body are at rest, i.e.,

Vl(t = 0) = vo(t = 0) = 0 (6.54)

We separate the Newtonian gravitational potential Ux due to the sourcefrom that due to the other bodies in the system:

l/(x) = Ufa) + £ mjra (6.55)

where rx = |x � xx\, ra = |x � xo|, and Ut is assumed for simplicity to bespherically symmetric.

The proper distance between the test body and the source is given by[see Equation (3.41)]

rp = £ [1 + yU(x(X)) + O(4)]|dx/dA|dk (6.56)

where to sufficient accuracy we may choose a straight coordinate line tojoin the two points:

x(l) = xo(l - X) + \tX, 0 < X < 1 (6.57)

Then

Neglecting the variation of the external gravitational potential across theseparation r01 leads to

rP = rj\ +yE- ) + ? \T U^da (6.59)

The proper distance rp is to be kept constant by the four-acceleration A,thus

drp/dt s d\/dt2 = 0 (6.60)

with the result, at t � 0,

)

where we have used the fact that Vj = v0 = 0 at t = 0, and have neglectedtime derivatives of the external potential. For the rest of this discussion,it is sufficient to drop the final term in Equation (6.59) (it leads only toterms that we previously decided to ignore) and to treat the coefficientof r01 as a constant. Thus,

( ) (6-62)


We now assume that the source follows a geodesic of spacetime, butthat the four-acceleration of the test body is A. Thus,

^source^source;v *A

«,VeS,«rest;v = A", uJU,^ = 0 (6-63)In PPN coordinates, Equation (6.53) may be written, at t = 0,

dt ' uuv o / \dt

,4° = 0 (6.64)

where, for the test body,

j ) = 1 - 2l/1(x0) - 2 £ mjrla + O(4) (6.65)

where we have again ignored the variation of the external potential inevaluating it at xt instead of at x0. We make use of the PPN Christoffelsymbols (Table 6.1) evaluated for the external point masses [substitutep = p*{l � jv2 � 3yU), \ap* d3x = ma] and use the Newtonian equationsof motion to simplify any post-Newtonian terms. We retain only theterms discussed above; for illustration we also keep the Newtonian tidalforce. Substituting Equations (6.64) into (6.61) yields, finally,

A - x 1 0 _ marloeJek(3n{an\. ~ $Jk)

_ ^ -3'10 a* l "la

'10 L a* I r l a

+ � � Vt/f'(x0) |"1 a 2wX - { X ^" ' "" ' " l (6-66)r10 1_Z �#] ?"la J

where

- XI

p*(x',t)d3x'9

-x)(xo-x)dx ( 6 ^ 7 )

For a spherically symmetric source, it is possible to show straightforwardlythat

O(6),

VjUfl(x0) = (mJr\o){Moekel - 2x%5l)i)l - 2x%dl» - 4o<5*') + O(4) (6.68)

where mt and It are the rest mass and spherical moment of inertia of thesource, given by

mx = I p*d3x, / t = I p*r2d3x (6.69)

We must now compute the invariant radial unit four-vector Er. Itscomponents at x0 are simply those of the tangent vector to the curve x(A)joining the two bodies,

E} = adx\k)ldk = - oxJ01, £r° = 0 (6.70)

The normalization a is obtained from

ErvE; == 1 = a2|x01|2 1 + 2V X ~ I (6-71)\ 0*1 ~\aj

where we have retained only the necessary terms. Thus

(6-72)

Then the invariant radial component of the four-acceleration A is

(6.73)

The final result is (Will, 1971d, 1973; Nordtvedt and Will, 1972)

A E, = X mar10[3(nlo � e)2 - l>r . 3

� T(WI � a2 � ot3)w1 � 5 a 2 ( w i ' e ) + t z^ (nio ' e 'fl#l '"la

The first term in Equation (6.74) is simply the Newtonian tidal accelera-tion. From the second term we may read off the locally measured gravita-tional constant,

H ^ y i ^ ) ^ (6-75)where

U& = X man{an\Jrla, UeU = [/£, (6.76)

Here, we see a direct example of the possibility of violations of theStrong Equivalence Principle, via preferred-frame or preferred-locationeffects in local Cavendish experiments. The preferred-frame effects dependupon the velocity Wj of the source relative to the universe rest frame,and are present unless the PPN preferred-frame parameters a1; a2, and<x3 all vanish. The preferred-location effects depend upon the gravitationalpotentials Unt and (/£*, of nearby bodies, and are present in generalunless the PPN parameters satisfy £, = (4)3 � y � 3 � £2) = 0- ^n t n e

next section we shall develop a conserved energy formalism for thespecial case of semiconservative theories of gravity that will reveal adirect connection between violations of local Lorentz and position in-variance in Cavendish experiments, and the violations of GWEP describedin Section 6.2.

We note here that general relativity predicts

GL = 1 (6.77)

6.4 N-Body Lagrangians, Energy Conservation, and theStrong Equivalence PrincipleIn the previous two sections we showed that some metric theories

of gravity may predict violations of GWEP and of LLI and LPI forgravitating bodies and for gravitational experiments. In the special caseof theories of gravity that possess conservation laws for energy and

momentum, namely semiconservative theories, it is possible to derive adirect relationship between these violations. The method is the same asthat developed in Section 2.5: derive a conserved energy expression for acomposite system in a quasi-Newtonian form, from which one can readoff the anomalous inertial and passive gravitational mass tensors Sm[J and5m'J, respectively. The use of cyclic gedanken experiments, parallel tothose used in Section 2.5, then reveals that violations of GWEP as wellas of LLI and LPI depend upon these anomalous mass tensors.

The derivation of these results proceeds as follows (Haugan, 1979): Wefirst restrict attention to semiconservative theories of gravity, thus <x3 =d = £2 = £3 = £4 == 0, and to systems in which the basic particles arepoint masses. We then build composite bodies out of point masses movingin their mutual gravitational fields. We work in a PPN coordinate frameat rest with respect to the universe rest frame. The equations of motionfor the particles then consist of the standard Newtonian accelerationplus the post-Newtonian n-body acceleration anbody, Equation (6.34) withw = 0 and with semiconservative PPN parameters,

'ab Lb*a 'ab L 'ab 'ab

c*ab rbc c*ab rac

a2)c*ab rac

- i(4? + 4 + ax)va � \b + i(2y + 2 + a 2 )^ - f (1 + a2)(>

'ab c±ab 'be

-11 5 0* - Wto I * fir - Tr \' rb

b*arab

'ab

'a

- (2y

4 + a i)vo - (4y + 2 + ax - 2a2)vfc]^ (6.78)

It is then possible to show straightforwardly that these equations ofmotion can be derived from the Euler-Lagrange equations obtained byvarying the trajectory xq(t), vq(t) of the qth particle in the action

^ (6.79)

where

3 + a i - a2)va � vfc - i ( l + a2)(vo � nab)(v6 � fij

Consider a system consisting of a body of mass mQ and a compositebody made up of bodies of mass ma. We assume that m0 » Xom«> anc*that the massive body is situated at rest at the origin a distance |X| fromthe composite body, where |X| is large compared to the size of the compos-ite body. Because it is more massive, the distant body may be assumed toremain at rest, thereby providing an external potential in which thecomposite body resides and moves. (We ignore coupling of the body toinhomogeneities in the external potential.) We now make a change ofvariables in L from xa to center-of-mass and relative variables X and xa,respectively, where

X = m~1 £ maxa, m = Ym<"a a

xa=xa-X (6.81)

We also have

\a = dxjdt, V = dX/dt (6.82)

A Hamiltonian H is then constructed from L using the standard technique

PJ = 8L/dVJ, pJa = dL/dvJ

a,

H = PJVJ + X rfpl - L (6.83)

The result is

P2 mm0 v-i Pa 1 *** *�*H = m + � + >2m R ~ 2ma 2 % rab

ab rab K ab rab

ab 'ab ab m 'ab

+ Jd + «2) - I ? (».,' P)(«.. � P.) + O(p') + O(P*) (6.84)m aft 'ab

where /? = |X|, n = X/R, and nofc = xab/fab. We have neglected post-Newtonian terms O(p4) and O(P*) in H that do not couple the internalmotion and the center-of-mass motion of the composite system. We nowaverage H over several timescales of the internal motions of the compositesystem, and make use of virial theorems for the internal variables,

+ mO(4)\ (6.85)

As in Section 2.6, we argue that although the post-Newtonian terms inEquation (6.85) may depend on P or X, this dependence does not affectthe form of H. The resulting average Hamiltonian is then rewritten interms of V using the equation V = 5<H>/5P. The result for the conservedenergy function is

£ c = M + \{M5ij + [(at - a2)Q<5y + a2Q0]} VW

- {M8'J + [(4j8 - y - 3 - 3£)Q<5ij' - ^QiJ2}mon'nj/R (6.86)

where

p« 1 V m - m > \2 m a l ab T ab I

(6.87)ab

By comparing Equation (6.86) with Equations (3.77) and (3.78) we mayread off the anomalous mass tensors

dm\J = (a! - tx2)Q8iJ + <x2QiJ,

3my = (4/J - y - 3 - 3{)IM<> - &J (6.88)

Substituting these results into Equation (3.80) yields

3ak = M-l\jAP - y - 3 - 3£)Q<5U - £&J~\(d/dXk)(mon'nJ/R)

+ AT '[(o^ - a2)iWw + oc2Qk}]m0x

J/R3 (6.89)

This is in complete agreement with the GWEP-violating terms in ajSjewt.Equations (6.40), (6.43), and (6.44) if we substitute the semiconservativevalues of the PPN parameters, and take into account that the potentialU'm is that due to a single distant point mass, i.e.,

Uim = monlnm/R (6.90)

To determine the influence of the internal structure of the compositebody on its center-of-mass motion, we fixed its structure and focussed onthe explicit P and X (or V and X) dependence of H. Now, to study theeffect of a body's motion on its internal structure, and thereby obtain anexpression for GL, we must fix the center-of-mass motion (P,X), andfocus on the explicit p and x dependence of H. Using the Newtonianvirial theorem [Equation (6.85)] to simplify the post-Newtonian termsin H, we obtain the conserved energy function

Ec = M + i{M8iJ - [(<*! - oc2)Qdij + a2Qy] }PiPj/M2

- {M5ij + [(4)3 - y - 3 - 3£)Q<5>V - ^QiJ]}moni«J7^ (6-91)

where M and Q'v are given by Equation (6.87). Notice that the quantitiesin square brackets are precisely 5m[J and 5m^, of Equation (6.88), but thatthe sign in front of dm[J is opposite to that in Equation (6.86) (a result ofexpressing Ec in terms of P rather than V).

Let us suppose for simplicity that the composite body is composed oftwo point masses in a local Cavendish experiment. Then with Ec writtenin the above form, it is possible to show straightforwardly that the effectiveforce between the two particles is given by

F=- (V£ c )p , X f U e d (6.92)

Then the effective local gravitational constant is

- [(4/3 - y - 3 - 3£)8iJ - ^e^monW/R (6.93)


where e = x12/r12, and Q'J = �mlm2e'eJ/r12. This is precisely Equation(6.75), with P/M = w1; mon

lnJ/R == U'Jxt, and with lx = 0. Again, we seethe explicit connection between violations of GWEP and violations ofLLI and LPI, for the case of semiconservative theories of gravity.

6.5 Equations of Motion for Spinning BodiesThe motion of spinning bodies (gyroscopes, planets, elementary

particles) in curved spacetime has been a subject of considerable researchfor many years. This research has been aimed at discovering (i) how abody's intrinsic angular momentum (spin) alters its trajectory (deviationsfrom geodesic motion), and (ii) how a body's motion in curved spacetimealters its spin.

No really satisfactory solution is available for the first problem, outsideof approximate solutions, or solutions in special spacetimes, because ofthe difficulties in defining rigorously a center of mass of a spinning bodyin curved spacetime. The most successful attempts at a solution havebeen made by Mathisson (1937), Papapetrou (1951), Corinaldesi andPapapetrou (1951), Tulczyjew and Tulczyjew (1962) and Dixon (1979).The central conclusion of these calculations has been that the intrinsicspin S1" (i.e., J"v evaluated in the body's "center-of-mass" frame) of abody should produce deviations from geodesic motion of the form

mSa* ~ Sv VK? a (6.94)

where W is the body's four velocity, and R*lX is the Riemann curvaturetensor. However, these calculations differ greatly in details and inter-pretation. For a spinning body moving with velocity v in a Newtoniangravitational potential U ~ M/r, these deviations are, in order of magni-tude:

5a ~ (|S*|/m)|v|(M/r3) ~ ( i 'A/rHM/r) 1 ' 2^ (6.95)

where b is the radius of the body, and k its rotational angular velocity.For a planet rotating near break-up velocity (X2 ~ m/b3), we have

Sa £ (m/b)1'2(M/r)1'2(b/r)aIhwl % 10 ~J 2aNewt (6.96)

and for a 4 cm-radius gyroscope orbiting the Earth (frequency 200 rps),

5a £ 10~20 aNewt (6.97)

Thus, for the most part, spin-induced deviations from geodesic motioncan be ignored in the solar system. In our derivation of massive-body


equations of motion (Section 6.2), we ignored the effects of tidal gravita-tional forces (Riemann curvature tensor); and thus our equation ofmotion, (6.31), does not include the effects of spin.

Even for a rapidly rotating neutron star such as the binary pulsar(b ~ 10 km, A ~ 102 Hz, m ~ lmG, r ~ 106 km),

(5a;glO-10aNewt (6.98)

and can be ignored (see Chapter 12).It is problem (ii), the effects of a body's motion on its spin, which is

better understood. All calculations to date have shown that, as long asthe direct effects of tidal gravitational forces (Riemann curvature tensor)on the spinning body can be neglected, the spin S is Fermi-Walkertransported along the body's world line. Here the four-vector S has thecomponents

S"s^Vi,> u"S,, = 0 (6.99)

The equation of Fermi-Walker transport is then

uvS?v = ul'id'S,,) (6.100)

where a" is the body's four-acceleration, given by

a" = uvufv (6.101)

The reader is referred to MTW, Section 40.7 for further discussion ofFermi-Walker Transport. The following derivation is patterned after thatsection.

It is convenient to analyze Equation (6.100) in a local Lorentz framewhich is momentarily comoving with the body. The basis vectors of thisframe are related to those of the PPN coordinate system by a Lorentztransformation plus a normalization, and are given by

e% = W,

e°j =vj + O(3),

4 = (1 - yU)8) + %Vjvk + O(4) (6.102)

where all quantities in Equation (6.102) are assumed to be evaluatedalong the world line of the body. Thus, because of Equation (6.99), thespin is a purely spatial vector in this frame, i.e.,

S6 s egS, = !«� = 0 (6.103)


We now calculate the precession of the spatial components of the spin Sj.Since efu^ = 0, we have, from Equation (6.100),

0 = efifS^ = i?Sj.v - SMuv4v (6-104)

and since Sj is a scalar (scalar product of two vectors), we have

uvS/;v = wvS;>v = dS}/dT (6.105)

The second term in Equation (6.104) is most easily evaluated in the PPNcoordinate frame. Using Equation (6.102), we first obtain relations betweenSM and Sf

S0=-VJSJ+O(3)SJ,

Sj=Sj+O(2)S} (6.106)

Then after some simplification, we get, to post-Newtonian order,

dSj/dx = SlVuak] + g0lKSi - (2y + l)vuU,k{\ (6.107)

This can be written in three-dimensional vector notation

dS/dx = ft x S,

ft = -%\ x a - ^V x g + (y + i)v x Vt/,

g =

In Equation (6.108) it does not matter whether the vectors entering intoft are evaluated in the PPN coordinate frame or in the comoving frame,since their spatial basis vectors differ only by terms of O(2). We havecalculated the precession of the spin relative to a comoving frame whichis rotationally tied to the PPN coordinate frame, and whose axes arefixed relative to the distant galaxies. Thus, we have calculated the spin'sprecession angular velocity ft relative to a frame fixed with respect tothe distant galaxies. We shall discuss the observable consequences ofthis precession in Chapter 9.

The Classical Tests

With the PPN formalism and its associated equations of motion in hand,we are now ready to confront the gravitation theories discussed in Chapter5 with the results of solar system experiments. In this chapter, we focuson the three "classical" tests of relativistic gravity, consisting of (i) thedeflection of light, (ii) the time delay of light, and (iii) the perihelion shiftof Mercury.

This usage of the term "classical" tests is a break with tradition. Tradi-tionally, the term "classical tests" has referred to the gravitational red-shift experiment, the deflection of light, and the perihelion shift of Mercury.The reason is largely historical. These were among the first observableeffects of general relativity to be computed by Einstein. However, inChapter 2 we saw that the gravitational red-shift experiment is really nota test of general relativity, rather it is a test of the Einstein EquivalencePrinciple, upon which general relativity and every other metric theory ofgravity are founded. Put differently, every metric theory of gravityautomatically predicts the same red-shift. For this reason, we have droppedthe red-shift experiment as a "classical" test (that is not to deny its im-portance, of course, as our discussion in Chapter 2 points out). However,we can immediately replace it with an experiment that is as importantas the other two, the time delay of light. This effect is closely related to thedeflection of light, as one might expect, since any physical mechanism inMaxwell's equations (refraction, dispersion, gravity) that bends light canalso be expected to delay it. In fact, it is a mystery why Einstein did notdiscover this effect. It was not discovered until 1964, by Irwin I. Shapiro.The simplest explanation seems to be that Shapiro had the benefit ofknowing that the space technology of the 1960s and 1970s would makefeasible a measurement of a delay of the expected size (200 us for a round

Classical Tests 167

trip signal to Mars). No such technology was known to Einstein. Hewas aware only of the known problem of Mercury's excess perihelionshift of 43 arcseconds per century, and of the potential ability to measurethe deflection of starlight. But the lack of available technology may notbe the whole story. After all, Einstein derived the gravitational red-shiftat a time when the hopes of measuring it were marginal at best (a reli-able measurement was not performed until 1960), and other workers suchas Lense and Thirring, and de Sitter derived effects of general relativity,with little or no hope of seeing them measured using the technology ofthe day. Why then, did no one at the time take the step from deflectionto time delay, if only as a matter of principle?

Nevertheless, despite its late arrival, the time delay deserves a place inthe triumvirate of "classical" tests, not the least because it has given oneof the most precise tests of general relativity to date!

We begin this chapter with the deflection of light (Section 7.1), turn tothe time delay (Section 7.2), and finally to the perihelion shift of Mercury(Section 7.3).

7.1 The Deflection of LightAn expression for the deflection of light can be obtained in a

straightforward way using the PPN photon equations of motion, (6.14)and (6.15). Consider a light signal emitted at PPN coordinate time te

at a point xe in an initial direction described by the unit vector ft, wheren n = l. Including the post-Newtonian correction xp, the resultingtrajectory of the photon has the form

x°(t) = t,

x(t) = x£ + %t - O + xp(t) (7.1)

where we have imposed the boundary condition xp(te) = 0. We decomposexp into components parallel and perpendicular to the unperturbedtrajectory:

Xp(f)|| = ft � Xp(t),

x p «i = xp(t) - n[n � xp(t)] (7.2)

Equations (6.14) and (6.15) then yield

^ (7.3)

= (1 + y){Uj - n^n � Vt/)] (7.4)


For simplicity, we assume that the Newtonian gravitational potential Uis produced by a static spherical body of mass m at the origin (Sun), i.e.,

Along the unperturbed path of the photon, U then has the form

mr(t) fi(t - te)\

(7.5)

(7.6)

To post-Newtonian order, then, Equation (7.4) can be integrated alongthe unperturbed photon path using Equation (7.6) with the result

where

ddt'

d =

r (

M

n X (X, xft)

mA lx(t) � i} d> \ r(t)

a xe ft (7.7)

(7.8)

Note that d is the vector joining the center of the body and the point ofclosest approach of the unperturbed ray (see Figure 7.1). Equation (7.7)represents a change in the direction of the photon's trajectory, toward thesun (in the direction -d). We then have

Consider an observer at rest on the Earth (©) who receives the photonfrom the source and a photon from a reference source located at a different

Figure 7.1. Geometry of light-deflection measurements.

ReferenceSource

Source

Earth

Classical Tests 169

position on the sky, xr. The angle 9 between the directions of the twoincoming photons is a physically measurable quantity, and can be givenan invariant mathematical expression. The tangent four-vectors feM =dx"/dt and fefr) = dx$r)/dt of the paths x"(t) and x("r)(t) of the two incomingphotons are projected onto the hypersurface orthogonal to the observer'sfour-velocity t / using the projection operator

PI = K + "X (71°)The inner product between the resulting vectors is related to the cosineofflby

If we ignore the velocity of the Earth, which only produces aberration,then Equation (7.11) simplifies to

coSe=l-(g00)-1gllvk%) (7.12)

By substituting Equations (7.1) and (7.9) into Equation (7.12) we get, topost-Newtonian accuracy,

'*

where

M M (x, x 8,) (7.14)

It is useful to note that, to sufficient post-Newtonian accuracy in Equation(7.13),

d = n x (xe x fl), dr = nr x (xe x fir) (7.15)

We now define the angle 60 to be the angle between the unperturbed pathsof the photons from the source and from the reference source, i.e.,

cos0osfl-ii r (7.16)

and we define the "deflection" of the measured angle from the unperturbedangle to be

d6 = 6-d0 (7.17)

There are two interesting cases to consider. This first is an idealized situa-tion that leads to a simple formula. We suppose that the Sun itself is the

reference source, then, dr = 0, the second term inside the braces in Equa-tion (7.13) vanishes,

(7.18)

and

d \ r e re

For a photon emitted from a distant star or galaxy,

re»r®, x e - n / r e ^ - l (7.20)

Also, to sufficient accuracy,

x e � fi/r® ~ nr � n = cos 90 (7.21)

thus,

(! + ,) * ( . + - * )

For general relativity (y = 1) this is in agreement with results obtainedby Shapiro (1967) and Ward (1970).

It is interesting to note that the classic derivations of the deflection oflight that used only the principle of equivalence or the corpuscular theoryof light (Einstein, 1911, Soldner, 1801) yield only the "1/2" part of thecoefficient in front of (Am/d)(\ + cos0o)/2 in Equation (7.22). That doesnot invalidate these calculations however; they are correct as far as theygo. But the result of these calculations is the deflection of light relativeto local straight lines, as denned for example by rigid rods; however,because of space curvature around the Sun, determined by the PPNparameter y, local straight lines are bent relative to asymptotic straightlines far from the Sun by just enough to yield the remaining factor "y/2".The first factor "1/2" holds in any metric theory, the second "y/2" variesfrom theory to theory. Thus, calculations that purport to derive the fulldeflection using the equivalence principle alone are incorrect (see Schiff,1960a, and the critique by Rindler, 1968).

The deflection is a maximum for a ray which just grazes the Sun, i.e.,for 60~0,d^Ro^ 6.96 x 105 km, m = mQ = 1.476 km. In this case,

<50max = I d + 7)1"75 (7.23)

The second case to consider is more closely related to the actual methodof measuring the light deflection using the techniques of radio interfero-

Classical Tests 171

metry. There one chooses a reference source near the observed source andmonitors changes 80 in their angular separation. If we define $ and <S>r

to be the angular separation between the Earth-Sun direction and theunperturbed direction of photons from the two sources, as in Figure 7.1,then

cos O = x e � ft/r9, cos <J>r = x e � nr/rm (7.24)

Assuming again that the two sources are very distant, we obtain

/ I + y\r4m /cos*, � cos<J>cos0o\ / I + cos$\= )\ ) )T \ sin*sin90

Am /cos <Dr cos 0O � cos 4>\ / I + cos<J>A~]~ T \ sin*rsin0o ) V 2 ) \ K1J5)

If the observed source direction passes very near the Sun, while the refer-ence source remains a decent angular distance away, we can approximate$ « <br, and thus,

60 =s <Dr - 4> cos x + O(*2/^r) (7.26)

where / is the angle between the Sun-source and Sun-reference directionsprojected on the plane of the sky (Figure 7.1). The resulting deflection is

This result shows quite clearly how the relative angular separation betweentwo distant sources may vary as the lines of sight of one of them passesnear the Sun (d ~ RQ, dr » d, % varying).

The prediction of the bending of light by the Sun was one of the greatsuccesses of Einstein's general relativity. Eddington's confirmation of thebending of optical starlight observed during a total solar eclipse in thefirst days following World War I helped make Einstein famous. However,the experiments of Eddington and his co-workers had only 30% accuracy,and succeeding experiments weren't much better: the results were scatteredbetween one half and twice the Einstein value, and the accuracies werelow (for reviews, see Richard, 1975; Merat et al., 1974; Bertotti et al., 1962).The most recent optical measurement, during the solar eclipse of 30 June1973 illustrates the difficulty of these experiments. It yielded a value

|(1 + y) = 0.95 ± 0.11 [lo- error] (7.28)

(Texas Mauritanian Eclipse Team, 1976 and Jones, 1976). The accuracywas limited by poor seeing (caused by a dust storm just prior to the


eclipse, and by clouds and rain during the follow-up expedition inNovember, 1973) that drastically reduced the number of measurable starimages. There were also variable scale changes between eclipse- andcomparison-field exposures. Recent advances in photoelectric andastrometric techniques may make possible optical deflection measure-ments without the need for solar eclipses (Hill, H. 1971).

The development of long-baseline radio interferometry has altered thissituation. Long-baseline and very-long-baseline (VLBI) interferometrictechniques have the capability in principle of measuring angular sepa-rations and changes in angles as small as 3 x 10 ~4 seconds of arc. Coupledwith this technological advance is a series of heavenly coincidences: eachyear, groups of strong quasistellar radio sources pass very close to theSun (as seen from the Earth), including the group 3C273, 3C279, and3C48, and the group 0111 + 02, 0119 + 11 and 0116 + 08. The angularposition of each quasar determines a phase in the radio signal at theoutput of the radio interferometer that depends on the wavelength of theradiation and on the baseline between the radio telescopes. The angular

Figure 7.2. Results of radio-wave deflection measurements 1969-75.

Value of i (1 + 7)

0.88 0.92 0.96 1.00 1.04 1.08

Ia,xw

1969

1970

1971

1972

1973

1974

1975

i i I i i i rRadio Deflection Experiments

Muhleman et al. (1970)

Seielstad et al. (1970) |

Hill (1971) I�

Shapiro (quoted in Weinberg, 1972)

Sramek (1971) | � �

Sramek (1974) I

Riley(1973)

Weileretal. (1974) t-

Counselman et al. (1974)

Weileretal. (1974)

Fomalont and Sramek (1975)

Fomalont and Sramek (1976)

i I r

� i

5 10 2040=o

Value of Scalar�Tensor GO

Classical Tests 173

separation between a pair of quasars is determined by a difference inphases. As the Earth moves in orbit, changing the lines of sight of thequasars relative to the Sun, the angular separation 89 varies [Equation(7.25)], resulting in a variation in the phase difference. The time variationin the quantities d, dr, d>, and 3>r in Equation (7.25) is determined using anaccurate ephemeris for the Earth and initial directions for the quasars,and the resulting prediction for the phase difference as a function of timeis used as a basis for a least-squares fit of the measured phase differences,with one of the fitted parameters being the coefficient ^(1 + y). A numberof measurements of this kind over the past decade have yielded an accuratedetermination of ^(1 + y), which has the value unity in general relativity.Those results are shown in Figure 7.2.

One of the major sources of error in these experiments is the solarcorona which bends radio waves much more strongly than it bent thevisible light rays that Eddington observed. Advancements in dualfrequency techniques have improved accuracies by allowing the coronalbending, which depends on the frequency of the wave, to be measuredseparately from the gravitational bending, which does not. Fomalont andSramek (1977) provide a thorough review of these experiments, and discussthe prospects for improvement.

7.2 The Time Delay of LightBecause of the presence of the gravitational field of a massive

body, a light signal will take a longer time to traverse a given distancethan it would if Newtonian theory were valid. An expression for this"time delay" can be obtained simply from Equation (7.3). Integrating theequation using Equation (7.6), we obtain

] (729)Then from Equation (7.1), the coordinate time taken to propagate fromthe point of emission to x is given by

^ l l ^ ] (7.30)For a signal emitted from the Earth, reflected off a planet or spacecraftat xp, and received back at Earth, the roundtrip travel time At is given by

At - 2|xe - xp| + 2(1 + y>»to[(r« + * ' - y ' - X ' - * ) ] (7.31)


where ft is the direction of the photon on its return flight. Here we haveignored the motion of the Earth and planets during the round trip of thesignal. To be completely correct, the round trip travel time should beexpressed in terms of the proper time elapsed during the round trip, asmeasured by an atomic clock on Earth; but this introduces no new effects,so we will not do so here. The additional "time delay" 8t produced bythe second term in Equation (7.31) is a maximum when the planet is onthe far side of the Sun from the Earth (superior conjunction), i.e., when

xffi � n ~ r$, xp � n ~ � rp, d =* solar radius (7.32)

then

5t = 2(1 + y)mln(4r9rp/d2)

= i(l + y) [240 ps - 20 JIS In (^ -Y (f\\ (7.33)

where R o is the radius of the Sun, and a is an astronomical unit. Forfurther discussion of the time delay see Shapiro (1964,1966a,b), Muhlemanand Reichley (1964), and Ross and Schiff (1966).

In the decade and a half since Shapiro's discovery of this effect, anumber of measurements of it have been made using radar ranging totargets passing through superior conjunction. Since one does not haveaccess to a "Newtonian" signal against which to compare the round triptravel time of the observed signal, it is necessary to do a differentialmeasurement of the variations in round trip travel times as the targetpasses through superior conjunction, and to look for the logarithmicbehavior. To achieve this accurately however, one must take into accountthe variations in round trip travel time due to the orbital motion of thetarget relative to the Earth [variations in |x e � xp| in Equation (7.31)].This is done by using radar-ranging (and possibly other) data on thetarget taken when it is far from superior conjunction (i.e., when the time-delay term is negligible) to determine an accurate ephemeris for the target,using the ephemeris to predict the PPN coordinate trajectory xp(t) nearsuperior conjunction, then combining that trajectory with the trajectoryof the Earth xffi to determine the quantity |x$ � xp| and the logarithmicterm in Equation (7.31). The resulting predicted round trip travel timesin terms of the unknown coefficient 5(1 + y) are then fit to the measuredtravel times using the method of least squares, and an estimate obtainedfor |(1 + y). [This is an oversimplification, of course. The reader isreferred to Anderson (1974) for further discussion.]

Classical Tests 175

Three types of targets have been used. The first type is a planet, suchas Mercury or Venus, used as a passive reflector of the radar signals("passive radar"). One of the major difficulties with this method is thatthe largely unknown planetary topography can introduce errors in roundtrip travel times as much as 5 /zs (i.e., the subradar point could be amountaintop or a valley), which introduce errors in both the planetaryephemeris and, more importantly, in the round trip travel times at superiorconjunction. Several sophisticated attempts have been made to overcomethis problem.

The second type of target is an artificial satellite, such as Mariners 6and 7, used as active retransmitters of the radar signals ("active radar").Here topography is not an issue, and the on-board transponders permitaccurate determination of the true range to the spacecraft. Unfortunately,spacecraft can suffer random perturbing accelerations from a variety ofsources, including random fluctuations in the solar wind and solar radia-tion pressure, and random forces from on-board attitude-control devices.These random accelerations c in cause the trajectory of the spacecraftnear superior conjunction to differ by as much as 50 m or 0.1 us fromthe predicted trajectory in an essentially unknown way. Special methodsof analyzing the ranging data ("sequential filtering") have been devised toalleviate this problem (Anderson, 1974).

The third target is the result of an attempt to combine the transpondingcapabilities of spacecraft with the imperturbable motions of planets byanchoring satellites to planets. Examples are the Mariner 9 Mars orbiterand the Viking Mars landers and orbiters.

In all of these cases, as in the radio-wave deflection measurements, thesolar corona causes uncertainties because of its slowing down of the radarsignal. Again, dual frequency ranging helps reduce these errors, in fact, itis the corona problem that provides the limiting accuracy for the mostrecent time-delay measurements.

The results for the coefficient |(1 + y) of all radar time-delay measure-ments performed to date are shown in Figure 7.3. Recent analyses ofViking data have resulted in a 0.1% measurement (Reasenberg et al.1979).

From the results of light-deflection and time-delay experiments, we canconclude that the coefficient ^(1 + y) must be within at most 0.2% ofunity. Most of the theories shown in Table 5.1 can select their adjustableparameters or cosmological boundary conditions with sufficient freedomto meet this constraint. Scalar-tensor theories must have co > 500 to bewithin 0.1% or w > 250 to be within 0.2% of unity.


Value of i (1+7)

0.88 0.92 0.96 1.00 1.04 1.08I I I T

Time�Delay Measurements

Passive Radar to Mercury and Venus

T I i

Shapiro (1968) �Shapiro etal. (1971)

Active Radar

Mariner 6 and 7 Anderson et al. (1975)

Anchored Spacecraft

Mariner 9 Anderson et al. (1978),Reasenberg and Shapiro (1977)

Viking Shapiro et al. (1977)Cain etal. (1978)Reasenberg etal. (1979)

i

- � 1

(±0.001)

5 10 2040°°Value of Scalar-Tensor co

Figure 7.3. Results of radar time-delay measurements 1968-79.

7.3 The Perihelion Shift of MercuryThe explanation of the anomalous perihelion shift of Mercury's

orbit was another of the triumphs of general relativity. However, between1967 and 1974, there was considerable controversy over whether theperihelion shift was a confirmation or a refutation of general relativitybecause of the apparent existence of a solar quadrupole moment thatcould contribute a portion of the observed perihelion shift. Although thiscontroversy has abated somewhat, the question of the size of the solarquadrupole moment has yet to be conclusively answered.

The PPN prediction for the perihelion shift can be obtained from thePPN equation of motion [Equation (6.31)]. We consider a system of twobodies of inertial masses n^ and m2, and self-gravitational energies Q^and Q2 � The first body has a small quadrupole moment Q'{. We assumethat the entire system is at rest with respect to the universe rest frame(w = 0) and that there are no other gravitating bodies near the system.In Chapter 8, we shall return to the effects of motion and of distantbodies (preferred-frame and preferred-location effects) on the perihelion

Classical Tests 111

shift. For the moment we ignore them. We work in a PPN coordinatesystem in which the center of mass of the system is at rest at the origin.

Making use of the fact that each body is nearly spherical, Qf mwe obtain from Equation (6.31) the acceleration of each body

a, = ! - - ^ F(2y + 20 ^

4 + ai)v! � v2 - i(2y + 2 + a2 + a3)t>l

+ f(1 + «2)(v2 � n)2l - ^ � £(2y + 2* , - (2y + l)v2j v,

+ Y7^' (4y + 4 + ai)Vl - (4r + 2 + ai - 2 a >2 k ,

a2 = {l<-*2;x-^ -x} (7.34)

where x = x21, n = x/r. Including the Newtonian contribution of thequadrupole moment in the quasi-Newtonian potential produced bybody 1, we have

xJ

(UJi = (mA)2 p-,

(Uj)2 = -(mA)x ^-\^r(^nknlW - 25H1) (7.35)

where (mA)j and (mA)2 are the active gravitational masses, given byEquation (6.51). For a body which is axially symmetric about an axiswith direction e, Qf can be shown to have the form

Qf = mxR\J2W^k - 3^2*) (7.36)

where J2 is a dimensionless measure of the quadrupole moment, givenby

J2={C- A)/mR2 (7.37)

where

C = [moment of inertia about symmetry axis],

A = [moment of inertia about equatorial axis],

R = [radius] (7.38)


(The subscript 2 on J2 denotes that it is associated with the quadrupole,or / = 2 moment of the body.)

Since the center of mass of the system is at rest, we may, to sufficientaccuracy in the post-Newtonian terms in Equation (7.34), replace vx

and v2 by

Vi = -(m2/m)v, v2 = (mi/m)v (7.39)

where

v = v 2 - v 1 , m = m1+m2 (7.40)

We also define the reduced mass

fi = mlm2/m (7.41)

Then the relative acceleration a 2 - a , 5 a takes the form

a = - ^ + * "1*^2(1) [ 1 5 ( g . ft)2fi _ 6 ( e . m

ocl+a2+ oc3)^v2 + | (1 + a2)-^-(v � n)m m

^ [ ^ ] (7.42)where

m* = (mP/w)2(mA)1 + (mp/m

= m(l + [self-energy terms for bodies 1 and 2]). (7.43)

The self-energy terms from Equation (6.51) that appear in the aboveexpression are at most ~10~ 5 for the Sun, and are constant. Thus thedifference between m* and m is unmeasurable, so we simply drop the (*)in Equation (7.42).

We consider a planetary orbit with the following instantaneous orbitelements (see Smart, 1953, for detailed discussion of the definitions):inclination i relative to a chosen reference plane, the angle Q from a chosenreference direction in the reference plane to the ascending node, the angleco of perihelion from the ascending node measured in the orbital plane,the eccentricity e and semi-major axis a. The sixth element T, the time ofperiastron passage, is an initial'condition and is irrelevant for our purposes.

Classical Tests 179

For the solar system, the reference plane is chosen to be the plane of theEarth's orbit (ecliptic) and the reference direction is the Earth-Sundirection at spring equinox.

Following the standard procedure for computing perturbations oforbital elements [Smart (1953), Robertson and Noonan (1968)], weresolve the acceleration a [Equation (7.42)] into a radial qomponent M,a component "W, normal to the orbital plane, and a component £f normalto Si and iV, and calculate the rates of change of the orbital elementsusing the formulae [in the notation of Robertson and Noonan (1968)]:

da, p® ^ip + r) . , iTr . / �-r-= - -r� cos 4> + �-, sin 4> � cot i sin(<w + <p), (7.44)at he he h

di Wr~ = � cos(co + <t>), (7.47)

r sm{w + <f>)i ; : � ~ � (7.48)dt h sm i

where h is the angular momentum per unit mass of the orbit, <j> is the angleof the planet measured from perihelion, and p is the semi-latus rectumgiven by

p = a(l - e2) (7.49)

The variables r and 0 are related to the instantaneous orbit elements bythe definitions

r == p(l + ecos 4>)~l,

r2 d4>/dt = h = (mp)1'2 (7.50)

Now, because observations of the planets are made with reference togeocentric coordinates, the perihelion measured is the perihelion relativeto the equinox, <o, given by

(o = co + Qcos i (7.51)

Then the rate of change "of c5 is given by

deb pM Sr(p + r)� = -y� cos 4> H T sm $ (7.52)dt he he

where we have used the fact that, for all the planets, i is small, so thatsin i« 1. For the perturbing acceleration in Equation (7.42) (we drop thesubscript " 1 " on m, R and J2)

-11 (2y + 20) - - yv2 + (2y + 2)(v � ii)2

(2 + «! - 2f2) £ - i(6 + «! + a2 + a3) �

y = - 3(m/?2J2/r4)(e � n)(e � 2)

+ j j (v � n)(v � X)|~(2y + 2) - £ (2 - at + «2) 1 <7-53)

where 2 is a unit vector in the plane of the orbit in the direction of theorbital motion, normal to ii. For Mercury's orbit, the solar symmetry axisis essentially normal to the orbital plane, hence e � n ^ 0. Then substitutingEquations (7.53) into (7.52) and integrating over one orbit using Equation(7.50) yields

AS = (67tm/p)[i(2 + 2y - 0)

- a2 + a3 + 2£2)fi/m + J2(R2/2mp)-] (7.54)

This is the only secular perturbation of an orbital element produced by thepost-Newtonian terms in Equation (7.42); however the quadrupole termscan be shown to produce secular changes in i and Q proportional to sin 0and sin 0/sin i respectively, where 6 is the tilt of the Sun's symmetry orrotation axis relative to the ecliptic (9 « 7°). The elements a and e sufferno secular changes under either of these perturbations.

The first term in Equation (7.54) is the classical perihelion shift, whichdepends upon the PPN parameters y and p. The second term dependsupon the ratio of the masses of the two bodies (Will, 1975); it is zero inany fully conservative theory of gravity (al = a2 = a3 = £2 = 0); it is alsonegligible for Mercury, since /x/m ~ m^/mo ~ 2 x 10"7. We shall dropthis term henceforth. The third term depends upon the solar quadrupolemoment J2. For a Sun that rotates uniformly with its observed surfaceangular velocity, so that the quadrupole moment is produced by centri-

Classical Tests 181

fugal flattening, one may estimate J2 to be ~1 x 10~7. Normalizing J2

by this value and substituting standard orbital elements and physicalconstants for Mercury and the Sun (Allen, 1976), we obtain the rate ofperihelion shift c5, in seconds of arc per century,

5 = 42795 Ape"1

Xv = [i(2 + 2y - P) + 3 x IO-V2/IO-7)] (7.55)

The measured perihelion shift is accurately known: after the effects ofthe general precession of the equinoxes (~5000" c""1) and the perturbingeffects of the other planets (280" c~1 from Venus, 150" c'1 from Jupiter,100" c~1 from the rest) have been accounted for, the remaining perihelionshift is known (a) to a precision of about one percent from optical obser-vations of Mercury during the past three centuries (Morrison and Ward,1975), and (b) to about 0.5% from radar observations during the pastdecade (Shapiro et al., 1976). Unfortunately, measurements of the orbitof Mercury alone are incapable at present of separating the effects ofrelativistic gravity and of solar quadrupole moment in the determinationof Xp. Thus, in two recent analyses of radar distance measurements toMercury, J2 was assumed to have a value corresponding to uniform rota-tion (effect on Xp negligible), and the PPN parameter combination wasestimated. The results were

1 fl.005 ± 0.020(1966-1971 data: Shapiro et al., 1972)s( + 7 P) ~ | 1 0 0 3 ± 0.005(1966-1976 data: Shapiro et al, 1976)

(7.56)

where the quoted errors are 1CT estimates of the realistic error (taking intoaccount possible systematic errors).

The origin of the uncertainty that has clouded the interpretation ofperihelion-shift measurements is a series of experiments performed in 1966by Dicke and Goldenberg (see Dicke and Goldenberg, 1974, for a detailedreview). Those experiments measured the visual oblateness or flatteningof the Sun's disk and found a difference in the apparent polar and equato-rial angular radii of AR = (43'.'3 ± 3'.'3) x 10"3. By taking into accountthe oblateness of the surface layers of the Sun caused by centrifugalflattening, this oblateness signal can be related to J2 by (Dicke, 1974)

J2 = §(A*/KG) - 5.3 x 10"6 (7.57)

which gives (i?G = 959")

J2 = (2.47 ± 0.23) x 10"5 (Dicke and Goldenberg, 1974) (7.58)


A value of J2 this large would have contributed about 4" c~1 to Mercury'sperihelion shift, and thus would have put general relativity in seriousdisagreement with the observations, while on the other hand supportingBrans-Dicke theory with a value co ^ 5, whose post-Newtonian contri-bution to the perihelion shift would thus have been 39" per century.

These results generated considerable controversy within the relativityand solar physics communities, and a mammoth number of papers wasproduced, both supporting and opposing solar oblateness. One recurringline of argument in opposition to the Dicke-Goldenberg result was thattheir method of measuring the difference in brightness between the solarpole and the solar equator of an annulus of the solar limb produced aroundan occulting disk placed in front of the Sun, could equally well be inter-preted by assuming a standard solar model (with a small J2 ~ 10 ~7

produced by centrifugal flattening) with a temperature difference on thesolar surface between the equator and the pole, leading to a brightnessdifference indistinguishable from that due to a geometrical oblateness.Such a brightness difference, it was suggested, could also be produced byan equatorial excess in the number of solar faculae. Refutations of thesearguments by Dicke and his supporters, and counter-refutations aboundedin the literature.

The controversy abated somewhat in 1973, when Hill and his collabo-rators performed a similar visual oblateness measurement that yieldedAR = (9'.'2 ± 6'.'2) x 1(T3 or

J2 = 0.10 ± 0.43 x 10"5 (Hill et al., 1974) (7.59)

an upper limit five times smaller than Dicke's value. (See also Hill andStebbins, 1975). The disagreement between these two observational resultsremains unresolved.

One of the major difficulties in relating visual solar oblateness resultsto J2 is that a considerable amount of complex solar physics theory mustbe employed. There is, however, a way of determining J2 unambiguously,namely by probing the solar gravity field at different distances from theSun, thereby separating the effects of J2 from those of relativistic gravi-tation through their different radial dependences [see Equation (7.42)].One method would compare the perihelion shifts of different planets. Butthe perihelion shifts of Venus, Earth, and Mars are not known to sufficientaccuracy, although Shapiro et al. (1972) pointed out that several moreyears of radar observations of the inner planets may permit such a com-parison. Another method would take advantage of Mercury's orbitaleccentricity (e ~ 0.2) and search for the different periodic orbital pertur-

Classical Tests 183

bations induced by J2 and by relativistic gravity. The accuracy requiredfor such measurements would necessitate tracking of a spacecraft in orbitaround Mercury, but preliminary studies have shown that J2 could bedetermined to within a few parts in 107 (Anderson et al., 1977, Wahr andBender, 1976). Finally, and most promisingly, a mission currently understudy by NASA for the 1980s known as the Solar Probe, a spacecraft ina high-eccentricity solar orbit with perihelion distance of four solar radii("Arrow to the Sun"), could yield a measurement of J2 with a precision ofa part in 108 (Nordtvedt, 1977, Anderson et al., 1977). Such missions wouldalso lead to improved determinations of y and /?. The possibility of deter-mining y and j8 from measurements of the precessions of the pericentersof the inner satellites of the gas giant planets has recently been consideredby Hiscock and Lindblom (1979).

8

Tests of the Strong Equivalence Principle

The next class of solar system experiments that test relativistic gravi-tational effects may be called tests of the Strong Equivalence Principle(SEP). That principle states that (i) WEP is valid for self-gravitatingbodies as well as for test bodies (GWEP), (ii) the outcome of any localtest experiment, gravitational or nongravitational, is independent of thevelocity of the freely falling apparatus, and (iii) the outcome of any localtest experiment is independent of where and when in the universe it isperformed. In Section 3.3, we pointed out that many metric theories ofgravity (perhaps all except general relativity) can be expected to violateone or more aspects of SEP. In Chapter 6, working within the PPN frame-work, we saw explicit evidence of some of these violations: violations ofGWEP in the equations of motion for massive self-gravitating bodies[Equations (6.33) and (6.40)]; preferred-frame and preferred-locationeffects in the locally measured gravitational constant GL [Equation (6.75)];and nonzero values for the anomalous inertial and passive gravitationalmass tensors in the semiconservative case [Equation (6.88)].

This chapter is devoted to the study of some of the observable conse-quences of such violations of SEP, and to the experiments that test forthem. In Section 8.1, we consider violations of GWEP (the Nordtvedt ef-fect), and its primary experimental test, the Lunar Laser-Ranging"E6tvos"experiment. Section 8.2 focuses on the preferred-frame and preferred-location effects in GL. The most precise tests of these effects are obtainedfrom geophysical measurements. In Section 8.3, we consider preferred-frame and preferred-location effects in the orbital motions of planets.Perihelion-shift measurements are important tests of such effects. Anotherviolation of SEP would be the variation with time of the gravitationalconstant as a result of cosmic evolution. Tests of such variation are de-

Tests of the Strong Equivalence Principle 185

scribed in Section 8.4. In Section 8.5, we summarize the limits on thevalues of the PPN parameters y, /?, £, al5 a2, and <x3 that are set by theclassical tests and by tests of SEP, and discuss the consequences forthe metric theories of gravity described in Chapter 5.

8.1 The Nordtvedt Effect and the Lunar Eorvos ExperimentThe breakdown in the Weak Equivalence Principle for massive,

self-gravitating bodies (GWEP), which many metric theories predict, hasa variety of observable consequences. In Chapter 6, we saw that this vio-lation could be expressed in quasi-Newtonian language by attributing toeach massive body inertial and passive gravitational mass tensors m{k andm£* which may differ from each other. The quasi-Newtonian part of thebody's acceleration may be written [see Equation (6.43)]

(mi)i*Kftew« = (mP)lrUlJ (8.1)

where Ulra is a quasi-Newtonian gravitational potential, and {fh^f and(fhp)'" are given by

(«i)? = ma{o-}k[l + (ax - a2 + C ^ / m J + (a2 - Ci + C2)OfM,},(mP)'a

m = ma{5"»[l + (4/8 - y - 3 - 3£)Qa/ma] - {Qf/m.} (8.2)

where Qa and Q£* are the body's internal gravitational energy and gravita-tional energy tensor (see Table 6.2), and ma is the total mass energy of thebody.

Now, most bodies in the solar system are very nearly spherically sym-metric, so we may approximate

a* * &a5Jk (8.3)

Any "Nordtvedt" effects that arise from the anisotropies in QJ* in Equa-tion (8.2) are expected to be too small to be measurable in the foreseeablefuture (see Will, 1971b, for an example). With the above approximationwe write the quasi-Newtonian Equation (8.1) in the form

Wkw. = K M . U , . (8.4)

where

(mp/ifi). = 1 + (40 - y - 3 - 4ft - ax + §<x2 - Ki - K ^ M . ,U= Z (mAV\,V (8-5)

The most important consequence of the Nordtvedt effect is a polariza-tion of the Moon's orbit about the Earth [Nordtvedt (1968c)]. Because


the Moon's self-gravitational energy is smaller than the Earth's, the Nordt-vedt effect causes the Earth and Moon to fall toward the Sun with slightlydifferent accelerations. Including their mutual attraction, we have [fromEquations (8.4) and (8.5), and neglecting quadrupole moments],

(8.6)

where X and Xo are vectors from the Sun to the Earth and Moon, respec-tively, and x is a vector from the Earth to the Moon (Figure 8.1). Therelative Earth-Moon acceleration a, denned by

® (8.7)

Figure 8.1. (a) Geometry of the Earth-Moon-Sun system.(b) The Nordtvedt effect - a polarization of the Moon's orbit with theapogee always directed along the Earth-Sun line.

Sun

Earth

Moon

(a)

(b)


is then given by

a = ~m*x/r3 + i/[(Q//n)© - (Q/m)J/noX/K3

+ {m^m^m^X/R3 - XJR3) (8.8)

where

G s (mA)©,

^+1^-1^-^2 (8-9)The first term in Equation (8.8) is the Newtonian acceleration between

the Earth and Moon and the second term is the difference between theEarth's and Moon's acceleration toward the Sun (Nordtvedt effect). Thethird term is the classical tidal perturbation on the Moon's orbit; sinceit is a purely nonrelativistic perturbation, we will not consider it for themoment. Hence, the equation of motion of the Moon relative to the Earth,including the perturbation arising from the Nordtvedt effect, is

a = -m*x/r3 + ff[(O/m)e - (n/m\~}mQX/R3 (8.10)

We assume that the Moon's unperturbed orbit is circular with angularvelocity co0 and in the x-y plane, and also that the orbit of the Eartharound the Sun is circular with angular velocity a>s in the same plane.We work in an inertial PPN coordinate system centered at the Sun. Thenthe acceleration a and the angular momentum per unit mass of the Earth-Moon orbit are given by

a = d2x/dt2, h = x x (dx/dt) (8.11)

and the following relations hold

d2r/dt2 = x � a/r + h2/r\

dh/dt= (x x a) (8.12)

where r = \x\. Thus, by making use of Equation (8.10) and by defining

da s ij[(|n|/m)e - (p\/m\-]mQ/R2 (8.13)

we obtain

d2r/dt2 = -m*/r2 + h2fr3 + SacosAt,

dh/dt= -rSa sin At (8.14)

where

cosAt s �n � x/r, sin At = �(n x x/r)z, A = co0 � ws (8.15)

where n = X/R. Note that At is the angle between the Earth-Sun andEarth-Moon directions. We next linearize about a circular orbit:

r = ro + 3r, h = ho + 5h (8.16)

and use m*jr% = hl/r% = co2,. Integration of the resulting equations yields

5h = (r0/A)5acosAt, (8.17)

(8.18)

Equation (8.18) represents a polarization of the Earth-Moon systemby the external field of the Sun. This polarization of the orbit is alwaysdirected toward the Sun if r\ > 0 (away from the Sun if r\ < 0) as it rotatesaround the Earth (see Figure 8.1).

Using Equations (8.13) and (8.18) and the values mQ/R2 st 5.9 x 10" 6

km s~2, w° ^ 13.4OJS * 2.7 x 1(T6 s"1, (Q/m)e * - 4.6 x 1(T10, and(Q/m)a =* -0.2 x 10~10 (Allen, 1976), we obtain

dr =s 8.0»/ cos(<o0 - cos)t m (8.19)

Actually, a more accurate calculation would take into account the effectof the Nordtvedt perturbation on the tidal acceleration term in Equation(8.8) and that of the tidal perturbation on the Nordtvedt term; this modi-fies the coefficient of 8r by a factor of approximately 1 + 2cos/co0 ^ 1.15,giving

8r ^ 9.2/7 cos(a>0 - ojs)t m (8.20)

Since August, 1969, when the first laser signal was reflected from theApollo 11 retroreflector on the Moon, the Lunar Laser-Ranging Experi-ment (LURE) has made regular measurements of the round trip traveltimes of laser pulses between McDonald Observatory in Texas and thelunar retroreflectors, with accuracies of 1 ns (30 cm) (see Bender et al,1973, Mulholland, 1977). These measurements were fit using the methodof least squares to a theoretical model for the lunar motion that tookinto account perturbations due to the other planets, tidal interactions,and post-Newtonian gravitational effects. The predicted round trip traveltimes between retroreflector and telescope also took into account thelibrations of the Moon, the orientation of the Earth, the location of theobservatory, and atmospheric effects on the signal propagation. The"Nordtvedt" parameter, //, along with several other important parametersof the model were then estimated in the least-squares method.


An important issue in this analysis is whether other perturbations ofthe Earth-Moon orbit could mask the Nordtvedt effect. Most perturba-tions produce effects in 5r, which, when decomposed into sinusoidal com-ponents, occur at frequencies different from that of the Nordtvedt term(e.g., at angular frequencies co0, 2A), and thus can be separated cleanlyfrom it using a multi-year span of data. However, there is one perturba-tion, due to the tidal term that we neglected in Equation (8.8), that doeshave a component at the frequency A. To see this, we expand X and Xo

about Xc, the center of mass of the Earth-Moon system, using

Xo = Xc + (roe/m*)x, X = Xc - (mjm*)x (8.21)

where we now ignore all post-Newtonian self-energy corrections to masses.Then the tidal acceleration in Equation (8.8) becomes

a I l X f e i y e ]

I - 2mc/m*) ~ [nc - 5(nc � e)% + 2(nc � S)e] (8.22)Kc

where fic = Xc/Rc, e = x/r. It is the second term, of order {r^/Rf), in theabove expression that leads to a perturbation in Sr of frequency A. Apply-ing Equations (8.12), (8.15), (8.16), and integrating, it is possible to showstraightforwardly that

2mA8 [col-A

+ [terms proportional to cos2At, cos 3At. . . ] (8.23)

Using the expressions

col = rn*/r30, cof = mQ/R? (8.24)

we may rewrite Equation (8.23) in the useful form

where Q = toj(o0 x 0.075. Again, a more accurate computation, takinginto account the mutual effect of the two terms in Equation (8.22), modifiesEquation (8.25) by corrections that depend upon Q, with the result (Brown,1960)

where

F(Q) = 1 + (81/15)Q + � � � ^ 1.64 (8.27)

Substituting numerical values (Allen, 1976) yields

<5>"tidai 110 cos At km (8.28)

Although this term is ten thousand times larger than the nominal am-plitude of the Nordtvedt effect, it turns out, fortunately, that the param-eters that appear in Equation (8.26) are known with sufficient accuracythat the tidal term can be accounted for to a precision of about 2 cm. Thevalues of Rc, mQ/m*, and Q. are known to sufficient precision from otherdata, while the values of mjm* and r0 are estimated using the laser-ranging data via their effects on the lunar orbit at frequencies other than A.

Two independent analyses of the data taken between 1969 and 1975were carried out, both finding no evidence, within experimental uncer-tainty, for the Nordtvedt effect. Their results for n were

_ fO.OO ± 0.03 [Williams et al. (1976)],n ~ (0.001 + 0.015 [Shapiro et al. (1976)] (8.29)

where the quoted errors are la, obtained by estimating the sensitivity ofn to possible systematic errors in the data or in the theoretical model.The formal statistical errors that emerged from the data analysis weretypically much smaller, of order <x(>7)fOrmai ~ + 0.004.

This represents a limit on a possible violation of GWEP for massivebodies of 7 parts in 1012 (compare Table 2.2). For Brans-Dicke theory,these results force a lower limit on the coupling constant a> of 29 (2a,Shapiro result).

Improvements in the measurement accuracy and in the theoreticalanalysis of the lunar motion may tighten this limit by an order of mag-nitude (Williams et al., 1976), while a comparable test of the Nordtvedteffect may be possible using the Sun-Mars-Jupiter system (Shapiro et al.,1976). Other potentially observable consequences of the Nordtvedt effectinclude shifts in the stable Lagrange points of Jupiter (measurable byranging to the Trojan asteroids), and modification of Kepler's third law(Nordtvedt, 1968a, 1970a, 1971a,b).

8.2 Preferred-Frame and Preferred-Location Effects:Geophysical TestsIn Section 6.3, we found that some metric theories of gravity pre-

dict preferred-frame and preferred-location effects in the locally measuredgravitational constant GL, measured by means of Cavendish experiments.


These effects represent violations of SEP. Unfortunately, present-dayCavendish experiments are only accurate to about one part in 105 in abso-lute measurements of GL (Rose et al, 1969), and so cannot discern thepost-Newtonian corrections to GL in Equation (6.75).

However, there is a "Cavendish" experiment that can detect correctionsin GL, one in which the source is the Earth and the test body is a gravi-meter on the surface of the Earth. A gravimeter is a device that measuresthe force required to keep a small "proof" mass stationary with respectto the center of the Earth. This is exactly the physical situation assumedin our derivation of GL in Section 6.3. Because of uncertainties in ourknowledge of the internal structure and composition of the Earth, it isimpossible to determine the absolute value of GL by this method withsufficient precision to detect post-Newtonian effects. Instead, gravimetersare powerful tools for measuring variations in the gravitational force. InNewtonian geophysics, these variations are known as solid-Earth tides;in post-Newtonian geophysics, measurements of these variations can testfor variations in GL, with high precision.

We therefore shall apply Equation (6.75) for GL to a gravimeter "Cav-endish" experiment, and shall focus on the post-Newtonian terms thatvary with time. A detailed justification of the application of Equation(6.75) to this situation is given by Will (1971d). Recall that

GL = 1 - [4)8 - y - 3 - C2 - «3 + //mr2)] Uext

- Aw.n - ^/m r2Vw. � z\2

2U - 3//mr2)(we � e)2 + «1 - 3//mr2)t/fxte^ (8.30)

where /, m, and r are the spherical moment of inertia, mass, and radius ofthe Earth, e is a unit vector directed from the gravimeter to the center ofthe Earth, and

U{kn = Z manJ9an%JrBa, Uext = UHt (8.31)

Consider the first post-Newtonian term in Equation (8.30). Because ofthe Earth's eccentric orbital motion, the external potential produced bythe Sun varies yearly on Earth by only a part in 1010, too small to be de-tected with confidence by Earth-bound gravimeters or Cavendish experi-ments. The time-varying effects of other bodies (planets, the galaxy) areeven smaller.

Next, consider the preferred-frame terms. The Earth's velocity we ismade up of two parts, a uniform velocity w of the solar system relative to

the preferred frame, and the Earth's orbital velocity v around the Sun, thus

w | = w2 + 2w � v + v2,

(we � 6)2 = (w � e)2 + 2(w � e)(v � e) + (v � e)2 (8.32)

So because of the Earth's rotation (changing e) and orbital motion (chang-ing v), there will be variations in the gravimeter measurements of GL,given by (we retain only terms which vary with amplitude larger than

a3 - ax)w � v

+ i<x2[(w � g)2 + 2(w � e)(v � e) + (v � e)2] (8.33)

where we have used the fact that, for the Earth,

I C* mr2/2 (8.34)

Finally, we consider the preferred-location term. According to our dis-cussion of the PPN formalism (see Section 4.1) the potential I/£, mustinclude all local gravitating matter that is not part of the cosmologicalbackground used to establish the asymptotically Lorentz PPN coordinatesystem. Therefore it must include the Sun, planets, stars, the galaxy, andpossibly the local cluster of galaxies. In this case, [/£, is dominated by ourgalaxy (Ua ~ 5 x 10~7), followed by the Sun (UQ ~ 1 x 10~8), thus,

AGJGh = -K t / G ( e � eG)2 - ^Uo(e � e0)2 (8.35)

In order to compare this variation in G with gravimeter data, we mustperform a harmonic analysis of the terms in Equations (8.33) and (8.35).The frequencies involved will be the sidereal rotation rate of the Earth Q,due to the changing direction of e relative to the fixed direction of w andeG, and its orbital sidereal frequency co due to the changing direction of vrelative to w, along with harmonics and linear combinations of these fre-quencies. We work in geocentric ecliptic coordinates, and assume a circu-lar Earth orbit, with the Earth at vernal equinox at t = 0. Then,

e 0 = cos cotex + sin a>tey,

v = i;(sin cotex � cos atey),

w s w[cos /^(cos Xwex + sin lwey) + sin /fwez],

eG = cos /SG(cos AGex + sin lGey) + sin /?Ge2 (8.36)

The latter two equations define the ecliptic coordinates (lw,/?w) and(^�G,PG)-

F o r a gravimeter stationed at Earth latitude L,

e = cosLcos(Qt � e)ex + [cosLsin(lQt � e)cos0 + sinLsin0]ey

- [cos L sin(Qt � e) sin 6 � sin L cos 0]ez (8.37)

where £ is related to the longitude of the gravimeter on the Earth, and 0 isthe "tilt" (23^°) of the Earth relative to the Earth's orbit (ecliptic). Equa-tions (8.36) and (8.37) give

w � v = wv cos fiw sin(a>t � Xw), (8.38)

(w � e)2 = w2[i + | ( i - sin2 <5J(i - sin2 L)

+ \ sin 2<5W sin 2L cos(fit � £ � ocw)

+ icos2<5H,cos2Lcos2(Qf - £ - <xj], (8.39)

(w � e)(v � e) = wi;{yCOS)S)1,sin(a»t - AJ

+ (i � sin2 L)[^cos jSw sin(cot � X^ + § sin <5W sin 0 cos cot]

+ j sin dw(l � cos 6) sin 2L sin[(Q + co)t � e]

� 5Cos<5wsin0sin2Lcos[(Q + (o)t � & � aw]

� \ sin ^w(l + cos 6) sin 2L sin[(Q � a>)t � s]

� jcos 6W sin 9 sin 2Lcos[(Q � a>)t � £ � aw]

\ - cos^)cos2Lsin[(2fi + a»)f - 2E - aw]

l + cos0)cos2Lsin[(2Q - co)t - 2E - aw]},

(8.40)

(v � e)2 = i>2{± + | ( i - sin2 L)(i - ^sin2 0)

� | ( i � sin2L)sin20cos2cof + |sin20sin2Lsin(Qf - e)

� i sin 0(1 - cos 9) sin 2L sin[(fi + 2co)t - e]

+ jsin 9(1 + cos 0) sin 2L sin[(Q - 2co)t - s]

+ \ sin2 9 cos2 L cos 2(Qf - s)

� i ( l - cos0)2cos2Lcos[2(Q + co)t - 2e]

� | (1 + cos 9)2 cos2 Lcos[2(Q - co)t - 2E]}, (8.41)

(SG � e)2 = i + | ( i - sin2 «5G)(i - sin2 L)+ -j sin 2<5G sin 2Lcos(Q( � s � aG)

+ \ cos2 ^G cos2 L cos 2(Q{ - £ - aG), (8.42)

(eo � e)2 = H | ( i - sin2 L)(i - ^sin2 9)

+ i(j - sin2L)sin29cos2cot + i sin 29 sin 2L sin(Qt - E)

+ | sin 0(1 - cos 0) sin 2L sin[(Q + 2co)t - e]

� 5 sin 0(1 + cos 0) sin 2L sin[(O - 2co)t � e\

+ i sin2 0 cos2 L cos 2(Q( - E)

+ | (1 - cos0)2cos2Lcos[2(Q + co)t - 2e]

+ i ( l + cos0)2cos2Lcos[2OQ - co)t - 2e] (8.43)

where we have used both the ecliptic coordinates (Xw, /?w), (XG, fic) and theequatorial coordinates (OLW,5W), (<XG,<SG) (Smart, 1960) corresponding tothe directions of w and eG in order to simplify the various expressions.These coordinate systems are related by

sin 8 = sin /? cos 9 + cos /? sin 9 sin X,

cos 8 cos a = cos fi cos X,

cos 8 sin a = � sin f$ sin 9 + cos /? cos 9 sin A (8.44)

Equations (8.38)-(8.43) reveal four different types of variations in GL.(i) Semidiurnal variations: These are the terms that vary with fre-

quency around 2Q: 2fi, 2Q, + co,2Q- a>, 2(Q. + co), 2(Q - co); i.e., thathave periods around twelve hours (co « Q) and vary with latitude accord-ing to cos2 L. These variations are completely analogous to the twelvehour solid-Earth tides produced by the Sun and Moon, called "semidiur-nal sectorial waves" [Melchior (1966)]. The true gravimeter measure-ments for these tides are affected not only by the variation in G, but alsoby the displacement of the Earth's surface relative to the center of theEarth, and by the redistribution of mass inside the Earth. This variationin gravimeter readings is related to the variation in G by

(AfifMemidiurnal = 1.16(AG/G)semidiurnal (8.45)

where the factor 1.16 is a combination of "Love numbers," which dependon the detailed structure of the Earth (Melchior, 1966). A more accuratecalculation of Ag/g would take into account the fact that in the Earth'sinterior the perturbing force generated by the variations in GL is propor-tional to pV U, whereas the tidal perturbing force is proportional to thedistance from the center of the Earth. If the Earth's density were uniform,then pWU would be proportional to r and the Love numbers would be thesame as in the Newtonian tidal case. However, in Newtonian tidal theory,the Love number for gravimeter measurements, (1.16), is not very sensi-tive (+ 5%) to variations in. the model for the Earth, thus we do not expectit to be sensitive to a different disturbing force law.

(ii) Diurnal variations: These are the terms that vary with a frequencyaround Q: Q, SI + co, fi � co, Q + 2co, Q � 2co; i.e., have periods around24 hours, and vary with latitude according to sin 2L. These variations areanalogous to the 24 hour "diurnal tesseral waves" of the solid Earth, andgive gravimeter readings related to the variation in G by the same factor:

(A<7M,iurnal = 1.16(AG/G)diurnaI (8.46)

(iii) Long-period zonal variations: These are the variations with fre-quencies co and 2co, and with latitude dependence (5 � sin2 L), that areanalogous to the long-period tides produced by the Sun and Moon, called"long-period zonal waves." These long-period zonal waves produce varia-tions in the Earth's moment of inertia, which in turn cause variations inthe rotation rate of the Earth. These rotation-rate variations are relatedto the amplitude of the zonal variations by (Mintz and Munk, 1953;Melchior, 1966)

(AQ/QL^, = 0.41^zonal (8.47)

where Azonai is related to the zonal variations in G in Equations (8.40),(8.41), and (8.43) by

(AG/G)zonal = AnJk - sin2 L) (8.48)

(iv) Long-period spherical variations: These are the variations [Equa-tions (8.38) and (8.40)] which have frequency <x>, but no latitude depen-dence; they represent a yearly variation in the strength of G, and have nocounterpart in Newtonian tidal theory. These variations produce a purelyspherical deformation of the Earth, as opposed to the sectorial, tesseral,and zonal waves which produce purely quadrupole deformations. Thisyearly spherical "breathing" of the Earth as G varies causes a variationin the Earth's moment of inertia, which in turn causes a variation in therotation frequency, given by

(AQ/Q)spherioal= -(AJ//)spherical (8.49)

However, because this effect has no counterpart in Newtonian tidaltheory, there is no Love number factor to relate A/// to AG/G. Insteadwe must do an explicit calculation to determine the factor.

We assume the Earth is spherically symmetric and momentarily atrest with respect to the PPN coordinate frame. Since we are focusingon long-period variations of GL (1 yr), we can assume that the Earth isin hydrostatic equilibrium at each moment of time, and changes onlyquasistatically. Then, from Equations (6.52) and (6.75), or from the PPNperfect-fluid equation of motion, Equation (6.29), keeping only the termsleading to significant long-period spherical perturbations, we find thatthe equation of hydrostatic equilibrium may be written

-T- = P -jr; [1 + i(<*2 + «3 - «i)w©] - j<x2w}@w%p �^ (8.50)


For a spherically symmetric body, it is straightforward to show that

dU__ m(r)~3r~~ ~~P~'

^ (851)

where

m(r) = 4TT P pr2 dr, I(r) = 4n f' pr4 dr (8.52)JO %J0

Substituting Equations (8.32), (8.38), (8.40), and (8.51) into Equation (8.50),and keeping only the spherical terms yields

GL(t) = 1 + (a3 + | a 2 - a t)wt; cos j8w sin(co£ - ^ J (8.54)

Using m(r) instead of r as independent variable, we may integrate Equa-tion (8.53), to obtain

f =? (8.55)

where m e is the mass of the Earth. By definition, p must vanish at thesurface of the Earth, i.e., pirn®) = 0. As GL(t) changes, the pressure distri-bution changes, causing a change £ = ^e in the position of each elementof matter. For a given shell of matter, the mass inside that shell is con-stant, by conservation of mass. Then if GL changes by AGL, we get fromEquations (8.52) and (8.55),

Am = 0,

Ap = p(AGL/GL) + O(f) (8.56)

But the volume of each element of matter changes, and this change canbe related to the pressure change using the bulk modulus K (we ignoretemperature changes)

Ap = - K(AV/V) = - KV � S, = - (K/r2)(r20,r (8.57)

Integrating Equation (8.57) and combining with Equation (8.56), we obtain

£ ('/'V2 d' O(£2) (8.58)

The spherical moment of inertia is given by

(8.59)

and the change caused by the displacement of each shell of matter is

A / = 2 [M r^dm (8.60)

Combining Equations (8.58) and (8.60) yields

A7 = - 8TT ^ J* prdr £ (p'/Ky2 dr' (8.61)

Numerical integration of this expression for a reasonable Earth modelyields

A/ / /= -0.17AGL/GL (8.62)

(see Lyttleton and Fitch, 1978; Nordtvedt and Will, 1972).We now substitute numerical values for the quantities that appear in

Equations (8.35)-(8.43). For the galaxy,

Ua * 5 x 10-7, AG = 266°, fio = - 6 ° ,

<xG = 265°, 5G = -29° (8.63)

For the velocity w of the solar system relative to the preferred frame, weuse the results of the most recent measurements of the anisotropy of the3 K microwave background. Our motion through this radiation causesthe measured effective temperature to be Doppler shifted differently inthe front and back directions. From measurements taken using a 33 GHzDicke radiometer flown on a U-2 aircraft (to get above a substantialamount of the Earth's atmosphere), Smoot et al. (1977) obtained a valuew = 390 ± 60 km s~1 in the direction <xw = 165° ± 9°, 5W = 6° ± 10°. Weshall adopt the values

<xw^165°, K-&, ^ 1 6 4 ° , j6w^0° (8.64)

We also have

i > ^ 3 0 k m s - \ 0 = 23.5° (8.65)

Using these values, we first compute the amplitudes of the dominantcomponents of the Earth tides, as listed in Table 8.1 (unconnected for Lovenumbers). For comparison, Table 8.1 also gives the amplitudes of thetidal potential for the dominant Newtonian tides in the frequency bandsof interest.


Table 8.1. Amplitudes of earth tides

Angularfrequency0

Doodsonlabel

PPN tidal amplitude(108 Ag/g)"

(a) Semidiurnal tides (latitude dependence cos2 L)2fi-3co2fl-2co2fi -co

2fi

2Q + m2Q + 2a>

T2

s2

K

0_

2.9 a2

f 17 a2

(9.6 {

-

(b) Diurnal tides (latitude dependence sin 2L)Q-2coii-w

an + wQ + 2co

Pi

s,�

0.7 a2

("3.5 a2

0.6 a2

�

Predicted Newtonianamplitude (108 Ag/g)*

0.142.40.02

0.67

00

1.30.03

4.1

0.030.06

" The angular frequencies of the Earth's rotation and the Earth's orbit are Q, and a>,respectively.b Amplitudes are uncorrected for Love numbers. An entry of zero denotes preciseabsence of a tide at that frequency, while an entry of a dash denotes that the nominalamplitude is smaller than 10 ~9 g.

Recent advances in superconducting techniques in the design and con-struction of gravimeters have resulted in highly stable devices capable ofmeasuring periodic changes in the local gravitational acceleration g assmall as 10" n g. Using such superconducting gravimeters, Warburtonand Goodkind (1976) have analyzed an 18 month record of gravimeterdata taken at Pinon Flat, California (33°59 N, 116?46W) in search ofanomalous PPN tidal amplitudes. From a harmonic analysis of therecord, they obtained amplitudes and phases of the tides at the frequenciesshown in Table 8.1. They then subtracted (vectorially) from these mea-sured tides the predicted Newtonian tides (corrected by an accuratelyknown Love number factor of 1.160). The remaining amplitudes andphases, known as "load vectors," are thought to be due primarily to thecomplex effect of ocean tides, which can influence gravimeter readings evenat the centers of continents. To take this "ocean loading" into account,they assumed that the anomalous load vectors at the diurnal Pt harmonicand at the semidiurnal T2 (or S2) harmonic, where the PPN effect isnegligible or absent, were entirely due to ocean loading. Since the effect ofocean loading is not believed to be strongly frequency dependent over

the narrow (few cycles per year) frequency bands under consideration, thePx and T2 load vectors were simply subtracted from the Kt and fromthe R2 and K2 load vectors, respectively. Small corrections for barometriceffects were also made. The remaining load vectors had amplitudes smallerthan 3 x 10"10 g for Ku 1 x 1(T10 g for K2, and 1 x 1 0 " u g for R2.(Compare with the PPN amplitudes in Table 8.1.) Furthermore, thephases of the remaining load vectors did not agree with the relationshipsamong the phases predicted by Equations (8.39)-(8.43). The result wasupper limits on the PPN parameters <x2 and t, given by

| a 2 | < 4 x l ( T 4 , | £ | < H T 3 (8.66)

The other important post-Newtonian geophysical effect is the possi-bility of periodic (co, 2co) variations in the Earth's rotation rate producedby the zonal and spherical variations in GL. The zonal variations haveamplitudes [see Equations (8.40), (8.41), and (8.43)]

{AGJGh)mnil ~ 3 x 10~8a2[frequency co],

~ 3 x 10" 10a2 [frequency 2a>],

~ 3 x 10-10£[frequency 2co] (8.67)

However, because of the tight limits on a2 and £ set by gravimeter data,we shall ignore these variations. The spherical variations [Equation (8.54)]have amplitude

(AGL/GL)spherical * 1.2 x 10"7[a3 + | a 2 - a t ] [frequency co] (8.68)

resulting in annual variations of the Earth's moment of inertia [Equa-tion (8.62)] with amplitude

|A///| =* 2.0 x 10-8[a3 + | a 2 - a,] (8.69)

Now, the observed annual variations in the Earth's rotation rate, of am-plitude |Afl/fi| 2^4 xl0~9can be accounted for as an effect of seasonalvariations in the angular momentum Jwind of atmospheric winds, to alevel of 4 parts in 1010 (Rochester and Smylie, 1974). Then, from con-servation of angular momentum, we have

A7T Q i/Q

< 4 x l O ~ 1 0 (8.70)

Thus, comparing Equations (8.69) and (8.70) we obtain

(8.71)

Theory and Experiment in Gravitational Physcis 200

8.3 Preferred-Frame and Preferred-Location Effects:Orbital TestsThere are a number of observable effects of a preferred-frame

and preferred-location type in the orbital motions of bodies governed bythe H-body equation of motion, (6.31). The most important of these effectsare perihelion shifts of planets in addition to the "classical" shift discussedin Section 7.

To determine these effects, we consider a two-body system whose bary-center moves relative to the universe rest frame with velocity w, and thatresides in the gravitational potential UG of a distant body (the galaxy isthe dominant such body). In the n-body equations of motion, (6.31), weshall ignore all the self-acceleration terms except the term (6.39) that de-pends on a3 and w. We shall also ignore the Newtonian acceleration, theNordtvedt terms, and all the post-Newtonian terms that were included inthe classical perihelion-shift calculation. Thus, from Equations (6.32),(6.33), and (6.39) we have the additional accelerations

+I(a1-a2-a3)w2+ia1w � v1+|(a1-2a2-2a3)w � v2

+fa2(wn)2 (w � n)(v2 � n)

. ^ J!° [2(fiG � x)nG - 3x(nG � n)2] + a2 - | (x � w)v2r rG r

i^ x . r a i V _(a,-2a,)i2 r3 L " l V l ( a i 2>

(8.72)

where x = x21, n = x/r, ra = |x1G|, nG = x1G/rG. In obtaining Equa-tion (8.72) we have ignored terms of order mGr/rQ, mGr2/rG, and so on.The first two terms inside the braces in Equation (8.72) are constant,therefore they can simply be absorbed into the Newtonian accelerationby redefining the gravitational constant [they are related to the constantcorrections to GL in Equation (6.75)]. Since our two-body system willconsist of the Sun and a planet, we can ignore Q/m for the planet. If body1 is chosen to be the Sun, then the relative acceleration <5a = 5a2 � d*i

is given by

<5a = � \^a.l(dm/m)yi � v + |a2(w � ft)2]

+ ZULU* [2(flG � x)nG - 3x(nG � n)2]r ro

j - � [^a^m/mjv + a2w]w + %tx3(Q/m)QY/ x <o (8.73)

where we have made use of Equations (7.39) and (7.40), and whereSm = my � m2 .

Following the method described in Section 7.3, we calculate the secularchange in the perihelion position. We assume that m2«m1, that e«l, andthat co is perpendicular to the orbital plane, then to zeroth order in e,we obtain for the secular change in a> over one orbit,

A<3= - 2

(8.74)4 rG

w 2 \ m JQ\ me

where vvP, wQ, «P, and nQ are the respective components of w and flG m

the direction of the planet's perihelion (wP, nP) and in the direction atright angles to this (wQ,«Q) in the plane of the orbit. The perturbations inEquation (8.73) can also be shown to produce secular changes in e, i, andQ. We now evaluate this additional perihelion shift for Mercury andEarth, using standard values for the orbital elements (Allen, 1976), nu-merical values for the Sun's gravitational energy and rotational angularvelocity

^ 4 x 10~6, |t»|Q =* 3 x 10~6 s"1 (8.75)

the direction of the galactic center, and our adopted value for w (seeSection 8.2.). Including the "classical" contributions (Section 7.3), theresult, in seconds of arc per century, is

= 43.0[i(2y + 2 -

- 123a! + 92a2 + 1.4 x 105a3

= 3.8[i(2y + 2 - 0)] - 198at + 12a2 + 2.4 x 106a3 + 14£ c" l (8.76)

Note that the effect of J2 on the Earth's perihelion shift is below theexperimental uncertainty. The measured perihelion shifts are

(^®)meas^3'.'8 + 0'.'4C-1 (8.77)

By combining Equations (8.76) and (8.77), eliminating the term involvingy and /?, and treating J2 as an experimental uncertainty with maximumvalue given by Hill's observations, \J2\ < 5 x 10~6 (Section 7.3), we ob-tain the following limit on the parameters at, a2, a3, and £

|49at - a2 - 6.3 x 105a3 - 2.2£| < 0.1 (8.78)

It is clear that <x3 must be extremely small,

| a 3 | < 2 x K T 7 (8.79)

otherwise there would be major violations of perihelion-shift data.Nonzero values of OLU a2, a3, or t, can also lead to periodic perturba-

tions in orbits, most notably in the lunar orbit, with nominal amplitudesranging from 70 km, for terms dependent upon oc3, to several meters, forterms dependent upon a1; a2, or £. For a partial catalogue of these effects,see Nordtvedt and Will (1972) and Nordtvedt (1973). In Section 9.3, weshall obtain an even tighter limit on a3 than that shown in Equation (8.79)by considering the effect of the acceleration term equation, (6.39), on themotion of pulsars.

8.4 Constancy of the Newtonian Gravitational ConstantMost theories of gravity that violate SEP predict that the locally

measured Newtonian gravitational constant may vary with time as theuniverse evolves. For the theories listed in Table 5.1, the predictions forG/G can be written in terms of time derivatives of the asymptotic dy-namical fields or of the asymptotic matching parameters. Other, moreheuristic proposals for a changing gravitational constant, such as thosedue to Dirac cannot be written this way. Dyson (1972) gives a detaileddiscussion of these proposals. Where G does change with cosmic evolu-tion, its rate of variation should be of the order of the expansion rate ofthe universe, i.e.,

G/G = oH0 (8.80)where Ho is the Hubble expansion parameter whose value is Ho cz55 km s"1 Mpc"1 s ( 2 x 1010 yr)~\ and a is a dimensionless parameterwhose value depends upon the theory of gravity under study and uponthe detailed cosmological model.

For very few theories has a systematic study of values of a beencarried out. For general relativity, of course, G is precisely constant{a = 0). For Brans-Dicke theory a ranges from a 2= � 3qo(a> + 2)"1 forq0 « 1 to a =s -(co + 2)"1 for q0 = i (flat Friedman cosmology) to a ^� 3.34<jJ/2(ct) + 2)"1 for <j0 » 1, where q0 is the deceleration parameterof the cosmology [see Section 16.4 of Weinberg (1972) for review andreferences]. In Bekenstein's variable-mass theory, generic cosmologicalmodels with chosen values of r and q (see Section 5.3) evolve to states atthe current epoch in which a < 5 x 10""3 (Bekenstein and Meisels, 1980).But for most other theories, detailed computations of this sort have notbeen performed (see Chapter 13).

However, several observational constraints can be placed on G/G,using methods that include studies of the evolution of the Sun, observa-tions of lunar occultations (including analyses of ancient eclipse data),planetary radar-ranging measurements, lunar laser-ranging measure-ments, and yet-to-be-performed laboratory experiments. The present sta-tus of these experiments is summarized in Table 8.2 [for a review of some ofthese methods see Halpern (1978)]. Some authors, chiefly Van Flandern(1975,1978), have claimed that the nonzero results for o shown in Table 8.2are significant and support the hypothesis of a varying gravitational con-stant, while others, notably Reasenberg and Shapiro (1978) have arguedthat unavoidable errors in the models used in the numerical estimation

Table 8.2. Tests of the constancy of the gravitational constant

Method a = (G/G) x (2 x 1O10 yr) Reference

Solar evolutionLunar occultations

and eclipses

Planetary and spacecraftradar

Viking radarLunar laser rangingLaboratory experiments

H<2|CT| < 0.8

a = -(0.6 ±0.3)<r= -(0.5 + 0.3)cr= -(2.5 ±0.7)H<8W<3

\a\ < 0.6

Chin and Stothers (1976)Morrison (1973)Van Flandern (1975, 1976, 1978)Muller(1978)Newton (1979)Shapiro et al. (1971)Reasenberg and Shapiro (1976,

1978), Anderson et al. (1978)Anderson (1979)Williams et al. (1978)Braginsky and Ginzberg (.1974),

Braginsky et al. (1977),Ritterand Beams (1978)

' Experiments yet to be performed.

Theory and Experiment in Gravitational Physcis 204

of parameters such as G/G may seriously degrade such estimates. Thelaser-ranging and radar-ranging results are regarded as being consistentwith G/G = 0. Reasenberg and Shapiro (1976) have pointed out that,because the errors in the radar observations of G/G decrease as T~5/2

where T is the time span of the observations, one can expect from thatmethod an accuracy of A|G/G| < 10" l l yr"1 by 1985. Anderson et al.(1978) and Wahr and Bender (1976) have shown that radar observationsof Viking or of a Mercury orbiter over two-year missions could yield

8.5 Experimental Limits on the PPN ParametersWe now summarize the results of the solar system experiments

described in Chapters 7 and 8, in the form of a set of limits on the PPNparameters. For the purposes of this summary, we shall consider onlysemiconservative theories of gravity, i.e., theories for which <x3 = £i =£2 = (3 = £4 = 0. Our reasons are the following: (i) we wish to keepthings simple; (ii) all currently interesting metric theories of gravity areLagrangian based, and are thus automatically semiconservative; (iii) wehave already seen that |a3| < 2 x 10~7; and (iv) decent experimental limitson the parameters Ci, (i> Cs, and £4 are hard to obtain, the only knownexceptions being a limit |£3| < 0.06 from the Kreuzer experiment, and apossible limit on |£2| from the binary pulsar (see Chapter 9 for discussionof these tests).

We thus have the la experimental limits

y = 1.000 ± 0.002 [Viking time delay], (8.81)

\{2y + 2 - j8) = 1.00 ± 0.02 [perihelion shift, Hill's

value for J 2] , (8.82)

\40 - y - 3 - ^ - <*! + fa2| < 0.015 [lunar laser ranging], (8.83)

|£ |<10" 3 [Earth tides], (8.84)

|a2| < 4 x 10"4 [Earth tides] (8.85)

|fa2 - ax| < 0.02 [Earth rotation rate], (8.86)

|49ax � a2 � 2.2^| < 0.1 [anomalous perihelionshifts] (8.87)

One useful way to represent these results pictorially is to construct"PPN theory space," a five-dimensional space whose axes are the fivesemiconservative PPN parameters. A given theory, with chosen valuesfor its adjustable constants and matching parameters, occupies a point in

this space. If we choose as variables y � 1, /?� 1, £, al9 and a2, thengeneral relativity occupies the origin, scalar-tensor theories with co > 0occupy the left hand (y - l)-(j8 - 1) plane, Rosen's bimetric theoryoccupies the a2 axis, and so on (see Figures 8.2 and 8.3). The results ofsolar system experiments can be viewed as "squeezing" the availabletheory space into smaller and smaller portions. For example, Figure 8.2shows the y-fi-% subspace of PPN theory space, and indicates the con-straints imposed by time delay, lunar-laser ranging, perihelion shift, andEarth tide measurements. The resulting available theory space is the "pillbox" around the origin (general relativity) shown. Figure 8.3 shows thea t -a2 plane, and indicates the constraints placed by Earth tide, Earthrotation rate, and perihelion-shift measurements.

Figure 8.2. The (y � 1)-(P � l)-£ space. Brans-Dicke theory occupiesthe negative (y � l)-axis(/? = 1), while the generalized scalar tensor theoriesof Bergmann, Wagoner, Nordtvedt, and Bekenstein occupy the half-plane(y � 1) < 0. The numbers on the negative (y � 1) axis are the correspondingvalues of co. General relativity resides at the origin. Shown are limits onthe PPN parameters placed by the Viking time delay (dotted lines), lunar-laser ranging (dashed lines), and perihelion shift (dot-dashed lines) mea-surements. The remaining available PPN theory space is the box shown,of thickness 2 x 10" 3 in the � direction.

/ 0.03;Scalar-Tensor(BWN, Bekenstein)

/ 0.02:

/

0-

25 :;

Brans�Dicke

For specific theories discussed in Chapter 5, these constraints can betranslated into constraints on adjustable constants or matching param-eters if the theory is to hope to remain viable. From the la constraintslisted above and from the formulae given in Chapter 5, we obtain thelimits

(i) Scalar-tensor theories: co > 500, A < 10"3

(ii) Will-Nordtvedt theory: K2 < 4 x 10" 4

(iii) Hellings-Nordtvedt theory: |coX2| < 2 x 10~4, co2K2 < 5 x 10~4

(iv) Rosen's bimetric theory: \co/c1 � 1| < 4 x 10" 4

(v) Rastall's theory: K2 < 3 x 10"2 (8.88)

Because many theories can be made to agree within experimental errorwith all solar system tests performed to date, we shall ultimately be forced,beginning in Chapter 10, to turn to new arenas for testing relativisticgravitation.

a/

IiIj Hellings-Nordtvedt: I 7 - IK0.002

Figure 8.3. The a!-a2 plane. The Rosen, Rastall, and Will-Nordtvedttheories occupy parts of the a2-axis shown. The Hellings-Nordtvedt theory,constrained by Viking time-delay measurements of y, occupies the shadedregion. General relativity and scalar-tensor theories (ST) reside at theorigin. Shown are limits placed by Earth tide (dotted lines), perihelionshift (dashed line), and Earth rotation rate (dot-dashed line) measurements.

Other Tests of Post-Newtonian Gravity

There remains a number of tests of post-Newtonian gravitational effectsthat do not fit into either of the two categories, classical tests or testsof SEP. These include the gyroscope experiment (Section 9.1), laboratoryexperiments (Section 9.2), and tests of post-Newtonian conservation laws(Section 9.3). Some of these experiments provide limits on PPN param-eters, in particular the conservation-law parameters Ci, d> £3* £4. thatwere not constrained (or that were constrained only indirectly) by theclassical tests and by tests of SEP. Such experiments provide new infor-mation about the nature of post-Newtonian gravity. Others, however,such as the gyroscope experiment and some laboratory experiments, allyet to be performed, determine values for PPN parameters already con-strained by the experiments discussed in Chapters 7 and 8. In some cases,the prior constraints on the parameters are tighter than the best limitthese experiments could hope to achieve. Nevertheless, it is important tocarry out such experiments, for the following reasons:

(i) They provide independent, though potentially weaker, checks of thevalues of the PPN parameters, and thereby independent tests of gravitationtheory. They are independent in the sense that the physical mechanismresponsible for the effect being measured may be completely different thanthe mechanism that led to the prior limit on the PPN parameters. Anexample is the gyroscope test of the Lense-Thirring effect, the dragging ofinertial frames produced purely by the rotation of the Earth. It is not apreferred-frame effect, yet it depends upon the parameter <xx.

(ii) The structure of the PPN formalism is an assumption about thenature of gravity, one that, while seemingly compelling, could be incorrect.This viewpoint has been expounded by Irwin Shapiro (1971) and others.They argue that one should not prejudice the design, performance, and


interpretation of an experiment by viewing it within any single theoreticalframework. Thus, the parameters measured by light-deflection and time-delay experiments could in principle be different according to this view-point, while according to the PPN formalism they must be identical[i(l + y)]- We agree with this viewpoint because although theoreticalframeworks such as the PPN formalism have proved to be very powerfultools for analyzing both theory and experiment, they should not be usedin a prejudicial way to reduce the importance of experiments that haveindependent, compelling justifications for their performance.

(iii) Any result in disagreement with general relativity would be ofextreme interest.

9.1 The Gyroscope ExperimentSince 1960, when Leonard Schiff proposed it as a new test of

general relativity, much effort has been directed toward the gyroscopeexperiment (Schiff, 1960b,c; Everitt, 1974; Lipa et al, 1974). The object ofthe experiment is to measure the precession of a gyroscope's spin axis Srelative to the distant stars as the gyroscope orbits the Earth. Accordingto the PPN formalism, this precession is given by (see Section 6.5)

dS/dt = ft x S,

ft = _ i v x a - |V x g + (y + |)v x VC/,

g = QofiJ (9.1)

where a is the spatial part of the gyroscope's four-acceleration, which iszero for a body in free-fall orbit. In a chosen PPN coordinate system,Equation (9.1) along with the expression for gOj in Table 4.1 yields

ft = i(4y + 4 + at)V x V - |a tw x\U + (y + frr x \U (9.2)

where w is the velocity of the coordinate system relative to the universerest frame, and where

V = Vjej (9.3)

For a system of nearly spherical bodies of masses ma, angular momentaJa, and velocities va, we have

V = £ mavjra - | £ xa x Ja/r3a + O(r8~

3) (9.4)

Other Tests of Post-Newtonian Gravity

where xa is the vector from the ath body to the gyroscope. Then

" = (7 + i) £ (v - va) x \{mjra)

209

- i(y + i + i«i) I [J. - 3fia(fifl � Jfl)]/rfl3

- i*i I (w + vj x V(m>a) - ± £ va x V(ma/ra) (9.5)a a

where na = xa/ra.The first term in Equation (9.5) is called the geodetic precession, a

consequence of the curvature of space near gravitating bodies. For acircular orbit around the Earth, the Earth's potential (a = ©) leads toa secular change in the direction of the gyroscope spin given, over oneorbit, by

as = -2n{y + i)(me/a)(S x h) (9.6)

Figure 9.1. Precession of gyroscopes in a polar Earth orbit. The gyro-scope with its axis in the plane of the orbit undergoes a geodetic preces-sion, while the gyroscope with its axis normal to the orbital plane suffersa precession due to the dragging of inertial frames.

8"/year /

where a is the orbital radius, and h is a unit vector normal to the orbitalplane. For a gyroscope whose initial direction lies in the orbital plane,the angular precession 39 (= |5S|/|S|) per year is given by

«)5/2 yr~' (9-7)

There is also a correction of ~0'.'01 yr~' due to the Earth's oblateness.Another secular contribution comes from the Sun's potential (a = Q),given by

( geodetic)© =* O'.'Q2ft(2y + 1)] yr~x (9.8)

where we have assumed a circular orbit for the Earth around the Sun.The second term in Equation (9.5) is known as the Lense-Thirring

precession or the "dragging of inertial frames" (for further discussion ofthis effect, see MTW Sections 19.2 and 33.4). For a circular orbit aroundthe Earth, it leads to a secular precession per orbit given by

<SS = i(y + 1 + i a i ) (P /a 3 ) [ J e ~ 3h(B � J e ) ] x S (9.9)

where P is the orbital period of the satellite. For a gyroscope in a polarorbit (fi � J@ = 0) or an equatorial orbit (fi � Jffi = |J®|), the precession isgiven by

<5SPOL = i(y + 1 + ia1)(P/a3)J® x S,

SSm = - i ( y + 1 + £ a i)(P/a3)J e x S (9.10)

with angular precessions, in arcseconds per year

50poL * 0'.'05[±(y + 1 + ia i)](/le/fl)3sin0 yr"1 ,

d0EQ ~ O'.'ll[i(y + 1 + W K / V a ) 3 sin «£ yr"1 (9.11)

where <j> is the angle between the spin vectors of the Earth and gyroscope.The third term in Equation (9.5) is a preferred-frame effect, dependent

upon the velocity of the ath body relative to the universe rest frame. Foran Earth-orbiting satellite, the dominant effect comes from the solarterm (a = O), leading to periodic precession of the form

(5S = - i a ^ W o x v e ) x S (9.12)

where vffi is the Earth's orbital velocity around the Sun and wG = w + v 0 .This leads to a periodic angular precession with a one year period, withamplitude

<50p.F. £ 5 x l O " 3 ' ^ (9.13)

Other Tests of Post-Newtonian Gravity 211

Since the ultimate goal of the experiment is to measure precessions to10" 3 arcseconds per year, this latter effect is probably too small to be ofinterest.

The last term in Equation (9.5) would appear to be anomalous, sinceit depends upon the velocity of each body va with respect to our arbitrarilychosen PPN coordinate frame. However, this is simply a result of thefact that the spin precession dSj/dx that we have calculated is not a trulymeasurable quantity, since the basis vectors es were not tied to physicalrods and clocks. A correct physical choice, and one that is closely relatedto the actual experimental method, is to use the directions of distant starsas basis directions (Wilkins, 1970). From Equations (7.1) and (7.9), thetangent vector to the trajectory of an incoming photon in the PPNcoordinate frame is given by

- (1+ y)l/] + (1 + y)93 (9.14)

where |n|2 = 1 is a unit spatial vector in the direction of the unperturbedtrajectory from the chosen star, and where 2> is equal to the right-handside of Equation (7.7), summed appropriately over all the gravitatingbodies in the system, and gives the gravitational deflection of the incomingsignals. We now project A onto the inertial basis of Equation (6.102), andnormalize the spatial components, so that X/(^j)2 = 1, to obtain

Aje;= n - n x (v x n)(l + v � n) - ^v x (n x v) + (1 + y)@ (9.15)

We now wish to show that the precession of the components of J on thisbasis is independent of the velocity of the PPN coordinate frame. InEquation (9.5), only the final term has this dependence, so we write it inthe form

- i I (v. - vB) x V(mjra) - hs * di/dt (9.16)a

where vB is the velocity of the solar system barycenter relative to the PPNcoordinate frame, and where we have used the fact that, for a freelyfalling gyroscope,

dx/dt = £ V mjra (9.17)a

The first term is now independent of the coordinate frame and so maybe dropped. The second term may be integrated immediately to obtain

<5SB = -i(vB x «5v) x So (9.18)

where the subscript B denotes that we retain only the terms that dependon vB. Then the change in the components of S with respect to A is given by

5(S;A;)B = <5SB � A + S � <5AB

= - [i(vB x <5v) x So] � A + i S 0 � [vB(<5v � A) - c5vvB � A]

= 0 (9.19)

Thus, as expected, there is no physically measurable dependence on thecoordinate-system velocity. In any case, the final term in Equation (9.5)produces only periodic precessions of negligible amplitude.

A variety of technical problems has caused the gyroscope experimentto be almost a quarter of a century in the making, from its inception in1960 to projected launch, in the middle 1980s. Among the more difficulttechnological hurdles that have had to be overcome in order to producea spaceworthy experiment that can measure gyroscope precessionsaccurate to 10"3 arcseconds per year, or equivalently to 10"1 6 rad/s,include:

(i) Fabrication of a gyroscope that is spherical and homogeneous to apart in a million. For this purpose, a 2 cm radius quartz sphere is used. Thisconstraint is necessary to reduce torques on the gyroscope. Even if thisconstraint is satisfied, there must be no residual gravitational forces onthe gyroscope larger than 10~9 g. This necessitates a drag-free satellite.

(ii) Readout of the direction of the spin axis. Conventional methods ofdetermining the spin direction of the gyroscope require violations of itssphericity and homogeneity, and thus introduce unacceptable torques.Thus a "London moment" readout method has been adopted. Thegyroscope is coated uniformly with a superconducting film. When spin-ning, the superconductor develops a magnetic dipole moment M parallelto its spin axis. Any change in the direction of M can be determined bymeasuring the current induced in a superconducting loop surroundingthe gyroscope. For this method to be viable, however, it was necessaryto develop a magnetic shield that could reduce the ambient magneticfield below 10 ~7 G, otherwise the gyroscope could contain trappedmagnetic flux of sufficient size to produce anomalous readout signals. Bycomparison, the ambient magnetic field of the Earth is about 0.5 G.

(iii) Determination of basis directions. The precession of the gyroscope'sspin axis is measured relative to the direction of a chosen reference star,as observed by a telescope mounted on the gyroscope housing. Thisdirection must be monitored to better than 10" 3 arcseconds per year, sothe design of a suitable optical system has been a major problem.


Further details of the experimental problems and progress are foundin Lipa and Everitt (1978) and Cabrera and Van Kann (1978).

A variant of the gyroscope experiment has recently been proposed byVan Patten and Everitt (1976) in which the "gyroscope" is itself the orbitof a satellite around the Earth. The dragging of inertial frames causes theplane of the orbit to rotate about an axis parallel to the Earth's rotationaxis. Assume the Earth is at rest, and rotates with angular momentum J.The substitution of Equation (9.4) for F, into the equations of motion(Section 4.2) yields the additional acceleration on a body near the Earth

da = -i(4y + 4 + a,) \ [2v x J - 3(v � n)(ii x J) + 3nv � (n x J)] (9.20)

where v is the body's velocity, and fi = x/r. For an orbit characterizedby inclination i relative to the plane normal to J, angle of the ascendingnode Q and orbit elements p, e, and co, the use of the orbit perturbationEquations (7.47) and (7.48) yields, over one orbit

5i = 0,

8Q = 2n{y + 1 + i^pl/imp3)1'2 (9.21)

Thus the "spin" vector S orthogonal to the orbital plane precesses aboutthe direction of J according to

dS/dt = ft x S (9.22)

where

ft = (y + 1 + iut)Ja- 3(1 - e2)- 3/2 (9.23)

For a body in a nearly circular orbit, this yields an annual angular pre-cession

5Q = O'.'22rj(y + 1 + U^jRJa)3 yr" l (9.24)

In order to eliminate the effects of other sources of precession (such asthe quadrupole moment of the Earth) two satellites counterrotating innearly identical orbits are necessary. With the use of drag-free satellitesand with two to three years of orbit data, an experiment with resultswithin 3% accuracy may be possible.

9.2 Laboratory Tests of Post-Newtonian GravityBecause the gravitational force is so weak, most tests of post-

Newtonian effects in the solar system require the use of the Sun andplanets as sources of gravitation. One disadvantage of such experiments

is that the experimenter has no control over the sources, and so is unableto manipulate the experimental configuration to test or improve thesensitivity of the apparatus, or at the very least, to repeat the experiment.Despite this disadvantage of solar system-sized experiments, the weaknessof post-Newtonian gravity has effectively prohibited laboratory experi-ments, with one exception.

That exception is the Kreuzer experiment (Kreuzer, 1968) that comparedthe active and passive gravitational masses of fluorine and bromine.Kreuzer's experiment used a Cavendish balance to compare the New-tonian gravitational force generated by a cylinder of Teflon (76% fluorineby weight) with the force generated by that amount of a liquid mixtureof trichloroethylene and dibromomethane (74% bromine by weight) thathad the same passive gravitational mass as the cylinder, namely theamount of liquid displaced by the cylinder at neutral buoyancy. In theactual experiment, the Teflon cylinder was moved back and forth in acontainer of the liquid, with the Cavendish balance placed near thecontainer. Had the active masses of Teflon and displaced liquid differedat neutral buoyancy, a periodic torque would have been experienced bythe balance. The absence of such a torque led to the conclusion that theratios of active to passive mass for fluorine and bromine are the same to5 parts in 105, that is

(mA/mP)F{ - (mA/wP)Br

(mA/mP)Br< 5 x 10- 5 (9.25)

[For further discussion of Kreuzer's experiment, see Gilvarry and Muller(1972) and Morrison and Hill (1973)].

If the active mass were to differ from the passive mass for these sub-stances, the major contribution to the difference would come from thenuclear electrostatic energy (as it does, say in the Eotvos experiment).Since Ee/m ~ 10"3 , one could regard such effects as post-Newtoniancorrections. However, the perfect-fluid PPN formalism of Chapter 4 ispoorly suited to a discussion of nuclear matter. A better approximationis one in which the PPN metric is generated by charged point masses,with gravitational potentials generated by masses, microscopic velocities,charges, and so on. Using this metric, one can calculate the active topassive mass ratio of a bound system (nucleus) of point charges, with theresult, for a spherically symmetric body (Will, 1976a),

mjmv = 1 + T£(£e/mP) (9.26)

where Ee is the electrostatic energy of the system of charges and £ is acombination of PPN parameters derived from the charged-point-massmetric. However, it can be shown that if the perfect-fluid PPN metric ofTable 4.1 is simply a macroscopic average of the point-mass metric (asone would expect in most reasonable theories of gravity), then the com-bination of charged-point-mass parameters that makes up e is preciselythe same as the fluid PPN parameter £3. Thus, in any such theory ofgravity,

mJrn? = 1 + Ka^ /HO (9.27)

(For further details, see Will, 1976a). The semiempirical mass formula(see Equation 2.8) yields

mjmp = 1 + 3.8 x 10~4£3Z(Z - l )^" 4 ' 3 (9.28)

For fluorine Z = 9, A = 19, and bromine Z = 35, A = 80, Equations(9.25) and (9.28) yield

|C3| < 6 x 1(T2 (9.29)

This generalizes and corrects a previous result of Thorne et al. (1971).Advancing technology may make several laboratory post-Newtonian

experiments possible in the coming decades (Braginsky et al., 1977). Theprogress that makes such experiments feasible is the development ofsensing systems with very low levels of dissipation, such as torque-balancesystems made from fused quartz or sapphire fibers at temperatures<;0.1 K, massive dielectric monocrystals cooled to millidegree tempera-tures, and microwave cavities with superconducting walls. Among someof the experimental possibilities are a measurement of the gravitationalspin-spin coupling of two rotating bodies; searches for time variationsof the gravitational constant, preferred-frame, and preferred-locationeffects; and a measurement of the dragging of inertial frames by a rotatingbody. The reader is referred to Braginsky et al. (1977) for detailed discus-sion and references.

9.3 Tests of Post-Newtonian Conservation LawsOf the alternative metric theories of gravity discussed in detail

in Chapter 5, all are Lagrangian based, that is, all possess integral con-servation laws for energy and momentum. In the post-Newtonian limit,their PPN parameters satisfy the semiconservative constraints

a3 = Ci = £2 = £3 s U = 0 (9.30)

What is the experimental evidence for these constraints?

In Chapter 8, we obtained the upper limit

| a 3 | < 2 x l ( T 7 (9.31)

from perihelion-shift data. The effect there was a combined preferred-frame effect and self-acceleration of a massive body, in particular of theSun.

However, this limit can probably be tightened considerably, althoughwith somewhat less rigor, by applying the self-acceleration term, Equation(6.39) to pulsars. For these bodies, assumed to be rotating neutron stars,|Q/m| ~ 0.1, and 2 s"1 < \co\ < 200 s"1, thus their self-acceleration hasthe form

Keifl <* 6 x 103|a

where v is the pulsar frequency, and 9 is the angle between the pulsarspin axis and its velocity relative to the universe rest frame. Althoughstrictly speaking, the post-Newtonian limit does not apply to pulsars, wefeel this is a reasonable estimate of the size of the effect in any theory witha3 # 0. This acceleration will cause a change in the pulse period Pp givenby

= aself-n (9.33)

where n is a unit vector along the line of sight to the pulsar. Thus,

- 2 x

independently of Pp, where O is the angle between aseIf and the line ofsight ii. For the 90 pulsars reported by Manchester and Taylor (1977)whose values of dPJdt have been measured, those values range between4 x 10"13 (Crab Pulsar) and 1 x 10"18 (PSR 1952 + 29), with half ofthem lying between 10"14 and 10"15. In all cases, dPJdt > 0, i.e., allpulsars are slowing down. Now for the 40 or so pulsars with 10"14 >dPp/dt > 10"15, it is extremely unlikely that either sin6 = 0 or cos® = 0for all of them, furthermore if a3 # 0, we would expect as many pulsarswith dPp/dt < 0 as with dPp/dt > 0, assuming their spin directions wereoriented randomly. Thus, a conservative limit on <x3 can be obtained bysetting sin 0 = cos = 5 in Equation (9.34), and imposing the 10"14

upper limit on an anomalous dPp/dt, giving

|a3l < 2 x 10"10 (9.35)

There may be one promising way to set a limit on the parameter £2

involving an effect first pointed out, incorrectly, by Levi-Civita (1937).The effect is the secular acceleration of the center of mass of a binarysystem. Levi-Civita pointed out that general relativity predicted a secularacceleration in the direction of the periastron of the orbit, and found abinary system candidate in which he felt the effect might one day beobservable. Eddington and Clark (1938) repeated the calculation usingde Sitter's (1916) n-body equations of motion. After first finding a secularacceleration of opposite sign to that of Levi-Civita, they then discoveredan error in de Sitter's equations of motion, and concluded finally that thesecular acceleration was zero. Robertson (1938) independently reachedthe same conclusion using the Einstein-Infeld-Hoffmann equations ofmotion, and Levi-Civita later verified that result. In fact, the secularacceleration does exist, but only in nonconservative theories of gravity;that is, it depends on the PPN parameters a3 and £2 (Will, 1976b).

The simplest way to derive this result is to treat the two-body systemas a single composite "body" in otherwise empty space, and to focus onthe self acceleration in the equation of motion, (6.32). For two point masses,Equation (6.32) and the formulae in Table 6.2 give

i(a3 + Ci){mim2x/r3)(vl - v2)

+ C1(w1rn2/r3)[v2(v2 � x) - v ^ � x) - fx(v2 � ft)2 + f x ^ � ft)2]

i ~ tn2)x/r4' + a3m1m2(w + V) � vx/r3 (9.36)

where x = x2 - \ u r = |x|, ft = x/r, v = v2 - v ls V is the center-of-massvelocity with respect to the PPN coordinate system, va = va � V, and

ms £ mJil+ffi-fa/r) [b # a] (9.37)a = l , 2

Substitutingvx s -(m2/m)v, v2 s {rnjnifs (9.38)

along with the expressions appropriate for a Keplerian orbit

x = p(l + e cos 4>)~l{ex cos</> + ej,sin<£),

v = (m/p)1/2[ �exsin0 + ey(e + cos<£)],

r2 dWdt = (mp)112 (9.39)and averaging (a)self over one orbit, we obtain

<(«)self > = («3 + C2)

- i a 3 ( l - e2y\Q/m)(Y/ + V) x 0 (9.40)

where

eP = � ex = [unit vector in the direction of the periastron of mx],

(o = (2n/P)ez = (mean angular velocity vector of orbit],

Q = <� m1m2/r) = � m1m2/a (9.41)

The second term in Equation (9.40) is the same as the term in Equation(6.39) except for the numerical factor (3 compared to j ( l � e 2)" 1] , whicharises from the difference in averaging for a stationary, nearly sphericalbody, and for a binary system. However, because of the limit we havealready obtained for a3, its effects on the self acceleration of a binarysystem will be negligible. Thus, we shall set a3 = 0, leaving

<(a)self > = C2

In the solar system, this has effects that are utterly unmeasurable. Forexample, the self acceleration of the Earth-Moon binary system producesa perihelion shift for the Earth of the order dcom ~ 10~5 per century.A more promising testing ground for this effect would be a close binarysystem, such as iBoo, with mt = 1.35mo, m2 = 0.68mQ, P = 0.268 day.The resulting change in the periods (inverse frequencies), say, of the spectrallines of the stars in iBoo would be

P~l(dPJdt) = 8.8 x 10-7£2e(l - e2)"3 / 2 sin co sin i yr"1 (9.43)

where i is the inclination of the orbit relative to the plane of the sky, andw is the angle of periastron. Unfortunately, because of Doppler broad-ening, the frequencies of spectral lines are not known to sufficient accuracyto make such a change observable.

However, the discovery of the binary pulsar (Chapter 12) has changedthe situation. The characteristics of the orbit are very similar to that ofiBoo, however the pulsar provides a much more precise and stable timestandard than do spectral lines. This enables one not only to measurechanges in the pulse period with high accuracy, but also to determinethe parameters of the orbit and thereby the change of the oribit periodPb with high accuracy. The results are (Table 12.1)

P^dPJdt = (4.617 ± 0.005) x 10~9 yr"1

Pb- l dPJdt = - (2.4 ± 0.4) x 10 - 9 yr - J (9.44)

However, because the binary pulsar is a "single-line spectroscopic binary,"the individual masses are not known from the velocity curve data (we

shall see that they can be determined if one assumes a particular metrictheory of gravity), rather, the known quantities are (see also Table 12.1)

e =* 0.62, Pb =* 27907 s

ft = (m2sini)3/m2 c 0.13mo

co si 179° + 4.23°(t - to)/(l yr) (9.45)

where / t is known as the mass function, and where t0 � [September,1974]. Then the predicted period change for both the pulsar and theorbit is given by

where X = mjm2 = wpu,sar/mcompanion, and where we have used the factthat, from Equation (9.45),

sin co st - 7 x 10"2(t - to)/(l yr), t - t0 < 10 yr (9.47)

Note too, that the second derivatives of the periods are given by

p ; 1 d2Pp/dt2 = p ^ 1 d2Pb/dt2

Now, from data covering a time span of several years, the error onPp 1 dPJdt was found to be 10"x 1 y r " l . In other words, P;1 dPJdt didnot change by more than 10"1X yr"1 in a year, that is,

\p-^d2Pvjdt2\ < lO-^yr"2 (9.49)

Assuming that the secular acceleration is responsible for no more thanthis amount, in other words, that there is no fortuitous cancellationbetween this effect and other sources of period change (Section 12.1), weobtain from Equations (9.48) and (9.49) the limit

|C2| < 2 x 10"4(mo/m)2/3|(l + X)2/4X(1 - X)\ (9.50)

Now, without assuming a particular metric theory of gravity, we do notknow the values of m and X, so the limit on £2 is uncertain (if the massesare equal, for example, X � 1, and there is no secular acceleration, byvirtue of symmetry).

If, for example, we assume that general relativity is valid except forthe sole possibility of a violation of momentum conservation manifested

by £2 # 0, then we can use the values of m and X obtained from periastron-shift data and from the gravitational red shift-second-order Dopplershift data (see Chapter 12 for details),

m ^ 2.85mo, X ~ 1.007 ± 0.1 (9.51)

Although the data are not yet sufficiently accurate to exclude X = 1, itis of interest to substitute the nominal value of X into Equation (9.50) toobtain |£2| < 10"2. As long as \X - 1| > 10" 3, we will still have |£2| < 0.1.

Of the remaining three conservation-law parameters, only £3 has beentested experimentally, as we saw in the previous section where we obtainedthe limit |£3| < 0.06 from the Kreuzer experiment. No feasible experimentor observation has ever been proposed that would set direct limits onthe parameters £1 or £4. Note, however, that these parameters do appearin combination with other PPN parameters in observable effects, forexample in the Nordtvedt effect (see Section 8.1).

10

Gravitational Radiation as a Tool forTesting Relativistic Gravity

Our discussion of experimental tests of post-Newtonian gravity inChapters 7, 8, and 9 led to the conclusion that, within margins of errorranging from 1% to parts in 10"7 (and in one case even smaller), thepost-Newtonian limit of any metric theory of gravity must agree withthat of general relativity. However, in Chapter 5, we also saw that mostcurrently viable theories of gravity could accommodate these constraintsby appropriate adjustments of arbitrary parameters and functions andof cosmological matching parameters. General relativity, of course, agreeswith all solar system experiments without such adjustments. Nevertheless,in spite of their great success in ruling out many metric theories of gravity(see Sections 5.7, 8.5), it is obvious that tests of post-Newtonian gravity,whether in the solar system or elsewhere, cannot provide the final answer.Such tests probe only a limited portion, the weak-field slow-motion, orpost-Newtonian limit, of the whole space of predictions of gravitationaltheories. This is underscored by the fact that the theories listed in Chapter 5whose post-Newtonian limits can be close to, or even coincident with,that of general relativity, are completely different in their formulations,One exception is the Brans-Dicke theory, which for large co, differs fromgeneral relativity only by modifications of O(l/a>) both in the post-Newtonian limit and in the full, exact theory. The problem of testingsuch theories thus forces us to turn from the post-Newtonian approxi-mation toward new areas of "prediction space," new possible testinggrounds where the differences among competing theories may appear inobservable ways. The remaining four chapters will be devoted to thesenew arenas for testing relativistic gravity.

One new testing ground is gravitational radiation. Almost from theoutset, general relativity was known to admit wavelike solutions analogous


to those of electromagnetic theory (Einstein, 1916). However, unlike thecase with electromagnetic waves, there was considerable doubt as to thephysical reality of such waves. Eddington (1922) suggested that they mightrepresent merely ripples of the coordinates of spacetime and as such wouldnot be observable. This lingering doubt was dispelled conclusively in thelate 1950s by the work of Hermann Bondi and his collaborators, whodemonstrated in invariant, coordinate-free terms that gravitationalradiation was physically observable, that it carried energy and momentumaway from systems, and that the mass of systems that radiate gravitationalwaves must decrease (Bondi et al., 1962).

The pioneering work of Joseph Weber initiated the experimentalsearch for gravitational radiation. Although no conclusive evidence forthe direct detection of gravitational waves exists at present [see Douglassand Braginsky (1979) for a review], gravitational-wave astronomy mayultimately open a new window on the universe.

Virtually any metric theory of gravity that embodies Lorentz in variance,on at least some crude level, in its gravitational field equations, predictsgravitational radiation. Thus, the existence of gravitational radiation doesnot represent a particularly strong test of gravitation theory. It is thedetailed properties of such radiation that will concern us here.

While the post-Newtonian approximation may be described as theweak-field, slow motion "near-zone" limit, our discussion of gravitationalradiation will center on the weak-field, slow motion, "far-zone" limit. Inthis limit, one finds that metric theories of gravity may differ from eachother and from general relativity in at least three important ways: (i) theymay predict a difference between the speed of weak gravitational wavesand the speed of light (see Section 10.1); (ii) they may predict differentpolarization states for generic gravitational waves (see Section 10.2); and(iii) they may predict different multipolarities (monopole, dipole, qua-drupole, etc.), of gravitational radiation emitted by given sources (seeSection 10.3). The use of gravitational-wave speed and polarization astests of gravitation theory requires the regular detection of gravitationalradiation, a prospect that may be far off (see Douglass and Braginsky,1979). However, the multipolarity of gravitational waves can be studiedby analyzing the back influence of the emission of radiation on the source(radiation reaction) for different multipoles. One example is the changein the period of a two-body orbit caused by the change in the energy ofthe system as a result of the emission of gravitational radiation. Such atest is now possible in the binary pulsar (Chapter 12).

Gravitational Radiation: Testing Relativistic Gravity 223

10.1 Speed of Gravitational WavesThe Einstein Equivalence Principle demands that in every local,

freely falling frame, the speed of light must be the same - unity, if oneworks in geometrized units. The speed of propagation of all zero rest-massnongravitational fields (neutrinos, for example) must also be the same asthat of light. However, EEP demands nothing about the speed of gravi-tational waves. That speed is determined by the detailed structure of thefield equations of each metric theory of gravity.

Some theories of gravity predict that weak, short-wavelength gravita-tional waves propagate with exactly the same speed as light. By weak, wemean that the dimensionless amplitude /zMV that characterizes the wavesis in some sense small compared to the metric of the background spacetimethrough which the wave propagates, i.e.,

IIU/IICII«iand by short wavelength, we mean that the wavelength X is small comparedto the typical radius of curvature 0t of the background spacetime, i.e.,

|A/£| « 1

This is equivalent to the geometrical optics limit, discussed in Chapter 3for electromagnetic radiation. In the case of general relativity, for example,one can show (see MTW, Exercise 35.15) that the gravitational wavevector /" is tangent to a null geodesic with respect to the "background"spacetime, i.e.,

i*r$» = o, i% = owhere "slash" denotes covariant derivative with respect to the backgroundmetric. In a local, freely falling frame, where gfj = rj^, the speed of theradiation is thus the same as that of light. Gravitational radiation prop-agates along the "light cones" of electromagnetic radiation.

General relativityA simple method to derive this result in general relativity, which

can then be applied to other metric theories, is to solve the vacuum fieldequations, linearized (weak fields) about a background metric chosenlocally to be the Minkowski metric. Physically, this is tantamount tosolving the propagation equations for the radiation in a local Lorentzframe. As long as the wavelength is short compared to the radius ofcurvature of the background spacetime, this method will yield the same

results as a full geometrical-optics computation. We thus write

G^ = Vw + V (10.1)

Then the linearized vacuum field equations (5.15) take the form

� Av + K* - <** - K,� = ° (10-2)where indices are raised and lowered using r\. We choose a gauge (Lorentzgauge) in which

Then

D A* = °whose plane-wave solutions are

V = �O'"**' '"'X* = 0 (10.5)Thus, the electromagnetic and gravitational light cones coincide, i.e.,the gravitational waves are null.

Scalar-tensor theoriesThe linearized vacuum field equations are (see Section 5.1 for

discussion of notation)

(io.6)

1 4 , - ^ , - ^ = 0 (10.7)

we obtain

U,9 = D A , = 0 (10.8)

whose plane-wave solutions are proportional to e"|ix" where

/"'V = 0 (10.9)

So in scalar-tensor theories, gravitational waves are null.

Vector-tensor theoriesIn this case the linearized field equations are much more complex

than in the scalar-tensor theories, with the propagation of linearizedmetric disturbances (h^) being strongly influenced by the background

D V + h

Choosing a gauge in which

�-K t

= 0,

>0 V,^v


cosmological value K of the vector field. In general there are ten differentsolutions, each with its own characteristic speed and polarization. Forone of these solutions, for example (for derivation see Section 10.2) thespeed is

v2g = (1 - «K 2) / [1 - (o> - i, - t)K2] (10.10)

Rosen's bimetric theoryWe have already discussed weak gravitational waves in this

theory, in Section 5.5(g). The resulting speed was given by v\ = Cx/c0

where c1 and c0 are cosmological matching parameters (see Section 5.5for discussion). If we take into account not only the cosmological boundaryconditions but also a gravitational potential t/ext due to an externalgravitating body (galaxy, sun), with the wavelength of the radiationbeing short compared to the scale over which l/ext varies, then c0 and ct

may be replaced by co(l � 2l/cxt) and c t(l +2£/ext), where c0 and cx denotethe purely cosmological values, and thus

v2g = (c!/co)(l + 4[/ext) (10.11)

Therefore, the velocity of gravitational radiation may depend both uponcosmological parameters and on the local distribution of matter. Noticethat solar system limits on a2 constrain v2 to be within ~ 4 x 10~4ofunity.

RastalFs theoryThe (extremely complicated) linearized vacuum field equations

for the vector field K^ in the rest frame of the universe, where

K, = K8° + k^ (10.12)

yield three independent polarizations for k^, one having a differentvelocity than the other two. However, to first order in the cosmologicalmatching parameter K, which is constrained to be small by Earth-tidemeasurements (see Sections 5.5, 8.5), the velocities are the same,

Vg = 1 + %K2

and the polarizations for a wave traveling in the z-direction are given by

k ( 1 ) o c e 0 - e z , k(2)azex, k(3)ocey (10.14)

(These results are valid only in the universe rest frame.)Table 10.1 summarizes the velocities of gravitational waves in these

and other theories of grayity discussed in Chapter 5. Generally speaking,there are two ways in which the speed of gravitational waves may differ

Table 10.1. Properties of gravitational radiation in alterna-tive metric theories of gravity.

GravitationalTheory wave speed E(2) class

General relativityScalar-tensor theoryVector-tensor theoryRosen's bimetric theoryRastall's theory 1BSLL theory 1Stratified theories

11

various(cx/c0)

1/:2

+ iK2 + O(K3)+ K^o + °>i) + O(co2)

a

N2N 3

n'ni 5

n6n«" Speed is a complicated function of parameters.

from that of light. The first is through the cosmological matching param-eters, i.e.,

vg =* vgc (10.15)

where vgc denotes the cosmologically determined speed. The second isthrough the local distribution of matter. If we take into account a nearlyconstant, but noncosmological gravitational potential t/ext («1), thematching parameters may be modified by terms of O(l/ext), resulting ina speed

vg =* 1^(1 + a[/ext) (10.16)

Solar system experiments limit some of the parameters that appear inthe expressions for vgc, but only to accuracies of order 10~3. A crucialtest of such theories would be provided by high-precision measurementsof the relative speed of gravitational and electromagnetic waves (Eardleyet al., 1973). By comparing the arrival times for gravitational waves andfor light that come from a discrete event such as a supernova, one couldset a limit on the relative speeds that, for a source in the Virgo cluster(11 Mpc from Earth) for example, would yield

precision in measuring. . . . (10.17)

time lag, in weeks

Another possible way to test whether vg = 1 has been described byCaves (1980) within the context of Rosen's bimetric theory. If vg < 1,then high-energy particles are prevented from being accelerated to speedsgreater than vg by gravitational-radiation damping forces that accompanythe nearly divergent gravitational radiation flux emitted by a particle atvelocities near vg. The indirect observation of cosmic rays with energies

exceeding 1019 eV places a very tight upper limit, if this analysis is correct,on 1 - vg in Rosen's theory. Similar conclusions would be expected tofollow in any theory in which vg < 1.

10.2 Polarization of Gravitational Waves(a) The E{2) classification schemeGeneral relativity predicts that weak gravitational radiation has

two independent states of polarization, the " + " and " x " modes, to usethe language of MTW, (Section 35.6), or the + 2 and � 2 helicity states,to use the language of quantum field theory. However, general relativityis probably unique in that prediction; every other known, viable metrictheory of gravity predicts more than two polarizations for the genericgravitational wave. In fact, the most general weak gravitational wavethat a theory may predict is composed of six modes of polarization, ex-pressible in terms of the six "electric" components of the Riemann tensorROiOj that govern the driving forces in a detector (Eardley et al., 1973;Eardley, Lee, and Lightman 1973).

Consider an observer in a local freely falling frame. In the neighbor-hood of a chosen fiducial world line ^(t), construct a locally Lorentzorthonormal coordinate system {t,xj) with t as proper time along theworld line and^(f) as spatial origin ("Riemann normal coordinates").The metric has the form (MTW, Section 13.6)

9 m = "m + hjn (10.18)

where

* + O(|x|3),

* + O(|x|3),

% = - i / l a y ^ x * + O(|x|3) (10.19)

where R^a, are components of the Riemann tensor. For a test particlewith spatial coordinates x\ momentarily at rest in the frame, the accel-eration relative to the origin is

at = 1*66,? = ~ Kof6j* ; (10.20)

where Roio} a r e t n e "electric" components of Riem due to gravitationalwaves or other external gravitational influences. Note that despite thepossible presence of auxiliary gravitational fields in a given metric theoryof gravity, the acceleration is sensitive only to Riem. [This is not necessar-ily true if the body has self-gravitational energy, as has been emphasizedby Lee (1974).]


Thus, a gravitational wave may be completely described in terms ofthe Riemann tensor it produces. We define a weak, plane, nearly nullgravitational wave in any metric theory [Eardley, Lee, and Lightman(1973)] to be a weak, propagating vacuum gravitational field character-ized, in some local Lorentz frame, by a linearized Riem with componentsthat depend only on a retarded time u, i.e.,

R*yi = KpyM (10.21)

(henceforth we shall drop the caret on indices) where the "wave vector"!� which is normal to surfaces of constant u, defined by

is almost null with respect to the local Lorentz metric, i.e.,

rfvlX = e, |e| « 1 (10.23)

where e is related to the difference in speed, as measured in a local Lorentzframe at rest in the universe rest frame, between light and the propagatinggravitational wave, i.e.,

e = (c/vg)2 - 1 (10.24)

We now wish to analyze the general properties of Riem for a weak,plane, nearly null gravitational wave. To do this, it is useful to introduce,instead of the locally Lorentz orthonormal basis (t, xJ), a locally null basis.Consider a null plane wave (light, for instance) propagating in the + zdirection in the local Lorentz frame. The wave is described by functionsof retarded time u, where

u = t - z (10.25)

(we use units in which the locally measured speed of light is unity). Asimilar wave propagating in the � z direction would be described by func-tions of advanced time v, where

v = t + z (10.26)

We now define the vector fields I and n to be I = Fe^, n = n^e,,, where

These vectors are tangent to the propagation directions of the two nullplane waves. In the (t, xJ) basis they have the form

/" = (1,0,0,1), n" = i ( l , 0,0,-1) (10.28)

and are null with respect to 17, i.e.,

I'l'V = rt\( = 0 (10.29)

We also introduce the complex null vectors m and m, where the bar de-notes complex conjugation, denned by m = m"e,,, where

m" = (2)- ^(O,1, i,0), m" = (2)" 1 /2(0,1, - i,0) (10.30)

and where

m"mvjjpv = m"mvf/ v = 0 (10.31)

These null vectors obey the orthogonality relations

>fv= -2J("nv) + 2m("mv) (10.32)

In a Cartesian basis, they are constant. For the remainder of this section,we shall use roman subscripts (excluding i, j , k) to denote componentsof tensors with respect to the null tetrad basis I, n, m, in, i.e.,

Zarb_ = Za$1..<f1fiV... (10.33)

where a, b, c,... run over I, n, m, and m, while p, q, r,... run over onlyI, m, and m.

Because the null tetrad I, n, m, and m is a complete set of basis vectors,we may expand the gravitational wave vector Tin terms of them; however,since the gravitational wave is not exactly null, this expansion will dependin general upon the velocity of the observer's local frame relative to theuniverse rest frame. Choose a "preferred" observer, whose frame is atrest in the universe, and let /" in this frame have the form

I" = f ( l + e,) + enn" + emm" + ejn" (10.34)

where {£,,£�, £�,,£�} ~ O(E). However, this observer is free (i) to orient hisspatial basis so that the gravitational wave and his null wave are parallel,i.e., so that

V oc V, (10.35)

and (ii) to choose the frequency of his positively propagating null waveto be equal to that of the gravitational wave, i.e.,

7° = 1° (10.36)

Hence, em = em = 0, £, = - | f in , and

? � = / " - sj^l" - n") (10.37)

Now, because the Riemann tensor is a function of retarded time u alone,

Thus, using the orthogonality relations among the null tetrad vectors,

Rw = ° (10-39)The linearized Bianchi identities Rab[cdie] = 0 then yield

RatPq,n = O(BnR) (10.40)

which, except for a trivial nonwavelike constant, implies

Rabpq = ^ W = O(£nR) (10.41)

Thus the only components of Riem that are not O(en) are of the formRnpnq. There are only six such components and all other components ofRiem can be expressed in terms of them. They can be related to particulartetrad components of the irreducible parts of Riem; the Weyl tensor, thetraceless Ricci tensor, and the Ricci scalar (see MTW Section 13.5 fordefinitions). These components are called Newman-Penrose quantities,denoted T, <J>, and A, respectively, (Newman and Penrose, 1962). For ournearly null plane wave in the preferred tetrad, they have the form

(i) Weyl tensor:

¥ 0 ~ O(s2nR), V1 ~ O(enR),

V2=~i;RnM + O(£nR),

^ 3 = -iRnlnm + O(EnR),

*4 = -Rnmnm (10.42)

(ii) traceless Ricci tensor:

$ 0 0 ~ O(en2R), <D01 ~ O1 0 =* O02 ^ 3)20 =s O(enR),

®i2 = *2i = ¥ 3 + O(enR) (10.43)

(iii) Ricci scalar:

A = - i « F 2 + O(enR) (10.44)

To describe the six independent components of Riem we shall choosethe set W2, *P3, *P4, and <D22 (¥3 and *P4 are complex). The above results

Gravitational Radiation: Testing Relativistic Gravity

y y

o

231

(a)

(/

(b)

o

/)Im * 4

3

(c)

1

o

J)(d)

i\

(e)

�

*

\

\ ;

/ )

(0

Figure 10.1. The six polarization modes of a weak, plane gravitationalwave permitted in any metric theory of gravity. Shown is the displacementthat each mode induces on a sphere of test particles. The wave propagatesin the +z direction and has time dependence cos cot. The solid lirie is asnapshot at cot = 0, the broken line one at cot = n. There is no displace-ment perpendicular to the plane of the figure. In (a), (b), and (c) the wavepropagates out of the plane; in (d), (e), and (f), the wave propagates in theplane.

are valid for a gravitational wave as detected by the preferred observer.Now in order to discuss the polarization properties of the waves, we mustconsider the behavior of these components as observed in local Lorentzframes related to the preferred frame by boosts and rotations. However,we must restrict attention to observers who agree with the preferred ob-server on the frequency of the gravitational wave and on its direction;such "standard" observers can then most readily analyze the intrinsic

polarization properties of the waves. The Lorentz frames of these standardobservers are related by a subgroup of the group of Lorentz transforma-tions that leave \ unchanged. The most general such transformation ofthe null tetrad that leaves T [cf. Equation (10.37)] fixed is given by

I' = (1 - aa£n)\ - en(am + am) + O(£2),

n' = (1 � aa£B)(n + aal + am + am) + O(e2),

m' = (1 - aaeje'^m + al) - e^e'^n + am) + O(E2),

m' = (1 - aaejg-'^m + al) - £nae"'>(n + oan) + O(e2) (10.45)

where a is a complex number that produces null rotations (combinationsof boosts and rotations) asd cp is an arbitrary real phase (0 < q> < 2n)that produces a rotation about ez. The parameter a is arbitrary except forthe restriction

aa«en"1 (10.46)

This expresses the fact that our results are valid as long as the velocityof the frame, w, is not too close either to the speed of light or of the gravita-tional wave, whichever is less; note that for nearly null waves e~ * » 1and almost any velocity that is not infinitesimally close to unity is per-mitted, since

aa ~ w2/(l - w2) (10.47)

For exactly null waves en = 0, and arbitrary velocities w < 1 are permitted.Under the above set of transformations, the amplitudes of the gravita-

tional wave change according to

W2 = V2 + O(snR),

+ 6a2*F2) + O{snR),2af 3 + 6aa»P2 + O(snR) (10.48)

Consider a set of observers related to each other by pure rotationsabout the direction of propagation of the wave (a = 0). A quantity thattransforms under rotations by a multiplicative factor e's<? is said to havehelicity s as seen by these observers. Thus, ignoring the correction termsof O(£nR), we see that the amplitudes {^2,¥3,'F4,4)22} have helicities

T2:s = 0, O22:s = 0,

¥3:s=-l, ¥3:s=+l,

¥4:s=-2, ?4:5=+2 (10.49)

However, these amplitudes are not observer-independent quantities, ascan be seen from Equation (10.48). For example, if in one frame *F2 # 0,*P4 ^ 0, then there exists a frame in which ¥4 = 0. Thus, the presence orabsence of the components of various helicities depends upon the frame.Nevertheless, certain frame-invariant statements can be made about theamplitudes, within the small corrections of O(snR). These statementscomprise a set of quasi-Lorentz invariant classes of gravitational waves.Each class is labeled by the Petrov type of its nonvanishing Weyl tensorand the maximum number of nonvanishing amplitudes as seen by anyobserver. These labels are independent of observer.

For exactly null waves, the classes are:Class II6; *F2 # 0. All standard observers measure the same value for

*F2, but disagree on the presence or absence of all other modes.Class III5: *F2 = 0, ¥3 ^ 0. All standard observers agree on the

absence of *F2 and on the presence of ¥3, but disagree on the presenceor absence of *P4 and 22.

Class N3: f 2s ^ 5 0> * 4 # 0. $22 # 0. Presence or absence of all

modes is observer-independent.Class N2: *¥2 = ¥3 = 4>22 = 0, ¥ 4 =£ 0. Independent of observer.Class O1:

x¥2 = x¥3 = *¥4 = 0, <D22 # 0. Independent of observer.

Class Oo: *¥2 = x¥3 = x¥4 = 0>22 = 0. Independent of observer: No

wave.For nearly null waves, simply replace the vanishing of modes (=0)

with the nearly vanishing of modes [~O(enR)].This scheme, developed by Eardley et al. (1973), is known as the E(2)

classification for gravitational waves, since in the case of exactly nullplane waves (en = 0), the transformation equations, (10.45), are the "littlegroup" E(2) of transformations, a subgroup of the Lorentz group. TheE(2) class of a particular metric theory is defined to be the class of itsmost general wave.

Although we have confined our attention to plane gravitational waves,one can show straightforwardly (Eardley, Lee, and Lightman, 1973) thatthese results also apply to spherical waves far from an isolated sourceprovided one considers the dominant l/R part of the outgoing waves,where R is the distance from the source.

(b) E(2) classes of metric theories of gravityTo determine the E(2) class of a particular theory, it is sufficient

to examine the linearized vacuum field equations of the theory in thelimit of plane waves (observer far from source of waves). The resulting

classes for the theories discussed in Chapter 5 are shown in Table 10.1.Here, we present some examples. Some useful identities that can beobtained from Equations (10.32) and (10.41) are

Rnl = RnM + O(snR), Rm = 2Rnmnih + O(enR),

Rnm = Kinm + O(enR), R = - 2Rnl + O(snR) (10.50)

If Riem is computed from a linearized metric perturbation h^{u), then

and

W2 = A + O(snR), W3 =

^ 4 = ifc*» + O{anR), O22 ^

General relativity

The vacuum field equations are

R,v = 0

The waves are null (en = 0). Thus,

KM = Rnn,nm = Rnlnn, = 0

or

V2 = «P3 = O22 = 0

O(snR),

+ O(enR) (10.52)

(10.53)

(10.54)

(10.55)

The only unconstrained mode is *F4 ^ 0, so general relativity is of E(2)class N 2 .

Scalar-tensor theoriesIn a local freely falling frame, the linearized vacuum field equa-

tions are

,9 = o,

R = O(<p2) (10.56)

(see Section 5.3 for details), where <f>0 is the cosmological boundary valueof the scalar field (f>. The solution to the first of Equation (10.56) for aplane wave is

cp = (10.57)

where tj^lT � 0. Then, from Equation (10.56),

«(1,= -ôV/'-V» (10.58)

Thus, Rnn * 0, Rnl s Rnm = 0, thus,

V2 = V3 = 0, <D22#0, ¥ 4 # 0 (10.59)

and scalar-tensor theories are of class N 3 .

Vector-tensor theoriesIn a local freely falling frame in the universe rest frame, the

linearized vacuum field equations take the form

- coK2h00iltv - 2coKk0tllv - (co + \r\ -

- i(co

- (if -

?) - ifa -^ / » + (i, + TJK/CÔ = 0, (10.60)

(6 - |T)D A - efcf,, - i « X ^ ( n ^ - ^ ) + i(»? - t ) X D ^, - ftg,^) = 0 (10.61)

By substituting plane wave forms h^ = hû) and fcM = k^(u), we canturn the field equations into a set of algebraic equations for the amplitudeshMV and k^. We now project these equations onto the null tetrad I, n, m,and m, and obtain ten homogeneous algebraic equations for the tenunknowns hab, ka, with coefficients that depend upon the parameters a>,n, x, e, and K, and upon en [see Equation (10.37)]. These equations are ofthe form

[(1 - coK2)en(l - K ) - ±fo - t)K2]hmm = 0, (10.62)

«AihM + XAIKM + *A3km = 0, (10.63)

pBiLm + Pnhu + PB3K + PB4.h'nn + pBt'k, + pB6kn = 0 (10.64)

where A = 1,2,3, and B = 1 , . . . , 6. One mode is given by

hmm * 0,

en(l - i O = kin ~ r)K\l - coK2yl (10.65)

Since

-2en(l - hn) = £ = (vgy2 - 1 (10.66)

the speed of this mode is given by

In this case it can be shown that (except for special values of the parameters)the remaining amplitudes satisfy

Km = fej = K = km = 0,

eJL> - (1 - Ktfi . s 0,(1 - hn)hnl + (1 - a ^ = 0 = (1 - !£�)£� + 2£n/I"nn (10.68)

whose consequence is *P2 = *F3 = $22 = 0. Hence, this mode is N2. Thereare in principle nine other modes with timm = 0, each with its own speedand characteristic polarization, some as general as II6.

Rosen's bimetric theoryThe linearized field equations are of the form D,^MV

= 0 with norestrictions on the h^, hence all modes are nonzero in general, hence thetheory is of class II6.

Rastall's theorySince the linearized physical metric g in the universe rest frame

has the form [Equation (5.78)]

g0J = KCQ lkp

gik = Col8jk + Kco*k05ik (10.69)

where k^ = k^u), then we have, after transforming to local Lorentz co-ordinates,

h'tt <x k0 + kz, tilih oc kg,,

^ 0, hmih<xk0 (10.70)

However, from the solution of the linearized field equations discussed inthe previous section [Equation (10.14)], it is clear that for all solutions,n'u = O(snR), hence,

¥ 2 = O(enR) (10.71)

Notice that for thek (1) mode, only <J>22 °c timjh # 0, so this mode is O t ; forthek(2) andk (3) modes, *P3 oc hm # 0, so these modes are III5. The vanish-ing of *P4 a hmih is valid only in the universe rest frame, a result of the


special form of g^ there. The most general wave therefore is III5, henceRastall's theory is of class III5.

It is possible to show that the other theories discussed in Chapter 5 areof class II6 (see Table 10.1).

(c) Experimental determination of the E(2) classConsider an idealized gravitational-wave polarization experi-

ment. An observer uses an array of gravitational-wave detectors to deter-mine via Equation (10.20) the six electric components ROiOj of Riem foran incident wave (for discussion of possible devices and arrays see Eardley,Lee, and Lightman, 1973; Paik, 1977; and Wagoner and Paik, 1977). Letus suppose that the waves come from a single localized source with spatialwave vector k (which the observer may or may not know a priori). If theobserver expresses his data as a 3 x 3 symmetric "driving force matrix"

StJ(t) = Roioj (10.72)

then, for a wave with k = e2, Equations (10.28), (10.30), (10.42), and (10.43)give the following form for S,7 in terms of the wave amplitudes

-2/2Rex¥3

(10.73)2/2,

where the standard xyz orientation of the matrix elements is assumed.Now, if the observer knows the direction k a priori, either by associatingthe wave with an independently observed event such as a supernova, orby correlating signals detected at two widely spaced antennas (gravita-tional-wave interferometry), then by choosing a z-axis parallel tok, onecan determine uniquely the amplitudes as given in Equation (10.73), andthereby the class of the incident wave. Because a specific source need notemit the most general wave possible, the E(2) class determined by thismethod would be the least general class permitted by any metric theoryof gravity.

However, if the observer does not know the direction a priori, it is notpossible to determine the E(2) class uniquely, since there are eight un-knowns (six amplitudes and two direction angles) and only six observables(Sy). In particular, any observed StJ can be fit by an appropriate wave ofclass II6 and an appropriate direction. However, for certain observedSjj, the E(2) class may be limited in such a way as to provide a test of gravi-tational theory. For example, if the driving forces remain in a fixed plane

and are pure quadrupole, i.e., if there is a fixed coordinate system in which

n n o\-X 0 (10.74)

\0 0 0/

then the wave may be either II6 (unknown direction), or N2 (directionparallel to z axis of new coordinate system). If this condition is not ful-filled, the class cannot be N2. Such a result would exemplify evidenceagainst general relativity. Eardley, Lee, and Lightman (1973) provide adetailed enumeration of other possible outcomes of such polarizationmeasurements.

10.3 Multipole Generation of Gravitational Waves andGravitational Radiation DampingIt is common knowledge that general relativity predicts the low-

est multipole emitted in gravitational radiation is quadrupole, in the sensethat, if a multipole analysis of the gravitational field in the radiation zonefar from an isolated system is performed in terms of tensor spherical har-monics, then only the harmonics with / 2 are present (see Thorne, 1980for a thorough discussion of multipole-moment formalisms). For materialsources, this statement can be reworded in terms of appropriately dennedmultipole moments of the matter and gravitational-field distributionwithin the near-zone surrounding the source: the lowest source multipolethat generates radiation is quadrupole. For slow-motion, weak-fieldsources, such as binary star systems, quadrupole radiation is in fact thedominant multipole. (Some have argued that this is true for any slow-motion source, whether weak field or not. One exponent of this viewpointis Thorne, 1980.) The result is a gravitational waveform in the radiationzone given by

hmm = (2/R)Imm (10.75)

where R is the distance from the source, 7y is the moment of inertia of thesource, and dots denote derivatives with respect to retarded time. Thewaveform h^m is related to the measured electric components of Riem byEquation (10.52),

«-�*=-*4=-i«** (10.76)

The flux of energy at infinity that results from this waveform is given by

dE/dt = - J < W (10.77)

where Jy is the trace-free moment of inertia tensor of the system, given tolowest order in a post-Newtonian expansion by

/� = Jp( x , t){xtXj - !<50x2) d3x (10.78)

and where angular brackets denote an average over several periods ofoscillation of the source (for a recent discussion, see Walker and Will,1980a).

These comments apply to the asymptotic properties of the outgoingradiation field. However, we are interested not in the properties of the out-going radiation field (those were relevant for Sections 10.1 and 10.2), butin the back reaction of the source to the emission of the radiation. A vari-ety of computations have led to the conclusion that the energy flux at in-finity given by Equation (10.77) is balanced by an equal loss of mechanicalor orbital energy by the system, and that this energy loss can be derivedfrom a local radiation-reaction force (MTW Section 36.11)

F(react)= _ (2/5)mi\?Xj (10.79)

where the superscript (5) denotes five time derivatives (Walker and Will,1980b). However, one school of thought maintains that these conclusionshave not been satisfactorily derived from a fully self-consistent, approxi-mate solution of Einstein's equations (Ehlers et al., 1976). It is not the pur-pose of this section to enter into this controversy. Instead, we shall simplymake the assumption that in any semiconservative metric theory of grav-ity, there is an energy balance between the flux of gravitational-waveenergy at infinity and the loss of mechanical energy of the source, providedone averages over several periods of oscillation, and that the energy fluxcan be determined using a slow-motion, weak-field approximationscheme of a kind suggested by Epstein and Wagoner (1975).

If we now focus on binary systems with total mass m = mx + m2, re-duced mass ii = m^m^m, orbital separation r, and relative velocity v,quadrupole radiation within general relativity [Equation (10.77)] leadsto a loss of orbital energy at a rate (Peters and Mathews, 1963)

^=_/^A(12^-ll^)\ (10.80)dt \ r4 /

where r = dr/dt, and where angular brackets denote an average over anorbit. This loss of energy results in a decrease in the orbital period P givenby Kepler's third law,

(10.81)

Quadrupole radiation also leads to a decrease in the angular momentumof the system, and to a corresponding decrease in the eccentricity of theorbit (see Wagoner, 1975, for references and a summary of the formulae).Faulkner (1971) has pointed out that these effects of quadrupole gravita-tional radiation may play an important role in the evolution of ultrashort-period binary systems (see also Ritter, 1979). But probably the mostpromising test of the existence of quadrupole radiation has been providedby observations of period changes P in the binary pulsar (Chapter 12).

Unlike general relativity, however, nearly every alternative metrictheory of gravity predicts the presence in gravitational radiation of allmultipoles-monopole and dipole, as well as quadrupole and highermultipoles (Eardley, 1975; Will and Eardley, 1977; and Will, 1977). Forbinary star systems, the presence of these additional multipole contribu-tions has two effects on the energy-loss-rate formula, (10.80): (a) modifi-cation of the numerical coefficients in (10.80) and (b) generation of anadditional term (produced by dipole moments) that depends on the self-gravitational binding energy of the stars. The resulting formula for dE/dtmay be written in a form that contains dimensionless parameters whosevalues depend upon the theory under study. Two parameters, KV and K2,are denoted "PM parameters" because they refer to that part of dE/dt thatcorresponds to the Peters-Mathews (1963) result for general relativity. Aparameter KD refers to the dipole self-gravitational contribution, where, atleast schematically, we may write

{dE/dt)dipole = - iK f l< D � D> (10.82)

where D is the dipole moment of the self-gravitational binding energy Gla

of the bodies

D = £ Qaxo (10.83)a

(Within each specific theory of gravity the details are more complicatedthan this, however.) For a binary system, the result is

f = - (~£- iUw2 - K2f2) + i*fl®

2]) (10-84)

where <3 is the difference in the self-gravitational binding energy per unitmass between the two bodies.

In this section, we shall derive these results using a post-Newtoniangravitational radiation formalism developed by Epstein and Wagoner(1975) and Wagoner and Will (1976). However, because of the complexityof many alternative theories of gravitation beyond the post-Newtonian

approximation, it has proven impossible to devise a general formalismanalogous to the PPN framework, beyond writing Equation (10.84) witharbitrary parameters. But, we can provide a general description of themethod used to arrive at Equation (10.84) within a chosen theory of grav-ity, emphasizing those features that are common to many currently viabletheories. Later, we shall describe the specific computations within selectedtheories. The method proceeds as follows:

Step 1: Select a theory. Restrict the adjustable constants and cosmol-ogical matching parameters to give close agreement with solar systemtests (Chapters 7, 8, and 9).

Step 2: Derive the "reduced field equations." Working in the universerest frame, expand the gravitational fields about their asymptotic values,and, using any gauge freedom available, express the field equations in the"reduced" form

(�v~2d2/dt2 + V2) [terms linear in perturbations of fields]

= - ten [source] (10.85)

where vg, the gravitational-wave speed, is a function of adjustible con-stants and matching parameters, and where the "source" consists of matterand nongravitational field stress energies, and of "gravitational" stressenergies consisting of terms quadratic and higher in gravitational-fieldperturbations. If we denote the linear term by xjt (it can be a tensor of anyrank) and the source by x, then the solution of Equation (10.85) that hasoutward propagating disturbances at infinity is

ij/(x,t) = 4 J\(r - v;x\x - x'|,x')|x - x'l"1 d3x' (10.86)

For field points far from the source (R = |x| » r = "size" of source, \i//\ «1), we have

\j/(x,t)=4R~l jxit-v^R + v^t � x',x')d3x' + O(r/R)2 (10.87)

where n = x/R. If we assume that the motions of the source are sufficientlyslow (source within wave zone, r < k/2n = wavelength/27t « R), thenEquation (10.87) may be expanded in the form

\x(t-v-lR,x'){n-x')md2x' (10.88)m=O

For further use, we note that

f; = - V9 %^,o + O(r/R2) (10.89)


Step 3: Determine the energy loss rate in terms of \jt. Let us restrictattention to Lagrangian-based theories of gravity (such as the currentlyviable theories described in Chapter 5). Such theories possess conservationlaws of the form (see Section 4.4)

©?vv = 0 (10.90)

where ©*" reduces in flat spacetime to the stress-energy tensor for matterT"v. Hence, we can define quantities P" that are conserved for a localizedsource, except for a possible flux 0" j of energy-momentum far from thesource: when integration is performed over a constant-time hypersurface,we have

P" = J0«° d3x, dP"/dt = J0fo° d3x = - Js 0W' d2Sj (10.91)

where S is a closed two-dimensional surface surrounding the region ofintegration. For each theory, it turns out that 0'"' may be written

0"v = f(il/)T"v + t"v (10.92)

where /(if/) -> 1 as \j/ -* 0, and t"v is an expression at least quadratic in thefirst-order perturbations (i/0 of the gravitational fields. If we choose forS a sphere of radius R in the wave zone far from the source, we have for

dP°/dt = -R2j> tOinj dQ (10.93)

Substituting Equation (10.88) into the expression for t0J provided by theequations of the theory yields an expression for dP°/dt in terms of timederivatives i >0 of the gravitational fields, evaluated in the far zone.

Step 4: Make a post-Newtonian expansion of the "source" x (seeChapter 4 for discussion). For this purpose, use the near-zone, post-Newtonian forms for matter variables and gravitational fields obtainedin Chapter 5 in the solution for the post-Newtonian metric, appropriatelytransformed to the gauge adopted in Step 2. Depending on the nature of^i, the sources x are of even ("electric") order in the post-Newtonian sense[O(0),O(2),...] or odd ("magnetic") order [O(l),O(3),.. .]� For electricsources, x typically contains terms of the form

lelectric ~ P,pn,pv2,pU,p (10.94)

modulo total divergences whose moments [monopole, dipole, etc.; seeEquation (10.88)] can be shown to be negligible upon integration by

parts [see Epstein and Wagoner (1975) for discussion]. For magneticsources, x typically has the form (modulo divergences)

^magnetic ~ M P&U P^U, pvh2, PVJ, PV\ PWj (10.95)

Step 5: Simplify i// using integral conservation laws. Because \jj,Equation (10.88), involves time derivatives of integrated moments of thesource T, and since time derivatives of ij/ will ultimately be used, it is con-venient to employ the integral conservation laws obtained from Equa-tion (10.90) to extract from the integrated moments terms that are constantin time, linear in time, etc. Some of these terms reflect the imprints of themass, momentum, angular momentum, and center of mass of the sourceon the far-zone field, and do not contribute to gravitational radiation.Since these integral conservation laws are to be applied only to near-zoneintegrals, we neglect surface integrals such as the one in Equation (10.91)(retaining them would only yield higher-order corrections to the energyloss-rate formula). These integral conservation laws give the followinguseful results (valid in the near zone)

(d/dt)

(d2/dt2) §®°°xJxkd3x=2

(d/dt) J0'o(fi � x)d3x = j®Jknkd3x (10.96)

Notice that, because we are dealing with semiconservative theories ofgravity, 0"v is not necessarily symmetric, so we have retained the con-tributions of the antisymmetric parts of 0"v where necessary. However,as we saw in Section 4.4, these terms depend upon the PPN parameters<*! and a2 and so they will be small if we impose the experimental con-straints on aj and a2 discussed in Chapter 8, or will be zero if we adopta version of the theory with a t = a2 = 0 (i.e., a fully conservative version).

Step 6: Apply to binary systems. We consider a system made up oftwo bodies that are small compared to their separations (d « r); that is,we ignore all tidal interactions between them. We may thus treat eachbody's structure as static and spherical in its own rest frame. We thenfollow the procedure of Section 6.2: for a given element of matter in bodya, we write

v = vfl [static structure], x = Xfl + x (10.97)

where

Xa = m-1 £ p*(l + II - if/)xd3x, ya = dXJdt,

ma = P°a= f p*(l + n-^O)d3x,

0 = Jo p*(x', t)|x - x' |" ' d3x' (10.98)

We note that ma is conserved to post-Newtonian order, as long as tidalforces are neglected. The full Newtonian potential U for spherical bodiesis given by

U(x, t)= Ua+ £ mb\x - Xb\ ~J [x inside body a]

b*a

= X mb\x � Xj.1"1 [x outside body a] (10.99)b

Then the total energy of the system P° [cf. Equation (4.108)] is given by

( K U 0 0 )

where rafc = |Xa � Xb|. For a binary system we may evaluate the orbitalterms in Equation (10.100) to the required order using Keplerian equa-tions (see Section 7.3 for definitions of orbital elements);

r = rab = a(l - e2)(l + e cos <^)"r (10.101)

The result is

° (10.102)

where m = ma + mb, n = mamb/m, and where the semi-major axis a isrelated to the orbital period P to the necessary order by (P/2n)2 � a3/m.In the emission of gravitational radiation whose source is the orbitalmotion, the quantities ma and mb will be unchanged because of our neglectof tidal forces and internal motions. Invoking energy balance, we thus have

(dE/dt)Tadiation = dP°/dt (10.103)

We now use the above procedure to split the moments oft that determine\ji into orbital parts (v2 ~ m/r) and "self" parts associated with each body(£7 ~ II ~ p/p ~ m/d » m/r). In terms of the quantities m/r and m/d, wefind that electric ij/ fields have the schematic form [Equations (10.88),


(10.94), (10.96)]

^eiectnc ~ 4(m/R) | [constant]

[ml r /mVl [mm]r , J J

~ + I ~ I + ~~ 3 Lm o n oP°' e anc* quadrupole]

-r) r b i v J[dipole]

(10.104)

and magnetic \jt fields have the form [Equations (10.88), (10.95), (10.96)]

� magnetic ~ 4(m/R) \ [constant]

+ � � � j (10.105)

Because the energy flux tOi [Equation (10.93)] typically depends on (i/'.o)2*the constant terms in Equations (10.104) and (10.105) do not contributeto the radiation. In Equations (10.104) and (10.105) it is the (m/r) termthat yields the PM contribution, since t/^0 ~ (m/R)2(m/r2)2v2. The termsof O(m/r)2 and O(m/r)3'2 in Equations (10.104) and (10.105) are post-Newtonian corrections of a kind discussed by Wagoner and Will (1976)for general relativity. The terms of O[(w/r)(m/d)] effectively renormalizethe masses that appear in the PM result by corrections of O{m/d).The terms of O[(m/d)(m/r)1/2] produce the dipole radiation of interest:(fo)2 ~ (m/R)2(m/d)2{mll2/r3l2)2v2 ~ (m/R)2(m/r2)2{rn/d)2. Cross termsproduce effects that are down from these by powers of (d/r) or that vanishon integration over solid angle [Equation (10.93)]. Hence, we retain onlyterms in \jj of order (m/r) and (m/d)(m/r)1/2.

In evaluating the "self" terms, we employ the standard virial theoremfor static spherically symmetric bodies:

3 japd3x + na = 0 (10.106)


where to the necessary order

Qfl= -^ap0d3x= -^ap{x)p{x')\x-x'\-1d3xd3x' (10.107)

Step 7: Calculate the average energy loss over one orbit, using New-tonian equations of motion to simplify the Newtonian PM contributionand the post-Newtonian dipole contribution to the radiation.

To illustrate this method, we shall now focus on three metric theories:general relativity, scalar-tensor theories, and Rosen's bimetric theory.For other theories, such as the BSLL theory and Ni's theory, see Will(1977).

General relativityBy defining

0"v = /i"v - \rf"h (10.108)

and choosing a gauge ("Lorentz" gauge) in which

0?vv = 0 (10.109)

where indices are raised and lowered using the Minkowski metric, onecan show that Einstein's equations are equivalent to the reduced fieldequations

� I | 0 " v = - 16TCT"V (10.110)

where

T"V = T"v + t"v (10.111)

with t"v a function of quadratic and higher order in 0^v and its derivatives.Because of the gauge condition, Equation (10.109), T"V satisfies

T VV = 0 (10.112)

Then

e»» = 4R-1 f; (l/ml)(d/dt)m {^(t-R, x')(n � x')md3x' (10.113)m=0 J

Because of the gauge condition, Equation (10.109), and the retardednature of 0"v, we need to determine only the 01J components, since


Now, because the source T"V for 6"v satisfies its own conservation lawtfv

v = 0, and is symmetric, we may make use of Equations (10.96), withTMV in place of 0*v to show that

6iJ = 2R-\d2ldt2)\ xoo(t - R, x)xixid3x

+ [quadrupole moments of xOj, TJ*] + � � �!� (10.115)

Notice that the monopole and dipole moments of zij have been re-expressed as second time derivatives of quadrupole moments. Since eachtime derivative (8/dt)x ~ v ~ (m/r)112, there can be no "dipole" contri-bution to 0** of the form (m/d)(m/r)112. Thus, the only contribution to thefield to the required order comes from the lowest order, "Newtonian"part of T00, namely

t00 = p [ i + O(2)] (10.116)

For a binary system, Equation (10.115) becomes

9iJ = 2R ~ \d2ldt2) X max\xi + O(m/r)3'2 (10.117)a

The conservation laws for T"V also imply that the center of mass of thesystem is unaccelerated, so, decomposing xa into center-of-mass andrelative coordinates to Newtonian order, using

X = m~i(mtx1 + m2x2), x = x2 � x t (10.118)

we obtain, modulo a constant,

6iJ = (2n/R)(d2/dt2){xixj) + O(m/r)312 (10.119)

Now, to determine the energy-loss rate, it is most convenient to usefor 0'"' the conserved quantity

= ( - g)(T»v + til), 0fL,v = 0 (10.120)

where ££L is the Landau-Lifshitz pseudotensor, given for example byMTW, Equation (20.22). (Actually, we could equally well have used T*"1

for this purpose, since one can show that both quantities yield identicalequations of motion for matter and identical integral conservation-lawresults as, for example, in Equation (10.91). The Landau-Lifshitz versionis simpler because t£L contains only first derivatives of O1"1.) Evaluatingt££ for use in Equation (10.93), using Equation (10.114), and defining the

transverse traceless (TT) part of 6lJ by

%PiJpklekl, p j = <s< - n% (10.121)

we obtain the energy-loss rate

dP°/dt = -(R2/32n) (Jj fl¥r,o0Tr,o da (10.122)

Substituting Equation (10.119) into Equation (10.122), performing theangular integrations, and averaging over several oscillations of the sourceyields

dp°/dt = -*<*;/«>. hj = I*(X,XJ - i v 2 ) (10-123)

Using the Newtonian equations of motion, d\/dt = � mx/r3, to evaluatethe time derivatives in Equation (10.123) to the required accuracy yieldsthe Peters-Mathews formula, Equation (10.80). Thus, for general rela-tivity Kt = 12, K2 = 11, KD = 0.

Scalar-tensor theoriesBy defining

f (10.124)

(see Section 5.1) and choosing a gauge in which

0?vv = 0 (10.125)

we can write the field equations for scalar-tensor theories in the form

� , 0 " v = - 16TE t"v, ��<? = - 16TI S (10.126)

where

S = - (6 + 4w)-1r[l - $0 - W^o - 2c»'(31["v + ^ " V y - a»'(3 + 2co)->�?�"]W , < ? ) 3 ) (10.127)

where co = a>(</>0), cu' = dco/d^l^, T = g^T"*, and indices on 0"v and<pf|1 are raised and lowered using i\. The quantity t"v is a function of qua-dratic and higher order in 0"v and q>. Now, because of the conservationlaw satisfied by �z'"', it is clear that 6Jk can be reexpressed as second timederivatives of quadrupole moments of x00, as in general relativity, andthus will not contribute any dipole terms. However, the source, S, does

lead to dipole terms, as follows. We first evaluate the post-Newtonianforms of 0"v and q> in the near zone. From the post-Newtonian limit ascalculated in Section 5.3, for instance, we obtain

600 = 2(1 + y)U + O(4), 60J = 2(1 + y)VJ + O(5),

0° = O(4), q> = (1 - y)4>0U + O(4) (10.128)

where y is the PPN parameter, given by

y = (1 + o>)/(2 + co), (10.129)

and where we have used Equation (5.38) to convert to geometrized units.Equations (10.127) and (10.128) then yield, to the necessary order

(10.130)

To simplify the source, S, and its moments, we use the post-Newtonianforms for conserved quantities in the near zone as given in Equation(4.107). Then, for a system containing compact objects, the general pro-cedure described above yields, for the far zone to the required accuracy,

6iJ = (1 + y)R-\d2ldt2) £ rnXxi + O(m/r)3'2,

n � P + 2[1 + 2ca'(3

- [1 + 4co'(3<ab

2o,'(3 + 2a,)- 2

+ O(m/r)312 (10.131)

For a binary orbit we obtain (modulo constants),

diJ = 2(1 + y)(fj,/R)(v'vj - mx'xj/r3)

cp=-(l- y)4>0{nlR){v2 - (n � v)2 + [1 + 4a.'(3 + 2©)"2~\m/r

+ m(a � x)2/r3 + 2[1 + 2co'(3 + 2a>)"2]G(n � v)} (10.132)

where S is given by

S = Q i M - Q2/m2 (10.133)

The most useful conserved quantity appropriate for determining theenergy flux is given by

0"* = ( - g # 0o \T>" + t£D (10.134)

where tfx is the scalar-tensor theory analogue of the Landau-Lifshitzpseudotensor, as given by Nutku (1969b) [for alternative conservedquantities, see Lee (1974)]. Evaluating t°{ a n d using Equations (10.114)and (10.121), we obtain

dP°/dt = -(R2/32n)(t>o <j> [^'T.O^TT.O + (4© + 6#o V.oP.o] <« (10.135)

Substituting Equation (10.132) into Equation (10.135) and integratingover solid angle yields Equation (10.84) with

Kl = 12 - 5/(2 + co), K2 = 11 - 45(1 + fa + ia2)/(8 + 4co),

KD = 2(1 + a)2/(2 + co) (10.136)

where2 co) (10.137)

Rosen's bimetric theoryFor simplicity, we choose the version of the bimetric theory

whose post-Newtonian limit is identical to that of general relativity, thatis, we choose c0 = c t (see Section 5.5). This is equivalent to assumingthat, far from the local system, both g and i; have the asymptotic formdiag(� 1,1,1,1). Our final results will then be valid up to corrections ofO(l � co/ci), which, according to Earth-tide measurements (limits ona2), must be small.

We then define

(10.138)

and write the field equations, (5.68), in the form

(10.139)

The post-Newtonian forms for 6"v in the near zone are (see Section 5.5)

00° = 41/ + Q(4), 6Oi = 4VJ, 0'-> = O(4) (10.140)

To the necessary order, Equation (10.139) then yields

T0 0 = p(l + n + v2 + 2U), T'J' = pvlv> + p5iJ,

tOj = p{v\\ + n + v2 + AU + pip) - 2V]~] (10.141)

The conserved quantity associated with the bimetric theory is

The near-zone conserved quantities 0 0 0 and 0 O j can be determined fromEquations (10.140) and (10.142) or taken directly from Equation (4.107),since we are using the fully conservative version of the theory. For asystem of compact objects, we then obtain

000 = 4R-1 \P° + n � P + X a + X mJ(n-v a)2

O(m/r)3'2,

+ E m.»i»i - | X Qa(n � v ja 3 a

+ O{m/r)312 (10.143)

For a binary orbit, we obtain (ignoring constant terms)

600 = 4(n/R){[ih � v)2 - m/r - m(n � x)2/r3] - <Sn � v},

601 = 4(|i/J?){[uJ'(fi � v) - mxJ(ii � x)/r3] - | S u J } ,

0iJ' = 4(/i/i?)[i;^-i + i®(B � v)5y] (10.144)

We now evaluate the energy flux tOj in the far zone using Equations(10.89), (10.138), and (10.142) and obtain

dp°/dt = - (R2/32TI) <j) {efte^o - i e oeiO) rfn (10.145)

Substituting Equation (10.144) into (10.145) and integrating over solidangle yields Equation (10.84), with

*i = - ¥ , *2= - ¥ , * D = - ¥ (10.146)

Other theoriesCalculation of the PM and dipole parameters within this for-

malism has been carried through for the BSLL theory and for Ni'sstratified theory (Chapter 5), restricting attention to those versions whosepost-Newtonian limits are identical to that of general relativity (see Will,1977, for details). The results are shown in Table 10.2.

We note the surprising result that, for all the theories listed in Table10.2, except scalar-tensor theories and general relativity, the dipoleradiation carries negative energy, i.e., increases the energy of the system(KD < 0), and that the PM radiation may carry either positive or negativeenergy, depending on the theory and on the nature of the orbit. It couldbe argued (and presumably will be argued by some) that this predictionalone should be sufficient grounds to judge each such theory unviable.However, this is a theorist's constraint that has little experimental foun-dation in the case of gravitation, and so we will restrict attention toobservational evidence for or against such an effect. Such evidence willbe provided by the binary pulsar (Chapter 12).

The only theory shown in Table 10.2 that automatically predicts nodipole radiation is general relativity. Scalar-tensor theories can alsoavoid dipole radiation for particular choices of the function co(</>). Forexample, if «(<£) = (4 - 3 )/(2<£ - 2) (Barker's constant G Theory), then1 + 2o>'(3 + 2a>)~2 = 0 = KD. In this case, to post-Newtonian order, thetheory satisfies the strong equivalence principle (Section 3.3); the locallymeasured gravitational constant GL is truly constant, and the theorypredicts no Nordtvedt effect (4/? � 7 � 3 = 0). The other theories inTable 10.2 violate SEP.

This suggests the general conjecture that a theory of gravity predictsno dipole gravitational radiation if and only if it satisfies SEP to theappropriate order of approximation. In Section 11.3, we shall see moredirectly how the violation of SEP can manifest itself in dipole gravitationalradiation.

It is also interesting to note the strong correlation between the sign ofthe energy carried by gravitational radiation and the E(2) class of thetheory, as summarized in Table 10.1. General relativity and scalar-tensortheories predict waves of the least general E(2) classes (N2 and N3), ofdefinite helicity (±2; ±2, 0), and of positive energy. The other theories

Table 10.2. Multipole gravitational radiation parameters in metric theories of gravity

PM parameters

Theory

Dipoleparameter Satisfies

SEP?Sign ofenergy

Definitehelicity?

General relativity

Scalar-tensor:

BWN, Bekenstein 12

12

2 + ca

5

2 + coBrans-Dicke 12

Vector tensor

Bimetric6:Rosen -21/2RastallBSLL -21/2

Stratified":Lee-Lightman-Ni �18

11 -

11

45

1145

-23/2a

73/8

�19

Yes

2

2 + coa

-20/3a

-125/3

-400/3

No

No

No

NoNoNo

No

Yes

Yes

Yes

No

NoNoNo

No

" Calculations have not been performed to determine these values.6 We adopt that version of each theory whose PPN parameters are identical to those of general relativity.


in Table 10.2 predict waves of more general classes, of indefinite helicity,and of negative or positive energy. It is perhaps not surprising that suchtheories predict indefinite sign for the emitted energy, since - accordingto quantum field theory - definite helicity, quantizibility, and positive def-initeness of energy go hand in hand. Whether or not a general conjecturealong these lines can be proved is an open question.

One of the drawbacks of the post-Newtonian method for derivingformulae for energy loss is that it assumes that gravitational fields areweak everywhere. This assumption is no longer valid in systems containingcompact objects (neutron stars or black holes), such as the binary pulsar.In the next chapter we shall describe a formalism that retains the essentialpost-Newtonian features of the orbital motion of such systems but thatpermits one to take into account the highly relativistic nature of anycompact objects in the system. Nevertheless, the basic conclusionssummarized in Table 10.2 will be unchanged.

11

Structure and Motion of Compact Objects inAlternative Theories of Gravity

Within general relativity, the structure and motion of relativistic, con-densed objects-neutron stars and black holes-are subjects that haveattracted enormous interest in the past two decades. The discovery ofpulsars in 1967, and of the x-ray source Cygnus XI in 1971, have turnedthese "theoretical fantasies" into potentially viable denizens of theastrophysical zoo. However, relatively little attention has been paid tothe study of these objects within alternative metric theories of gravity.There are two reasons for this. First, as potential testing grounds fortheories of gravitation, the observations of neutron stars and blackholes are generally thought to be weak, because of the large uncertaintiesin the nongravitational physics that is inextricably intertwined with thegravitational effects in the structure and interactions of such bodies.Examples are uncertainties in the equation of state for matter at neutron-star densities, and uncertainties in the detailed mechanisms for x-rayemission from the neighborhood of black holes. Second, compared withthe simplicity of the post-Newtonian limits of alternative theories andthe consequent availability of a PPN formalism, the equations for neutron-star structure and black hole structure are so complex in many theories,and so different from theory to theory, that no systematic study has beenpossible.

Neutron stars were first suggested as theoretical possibilities withingeneral relativity in the 1930s (Baade and Zwicky, 1934). They are highlycondensed stars where gravitational forces are sufficiently strong to crushatomic electrons together with the nuclear protons to form neutrons,raise the density of matter above nuclear density (p ~ 3 x 1014 g cm"3),and cause the neutrons to be quantum-mechanically degenerate. A typicalneutron-star model has m ^ lm0 , R =* 10 km.


However, they remained just theoretical possibilities until the discoveryof pulsars in 1967 and their subsequent interpretation as rotating neutronstars. Since that time much effort has been directed toward calculatingdetailed neutron-star models within general relativity, with particularinterest in masses, moments of inertia, and internal structure. Thesequantities are important in understanding both steady changes anddiscontinuous jumps ("glitches") in the observed periods of pulses frompulsars. The principle uncertainty in these computations is the equationof state of matter above nuclear density (for a review see Baym andPethick, 1979).

In a certain sense, black hole theory has a longer history than neutron-star theory, as it dates back to a 1798 suggestion by Laplace that suchobjects might exist in Newtonian gravitation theory (see Hawking andEllis, 1973, Appendix A). Within general relativity, two key events in thehistory of black holes were the discovery of the Schwarzschild metric(Schwarzschild, 1916) and the analysis of gravitational collapse acrossthe Schwarzschild horizon (Oppenheimer and Snyder, 1939). However,theoretical black hole physics really came into its own with the discoveryin 1963 of the Kerr metric (Kerr, 1963), now known to be the uniquesolution for a stationary, vacuum, and rotating black hole (with theSchwarzschild metric being the special case corresponding to no rotation).It was the discovery in 1971 of the rapid variations of the x-ray sourceCygnus XI by telescopes aboard the UHURU satellite that took blackholes out of the realm of pure theory. The source of x-rays was observedto be in a binary system with the companion star HDE 226868; analysisof the nature of the companion and of its orbit around the x-ray source,and detailed study of the x-rays, led to the conclusion that the unseenbody was a compact object (white dwarf, neutron star, or black hole) witha mass exceeding 9m© (Bahcall, 1978). Since the maximum masses ofwhite dwarfs and neutron stars are believed to be approximately 1.4m©and 4m©, respectively, the simplest conclusion was that the object wasa black hole. The source of the x rays was believed to be the hot, innerregions of an accretion disk around the black hole, formed by gas strippedfrom the atmosphere of the companion star. Since 1971, other potentialblack hole candidates in x-ray binary systems have been found, andstudies of the central regions of some galaxies and globular clusters haveindicated the possible existence of supermassive black holes [see Blandfordand Thorne (1979) for a review]. However, a crucial link in the chain ofargument that leads to the black hole conclusion for Cygnus XI is thatthe maximum mass of a neutron star is less than 4m©. There are three

Structure and Motion of Compact Objects 257

possible sources of uncertainty in this limit (the maximum mass of whitedwarfs is much more certain). The first is the equation of state. However,it has been possible to obtain bounds on the maximum mass of between3 and 5mQ using arguments that are independent of the details of thehigh-density equation of state (Hartle, 1978). The second is rotation.However, most analyses indicate that rotation cannot increase themaximum mass beyond about 20%. The third is the theory of gravitation.Although alternative theories of gravitation may have post-Newtonianlimits close to that of general relativity, their predictions for the highlynonlinear, strong-field regime of neutron-star structure may differmarkedly from those of general relativity. Indeed, some theories predictno maximum mass for neutron stars. Since the only present evidence forblack holes crucially depends upon the maximum-mass argument, theseresults within alternative theories are used by many authors as reasonsfor caution in making the black hole interpretation, rather than as testsof competing theories. As we shall see, some alternative theories do noteven predict black holes.

However, the discovery of the binary pulsar (Chapter 12) has made thestudy of neutron-star structure and motion an important tool for testinggravitation theory. The precise orbital data obtained for that systempermits for the first time the direct measurement of the mass of a neutronstar and the study of relativistic orbital effects (such as periastron shifts)in systems containing condensed objects. In alternative theories of gravity,the nonlinear gravitational effects involved in the neutron star can makesignificant differences in many relativistic effects, even though in the post-Newtonian limit, these effects would have been the same as in generalrelativity. Crucial tests of competing theories may then be possible.Discussion of these tests will be presented in Chapter 12; this chaptersets the framework for that discussion. In Section 11.1, we analyze theequations of neutron-star structure and, in Section 11.2, the equationsof black hole structure in alternative theories of gravitation. In Section 11.3we present a framework for discussing the motion of compact objects,such as neutron stars, in competing theories. As we noted above, verylittle systematic study of these issues has ever been carried out, so weshall merely present a few relevant examples.

11.1 Structure of Neutron StarsIn Newtonian gravitation theory, the equations of stellar structure

for a static, spherically symmetric star composed of matter at zero tem-perature (T � 0 is an adequate approximation for neutron-star matter)

are given by

dp/dr = p dU/dr [Hydrostatic equilibrium],

P = P(P) [Equation of state],(d/dr)(r2dU/dr)= -4nr2p [Field equation] (11.1)

where p(r) is the pressure, p{r) the density, and U(r) the Newtoniangravitational potential.

In any metric theory of gravity, it is simple to write down the equationscorresponding to the first two of these three equations, because they followfrom the Einstein Equivalence Principle (Chapter 2), which states thatin local freely falling frames the nongravitational laws of physics are thoseof special relativity, Tfv

v = 0, and p = p(p). Thus, we have in any basis,

Tfvv = 0, p = p(p) (11.2)

For a perfect fluid,

T"v = {p + p)«"uv + pg"" (11.3)

where we have lumped the internal energy pTl into p [compare Equation(3.71)].

It is useful to rewrite these equations in a form that parallels the firsttwo parts of Equation (11.1). For a static, spherically symmetric spacetime,there exists a coordinate system in which the metric has the form

ds2 = -e20ir)dt2 - TV{r)drdt + e2Mr)dr2

+ e2mr\dQ2 + sin2 0 d(f>2) (11.4)

For theories of gravity with a preferred frame, this coordinate systemmust be at rest in that frame. There still exists the freedom to change thet coordinate by the transformation

t = t'-f(r) (11.5)

where f(r) can be chosen to eliminate the off-diagonal term in the metric,namely

f(r) = J ' m(r)e ~ 2o<r) dr (11.6)

There is the further freedom to change the coordinate r by

r = 9(r') (11.7)

If the radial coordinate is chosen so that fi(r) = 0, the coordinates arecalled "curvature coordinates;" in such coordinates, 2nr measures theproper circumference of circles of constant r. In general relativity, theyare known as Schwarzschild coordinates. If the radial coordinate ischosen so that n(r) = A(r), the coordinates are called "isotropic coordi-nates."

However, in two-tensor theories of gravity, such as those with a back-ground flat metric q, these transformations are usually best carried outafter the solution to the field equations has been obtained. The reason isthat the above transformations will in general make the second tensor acomplicated nondiagonal function of r, which may result in worse com-plications in the field equations than those introduced by starting withthe general nondiagonal physical metric, Equation (11.4). For example,if the second tensor field is t\, the field equations may take their simplestform in coordinates in which

ij = diag(-l,l,r2,r2sin20) (11.8)

In such a coordinate system there is no freedom to alter <J>, A, T, or fx apriori.

Now, for hydrostatic equilibrium, the equations of motion Tfvv = 0

may be written in the form

where j runs over r, 6, (p. For spherical symmetry only the j = r compo-nent is nontrivial, and, using the fact that u = (e~0(r), 0,0,0), we obtain

dp/dr = �(p + p)d<b/dr,

P = p{p) (11.10)

Notice that in the Newtonian limit, p « p, $ =s �U and we recover thefirst two parts of Equation (11.1). Equation (11.10) is valid independentlyof the theory of gravity. The field equations for <J>, A, *F, and fi will dependupon the theory. In constructing a stellar model, boundary conditionsmust be imposed. These are

dp/dr\r=0 = 0,

p{R) = 0, R = [stellar surface],

e2A(r) � e2M ,. C i ( 1 L U )


The first of these conditions is a continuity condition for the matter, thesecond defines the stellar surface and its radius R. The remaining fourare asymptotic boundary conditions on the metric functions [see Equa-tion (5.6)]. They guarantee that in asymptotically Lorentz coordinates,and in geometrized units (Gtoda}, = 1),

0oo -» - 1 + 2m/r, gtJ - r\i} (11.12)

and thus that the Kepler-measured mass of the star will be m.Unfortunately, this exhausts the common features of the equations of

relativistic stellar structure, so we must now turn to specific theories.

General relativityIn curvature coordinates [*F(r) = /x(r) = 0], the field Equation

(5.14) takes the form

d/dr[r(l - e~2A)] = 8nr2p (11-13)

with the solution

e2A = ( l - 2 r n ( r ) / r r 1 (11.14)

where

, . . r r 2 dm 2 . . . . . .m(r) = 4JI t par, or � = 4nr p (11.15)

Jo dr

and

dO m + 4nr3p

dr r(r � 2m)(11.16)

This equation together with Equations (11.10), (11.14), and (11.15), andthe boundary conditions, Equation (11.11) (with c0 = c1 = 1), are sufficientto calculate neutron-star models, given an equation ofstate. These equa-tions are called the TOV (Tolman, Oppenheimer, and Volkoff) equationsfor hydrostatic equilibrium. They form the foundation for the studyof relativistic stellar structure within general relativity. For reviews ofneutron-star structure, see Baym and Pethick (1979), Arnett and Bowers(1977), and Hartle (1978).

Scalar-tensor theoriesUsing curvature coordinates [*P(r) = ft(r) = 0] and defining

e2A = [1 - 2 m ( r ) / r ] - 1 (11.17)

we can put the field equations for scalar-tensor theories, Equations (5.31)and (5.33), into the form

2m

drr(r- 2m)(l

1.18)

[Note that the equations quoted in Rees, Ruffini, and Wheeler (1975,p. 13) are in error.] The present value of G is related to the asymptoticvalue of <j>, by [see Equation (5.38)]

l (11.19)

For the special cases of Brans-Dicke theory (co = constant) and theVariable-Mass Theory [co(</>) satisfies Equation (5.40)], it has been shownthat for values of co consistent with solar system experiments (i.e., a> ;> 500),all features of neutron-star structure differ from those predicted by generalrelativity only by relative corrections of O(l/co) (see Salmona, 1967;Hillebrandt and Heintzmann, 1974; Bekenstein and Meisels, 1978).

Rosen's bimetric theory

In coordinates in which the flat background metric has the form

t, = diag( - Co \ el \ cl V2, ci V2 sin2 9) (11.20)

the field equations take the form

V 2 $ + |D~1e""2"2A|VxP - 2»PV|2 \

= 47tG(c0c1)1/2e+A+2''D1/2(p +

V2A + \D-le-2*-2A\\y¥ - 24*VA|2

1 / 2 * + A 2 1 / 2 - p + 2(p

= -4nG(c0c1)ii2e">+A+2>lD1i2(p - p),

Y - 2*PVA)

De2A~2" = 0

where

D = 1 + *p2e-2<I-2A ,

V = tr(d/dr),

V2 = r ~ 2{d/dr)r2(d/dr) (11.22)

Here, we see an example of the loss of freedom to vary the metric func-tions *P and pi a priori. Thus, for example, the substitutions *F = 0, ju s 0(Schwarzschild form of the metric) do not lead to a solution of the equa-tions for general matter distributions. However, the metric function *Palone is freely specifiable, for the following reason. If, instead of Equation(11.20) for IJ, we had chosen the equally valid flat metric whose line ele-ment is

dsLt = - c o ' [ A + f(r)drf + c^[dr2 + r2(dd2 + s in20#2)]

where /(r) is an arbitrary function of r, then had changed coordinates toput this metric into the form of Equation (11.20), the result would be tochange the function ¥ in g^ by an arbitrary amount. Thus, for example,we are free to choose *P = 0. This is consistent with the fact that *P = 0is a solution of the fourth field equation, (11.21). The free choice of *F ispart of the absolute, prior-geometric character of i\, and represents thefreedom to "tip" the null cones of i; relative to those of g. Different choicesof *F lead to physically different spacetimes (this point has been overlookedby most authors).

The simple choice T s O leads to the field equations

V20> = 47tG(coc1)1/2e*+3A(p + 3p),

V2A = -47tG(coc1)1/2e*+3A(p - p),

H = A (11.23)

Henceforth, we shall adopt the choice T s O . The boundary conditionson and A are given by

$ � 0, A � 0 (11.24)

Notice that the matching of the tensors i\ and g to the external worldinfluences the structure of the star (violation of SEP) via the effectivegravitational constant G^QCJ)1 '2 . We now recast the field equations intothe form

dfb/dr = Gom*(r)/r2,

dA/dr=-GomA(r)/r2 (11.25)

where

Go= (c0Ci)1/2G = 1 [geometrized units],

= An £ e*+3A(p - p)r2 dr (11.26)

Outside the star, r > R,

<D = _ MJr, A = MJr (11.27)

where M^ = m^R), MA = mA(R). In quasi-Cartesian coordinates, theexterior metric then has the form

ds2 = - exp( - 2MJr) dt2 + exp(2MJr)(dx2 + dy2 + dz2) (11.28)

A variety of numerical integrations of the field equations, (11.25), andthe hydrostatic equilibrium equations, (11.10), have been carried out usingvarious equations of state (see Rosen and Rosen, 1975; Caporaso andBrecher, 1977; and Will and Eardley, 1977). Generally, neutron stars withKepler-measured masses M& much larger than those permitted by generalrelativity are possible, with maximum masses ranging from ~8mo forsoft equations of state, to ~80mo for equations of state of the formp = p - p0 for p > p0 ~ 1014 g cm"3.

NVs stratified theoryIn coordinates in which

i/ = diag(-l,l,r2,r2sin20), (11.29)

we have [see Section 5.6(g)-(i)]

e 2* = / i(*X e2A = e2" = f2(<t>),

V=-Kr, X9 = K0 = O (11.30)

The field equation for Kr is

V2Kr - r-2Kr = -47te(/2//1D)1/2Xrp (11.31)

where D = 1 + K2(f1f2)~1. One immediate solution of this equation is

Kr = 0 (no tipping of null cones). Then, the field equation for (f> is given by

V20 = 27t(/1/3)1'2[p(/'1//1) - 3p(/'2//2)] (11.32)

where f\ = dfjd<f>, f2 = df2/d<j>. The boundary condition on Tzz^t 0. Outside the star, <p is given by

<p(r)= -MJr (11.33)

where

M, = 2n J* (fJiy'WJA) - 3p(/'2//2)]r2 dr (11.34)

Asymptotically, the functions fx and /2 are assumed to have the forms

M4) = c0 - 2c<\> + O(4>2),

/2(0) = c, + O(0) (11.35)

In coordinates in which g^ is asymptotically Minkowskian, g00 thenhas the form

goo=-l+2GoMJr (11.36)

where

Go = C2C\I2CQ 3/2 = 1 [geometrized units],

Mikkelson (1977) has numerically integrated these equations using reason-able equations of state, after first assuming a specific form for the func-tions fi(<j>) and f2(4>), designed to yield agreement with general relativityin the post-Newtonian limit. These forms were

with a being an adjustable parameter (note co = ct = c = 1). For a = 1,the maximum Kepler-measured mass was ~1.4mo; for a = 64, it was~840mo; and in the limit a-* oo, the maximum mass was unbounded.The stiffer the equation of state used, the larger the maximum masses.

Thus, neutron-star models in alternative theories of gravity can bevery different from their counterparts in general relativity, the knownexceptions to this rule being scalar-tensor theories. In particular, themaximum mass of a neutron star may be orders of magnitude larger thanthat in general relativity.

11.2 The Structure and Existence of Black HolesGeneral relativity predicts the existence of black holes. Black

holes are the end products of catastrophic gravitational collapse in whichthe collapsing matter crosses an event horizon, a surface whose radiusdepends upon the mass of matter that has fallen across it, and which is aone-way membrane for timelike or null world lines. Such world lines can

cross the horizon moving inward but not outward. The interior of theblack hole is causally disconnected from the exterior spacetime. There isnow considerable evidence to support the claim that any gravitationalcollapse situation, whether spherically symmetric or not, with zero netcharge and zero net angular momentum, results in a black hole, whosemetric (at late times after the black hole has become stationary) is theSchwarzschild metric, given in Schwarzschild coordinates by

ds2 = - (1 - 2M/r)dt2 + (1 - 2M/r)-ldr2 + r2(d62 + sin20#2) (11.39)(If the collapsing body has net rotation, the black hole is described bythe Kerr metric.) Much is now known about the theoretical propertiesof black holes within general relativity, and there are strong candidatesfor observed black holes in Cygnus XI and elsewhere. For reviews ofthis subject see Giacconi and Ruffini (1978) and Hawking and Israel(1979).

However, the existence of black holes is not an automatic byproductof curved spacetime. To be sure, curved spacetime is essential to theexistence of horizons as one-way membranes for the physical interactions,but whether or not a horizon occurs depends crucially on the field equa-tions that determine the curvature of spacetime. In the following examples,we shall illustrate this point. Throughout this section, we restrict our-selves to nonrotating, spherically symmetric systems.

Scalar-tensor theoriesAs one might have expected, scalar-tensor theories, being in some

sense the least violent modification of general relativity, predict blackholes. However, what is unexpected is that they predict black holes whosegeometry is identical to the Schwarzschild geometry. The reason is thatthe scalar field </> is a constant throughout the exterior of the horizon,given by its asymptotic cosmological value (f>0. Thus, the vacuum fieldequation, (5.31), for the metric is Einstein's vacuum field equation, andthe solution is the Schwarzschild solution. The scalar field has no effectother than to determine the value of the gravitational constant. (Thisresult also holds for rotating and charged black holes.) In Brans-Dicketheory, for instance, the most direct way to verify this is to use the vacuumfield equation for cp = cj) � (j)0, \3g<P = 0, and to integrate the quantity<pOg(p over the exterior of the horizon between two spacelike hyper-surfaces at different values of coordinate time. After an integration byparts, we obtain

J<P,*9'*(-9)ll2d*x - I j(<p2y*dZx = 0 (11.40)

Now the surface integrals over the spacelike hypersurfaces cancel becausethe situation is stationary; that over the hypersurface at infinity vanishesbecause <p~r~1 asymptotically; and that over the horizon vanishesbecause dEa is parallel to the generators of the horizon and is thus in adirection generated by the symmetry transformations of the black hole(Killing direction) whereas Op2)'" is orthogonal to that direction, sincethe derivatives of q> along symmetry directions must vanish. Thus,

= 0 (11.41)

But cptX is spacelike, since cp is stationary, so (pAqy* > 0 everywhere, andEquation (11.41) thus implies (px = 0. Further details and other argumentscan be found in Thorne and Dykla (1971), Hawking (1972), Bekenstein(1972), and Bekenstein and Meisels (1978).

Rosen's bimetric theoryFor the case *P = 0, the static spherically symmetric vacuum field

equations are [see Equation (11.23)]

fi = A, V20> = V2A = 0 (11.42)

with solutions

<D = -MJr, n = A = MJr (11.43)

and

ds2 = - exp( - 2MJr) dt2 + exp(2MJr)(dx2 + dy2 + dz2) (11.44)

There is no horizon in this spacetime, only a naked singularity at r = 0.Thus, at least within the subset of vacuum spacetimes specified by *F = 0,there are no black holes in Rosen's theory.

11.3 The Motion of Compact Objects:A Modified EIH FormalismIn Chapter 6, we derived the n-body equations of motion for

massive, self-gravitating bodies within the parametrized post-Newtonian(PPN) framework [see Equations (6.31)-(6.34)]. A key assumption thatwent into that analysis was that the weak-field, slow-motion limit of gravi-tational theory applied everywhere, in the interiors of the bodies as wellas between them. This assumption restricted the applicability of the equa-tions of motion to systems such as the solar system.

However, when dealing with a system such as the binary pulsar in whichthere is a neutron star with a highly relativistic interior, one can no longer


apply the assumptions of the post-Newtonian limit everywhere, exceptpossibly in the interbody region between the relativistic bodies. Instead,one must employ a method for deriving equations of motion for compactobjects that, within a chosen theory of gravity, involves (a) solving the full,relativistic equations for the regions inside and near each body, (b) solvingthe post-Newtonian equations in the interbody region, and (c) matchingthese solutions in an appropriate way in a "matching region" surroundingeach body. This matching presumably leads to constraints on the motionsof the bodies (as characterized by suitably denned centers of mass); theseconstraints would be the sought after equations of motion. Such a proce-dure would constitute a generalization of the Einstein-Infeld-Hoffmann(EIH) approach (see Einstein, Infeld, and Hoffmann, 1938).

Let us first ask what would be expected from such an approach withingeneral relativity. In the full post-Newtonian limit, we found that themotion of post-Newtonian bodies is independent of their internal struc-ture, i.e., there is no Nordtvedt effect. Each body moves on a geodesic ofthe post-Newtonian interbody metric generated by the other bodies, withproper allowance for post-Newtonian terms contributed by its own inter-body field. This is the EIH result. It turns out however, that this conclu-sion is valid even when the bodies are highly compact (neutron stars orblack holes). The only restriction is that they be quasistatic, nearly spher-ical, and sufficiently small compared to their separations that tidal inter-actions may be neglected. The effects of rotation (Lense-Thirring effects)are also neglected. This would be a bad approximation for a neutron starabout to spiral into a black hole, for example, but is a good approximationfor the binary pulsar (rpulsar/rorbit ^ 10"5).

Although this conclusion has not been proven rigorously, a strongargument for its plausibility can be presented by considering in more de-tail the matching procedure discussed above. We first note that the solu-tion for the relativistic structure and gravitational field of each body isindependent of the interbody gravitational field, since we can alwayschoose a coordinate system for each body that is freely falling and ap-proximately Minkowskian in the matching region and in which the body isat rest. Thus, there is no way for the external fields to influence the bodyor its field, provided we can neglect tidal effects due to inhomogeneitiesof the interbody field across the interior of the matching region. Onlythe velocity and acceleration of the body are affected. Now, provided thebody is static and spherically symmetric to sufficient accuracy, its externalgravitational field is characterized only by its Kepler-measured mass m,and is independent of its internal structure. Thus, the matching procedure

described above must yield the same result, whether the body is a blackhole of mass m or a post-Newtonian body of mass m. In the latter case,the result is the EIH equations of motion (see Section 6.2), so it must bevalid in all cases. A slightly different way to see this is to note that becausethe local field of the body in the freely falling frame is spherically sym-metric, depends only on the constant mass m, and is unaffected by theexternal geometry, the acceleration of the body in the freely falling framemust vanish, so its trajectory must be a geodesic of some metric. Themetric to be used is a post-Newtonian interbody metric that includespost-Newtonian terms contributed by the body itself, but that excludesself-fields. This conclusion has been verified for nonrotating black holesby D'Eath (1975), and for the Newtonian acceleration of post-post-Newtonian bodies by Rudolph and Borner (1978). D'Eath (1975) gives adetailed presentation of the matching procedure described above.

A key element of this derivation is the validity of the Strong EquivalencePrinciple within general relativity (see Chapter 3 for discussion), whichguarantees that the structure of each body is independent of the sur-rounding gravitational environment. By contrast, most alternative theo-ries of gravity possess additional gravitational fields, whose values in thematching region can influence the structure of each body, and therebyaffect its motion. Consider as a simple example a theory with an addi-tional scalar field (scalar-tensor theory). In the local freely falling coor-dinates, although the interbody metric is Minkowskian up to tidal terms,the scalar field has a value <Ao(0- In a solution for the structure of thebody, this boundary value of <j>0{i) will influence the resulting mass ac-cording to m = m((j)0). Thus, the asymptotic metric of the body in thematching region may depend upon its internal structure via the depen-dence of m on <j)0 (essentially, the matching conditions will depend uponm, dm/d(j),...). Furthermore, the acceleration of the body in the freelyfalling frame need not be zero, as we saw in Sections 2.5 and 3.3. If theenergy of a body varies as a result of a variation in an external parameter,we found, using cyclic gedanken experiments that assumed only conserva-tion of energy, that [see Equation (3.80)]

a - a8eodesic ~ \EB(X, V) ~ (dm/dWt (11.45)

Thus, the bodies need not follow geodesies of any metric, rather theirmotion may depend strongly on their internal structure.

In practice, the matching procedure described above is very cumber-some (D'Eath, 1975). A simpler method, within general relativity, for


obtaining the EIH equations of motion, is to treat each body as a "point"mass of inertial mass ma and to solve Einstein's equations using a point-mass Lagrangian or stress-energy tensor, with proper care taken to neglect"infinite" self fields. In the action for general relativity we thus write

JGR = (lenG)-1 JR(-g)l/2d*x - ^ m f l jdxa, (11.46)

where xa is proper time along the trajectory of the ath body. By solvingthe field equations to post-Newtonian order, it is then possible to derivestraightforwardly from the matter action an n-body EIH Lagrangian inthe form

JEIH = u ...xn,\u... yn)dt (11.47)

written purely in terms of the variables (xfl, vo) of the bodies. The result isEquation (6.80) with the PPN parameters corresponding to general rel-ativity. The n-body EIH equations of motion are then given by

� - f t - a «-l,...,n (11.48)dt dv'a dx'a

In alternative theories of gravity, the only difference is the possibledependence of the mass on the boundary values of the auxiliary fields. Inthe quasi-Newtonian limit (Sections 2.5 and 3.3) this was sufficient toyield the complete quasi-Newtonian acceleration of composite bodies in-cluding modifications (Nordtvedt effect) due to their internal structure.Thus, following the suggestion of Eardley (1975)1 we merely replace theconstant inertial mass ma in the matter action with the variable inertialmass ma{\j/A), where \j/K represents the values of the external auxiliaryfields, evaluated at the center of the body (we neglect their variation acrossthe interior of the matching region), with infinite self-field contributionsexcluded. The functional dependence of ma upon the variable i//A willdepend on the nature and structure of the body. Thus, we write

1 = JG ~ £ Jm-{^A[x.(Tj]} dxa (11.49)a

In varying the action with respect to the fields g^v and i A the variation ofma must then be taken into account. In the post-Newtonian limit, wherethe fields i//A are expanded about asymptotic values i//1^ according to

1 Parts of this section are developed from unpublished notes by DouglasEardley.

<AA = <AA * + <5<AA> it is generally sufficient to expand ma(i/jA) in the form

+ \ Z (5 W # k O ) # i ? V l M * B + � � � (11-50)A,B

Thus, the final form of the metric and of the n-body Lagrangian willdepend on ma and on the parameters dmjdxj/^, and so on. We shall usethe term "sensitivity" to describe these parameters, since they measure thesensitivity of the inertial mass to changes in the fields \j/ A. Thus, we shalldenote

s<A) = - 3(ln mJ/a^A31 ["first sensitivity"]

s<,AB>' = -d2(lnma)/#£>)#B3) ["second sensitivity"] (11.51)

and so on. The final result is a "modified EIH formalism."By analogy with the PPN formalism, a general EIH formalism can be

constructed using arbitrary parameters whose values depend on the the-ory under study, and, in this case, on the nature of the bodies in the system.However, to keep the resulting formalism simple, we shall make somerestrictions. First, we restrict attention to fully conservative theories ofgravity. Technically, this means any theory whose EIH Lagrangian ispost-Galilean invariant. Now, every Lagrangian-based metric theory ofgravity will possess an EIH Lagrangian (thus all the theories discussedin Chapter 5 fall into this class), however not every theory is fully conserva-tive. Only general relativity and scalar-tensor theories are automaticallyfully conservative. Other theories can be fully conservative, in their post-Newtonian limits, at least, only for special choices of adjustable constantsand cosmological matching parameters that make the PPN parameters<*! and oe2 equal to zero (Rosen's bimetric theory with co = cu for example).It is not known whether these choices are sufficient to guarantee that theEIH Lagrangian also be post-Galilean invariant. Nonetheless, the experi-mental upper limits on the PPN parameters OL1 and <x2 (Chapter 8) obtainedfrom searches for post-Newtonian preferred-frame effects make it unlikelythat the analogous effects in the EIH formalism will be of much interest.Therefore we shall adopt a fully conservative EIH formalism.

We shall also restrict attention to theories of gravity that have noWhitehead term in the post-Newtonian limit (i.e., £ = 0). The experimentalconstraints on £ (Chapter 8), from searches for galaxy-induced effects inthe solar system, likewise make the analogous effects in the EIH formalismof little interest.

Each body is characterized by an inertial mass ma, defined to be thequantity that appears in the conservation laws for energy and momentumthat emerge from the EIH Lagrangian. We then write, for the metric,valid in the interbody region and far from the system,

000 = - 1 + 2 £ «fl*ma|x - xa| ~ * + O(4),a

9OJ= 0(3),

9u = ( l + 2 1 7>a\* ~ xa| -*) Su (11.52)

where a* and y* are functions of the parameters of the theory and of thestructure of the ath body. For test-body geodesies in this metric, the quan-tities x*ma and â*wJa are the Kepler-measured active gravitationalmasses of the individual bodies and of the system as a whole. In generalrelativity, a* = y* = 1.

To obtain the EIH Lagrangian, we first generalize the post-Newtoniansemiconservative n-body Lagrangian [Equation (6.80)] in a natural way,to obtain

= - I ma{l - \^vl - i <2><] + \a,b rab

a*b

X h*J flj( flj] (i i-53)

where nab = xjrab. The quantities s/«\ st�, 9^,, ®ah, 9abc, <£ttb, and Sah

are functions of the parameters of the theory and of the structure of eachbody, and satisfy

âb � *(<.»)> Wab � �(abY «ab ~ *(ab),

In general relativity, all these parameters are unity. In the true post-Newtonian limit of semiconservative theories (with t, � 0), for structurelessmasses (no self gravity), the parameters have the values [compare Equa-tion (6.80)]

âb = 7(4? + 3 + at - a2), Sab= 1 + a2 (11.55)

In the fully conservative case, including contributions of the self-gravita-tional binding energies of the bodies, one can show to post-Newtonian

order, that

1, 9A = 1 + (4/? - y - 3)(QJma + Qb/mb),

3), <fa6 = l (11.56)

where Qa is the self-gravitational energy of the ath body.We now impose post-Galilean invariance on the Lagrangian in Equa-

tion (11.53). We make a low-velocity Lorentz-transformation from (t,x)to (T, £) coordinates, given by

x = { + (1 + |W2)WT + | (£ � w)w + O(4) x &

t = T(1 + W + fw4) + (1 + W)S � w + O(5) x T (11.57)

We required that L be invariant, modulo a divergence, i.e.,

L(l T) = L(x, f) dt/dt + # / d t (11.58)

for some function ^. From the transformation Equation (11.57), we have

v = v + w � jw2v � v(v � w) � jw(v � w),

r*1 = C,1 [1 + i(w � fi^)2 + w � fi> � n^,] (11.59)

where v = d£/dz, and n^b = %abl£,ab. Substituting these results into Equa-tions (11.53) and (11.58), and dropping constants and total time derivatives,yields

L«,T)= - I ma{\ - \a

2 + va2(va � w) + (v. � w)2]}

+i($ab + ^ab - T#ab)(w + 2va � W) } (11.60)

Thus the action is invariant if and only if

ab =

Furthermore, the scale of L and the constant term £ a ma are irrelevant,thus, we can always scale the values of ma so that .a?*,1' = 1, and choose

the constant to be £ a ma in terms of the rescaled mass. This merely guar-antees that the inertial mass obtained from the Hamiltonian constructedfrom L agrees with that obtained from the equations of motion. Thus,the final form of the modified EIH Lagrangian is

ad

yb - &Jya � HJfo � nj I (11.62)U)]Since our ultimate goal is to apply this formalism to binary systems

containing compact objects, such as the binary pulsar, let us now restrictattention to two-body systems. Denning & = <g12 and Si = @12, we ob-tain from L the two-body equations of motion [compare Equation (7.34)]

r3 [ r r

- 2vt � v2) -

^ (v2 - v j x � [(SF

� fl)2

a2 = {1^2,x-> - x } (11.63)

where a s dsjdt, x = x2 � xl5 r = |x|, n = x/r. It is possible to showstraightforwardly from these equations that if we define

ma=ma + \mav2a - \^mjnhjrah, a # b

X = (mjXj + m2x2)/("'i + "»2) (11.64)

then the "center of mass" X of the system is unaccelerated, i.e.,2 = 0 (11.65)

This agrees with the fully conservative nature of the EIH Lagrangianand justifies our identification of ma as the inertial rest mass of each body.If we now choose the center of mass to be at rest at the origin, X = X 2 0,then, to sufficient post-Newtonian accuracy we may write

x2 = [nti/m + O(2)]x (11.66)

As in Section 7.3, we define

y=v2-\u a = a2-a1;

m = ml + m2, n = m1m2/fn, dm = m2 � mt (11.67)

then the equations of motion for the relative orbit take the form

dm

r m

(11.68)f ( ) ^m J r

In the Newtonian limit of the orbital motion, we have

a = - m ^ x / r 3 (11.69)

with Keplerian orbit solutions

x = p(l + ecos^>)~1(exCOS0 + e,,sin</>),

r2 d4>ldt = hs CSmp)112, p = a{\ - e2),

v = (&m/p)il2[-exsin<l> + ey(cos<p + e)\

(Pb/2n)2 = a3/&m (11.70)

where a, p, and e are the semi-major axis, semi-latus rectum, and eccen-tricity, respectively; h is the angular momentum per unit mass; and P b isthe orbital period. In this solution we have chosen the x direction to bein the direction of the periastron. The post-Newtonian terms in Equation(11.68) can then be viewed as perturbations of the Keplerian orbit. Usingthe method of perturbations of osculating orbital elements outlined inSection 7.3, we find that the periastron advance is given by

1 (11.71)

where

0> � q)(% .). <§2 _ ^(/nj:^2 1 1 + w2®122)/m (11.72)

This is the only secular perturbation produced by the post-Newtonianterms in Equation (11.68). In the PPN limit, this result agrees withEquation (7.54), for fully conservative theories (with £ = 0).

In obtaining the modified EIH equations of motion, we assumed thatthe field equations obtained from Equation (11.49) were solved for theinterbody gravitational fields through post-Newtonian order. However,

those equations can also be solved for the gravitational-radiation fieldsin the far zone, and for the rate of energy loss via gravitational radiation.The method parallels that presented in Chapter 10, except that now, theself-gravitational corrections in the sources of the fields ip (ifr may includethe metric itself) are automatically taken into account to all orders viathe sensitivities s [see Equation (11.51)]. The only terms that we need toretain in order to determine the lowest order quadrupole and dipolecontributions to the energy loss rate are [compare Equations (10.104)and (10.105)]

�Aeiectric ~ 4(m/R) < - [1 + (s) + � � �] [monopole and quadrupole]

}, [dipole]

] [dipole]

+ � [1 + (s) + � � �] 1 [quadrupole] (11.73)

The only other differences from the post-Newtonian method described inSection 10.3 are the use of the conservation laws and Newtonian equationsof motion obtained from the modified EIH Lagrangian. We shall ulti-mately be interested primarily in the energy loss due to dipole gravitationalradiation, so it is useful to rewrite the dipole portion of Equation (10.84)using terms more suited to the modified EIH formalism, namely

where S is related to the difference in sensitivities between the two bodies.As an illustration of these methods, we shall again focus on specific

theories: general relativity, Brans-Dicke theory, and Rosen's bimetrictheory.

General relativityAs we have already seen, the EIH equations of motion for com-

pact objects within general relativity are identical to those of the full post-Newtonian limit. In other words, &ab = 0&ab = 3)abc = 1, independently ofthe nature of the bodies. Furthermore, the gravitational radiation pro-duced by the orbital motion is dominated by quadrupole radiation (no

dipole radiation), and the energy loss rate is the same as in the pure post-Newtonian case, obeying the Peters-Mathews formula, Equation (10.80)with Kj = 12, K2 = 11, KD = 0. (The same caveats regarding the rigor withwhich this result has been established apply here as in Section 10.3.)

Brans-Dicke theoryThe modified EIH formalism was first developed by Eardley

(1975) for application to Brans-Dicke theory. It makes use of the fact thatonly the scalar field (f> produces an external influence on the structure ofeach compact body via its boundary values in the matching region. Infact the boundary value of <t> is related to the local value of the gravitationalconstant as felt by the compact body by

(j) = G'1(4 + 2o})/(3 + 2co) (11.75)

Hence, we shall regard the inertial mass ma of each body as being a func-tion of G, or more specifically, of In G. Then, if post-Newtonian, interbodygravitational fields lead to variations in 4> away from its asymptotic (cos-mological) value <p0 according to 4> = 4>o + <?�>tnen w e m a v write

( 1 L 7 6 )

Defining the sensitivities sa and s'a of the inertial mass of body a to changesin the local value of G, following Equation (11.51), by

s'a=-[d2(\nma)/d(\nG)2]0 (11.77)

and dropping the 0 subscript, we obtain

ma(4) = mil + sa{cpl4>Q) - Wa - st + sa)(cp/ct>0)2 + O(W0o)3] (11.78)

The action for Brans-Dicke theory is then written

l z a (11.79)

where the integrals in the matter action are to be taken along the trajec-tories of each particle, and where infinite "self-fields" are to be ignored.

The resulting field equations are (compare Section 5.3)

Dg<l> = Y ^ [T - 24>dT/8(j>-] (11.80)

where

T*v = (-gy112 I m s (^« ' (u 0 ) -^ 3 (x - xfl),

dT/d<l> = - ( - 9 ) - 1 / 2 2 (5ma(</.)/^)(u°)-^3(x - x j (11.81)

The equations of motion take the form

T?vv - (dT/dtfrW = 0 (11.82)

Performing a post-Newtonian expansion following the method outlinedin Chapter 5, we obtain to lowest order

<p/4>0 = (2 + a ) - 1 X ma(l - lsa)/ra,a

g00 = - 1 + 2 X (ma/ra)[l - s./(2 + <»)] + O(4),a

9iJ = dJl + 2y X (ma/ra)[l + s./(l + ai)]J (11.83)

where rfl = |x � xo|, y = (1 + a;)/(2 + co), and we have chosen units inwhich G s l . Notice that the active gravitational mass as measured bytest-body Keplerian orbits far from each body is given by

W , = oi*ma = ma[l - sj(2 + <»)] (11.84)

In the full post-Newtonian limit, where sa =* �QJma, this agrees withEquation (6.49). If we define a "scalar mass" (ms)a by

K ) a = i ( 2 + oJ)-1ma(l-2sa) (11.85)

so that

( ) (11.86)

the metric can be written

S'oo = ~ 1 + 2 Xa

gtj = dv {l + 2 I [(mA)a - 2(ms)J/r.J (11.87)

From the active mass and the scalar mass, it is useful to define a "tensormass" (mT)o,

(mT)a = (mA)a - (ms)a = ( | ^ ) * . (11.88)

It can then be shown (Lee, 1974) that the tensor mass (mT)a is associatedwith a conservation law of the form

U-gMT**+ ni* = 0 (11.89)

where V� is a symmetric stress-energy "pseudotensor" given by Lee (1974).This result is consistent with the identification of ma as the inertial mass ofthe ath body.

The full post-Newtonian solution for g^ and cp may now be obtained,and the results substituted into the matter action, which, for the ath bodytakes the form

I.= - jma(cj>)dt(-g00 - 2g0Jvi - g,/^1'2 (11.90)

To obtain an n-body action in the form of Equation (11.47), we first makethe gravitational terms in /� manifestly symmetric under interchange ofall pairs of particles, then take one of each such term generated in la, andsum over a. The resulting n-body Lagrangian then has the form of Equa-tion (11.62) with

<§ab = 1 - (2 + coy s. + sb- 2Sasb),

®ab = 4(2y + 1) + i(2 + oi)- 1(so + sb- 2sasb),

9abc = 1 - (2 + Co)"x(2sfl + sb + se) + (2 + co)-2[(l - 4sa)sksc

+ (5 + 2(o)sB{sb + sc) - (s'a - s2a)(l - 2sb)(l - 2sc)] (11.91)

The quasi-Newtonian equations of motion obtained from the EIH La-grangian are

K = ~ I K x X ) [ l - (2 + coy l(sa + sb- 2Sasby] (11.92)b*a

In the full post-Newtonian limit, the product term s^ may be neglected,and the acceleration may be written

a. a -[(mp)./mj £ {m^Jrl (11.91)b*a

where (mA)b is given by Equation (11.84) and where

W . = m.[l - sj(2 + co)-] (11.92)

in agreement with our results of Section 6.2, Equation (6.49). However, ifthe bodies are sufficiently compact that sa ~ sb ~ 1, then because of theproduct term s^,,, it is impossible to describe the quasi-Newtonian equa-tions simply in terms of active and passive masses of individual bodies.

Roughly speaking, the sensitivity s ~ [self-gravitational binding en-ergy]/[mass], so s e ~ 10~10, so ~ 1(T6, swhitedwarf ~ 10~3. For neutronstars, whose equation of state is of the form p = p(p), a model is uniquelydetermined (for a given value of <w) by the local value of G and by the cen-tral density pc or the total baryon number JV. Now, the sensitivity s is to becomputed holding N fixed; it can then be shown that

dlnN\

For fixed equation of state and fixed central density, a simple scalingargument reveals that m and N scale as G~3/2, so

fdlnm\ (dlnm\ /SlnnA fdlnN\

Note that (d In m/d In N)G is the injection energy per baryon. Then it canbe shown that

S'NS = ( ! - sNS)(dsNS/d In m)G (11.95)

Equations (11.94) and (11.95) actually hold in any theory of gravity inwhich the local structure depends upon a single external parameter whoserole is that of a gravitational "constant." For a variety of neutron starmodels, Eardley (1975) has shown that s ranges from s ^ 0.01 for m =0.13mo to s ^ 0.39 for m = 1.41mo.

For black holes, we have seen (in Section 10.2) that the scalar field isconstant in the exterior of the hole, thus from Equation (11.83) sBH = | ;equivalently mBH scales as G~1/2. Note that the quasi-Newtonian equa-tion of motion for a test black hole and a companion (mBH « mc) is thusgiven, from Equation (11.92), by

= - [(3 + 2cu)/(4 + 2o))]mcx/r3 (11.96)

therefore the Kepler-measured mass experienced by a test black hole isthe tensor mass mT of the companion (Hawking, 1972).

The energy loss rate due to dipole gravitational radiation can be com-puted simply in this formalism (the PM radiation can also be calculated,but the result is not particularly illuminating). The wave-zone form of cp

is given, from Equations (11.78), (11.80), and (11.81) by

<P = y ^ ^ | E ma(l - 2sa)[l + ii � va + O(m/r)] | (11.97)

For a binary system, modulo constants, we obtain

9 = - [4 / (3 + 2co)]K~ V<»(n � v) (11.98)

where

©ss2-Sl (11.99)

Then, following the method of Section 10.3, we find that the rate ofenergy loss is given by Equation (11.74) with KD = 2/(2 + co).

Rosen's bimetric theoryIn Rosen's theory, the flat background metric ij can influence

the structure of a compact object, in spite of the fact that it is a non-dynamical field. In a coordinate system in which the physical metric g isasymptotically Minkowski, and thus in which i\ has the form

i; = diag(-Co1 ,cr1 ,cr1r2 ,c1-V2sin20) (11.100)

we found in Section 11.1 that the equations of structure for static starsdepend only on the quantity (CQCJ)1'2. This quantity, as we discoveredfrom the post-Newtonian limit of Rosen's theory (Equation 5.70), playsthe role of the local gravitational constant G. Let us now adopt the fullyconservative version of the theory, i.e., the version in which the cosmo-logical values of the matching parameters are c0 = ct = 1.

Consider a body moving with velocity v through some given post-Newtonian interbody field. In asymptotically Minkowski coordinates,the metric has the form

ds2 = -e2*'dt2 + ix'jdxJdt + e2A'(dx2 + dy2 + dz2) (11.101)

where <t7 ~ O(2), A7 ~ O(2), x'j ~ O(3), with the superscript / denotinginterbody values. Now to determine the structure of the body, we musttransform to coordinates x4 in which g has the Minkowski form in thematching region. But in this coordinate system, the components of i; areessentially the inverse of those of g (we ignore variations of the componentsacross the interior of the matching region); to post-Newtonian order wehave

We must also transform to coordinates xf in which the body is at rest.For |v| ~ O(l), we have, using Equation (4.49),

ri-oo = - e - 2 * ' [ l - 2v2(& - A') + 2X' � v],

i?8j=-z} + 2i;/<&/-Af),

mi= e-^'dtj + 2viVj(^ - A') - Iv^'n (11.103)

Now the nondiagonal components of nm are of O(3) or O(4), and by thenature of the local field equations, (11.21), for static spherically symmetricbodies, they contribute only quadratically, i.e., at O(6) or higher. Thus,to O(4) we can determine the local values of c0 and cx by

(cokcai = - tooo) ' 1 = e2<D*[l + 2»2(*/ - A1) - 2ZJ � v],

(cxkoa. = (iriu)-1 = e2A'[l - iv2(<D' - A') + h' � v] (11.104)

Thus, the local structure to O(4) is determined by the "local value of G,"given by

G = (coCi)1'2 = e*+A[l + f»2(* - A) - fv � z ] (11.105)

where we have dropped the superscript /. If we view the inertial mass ma

as a function of the interbody values of the metric coefficients (and of thevelocity), then we may write

ma{gj = m.{\- sfl[(D + A + §i>2(«> - A) - |v � Z ]

- Wa ~ s2a)(® + A)2 + � � �} (H-106)

where sa and s'a are given by Equation (11.77). The action for the theorycan then be written (see Section 5.5)

.igjdr. (H-107)

where we again ignore infinite "self-field" contributions to ma. In coordi-nates in which 17 = diag(-1,1,1,1), the field equations are

D ^ v - g'^g^yg^s = - IGnig/r,)1'2^ - k,vT) (11.108)

where

1 � * (0) + 1 (1).

Tfov, = (-g)'112 I M . t e ^ u 0 ) - 1 ^ - xfl),

a

Tft = -2{-g)-112 E (dmjdg^iu0)-^^ - xa) (11.109)


A post-Newtonian expansion using the methods of Chapter 5 yields

g00 = - 1 + 2 £ mjra + O(4),

gu = 6U [ l + 2 I m.(l - fsj/r.j (11.110)

hence

a? = l, 7fl* = l - | s f l (11.111)

We note that, unlike the case in Brans-Dicke theory, the active mass asmeasured by test-body Keplerian orbits is equal to the inertial mass ma.The full post-Newtonian metric can now be obtained (the function Amust also be determined to O(4) for use in ma), and the results for it andfor mjig^) substituted into the matter action

a(gjdxa (11.112)

The resulting n-body Lagrangian is of the EIH form, with

Values of sa and s for neutron stars range from sa m 0.05, s'a ss 0.07 forma =i 0.4 mo to sa ^ 0.6, s'a ^ 0.2 for m =: 12 m o (Will and Eardley, 1977).

Calculation of the dipole gravitational radiation energy loss rate pro-ceeds as in Section 10.3, but using Equations (11.106), (11.108), and(11.109), with the result given by Equation (11.74), with KD = -20/3 asbefore, and ® = s2 � s1.

12

The Binary Pulsar

The summer of 1974 was an eventful one for Joseph Taylor and RussellHulse. Using the Arecibo radio telescope in Puerto Rico, they had spentthe time engaged in a systematic survey for new pulsars. During thatsurvey, they detected 50 pulsars, of which 40 were not previously known,and made a variety of observations, including measurements of their pulseperiods to an accuracy of one microsecond. But one of these pulsars,denoted PSR 1913 + 16, was peculiar: besides having a pulsation periodof 59 ms - shorter than that of any known pulsar except the one in the CrabNebula - it also defied any attempts to measure its period to ± 1 us, bymaking "apparent period changes of up to 80 fis from day to day, andsometimes by as much as 8 us over 5 minutes" (Hulse and Taylor, 1975).Such behavior is uncharacteristic of pulsars, and Hulse and Taylor rapidlyconcluded that the observed period changes were the result of Dopplershifts due to orbital motion of the pulsar about a companion. By the endof September, 1974, Hulse and Taylor had obtained an accurate "velocitycurve" of this "single line spectroscopic binary." The velocity curve was aplot of apparent period of the pulsar as a function of time. By a detailedfit of this curve to a Keplerian two-body orbit, they obtained the followingelements of the orbit of the system: Ku the semiamplitude of the variationof the radial velocity of the pulsar; Pb, the period of the binary orbit; e, theeccentricity of the orbit; <a the longitude of periastron at a chosen epoch(September, 1974); ax sin i, the projected semi-major axis of the pulsarorbit, where i is the inclination of the orbit relative to the plane of the sky;and /j = (m2 sin i)3l{ml + m2)

2, the mass function, where mx and m2 arethe mass of the pulsar and companion. In addition, they obtained the "rest"period Pp of the pulsar, corrected for orbital Doppler shifts at a chosen


epoch. The results are shown in the first column of Table 12.1 (Hulse andTaylor, 1975).

However, at the end of September 1974, the observers switched to anobservation technique that yielded vastly improved accuracy (Taylor et al.,1976). That technique measures the absolute arrival times of pulses (asopposed to the period, or the difference between adjacent pulses) andcompares those times to arrival times predicted using the best availablepulsar and orbit parameters. The parameters are then improved by meansof a least-squares analysis of the arrival-time residuals. With this method,it proved possible to keep track of the precise phase of the pulsar overintervals as long as six months between observations. This was partiallyresponsible for the improvement in accuracy. The results of this analysisusing data up to August 1980 are shown in column 2 of Table 12.1(Taylor, 1980).

The discovery of PSR 1913 + 16 caused considerable excitement in therelativity community (to say nothing of the editorial offices of the Astro-physical Journal Letters), because it was realized that the system couldprovide a new laboratory for studying relativistic gravity. Post-Newtonianorbital effects would have magnitudes of order v2 ~ K\ ~ 5 x 10"7,m/r ~ fxlax sin i ~ 3 x 10~7, a factor often larger than the correspondingquantities for Mercury, and the shortness of the orbital period (~ 8 hours)would amplify any secular effect such as the periastron shift. This expec-tation was confirmed by the announcement in December, 1974 (Taylor,1975) that the periastron shift had been measured to be 4.0° ± 1.5° yr"1

(compare with Mercury!). Moreover, the system appeared to be a "clean"laboratory, unaffected by complex astrophysical processes such as masstransfer. The pulsar radio signal was never eclipsed by the companion,placing limits on the geometrical size of the companion, and the dispersionof the pulsed radio signal showed little change over an orbit, indicating anabsence of dense plasma in the system, as would occur if there were masstransfer from the companion onto the pulsar. These data effectively ruledout a main-sequence star as a companion: although such a star couldconceivably fit the geometrical constraints placed by the eclipse and dis-persion measurements, it would produce an enormous periastron shift(>5000° yr"1) generated by tidal deformations due to the pulsar's gravi-tational field (Masters and Roberts, 1975 and Webbink, 1975). Anothersuggested companion was a helium main-sequence star, which couldaccommodate the geometrical and periastron-shift constraints. Estimatesof the distance to the pulsar (5 kpc: Hulse and Taylor, 1975) and extinctionalong the line of sight (~ 3.3 mag: Davidsen, et al., 1975) indicated that such

Table 12.1. Measured parameters of the binary pulsar"

Parameter

Right ascension (1950.0)Declination (1950.0)Pulse periodDerivative of periodVelocity curve half-amplitudeProjected semi-major axisOrbital eccentricityOrbital periodMass functionLongitude of periastron (9/74)Periastron advance rateRed-shift-Doppler parameterSine of inclination angleDerivative of orbital periodReference

Symbol (units)

a5PP(s)PP(ss-x)K^kms" 1)ay sin i (cm)ePb(s)fi(fnQ)COcb (deg y r" l )

nS)sin iP^ss"1)

Value from perioddata (summer, 1974)

19h13m13s ± 4s

+16°00'24" + 60"0.059030 + 1< K T 1 2

199 ± 5(6.96 + 0.14) x 1010

0.615 + 0.01027908 + 70.13 + 0.01179° ± 1°

��

Hulse and Taylor (1975)

Value from arrival-timedata (9/74-8/80)

19h13m12'469 + K01416°O1'O8'.'15 + O'.'2O0.0590299952695 + 8(8.636 ± 0.010) x 10"18

***(7.0208 + 0.0012) x 1010

0.617138 + 827906.98157 ± 6

***178.867° + 0.002°4.226 + 0.0010.0044 + 0.0003<0.96(-2.1+0.4) x HT1 2

Taylor (1980)

" The entry of a dash (-) denotes that a determination of the parameter was beyond the accuracy of the data;the entry (***) denotes that an accurate value of the parameter is not needed.


a helium star would have an apparent magnitude mR ~ 21. A number ofsearches have failed to detect any such object within an error circle ofradius 0'.'5 around the radio pulsar position derived from the analysis ofthe arrival-time data (Kristian et al., 1976; Shao and Liller, 1978; Craneet al., 1979; and Elliott et al., 1980). Other possible companions to thepulsar are the condensed stellar objects: white dwarf, neutron star, orblack hole. None of these is expected to be observable optically, and thereis no evidence for a second (companion) pulsar in the system.

Attempts to delineate further the nature of the companion involvedconstructing scenarios for the formation and evolution of the system. Thefavored scenario appears to be evolution from an x-ray binary phase whoseend product is two neutron stars (Flannery and van den Heuvel 1975,Webbink 1975, and Smarr and Blandford 1976). However alternativescenarios have been constructed that lead to white dwarf companions(Van Horn et al. 1975, Smarr and Blandford 1976), black hole compan-ions (Webbink 1975, Bisnovatyi-Kogan and Komberg 1976, Smarr andBlandford 1976) and helium star companions (Webbink 1975, Smarr andBlandford 1976). As we shall see, the nature of the companion is crucialfor discussion of various relativistic and astrophysical effects in the system.

One of the most important of these effects is the emission of gravitationalradiation by the system, and the consequent damping of the orbit(Wagoner, 1975). The observable effect of this damping is a secular changein the period of the orbit. However, the timescale for this change, accordingto general relativity, is so long (~ 109 yr) that it was thought that 10 to 15years of arrival-time data would be needed to detect it. However, withimproved data acquisition equipment and continued ability to "keep inphase" with the pulsar, Taylor and his collaborators surpassed all expec-tations, and announced in December 1978 a measurement of the rate ofchange of the orbital period in an amount consistent with the predictionof gravitational radiation damping in general relativity (Taylor et al. 1979,Taylor and McCulloch 1980, Taylor 1980).

This chapter presents a detailed discussion of the confrontation betweengravitation theory and the binary pulsar. In Section 12.1, we develop anarrival-time formalism analogous to that used by the observers to analyzethe data from the binary pulsar, and we discuss the important relativisticgravitational effects (and some competing nonrelativistic effects) in thesystem. We encounter a new and unexpected role for relativistic gravita-tional theory: that of a practical, quantitative tool for measuring astro-physical parameters (such as the mass of a pulsar). In Section 12.2, we usegeneral relativity to interpret the data from the system. In Section 12.3, we

Binary Pulsar 287

interpret the data using alternative theories of gravitation, and discoverthat one theory, Rosen's bimetric theory, faces a killing test. In fact, weconjecture that for a wide class of metric theories of gravity, the binarypulsar provides the ultimate test of relativistic gravity.

12.1 Arrival-Time Analysis for the Binary PulsarBecause the pulsar is the only object seen to date in the system, the

analysis of its radio signal is equivalent to that of optical stellar systems inwhich spectral lines from only one of the members are observed. Suchsystems are known as "single-line spectroscopic binaries," and standardmethods exist for analyzing them. However, there are important differ-ences in the binary pulsar, including the possibility of large relativisticeffects, and the ability to measure directly the arrival times of individualpulses, instead of the pulse period. For this reason it is worthwhile todevelop a "single-line spectroscopic binary" arrival-time analysis tailoredto systems like the binary pulsar. Such an analysis was first carried out indetail by Blandford and Teukolsky (1976) and extended by Epstein (1977)(see also Wheeler, 1975).

We begin by setting up a suitable coordinate system. We choose quasi-Cartesian coordinates (t, x) in which the physical metric is of post-Newton-ian order everywhere except possibly in the neighborhood of the pulsar andits companion, and is asymptotically flat. The origin of the coordinatesystem coincides with a suitably chosen "center of mass" of the binarysystem. The "reference plane" (Figure 12.1) is denned to be a plane perpen-dicular to the line of sight from the Earth to the pulsar (plane of the sky),passing through the origin. The "reference direction" is the direction in thereference plane from the origin to the north celestial pole. At any instant,the orbit of each member of the binary system is tangent to a Keplerianellipse ["osculating" orbit; see Smart (1953) for discussion of theseconcepts and definitions]. This ellipse lies in a plane that intersects thereference plane along a line (line of ascending nodes) at an angle Q (angleof nodes) from the reference direction. The orbital plane is inclined at anangle i from the reference plane. The periastron of the osculating orbit ofthe pulsar occurs at an angle a> from the line of nodes, measured in theorbital plane. The other elements of the osculating relative orbit are thesemi-major axis a, the eccentricity e, and the time of periastron passageTo. Then the instantaneous relative coordinate position x = x2 � x t

(1 = [pulsar], 2 = [companion]) is given by

x = -a[(cos£ - e)eP + (1 - e2)1'2sin£§Q] (12.1)


To Observer

Figure 12.1. Geometry and orbit elements for the binary pulsar.

where eP is a unit vector in the direction of the periastron of the pulsar, andeQ is a unit vector at right angles to this in the orbital plane (measured inthe direction of motion of the pulsar). The quantity E is the eccentricanomaly, related to coordinate time t by

E - e sin E - {2n/Pb){t - To) (12.2)

where Pb is the binary orbit period. (For the purposes of this arrival-timeanalysis, it is more convenient to use E than the true anomaly 4>.) Therelative separation is given by

r = |x| = a(\ � ecos£) (12.3)

By solving the quasi-Newtonian limit of the modified EIH equations ofmotion (See Section 11.3), taking into account the modifications due to theself-gravity of the pulsar and its companion, we find [from Equation(11.70)]

PJ2n = (12.4)

where m = mt + m2 is the sum of the inertial masses of the bodies. Toquasi-Newtonian order, the center of mass [Equation (11.64)] may be

Binary Pulsar 289

chosen to be at rest at the origin, i.e.,

X = m~1(m1x1+m2x2)s0 (12.5)

then

Xj = � (m2/m)x, x2 = (m1/m)x (12.6)

Now, any perturbation of the orbit, whether relativistic or not, is to beviewed as causing changes in the orbit elements Q, i, a>, a, and e of theosculating Keplerian orbit; given a set of values of these elements at anyinstant, Equations (12.1), (12.2), and (12.6) define the coordinate locationsof the two bodies. The changes in the osculating elements produced byperturbations can be either periodic or secular.

We next consider the emission of the radio signals by the pulsar. Let xbe proper time as measured by a hypothetical clock in an inertial frame onthe surface of the pulsar. The time of emission of the Nth pulse is given interms of the rotation frequency v of the pulsar by

N = No + vr + ivt2 + £vt3 + � � � (12.7)

where No is an arbitrary integer constant, and v = dv/di\z=0, v =d2v/dt2\t=0. We shall ignore the possibility of discontinuous jumps("glitches") in the frequency of the pulsar. We ultimately wish to determinethe arrival time of the Nth pulse on Earth.

Outside the pulsar and its companion, the metric in our chosen coordi-nate system is given by Equation (11.52),

0OO = - 1 + 2 X «;»./|x - xa(0| + 0(4),<J=1,2

goj = O(3),

2 X y*mj\x - xa{t)\ + O(4)) (12.8)

where ma is the inertial mass of the ath body, and a* and y* are factors thattake into account the possibility of self-gravitational corrections to the"gravitational" masses if any of the bodies are compact (see Section 11.3for discussion). Because we are interested in the propagation of the pulsarsignal away from the system, we shall ignore the possibility of large,beyond-post-Newtonian corrections to the metric in the close neighbor-hood of the pulsar and the companion. The main result of such correctionswill be either constant additive terms in the arrival-time formula equation,(12.7), that can be absorbed into the arbitrary value of No, or constantmultiplicative factors (such as the red shift at the surface of the neutron

star) that can be absorbed into the unknown intrinsic value of v. Modulosuch factors, proper time x at the pulsar's point of emission can be relatedto coordinate time t by

dx = dt[\ - <x%m2jr - \v\ + O(4)] (12.9)

where we have dropped the constant contribution of the pulsar's gravita-tional potential a}'m1/|xem � xt|, and where we have ignored the differencein the potential and the velocity between the emission point and the centerx t of the pulsar. The two correction terms in Equation (12.9) are thegravitational red shift and the second-order Doppler shift. We can rewriteEquation (12.9) using Equations (12.1) and (12.2), which yield

v\ = <g(ml/m)(2/r - I/a) (12.10)

with the result (modulo constants)

dx/dt = 1 - afmjr - ^m\jmr (12.11)

Using Equations (12.2) and (12.3) we may integrate this equation to obtain

T = t - (m2/a)(<x£ + gm2/m)(PJ2n)e sin E (12.12)

Although the constants that have been dropped in integrating Equation(12.11) may actually undergo secular or periodic variations in time due toorbital perturbations and other effects (such as in a), the correction termin Equation (12.12) is already sufficiently small that such variations willhave negligible effect.

Now, the pulsar signal travels along a null geodesic. We can thereforeuse the method in Sections 6.1, 7.1, and 7.2 to calculate the coordinatetime taken for the signal to travel from the pulsar to the solar systembarycenter x0, with the result

tarr ~ * = |*o('arr) ~ * lW|

+ (a? + yf)m2 ln{2ro(tm)/[r(t) + x(t) � i]} (12.13)

where r0 = |xo|, n = xo/ro, and where we have used the fact that r0 » r.The second term in Equation (12.13) is the time delay of the pulsar signalin the gravitational field of the companion; the time delay due to thepulsar's field is constant to the required accuracy, and has been dropped.

Our ultimate goal is to express the timing formula equation, (12.7), interms of the arrival time farr. In practice, one must take into accountthe fact that the measured arrival time is that at the Earth and not atthe barycenter of the solar system, and will therefore be affected by theEarth's position in its orbit and by its own gravitational red shift and

Binary Pulsar 291

Doppler-shift corrections. In fact, it is the effect of the Earth's orbitalposition on the arrival times that permits accurate determinations of thepulsar position on the sky. It is also necessary to take into account theeffects of interstellar dispersion on the radio signal. These effects can behandled in a standard manner [see Blandford and Teukolsky (1976), forexample], and will not be treated here. Now, because r0 » r, we may write

\*o(tm) - XiW| = ro{tm) - x,(t) � n + O(r/r0) (12.14)

Combining Equations (12.13) and (12.14), and using the resulting formulato express xx(r) in terms of x1(tarr) to the required post-Newtonian order,we obtain

t = *arr ~ ^0 + *l(ttn ~ r0) � fi

+ (*i(tm - r0) � fi)(Xl(tarr - r0) � ft) + [O(3)tarr] (12.15)

where the time-delay term is [O(3)tarr]. We now choose the constant inEquation (12.2) so that

E - e sin E = (2n/Ph){tMt - r0) (12.16)

then, combining Equations (12.12), (12.15), and (12.16), substitutingEquations (12.1) and (12.6), and noting from Figure 12.1 that

eP � n = � sinisintu, eQ � n = � sinicosco (12.17)

we find, modulo constants,

T = tatI - s?(cosE - e) - (@ + ^ ) s in£

- (2n/Pb)(l - e cos E)~ l{d sin E-36 cos £)[j/(cos E - e)+& sin £ ]

+ [O(3)tarr] (12.18)

where

s4 = at sin i sin co, 0$ = (1 � e2)1/2a t sin i cos co,V = (w|/ma1)(a? + &m2/m)(PJ2n)e (12.19)

with

a, = (m2fm)a (12.20)

The timing formula then takes the form

N = N0 + vtarr - va/(cos£ - e) - \{08 + ^sinE

-v(2n/Pb){l-ecosE)-i(s/sinE-@cosE)[#?(cosE-e)+@sinE'}

- e) + & sin E] + |vfa3rr + � � � (12.21)

The quantity N is to be regarded as a function of the time tarr and ofthe parameters No, v, v, v, alsin/, a>, e, Pb, tQ, c£. From an initial guessfor the values of these parameters a prediction for the arrival time of agiven N is made. The difference between the predicted arrival time andthe observed arrival time is used to correct the parameters using themethod of least squares. Possible variations with time of the parametersresulting from perturbations of the system can also be determined, forexample, by substituting

CO-KO + <bt + � � �,

e -*� e + et + � � �,

P b - + P b + $ P b t + --- (12.22)

and so on, into Equation (12.21). (The factor \ in the formula for Pb comesfrom the formal definition of Pb in terms of osculating elements.)

We now turn to a discussion of the important measurable parametersand their interpretation.

(a) The pulsar periodThe terms linear, quadratic, and cubic in tarr in the timing formula,

Equation (12.21), determine the effective pulse period (at a chosen epoch)and its derivatives. The results of least-squares fits using data up toAugust 1980 were (Table 12.1)

p p = v-» = 0.0590299952695 ± 8 s,

Pp = -vv~2 = (8.636 ± 0.010) x 1(T18 s s"1 (12.23)

where the epoch was September, 1974. No determination of Pp has beenpossible to date except for the crude limit set by the fact that Pp has notchanged by more than the experimental error over timescales of one year,thus,

|P p | <6 x 1 0 " 2 8 s s - 2 (12.24)

Despite the fact that the pulsar's period is the second shortest known, its"spin-down rate," Pp , is anomalously small, i.e., Pp/Pp = (4.617 ± 0.005) x10"9 yr"1. The most popular explanation for this is that the pulsar hasa weak magnetic field, leading to small braking torques caused by magneticLorentz forces, thence a small Pp. However, its short period is a remnantof an earlier phase in the evolution of the system, during which accretionof matter onto the pulsar caused it to be "spun up," essentially to itspresent period [for discussion see Smarr and Blandford (1976), Bisnovatyi-Kogan and Komberg (1976)].

Binary Pulsar 293

(b) The Keplerian velocity curveThe terms � vs/(cosE � e) � v^tsinE in Equation (12.21) will

be referred to as the "Keplerian velocity curve." The time derivative ofthese terms yields simply the first-order Doppler shift of the pulsarfrequency, given by Av/v oc vt � ii. This variation in frequency is thequantity usually measured in spectroscopic binaries, and was the quantitymeasured in the binary pulsar until the method of arrival-time measure-ments was adopted in late 1974. By fitting the measurements of arrivaltimes to cos£ and sin is curves using Equation (12.21), a determinationof the parameters Pb, e, stf, and 88 can be made. From these parametersit is conventional to determine (i) the periastron direction at a given epoch:

tan w = (1 - e2)- 1/2J%s/ (12.25)

(ii) the projected semi-major axis of the pulsar:

ax sin i = [ y 2 + m\\ - e2yxY'2 (12.26)

(iii) the mass function of the pulsar:

A M * ! sin 03(iV2*r2 (12.27)

The observed values for these quantities are shown in Table 12.1. UsingEquations (12.4) and (12.20), we may also express fx in the form

/ 1 = Sr(m2 sin i)3/m2 (12.28)

These interpretations assume a fixed Keplerian orbit with constant valuesof the orbit elements. In reality, variations with time of the elements asgiven for example by Equation (12.22), make it necessary to treat theabove values as being valid at a chosen epoch, and to perform furtherleast-squares fits to determine rates of change such as tit, Pb, e, and so on.We shall discuss some of these quantities below.

(c) The periastron shiftBy substituting <x>-*a> + cat into the expressions for si and 88

in the Keplerian velocity curve, one can make an accurate determinationof cb. The best value to date is d> = 4?226 ± 0!001 yr~1.

There are several possible sources of periastron shift in the binarypulsar. The first is relativistic: in Section 11.3, we found that in a binarysystem with compact objects, the periastron shift rate in a fully conser-vative theory of gravity in the modified EIH formalism took the form


[from Equation (11.71)]

+ m23>122)/m (12.29)

\M here should not be confused with that defined by Equation (12.19).]Substituting the known values of Pb and e and using Equation (12.4), weobtain

d»rel = 2!lO(m/m0)2/3^«r 4/3 yr~* (12.30)

In general relativity 9 s ^ == 1.The second possible source is a noninverse square gravitational

potential produced by tidal deformation of the companion by the pulsar.The resulting rate is given by

d)tidal =* 30nk2Ph-\$m1/m2)(R2/a)5f(e),

/(e) = ( l - e 2 ) - 5 ( l + | e 2 + i e4 ) (12.31)

where R2 is the radius of the companion (Cowling, 1938). The quantityk2 is a dimensionless factor that depends on the mass distribution of thecompanion and is of order 10 ~2 for white dwarfs or helium stars. Inobtaining Equation (12.31), we have assumed that the companion is nota neutron star or a black hole in order to avoid the additional compli-cations of self-gravitational effects on the tidal effects. Because of the{R2/a)5 dependence, the tidal contribution of such objects would benegligible in any case. Substituting numerical values we obtain

where X = mxfm2. For a white dwarf companion (R2 5> 104km), thetidal effect is also negligible. Only for a "helium-star" companion, forwhich (fc2/10~2)(i?2/105 km)5 ~ 6(m2/mQf-6, can the tidal periastron ad-vance be significant (Roberts et al., 1976).

The third possible source of periastron shift is the noninverse squarepotential produced by rotational deformation of the companion. For abody that rotates with angular velocity Q about an axis that is inclinedby an angle 8 relative to the plane of the orbit, the result is

d)rol * 27t/c2Pb-1(3Wm2)(i?2/a)5(l + X-1)(n/n)2g(e)P2(cos9),

g(e) 3 (1 - e2r2, n = 2n/Pb (12.33)

This result is valid provided one assumes that the angular momentumof the companion is small compared to that of the orbit, an assumption

Binary Pulsar 295

that is valid for most reasonable companions [see Smarr and Blandford(1976) for discussion]. With numerical values, we find

, 5

\mo) K ^ ' V 1 0 ~ 2 A 1 0 5 km

x �

a O?83a(^m/mo)"2/3P2(cos0) yr"1 (12.34)

where

a = %k2(Cl2Rl/m2)(R2/l03 km)2 (12.35)

For stable, uniformly or differentially rotating white dwarf models, forexample, a may range from zero to ~ 15 (Smarr and Blandford, 1976).Notice that the rotationally induced periastron motion can be either anadvance [P2(cos 6) > 0], for example, when the spin axis is normal to theorbital plane, or a regression [P2(cos 9) < 0], as when the spin axis liesin the orbital plane.

If the companion is a neutron star, a black hole, or a nonrotating whitedwarf, then only the relativistic periastron precession is present. Theobserved advance shown in Table 12.1 then yields via Equation (12.30) arelation between the masses of the bodies:

m = 2.85mG^-3/2#2 (12.36)

where & and ^ are functions of mu m2, and the structure of the pulsarand possibly of the companion.

If the companion is a rotating white dwarf, only the relativistic androtational contributions are significant, thus we may write

(b = 2°10(/n/mG)2/3^«r4/3 + O°83a(m/mGr2/3«r2/3P2(cos0) yr"1

(12.37)

If the companion is a helium star with rotation axis perpendicular tothe orbital plane, all three sources of periastron precession may bepresent, with

2/3/m\2

\mQJ mQ

(12.38)

(d) The gravitational red shift and second-order Doppler shift:the way to weigh the pulsarThe term <^sin£ in the timing formula, Equation (12.21), repre-

sents the combined effects of the gravitational red shift of the pulsarfrequency produced by the gravitational field of the companion and ofthe second-order Doppler shift produced by the pulsar's motion. In sometheories of gravity, there is another effect that contributes to the timingformula at the same order as <if sin E and should be included here (Eardley,1975). That effect is the following: in theories of gravity that violate SEP,the local gravitational constant at the location of the pulsar may dependon the gravitational potential of the companion, i.e.,

GL = G0(l - »*m2/r) (12.39)

If the companion is a white dwarf or a helium star, for example, theparameter n* is simply the combination of PPN parameters n* = 4p �y � 3 (fully conservative theories with £ = 0), however, if the companionis a neutron star or a black hole, rj* could be more complicated and coulddepend upon the internal structure of the companion. As GL then variesduring the orbital motion, the structure of the pulsar, its moment ofinertia, and thence its intrinsic rotation frequency will vary, according to

_ K , * ^ (12.40)rv / GL r

where K determines the response of the moment of inertia to the changingG. The contribution of this variation to the timing formula is given by

J Av dt = - vKn*(m2/a)(Pb/2n)e sin E (12.41)

modulo constants. Thus the parameter W is actually given by

V = (mi/ffMiHaJ + « W m + Kn*){PJ2n)e (12.42)

With numerical values it takes the form

« a* 2.93 x Kr3(m2/m)(mAn0)2/3«r 1/3(<x| + <Zm2/m + Kn*)s (12.43)

However, were it not for the presence of periastron precession in thesystem, this parameter would be entirely unmeasurable, since for con-stant values of sd and 8$, the term %> sin E is degenerate with the twoKeplerian velocity curve terms, i.e., it cannot be separated from them ina least-squares fit (Brumberg et al. 1975, Blandford and Teukolsky,1975). However, the variation of co at 4° per year causes $4 and J1 them-selves to vary with approximately a 90-year period. Thus, over a suf-

Binary Pulsar 297

ficiently long time span (though much shorter than 90 years, fortunately),a separate determination of si, 88, and <� can be made. Using data throughAugust, 1980, Taylor (1980) in fact reports

<«?~4.4±0.3 x HT 3 s (12.44)

Equations (12.43) and (12.44) yield a further relation between the massesof the bodies. When combined with the mass relations provided by themass function [Equation (12.28)], and by the periastron shift [Equations(12.36), (12.37), or (12.38)], and with assumptions about the nature ofthe companion and about the theory of gravitation, they permit a unique(within experimental errors) determination of the masses m^ and m2 andof sin i. This is that unique new role of relativistic gravity alluded to inthe introduction to Chapter 12. Not only does a relativistic effect, theperiastron shift, yield a constraint on the masses of the bodies, it alsoenables the determination of a second relativistic effect, the red-shiftDoppler coefficient (�. Nowhere in astrophysics has relativistic gravityplayed such a direct, quantitive role in the measurement of astrophysicalparameters.

(e) Post-Newtonian effects and sin iIn Equation (12.15), we dropped the explicit term arising from

the time delay, and denoted it [O(3)farr]. There are additional terms inthe timing formula that are also of [O(3)£arr], produced by post-Newtoniandeviations of the orbital motion from a pure Keplerian ellipse. Withingeneral relativity, these terms have been analyzed in detail by Epstein(1977), and included in the data analysis by Taylor, et al. (1979). Theyprovide an independent means to determine the parameters of the system,especially the inclination angle i. This is a valuable consistency checkfor any interpretation of the data. In fact the data are just accurate enoughto be sensitive to these effects, and the limit on sin i quoted in Table 12.1was obtained from these terms. Note that this particular result is validonly in general relativity; the corresponding analysis of the [O(3)farr]terms using the modified EIH formalism has not been carried out.

(f) Decay of the orbit: a test for the existence ofgravitational radiationA variety of effects may cause the orbital period Pb of the system

to undergo a secular change with time, but the most important is theeffect of the emission of gravitational radiation. According to the quad-rupole formula of general relativity (see Section 10.3), a binary system


should lose energy to gravitational radiation at a rate given by Equation(10.80),

dE_ _ /n2m2

dt ~

= - ( 1 . 9 1 x l O - 9 ) ( ^ l T T - ^ y r - 1 (12.48)

where /i is the reduced mass of the system, and

F{e) = (1 + He2 + Me4)(l - e2)'1'2 (12.46)

The resulting rate of change of Pb is given, from Kepler's third law, by

Pb-1 dPJdt = -IE"1 dE/dt = -¥{nm2/aA)F{e) (12.47)

where E = �\\an/a. For the known parameters of the binary pulsar wefind

jn\5'* X_KmQ) (1

As we pointed out in Section 10.3, most theories of gravitation alter-native to general relativity predict the existence of dipole gravitationalradiation. Since the magnitude of the effect in binary systems dependsupon the self-gravitational binding energies of the two bodies, the binarypulsar provides an ideal testing ground. In general relativity, neutron-star binding energies can be as large as half their rest masses, and in othertheories even larger, so the dipole effect, if present, could produce morerapid period changes than the general relativistic quadrupole effect. Thepredicted energy loss rate is given by

ah/at = � 2KD\ & n m vs> /r }

where KD is a parameter whose value depends upon the theory in ques-tion, and S is related to the difference in "sensitivities" (s2 � Si) betweenthe two bodies, where sa is a measure of the self-gravitational bindingenergy per unit mass of the ath body. In general relativity, KD = 0. Therate of change of period is thus given by

l + |e2)(l - e 2r 5 / 2 (12.50)

where now E = � \ 'S^mja. For the parameters of the binary pulsar, weobtain

- "(3.09 x w - ^ J ^ j ^ y , - ' (,2.M,

Binary Pulsar 299

For a theory of gravity with |/cD| ~ 1, this can be several orders of magni-tude larger than the general relativistic quadrupole prediction, unless,for instance, the two bodies are identical, in which case there is no dipoleradiation, by virtue of the symmetry of the situation.

However, before these effects can be used as a reliable test for theexistence of gravitational radiation or as a test of alternative gravitationtheories, other possible sources of period change must be accounted for.Since we have previously discussed tests of gravitational theory involvingdetecting changes in the pulsar period as well as in the orbital period(see Section 9.3), we shall review possible sources of both.

(i) Tidal dissipation. Tides raised on the companion by the gravita-tional field of the pulsar will change both the energy of the orbit and therotational energy of the companion via viscous heating. The corres-ponding tides raised on the pulsar are negligible because of its small sizeand by the same token, if the companion is a neutron star or a black hole,tidal dissipation is negligible. For a companion with rotation axis normalto the plane of the orbit, the rate of change of the orbital period is given by(Alexander, 1973)

Ph 672TC( 1 - e 2 ) - 6 ^ ^ i 3 W fete2� (12.52)

Pb 25 v ' \m22)\a ) \ m2 ) \ ' nj

where n =� 2n/Pb is the orbital mean anomaly, </*> is an "average" coeffi-cient of viscosity of the companion given by

fir8dr (12.53)

where n is the local coefficient of viscosity in units of gem"1 s"1, andh(e2,Q/n) is a complicated function of e2 and Q/n of the following generalform

He2,0/n) = ht(e2) - (�i/n)h2(e

2) (12.54)

For circular orbits h1 = h2 = 1, however for the binary pulsar (e m 0.6)they could be an order of magnitude larger (but hx ^ h2). We note that ifQ < «(companion counter rotates relative to the orbit), tidal dissipationalways decreases the orbit energy and thus the period, whereas if Q/n >hx(e

2)/h2(e2) (companion rotates faster than the orbit by some factor of

order unity), dissipation increases the orbit energy (at the expense ofrotational energy) and causes the period to increase. Notice than even ifthe companion is in synchronous (tidally locked) rotation, Q = n, therecan still be tidal dissipation due to the time-changing deformation of the

companion resulting from the eccentric motion of the pulsar. Substitutingthe observed parameters of the system, we obtain

= - 2 xmo

R2 \ 9

T73T� </*>i3*(e2,n/»)yr~1 (12.55)105 km/

where </x\3 s 10"13<(^>. For standard molecular viscosity, (n~) ~ 1,i.e., <ju>13 ~ 10"13, and tidal dissipation is completely negligible. How-ever, if the source of viscosity is tidally driven turbulence (Press et al.,1975; Balbus and Brecher, 1976), 13 could be as large as unity. For ahelium star companion, PJPb could then be comparable to the generalrelativistic quadrupole radiation damping rate. For a white dwarf com-panion {R2 < 104 km), tidal dissipation is negligible unless the whitedwarf is very rapidly rotating (|Q| » n), and a very strong source of vis-cosity, such as magnetic viscosity «M>i3 ~ 103), is present (Smarr andBlandford, 1976).

(ii) Mass loss from the system. The emission of energy of various forms(particles, electromagnetic radiation, etc.) from the pulsar results in adecrease in its rotational kinetic energy, and thus in an increase in itspulse period, given by

Erol = -l{2nlPpfPJPp (12.56)

where / is the moment of inertia of the pulsar. The loss of mass energyfrom the pulsar leads to a change in the orbital period at a rate

PJPb = -\rhjm (12.57)

Now, if the emission of energy is dominated by relativistic particles(photons, for example) then most of the mass loss will occur at theexpense of rotational kinetic energy, i.e.,

£rot < m, (12.58)

Thus,

PJPb ~ 1 x 10"6 {ml2.MmQ)-HA5(PpIPp) (12.59)

where 745 = 7/1045 g cm2. Since the observed value for Pp/Pp is ~4 x10~9 yr~x (Table 12.1), then PJPb due to energy loss must be ~10~14

yr"1.(iii) Acceleration of the binary system. If the center of mass of the binary

system suffers an acceleration relative to that of the solar system, then

Binary Pulsar 301

both the orbital and pulsar periods will change at a rate given by

PJPb = Pp/Pp = r0 = a � n + r0-J [V - (v � n)2] (12.60)

where v and a are, respectively, the relative velocity and accelerationbetween the binary system and the solar system. The first term is theprojection of the acceleration along the line of sight, while the secondrepresents the effect of variation of the line of sight. Accelerations mayalso lead to observed second-time derivatives of periods, given by

PJPb = Pp/Pp = r0 + 2(P/P)0 r0 - Tr% (12.61)

where (P/P)o is the observed relative rate of change for the correspondingperiod.

One possible source of acceleration was discussed in Section 9.3, namelya violation of conservation of total momentum in some theories ofgravity. There, we used the observed limits on Pp/Pp to set a potentiallimit on the PPN conservation law parameter £2 [

m t n a t c a s e > t n e secondand third terms in Equation (12.61) are negligible compared to the first].

Another source is the differential rotation of the galaxy. If we assumethat the binary system (b) and the solar system (©) are in circular orbitsaround the galaxy with angular velocities Qb and Q 0 , distances from thegalactic center rb and rQ, and longitudes relative to the galactic centercf>b and <t>Q, then Equation (12.60) takes the form

Pp/Pp = PJPb = («o - 0^2r0Vo 'x [cos(^b - <t>0) - rQrbro2sin2(^b - <£0)] (12.62)

Estimates of the location and distance of the binary pulsar (Hulse andTaylor, 1975) yield

r o ~5kpc , r Q ~10kpc , r b ~8kpc , <£b-<£0~3O° (12.63)

Using the standard galactic rotation law, Q(r) ~ 250 (km s"1)/?-, we find

PJPb = K/Pp ~ 2 x lO"13 yr"1 (12.64)

This is too small to be of importance (Will 1976b, Shapiro and Terzian1976).

Another possible source of acceleration is a third massive body in thevicinity of the binary system. For a body of mass m3, and for a circularorbit with orbit elements a3, co3, and i3, we have

PJPb = PJPp~ - feY(-^~)a 3 s in i 3 cos(co 3 + 4>) (12.65)

where </> is the orbital true anomaly, and

Pb/Pb = Pp/Pp * ( ^ J ( - ^ L _ ) a3 sin i, sin(a>3 + </») (12.66)

It is then simple to show that if t] represents the observed upper limit on\PP/Pp\, then the contribution of a third body to period changes is limitedby

\PJPp\ = |Pb/Pb| < (7 x lO-^if/meY'^/lO-11 yr-2)4 '7

x |cot(0 + a>3)sm(4> + o)3)3/7| yr"1 (12.67)

where / is the mass function of the binary system relative to the thirdbody, given by

/ = (m3 sin i3)3(m + m3)"2 (12.68)

Since the observed valued of Pp/Pp has not changed by more than itsexperimental error in a year (Table 12.1), we may conclude that rj <10"11 yr~2. An explicit determination of r\ from the timing data thatimproves this limit would help to determine the likelihood that a thirdbody is responsible for part of the observed orbit period change.

(g) Precession of the pulsar's spin axisIf the pulsar is a rapidly rotating neutron star, it should experience

the same relativistic precession effects on its spin axis as does a gyroscopein orbit around the Earth (see Section 9.1). The dominant effects are thegeodetic precession due to the companion's gravitational field, and aLense-Thirring-type precession due to the companion's "magnetic" gravi-tational field generated by its orbital motion [see Equations (9.2) and(9.4)]. The Lense-Thirring precession due to the possible rotation of thecompanion is negligible. By substituting Equations (9.4) and (9.2) withJ = 0 into Equation (9.1), inserting the orbital elements for the binarypulsar, and averaging over an orbit, one finds

f = ftxSdt

il = (3n/Ph)[m22/ma(l - e

2)][i(2y + l) + f(y+ 1 +ia1)(mi/m2)]fi (12.69)

where y and at are PPN parameters and h is a unit vector normal to theorbital plane (Barker and O'Connell, 1975; Hari Dass and Radakrishnan,1975; and Rudolph, 1979). In obtaining this result we have ignored thepossibility of modified-EIH-formalism corrections to effective masses inalternative theories of gravity. The magnitude of ft is about one degreeper year (compare with an Earth-orbiting gyroscope in Section 9.1); note,

Binary Pulsar 303

however, that no precession occurs if the pulsar's spin axis is normal tothe plane of the orbit.

If precession does occur, it could be viewed as a means to test gravi-tational theory. However, it may be more fruitful to use the relativisticprecession as a means to probe the nature of the pulsar's emission mech-anism. As the pulsar precesses, the observer's line of sight intersects thesurface of the neutron star at different latitudes, thus it may be possibleto obtain two-dimensional information on the shape of the emitted beam,as well as to study the variation of spectrum and polarization with lati-tude. Unfortunately, in most pulsar models, the radio pulses are emittedin a pencil beam, so the pulsar might one day disappear altogether.

We now turn to the confrontation between the binary pulsar and gravi-tation theory. It is here that the philosophy of testing gravitation theorymust depart somewhat from that adopted in Chapters 2 through 9. There,we regarded experiments as "clean" tests of gravitational theory. Becausethe underlying nongravitational physics associated with solar system andlaboratory experiments was reasonably well understood, the experimentalresults could be viewed as limiting the possible alternative theories ofgravity, in a theory-independent way. The use of the PPN formalism wasa clear example of this approach. The result was to "squeeze theory space"in a manner suggested by Figures 8.2 and 8.3.

However, when complex astrophysical systems such as the binary pulsarare used as gravitational testing grounds, one can no longer be so certainabout the underlying physics. In such cases, a gravitation-theory-indepen-dent approach is not useful. Instead, a more appropriate approach wouldbe to assume, one by one, that individual theories are correct, thenuse the observations to make statements about the possible compatiblephysics underlying the system. The viability of a theory would thenbe called into question if the resulting "available physics space" weresqueezed into untenable, unreasonable, or ad hoc positions. Such a methodwould be most powerful for theories that make qualitatively different pre-dictions in such systems. We shall illustrate this philosophy of "squeezingphysics space" (using relativistic gravity to determine astrophysical param-eters) with general relativity, Brans-Dicke theory, and Rosen's bimetrictheory.

12.2 The Binary Pulsar According to General RelativityThe confrontation between relativistic gravity and binary pulsar

data takes its simplest and most natural form within general relativity.In general relativity, there are no EIH self-gravitational mass corrections

due to violations of SEP (see Section 11.3), and there is no dipole gravi-tational radiation. Thus, E g = aj H yf = 1 and KD = n* � 0. The rel-evant measured parameters of the system are then given by the followingexpressions

Mass Function:

fi = (m2 sin ifjm2

Orbital Period:

PJ2n = ( a» 1 / 2

Periastron Shift:

a; =

2?10(m/mG)2/3 yr""1, [black hole, neutron star,nonrotating white dwarf companion]

2?10(m/mo)2/3 + O?83a(m/mor2/3P2(cos0)yr-\

[rotating white dwarf companion]

(12.70)

(12.71)

(12.72)

(12.73)

"V»©/ 105km/

[aligned rotating helium star companion] (12.74)

Red-shift-Doppler Parameter:

^ = 2.93 x 10"31 �m\213

m

Gravitational Radiation Reaction (Pure quadrupole):

VAJgr.quad = -(1.91 x 10-9)(m/mG)5/3X(l + X)'2 yr"1

(12.75)

(12.76)

Together with the measured values shown in Table 12.1, these equa-tions determine constraints on the possible masses of the pulsar andcompanion, and on the inclination i. The most convenient way to displaythese constraints is to plot m1 vs. m2. The results are shown in Figure 12.2.One constraint is provided by the mass function fx and by the fact thatsin i < 1. The periastron shift constrains the system to lie along thestraight line BH-NS-WD if the companion is a black hole, neutron star,or nonrotating white dwarf [Equation (12.72)]. This line represents atotal mass m = 2.85 mQ. It is useful to remark that the maximum massof a nonrotating white dwarf is ~ 1.4 solar masses. If the companion is arapidly rotating white dwarf (with "U" denoting uniform rotation and

Binary Pulsar 305

"D" denoting differential rotation), the system could lie in the regionsdenoted U and D [Equation (12.73)]. The regions to the left of the BH-NS-WD line correspond to white dwarfs with spin axes aligned perpen-dicular to the orbital plane (6 = 0). In this case, the rotational contributionto d> is positive so the inferred system mass m must be less than 2.85m0.The regions to the right of the BH-NS-WD line correspond to whitedwarfs with spin axes in the orbital plane (0 = n/2). Here the rotationalperiastron shift is retrograde and thus m > 2.85mo. Values of the param-eter a that depends upon the structure and rotation rate of the whitedwarf were given by Smarr and Blandford (1976). Figure 12.2 also showsthe configuration if the companion is a helium star, tidally locked intothe orbital rotation rate (Q = n). Values for k2 and R2 for helium starswere given by Roberts, Masters, and Arnett (1976). The red-shift-Dopplerparameter then constrains the system to lie between the lines marked #.Finally, if we attribute all the observed orbit-period decay to gravitational-radiation damping, then the system must lie between the lines marked

Figure 12.2. The mx-m2 plane in general relativity. The shaded regionfits all the formal observational constraints. The point marked "a" is themost likely configuration.

©

ii

3.0

2.0

1.0

^H-NS

- N.

y

\\\

»',

, D-WD"

\HE

i

\\\

\\\

-^^^BH-NS-WD

i i

s

.-��

sin

I

i> 1

fo, ""I-D ^ - - '

gr.§C

1 1

1.0 2.0Mass of Pulsar m,/m0

3.0

Pb. This leaves the shaded region available. The most natural physicalinterpretation therefore seems to be that the companion is a black hole,neutron star, or nonrotating white dwarf (point "a" in Figure 12.2) of massm2 = 1.42 ± 0.07m©. The mass of the pulsar is then mx = 1.43 ± 0.07m©and the sine of the inclination angle (from the mass function) is sin i =0.72 + 0.04. This interpretation is also consistent with the constraintsin i < 0.96 obtained by taking into account in the timing formula post-Newtonian effects such as the time delay [Equation (12.13)] and periodicperturbations of the Keplerian orbit (Taylor, 1980). Before this interpre-tation can be accepted with confidence, however, some account must betaken of the possible nonrelativistic sources of orbit period change dis-cussed in the previous section, in particular tidal dissipation and a thirdbody.

Thus, barring these remote possibilities, general relativity leads to anatural physical configuration for the system, and the results support theconclusion that the measurement of Ph represents the first observation ofthe effects of gravitational radiation. They also lend support to the validityof the quadrupole formula (see Section 10.3) for radiation damping, atleast as a good approximation, and rule out the possibility that gravita-tional waves are composed of half-retarded plus half-advanced fields andtherefore carry no energy at all (Rosen, 1979).

12.3 The Binary Pulsar in Other Theories of Gravity(a) Brans-Dicke theoryBecause solar-system experiments constrain the coupling con-

stant a to be large (co > 500) we expect the predictions of scalar-tensortheories to be within corrections of order (1/co) of their general relativisticcounterparts for the binary pulsar. The self-gravitational mass renormal-izations merely introduce corrections of order (l/co) (see Section 11.3).Thus, the mt � m2 plane in scalar-tensor theories is largely indistinguish-able from that in general relativity. Even the added possibility of dipoiegravitational radiation does not seriously constrain either "physics" spaceor the coupling constant co. Substituting the value KD = 2/(2 + co) intoEquation (12.51) we find

(A/PbWie = - (1 x 10-9)(500/a>)(S/0.1)2(AVm©) yr"l (12.77)

For neutron star models with masses around 1.4m©, s ~ 0.39 (Eardley,1975). Thus, (PJPb) dipoie could be significant if the companion is a whitedwarf or a neutron star whose mass differs from that of the pulsar bygreater than ~ 10%. In such an event, it might be possible to push the

Binary Pulsar 307

coupling constant even higher than 500. However, the data can equallywell be fit (for <x> ~ 500) by a system with two nearly equal-mass neutronstars, or by one of the above possibilities with a small contribution toPb from some nonrelativistic source.

This is a case in which the theoretical predictions are sufficiently closeto those of general relativity, and the uncertainties in the physics stillsufficiently large that the viability of the theory cannot be judged reliably.We would expect roughly the same conclusions to be valid in generalscalar-tensor theories such as Bekenstein's VMT (see Section 5.3).

(b) Rosen's bimetric theoryIn the bimetric theory, however, the situation is very different.

The EIH self-gravitational mass corrections (of Section 11.3) lead toqualitative differences for two reasons. First, the correction terms in^, 0*, etc. are ~s, and second, s can be much larger for bimetric neutron-star models than for their general relativistic counterparts. Table 12.2

Table 12.2. EIH sensitivities, s, s', inRosen's bimetric theory."

Inertial massm(mQ)

normal starwhite dwarfneutron stars0.0970.1650.4090.6350.8651.1581.8682.3713.2174.5537.177

10.9312.3414.45C

s

~ 1 0 ~ 6

g l O " 3

0.0060.0180.0480.0710.0960.1280.2060.2580.3310.4100.4940.5610.5820.628

s'

~10~ 6

£io-3

b

b

0.0650.0990.1320.1740.2650.2940.2710.2230.1750.1520.222b

" For equations of state from Canuto (1975).Accuracy + 3 in last place.* Accurate value not computed.c Maximum mass.

shows values of s and s' for normal stars and white dwarfs, and for neutron-star models with inertial masses up to 14.5wo (Will and Eardley, 1977).From these values, we compute values for'S and 0, given by [see Equations(11.72) and (11.113)]

(12.78)

and plot the corresponding m1 � m2 plane for the bimetric theory, shownin Figure 12.3. [For simplicity, we have ignored the effect of changes inGL on the parameter <�, Equation (12.43). It is only significant if the com-panion is a neutron star {t\* � %s2), and is expected to modify <<? by onlyabout 20%.]

Figure 12.3. The mx-m2 plane in Rosen's bimetric theory. Note the scaleof masses is almost double that of Figure 12.2. The numbers shown arethe predicted values of Pb /Pb due to gravitational radiation, includingdipole gravitational radiation.

6.0 -

5.0 -

4.0 -

a,

<S 3.0

�s

s2.0 -

1.0 -

^ ^ \ ^ � �

D-WD JT>VC

' * ' � - ' ' ' + 1(^ - ' ^ + 1 0 ^ - ^

/"^U-WDI i

N. /+10-*

\� +10"'V-+10-6

sin i> 1

I i i

1.0 2.0 3.0 4.0Mass of Pulsar nij/nio

5.0

Binary Pulsar 309

In Figure 12.3, we notice that the companion cannot be a nonrotatingwhite dwarf, since such a configuration would violate the conditionsini < 1. If the companion is a neutron star, the system must lie alongthe curve "NS," with total inertial mass ~7m o . When the red-shift-Doppler constraint (curves 'V') is folded in, the theory is left with a majorproblem. Dipole gravitational radiation causes the system to gain energyand the period to increase at a rate

( i V n W i e =* +(2 x l(T5)(6/O.3)2(Mn0) yr"1 (12.79)

with specific values for various companions shown in several locationsin Figure 12.3. In order to agree with the observed value of PJPb ^� (2.4 + 0.4) x 10~9 yr~\ the theory must produce a mechanism (tidaldissipation, third body) to cancel this predicted increase and account forthe observed period decrease. The contrived and ad hoc nature of suchmechanisms deals a convincing blow to the viability of this theory.

(c) The ultimate test of gravitation theory?This result may, in fact, apply to many other theories, particularly

those with "prior geometry." In such theories, SEP is violated, and thedifferences between the theories and general relativity become larger thestronger the gravitational fields. Thus, one can expect qualitative EIHmass renormalizations similar to those in the bimetric theory. Further-more, all such theories are expected to predict dipole gravitationalradiation of magnitude comparable to that in the bimetric theory. So itis very likely that the binary pulsar data will be able to rule out a broadclass of alternative gravitation theories.

However, the class of "purely dynamical" theories has the propertythat the effects of the additional gravitational fields can usually be madeas small as one chooses, both in weak-field and in strong-field or gravita-tional-radiation situations, by choosing sufficiently weak coupling con-stants (co'1 -> 0 in Brans-Dicke, for instance). Thus, Brans-Dicke theory,with to 500, is consistent with the present binary pulsar data, eventhough it, too, predicts dipole gravitational radiation. Such theories, thatmerge smoothly and continuously with general relativity, can never betruly distinguished from it (as long as experiments continue to be consis-tent with general relativity). Except for such cases, the binary pulsar mayprovide the "ultimate" test of gravitation theory.

13

Cosmological Tests

Since the discovery by Hubble and Slipher in the 1920s of the recessionof distant galaxies and the inferred expansion of the universe, cosmologyhas been a testing ground for gravitational theory. That discovery wasthought at the time to be a great confirmation of general relativity fortwo reasons. First, general relativity, in its original form, predicted adynamical universe that necessarily either expands or contracts. Of course,Einstein had later modified the theory by introducing the "cosmologicalconstant" into the field equations in order to obtain static cosmologicalsolutions in accord with the current, pre-Hubble observations. To hisgreat joy, following Hubble's discovery, Einstein was allowed to drop thecosmological constant.

Second, was simply the fact that general relativity was capable of dealingwith the structure and evolution of the universe as a whole, a capabilitynot shared by Newtonian theory (unless special assumptions are made).However, this capability is more a consequence of the Einstein Equiv-alence Principle (alternatively of the metric-theory postulates) than aproperty of general relativity. Because of EEP, spacetime is endowedwith a metric g which determines the results of observations made usingnongravitational equipment (light rays, telescopes, spectrometers, etc.)and the motion of test bodies (galaxies). Via the field equations providedby each metric theory of gravity, the distribution of matter then determinesthe metric g, and thereby the entire physical spacetime in which observa-tions are made. By contrast, in Newtonian cosmology, space and timeare fixed a priori, and one is faced either with the problem of specifyingand interpreting the boundary of a finite universe or with the mathematicalproblems associated with an infinite universe in Newtonian theory [seeSciama (1975) for a discussion of Newtonian cosmology].

Cosmological Tests 311

Despite the success of general relativity in treating the expansion ofthe universe, there remained doubts. Chief among these was the "timescaleproblem." The early values for the Hubble constant Ho, the ratio betweenrecession velocity and distance, implied an age of the universe since thebeginning of the expansion ("big bang") that was shorter than the estimatedages of the stars (from stellar evolution theory) and of the radioactiveelements on the Earth. However, by the late 1950s, revisions in the extra-galactic distance scale (increase by a factor of five) and the consequentreduction of the Hubble constant increased the age of the universe to avalue greater than that of our galaxy, thus resolving the timescale problem.But the crucial confirmation of the big bang model came in 1965 withthe discovery of the 3K cosmic microwave background radiation (Penziasand Wilson, 1965). This discovery implied that the universe was oncemuch hotter and much denser than it is today [see Weinberg (1977) for adetailed account of the discovery and of its interpretation]. In particular,it made the steady-state theory of Bondi, Gold, and Hoyle untenable. Italso made it possible to resolve the discrepancy between the observedcosmic abundance of helium (20-30% by weight) and estimates of theproduction of helium in stars (a few percent at most). Calculations byPeebles (1966) and by Wagoner, Fowler, and Hoyle (1967) of nucleo-synthesis in a hot (109 K) big bang yielded helium abundances preciselywithin the observed range [for a review, see Schramm and Wagoner(1977)]. The hot big-bang model within general relativity is today thestandard working model for cosmology [for reviews of general relativisticcosmology, see Peebles (1971), Weinberg (1972), MTW, Sciama (1975)and Zel'dovich and Novikov (1983)].

However, cosmological models within alternative theories of gravityhave not undergone a systematic study with a view toward testing themin a cosmological arena. One reason is that in their exact, strong-fieldformulations, alternative theories are sufficiently different that it has notbeen possible to date to devise a general scheme, analogous to the PPNformalism, for classification, comparison, and confrontation with obser-vations. Also, cosmological observations are not "clean" tests of gravita-tion since much "dirty" astrophysics often goes into their interpretation.

But many alternative theories of gravity, even those whose post-Newtonian limits are identical to or close to that of general relativity, aredifferent enough in their full formulations that they may predict qualita-tively different cosmological histories. These may be sufficiently differentthat observational data such as the mere existence of the microwavebackground or the observed abundance of helium, however imprecise,


may suffice to rule out some theories in spite of the astrophysical andobservational uncertainties.

Section 13.1 outlines the general approach to be used in buildingcosmological models in alternative metric theories of gravity. In Section13.2, we present a brief and qualitative survey of what little is known atpresent about cosmology in such theories.

13.1 Cosmological Models in Alternative Theories of GravityWe begin by making two important assumptions about the nature

of the universe that should hold in any metric theory of gravity:Assumption 1: The Einstein Equivalence Principle (EEP) is valid.Assumption 2: The Cosmological Principle is valid.

As we saw in Chapter 2, the validity of EEP is equivalent to the adoptionof a metric theory of gravity. The cosmological principle states that theuniverse presents the same aspect to all observers at any fixed epoch ofcosmic time, or equivalently, that the universe is homogeneous andisotropic, at least on large scales (~ 100 Mpc). The cosmological principlemay be justified by noting the observations of isotropy of the universe,especially of the microwave background, and by assuming that we occupya typical, not special place in the universe (Copernican principle). Neitherof these two assumptions is open to much question (see, however, Elliset al., 1978) although there has been considerable study of cosmologicalmodels within general relativity that, while approximately isotropic today,were highly anisotropic in the past (for a review, see MacCallum, 1979).

Because of EEP, spacetime is endowed with a metric g in whose localLorentz frames the nongravitational laws of physics take their specialrelativistic forms. The cosmological principle then demands that the lineelement of g must take the Robertson-Walker form (MTW, Section 27.6)

ds2 =^gflvdxlidxv

= -dt2 + a(t)2[(l - kr2rldr2 + r2(d62 + sin2 0 # 2 ) ] (13.1)

where r, 9, and 4> are dimensionless coordinates, t is proper time asmeasured by an atomic clock at rest, a(t) is the expansion factor (units ofdistance), and k e {+1,0} is a constant. Each element of cosmic matter(galaxy) is assumed to be at rest in these coordinates. If k = 1, the universeis closed (i.e., has closed spatial sections), if k = � 1, the universe is open,and if k = 0, the universe is open, with Euclidean spatial sections. Thealternative form of the metric that was used in Section 4.1 to establishthe asymptotically flat PPN metric can be obtained from this by making

the transformation to the new radial coordinate r' given by

r = {r'lao){l + kr'2IAaly" (13.2)

where a0 is the value of a(t) at the present epoch.Although the present value of the scale factor a is difficult to measure,

its rate of variation with time is subject to observation. In particular onedefines the Hubble constant Ho and the deceleration parameter q0 by

Ho = (d/a)0, qo=- H« 2(a/a)0 (13.3)

where a dot denotes d/dt and the subscript "0" denotes present values.These parameters may be measured by a variety of techniques, such asthe magnitude-red-shift relation or the angular-size-red-shift relation fordistant galaxies. The present "best" values for these parameters are

Ho ^ 60 ± 2 0 km s"1 Mpc~\ -l<qo^2 (13.4)

The large uncertainty in q0 is a result of the uncertain effects of galacticevolution on the intrinsic luminosities of distant galaxies used as "standardcandles" in magnitude-red-shift measurements.

The validity of EEP also allows one to determine the behavior of thematter in the universe, independently of the theory of gravity. If weidealize that matter as a homogeneous perfect fluid, then the equationsof motion Tfv

v = 0 can be shown (MTW, Section 27.7) to yield the followingequations for the evolution of the mass-energy density p(t) and thepressure p(t):

P(t) = pmO|>oMt)]3 + Pro[ao/a(t)T,

Pit) = yrolao/a(t)Y (13.5)

where pm0 and pr0 denote the present mass-energy densities of matterand radiation, respectively. These equations will be valid for temperaturesless than about 1010 K, when the electrons and positrons annihilated.

We now turn to an outline of the recommended method for obtainingcosmological models in any metric theory of gravity.

Step 1: Use the cosmological principle to determine the mathe-matical forms in Robertson-Walker coordinates to be taken by all thedynamical and nondynamical fields of the theory. For the dynamicalfields listed in Section 5.1, these forms are

Metric: ds2 = -dt2 + a(t)2da2,

Scalar: 0(r),

Vector: K^dx* = K{t)dt,

Tensor: B^dx"dxv = co0(t)dt2 + (ox{t)do2 (13.6)

where

da2 = (1 - /cr2)"1 dr2 + r2{d62 + s i n 2 0# 2 ) (13.7)

For a nondynamical flat background metric r\ governed by the equationRiem(i/) = 0, the general form for its line element dy2 = tj^dx" dx" inRobertson-Walker coordinates is

dSf2 = - i(t)2 dt2 + %{i)2 da2 [k = - 1 ] ,

dSf2 = -i(t)2 dt2 + da2 [k = 0] (13.8)

where r(t) is a function of t, with i = dt/dr (as in Section 11.3, we shallignore the possibility of "tipping" of the t] cones relative to the g cones).Note that there is no solution for the case k = 1. Thus, it is very unlikelythat any theory of gravity with a flat background metric can have aclosed (k = 1) cosmological model for the physical metric g. For a non-dynamical cosmic time coordinate T, it is sufficient to assume that T =T(t). The matter variables have the form

p = p(t), p = p(t), u" = (1,0,0,0) (13.9)

Step 2: Substitute these forms into the field equations of the theory.Step 3: Set boundary conditions on the fields, in particular on their

present values <j>0,K0,x0,a0, etc. These values are related in general to suchmeasurable quantities as H0,q0, and k, as well as to the PPN parametersand the present rate of variation of G, or (G/G)o. Use the present experi-mental values or limits on these parameters to limit the class of cosmo-logical models to be considered.

Step 4: Integrate the field equations and the equations of motionbackward in time (using numerical methods as a rule), taking into accountpossible changes in the equation of state for the matter variables as theuniverse becomes hotter and denser (see MTW, Section 28 for discussion).

Step 5: The tests. Although cosmological data is sketchy and im-precise, there are two pieces of evidence about the early universe aboutwhich there seems to be little disagreement, the 3K cosmic microwavebackground radiation and the cosmic abundance of helium.

(i) The microwave background: There is now general consensus thatthe microwave background is the relic of a hotter, denser phase of theuniverse, where the temperature exceeded 4 x 103 K. No reasonablemechanism has yet been devised to produce the background during laterepochs (T < 4 x 103 K) that agrees both with the observed high degreeof isotropy of the radiation [after the effects of the Earth's motion (seeSection 8.2) have been subtracted] and with the close agreement of the

spectrum with that of a black body. Prior to the epoch T = 4 x 103 K, avariety of physical processes are consistent with the observed background,ranging from recombination of electrons and protons to form hydrogento the quantum evaporation of primordial "mini"-black holes (m <1015 g). Thus, in order to predict cosmological models with the microwavebackground, the theory must guarantee that the universe evolved from astate with T>4 x 103 K (p/p0 > 109, a/a0 < 10~3). An example of anunviable cosmological model would be one that contracts from someearlier dispersed state to a maximum density and temperature below theabove limits, then bounces and reexpands to the present observed state.Such a model would contain no reasonable explanation for the microwavebackground. A class of models in Rosen's bimetric theory has this prop-erty (see Section 13.2).

(ii) The helium abundance: It is also generally believed that stellarnucleosynthesis can account for only a small fraction of the observed20-30% abundance by weight of helium, and thus, most of the heliumwas produced in the early universe. Similar claims have been made forthe deuterium abundance (observed to be ~2 parts in 105 by weight),but in this case the contributions of galactic production (and destruction)and of chemical fractionation are more uncertain, so we shall focus onhelium [see Schramm and Wagoner (1977)]. Primordial nucleosynthesisrequires temperatures in excess of 109 K and baryon number densities> 10~6 cm"3, and therefore a viable cosmological model must predict astate at least this hot and dense. Furthermore, the fraction of heliumproduced is sensitive to the rate of expansion of the universe at the epochof nucleosynthesis. The reason is as follows: when nucleosynthesis occurs,essentially all the neutrons go into helium nuclei, so the abundance ofhelium depends only on the neutron-proton abundance ratio at the timetN of nucleosynthesis, i.e.,

X(He4) = 2(n/p)(l + n/p)-%N (13.10)

where X denotes the mass fraction and n/p is the neutron-proton densityratio. This ratio n/p is determined by two factors. First is the (n/p) ratioat the moment ("freeze out") when weak interactions are no longer fastenough to maintain the neutrons and protons in chemical equilibrium;at freeze out their ratio is thus given by (n/p)F � exp[(mn � mp)//c7V],where mn and mp are the proton and neutron rest masses and TF is thetemperature at freeze out. Second is the interval of time between freezeout and nucleosynthesis, during which the neutrons undergo free decay.The faster the expansion rate at a given temperature, the earlier the


weak interactions freeze out, thus TF is higher and (n/p) is closer to unity.In addition, the time between freeze out and nucleosynthesis is shorterand fewer neutrons decay. The result is a higher abundance of helium.The opposite occurs for a lower expansion rate. In some cosmologicalmodels, the expansion rate during nucleosynthesis can be expressedphenomenologically as

a^da/dt^ZQnp)1'2 (13.11)

where ^ is a parameter whose value is 1 in the standard model of generalrelativity, and p is the total mass-energy density. The resulting heliumabundance is given approximately by

X(He4) ~ 0.26 + 0.38 log £ (13.12)

for a present density p o ~10~ 3 O gmcm~ 3 (see Schramm and Wagoner,1977, for discussion). Thus, a value of £ greater than about 3 or less thanabout 5 would do serious violation to observed helium abundances. Sincethe scale a{t) of the universe was 109 times smaller at this epoch than atpresent, this is a very restrictive result for a generic theory of gravity.

Other possible tests of cosmological models, such as the question ofgalaxy formation or the problem of the observed ratio of the number ofphotons to the number of baryons (nY/nb ~ 108) are so poorly understoodwithin general relativity that they are unlikely to be useful tools fortesting alternative theories in the foreseeable future.

13.2 Cosmological Tests of Alternative Metric Theories of GravityFor specific theories of gravity, results for the confrontation

between theory and cosmology are sparse. No systematic study of cos-mological models in alternative theories has been carried out, and of thoseanalyses that have been performed within specific theories, few haveaddressed such questions as the microwave background and the heliumabundance. Thus, we shall confine ourselves to a brief list, without detailsand largely without comment, of those few results that are known.

General relativityThe "standard big bang model" (MTW, Section 28) agrees at

least qualitatively with all observations, although there remain problemswhen one pushes for more precision or more detailed comparison withobservation such as galaxy formation, the photon-to-baryon ratio puzzle,the initial singularity, the value of k, the abundances of deuterium andthe other light elements, the mean density of the universe, and so on.


Brans-Dicke theorySeveral computations (Greenstein 1968, Weinberg 1972) have

shown that a wide class of cosmological models in Brans-Dicke theoryare in qualitative agreement with all observations, including the heliumabundance. The models begin from a singular big bang as in generalrelativity, one difference being the uncertainty in the boundary conditionto be placed on the scalar field (f> at t = 0. However, choices can be madefor this boundary condition that yield results similar to those of generalrelativity for similar values of the present uncertain matter density p0.Moreover, the larger the value of a>, the closer the agreement with generalrelativity. In all cases, the present value of G/G is below the experimentaluncertainty (see Chapter 8).

Bekenstein's variable-mass theory (VMT)By contrast with Brans-Dicke theory, the VMT can have cosmo-

logical models that begin the expansion from a nonsingular "bounce"(which presumably was preceded by a contraction phase). Bekensteinand Meisels (1980) have studied a variety of such models that satisfy thefollowing constraints: at the initial moment of expansion, /($) is small(Equation 5.40), i.e., co{4>) ~ � § (required for the model to start from aminimum radius), and a c± 1016-1017 cm (appropriate for initial tem-peratures of ~ 101 * K). After numerical integration of the field equationsfor a variety of values of the curvature parameter k and the arbitraryconstants r and q (see Chapter 5), they reached the following conclusions:(i) Although the initial value of a> was quite small, its present value inmany models exceeded 500, thus yielding close agreement with all experi-mental tests, and with the predictions of general relativity for neutronstars, black holes, gravitational waves, the binary pulsar, etc. (ii) Thegravitational constant G decreased by between 36 and 40 orders ofmagnitude between the initial moment and the present, thereby accountingfor the "large number" puzzle that Gm^/hc^ 10"38, where mp is theproton mass. Because of the large variation in G, this ratio was initiallynear unity, (iii) Despite the large variation in G, the present value ofG/G, in most cases, was well below the experimental upper limits. Becausethe universe in these models began from a hot, dense (though nonsingular)state, it permits origins for the cosmic microwave radiation as naturallyas does general relativity. However, the helium abundance remains anopen question at this writing. At the time of nucleosynthesis (T ~ 109 K)the expansion rate would have been very different from that of generalrelativity, since ca was very small then (perhaps of order � f). Only a

detailed computation can determine whether there are VMT cosmologicalmodels that are consistent with the helium abundance.

Rosen's bimetric theoryBecause the theory has a flat background metric i/, there are no

closed (k = 1) cosmological models. The Euclidean (k = 0) models havebeen studied by Babala (1975) and by Caves (1977). There are only twoclasses of models that have a physically reasonable expansion phase. Oneclass expands from a singular state at a finite proper time in the past.These models make the definite prediction

{G/G)o ;> 0.51H0[l + 3Om0(l + «2o)~'] (13.13)

where Qm0 = 4npmO/3Hl, and a2o is the present value of the PPN param-eter a2. Experiments (Chapter 8) place the limit |a20| « 1, and obser-vations indicate Qm0 < 0.1 for Ho ^ 55 km s"1 Mpc"1. This predictioncould thus be tested by future measurements or limits on (G/G)o. Theother class of models have a bounce at a minimum radius given by (Caves,1977)

amjao £ [1 + (1 + a20)/3Qm0r2 £ <To (13.14)

too large to permit a natural origin of the microwave background. Theopen (k = � 1) models have, among other possibilities, expansion froma singular state at finite proper time in the past, and a similar expansionfrom a singular state at an infinite proper time in the past (Goldman andRosen, 1976). These models have not been meshed with the present valuesof the PPN parameters, Ho, or {G/G)o. Rosen (1978) has also studiedmodels in which the background metric t\ is not flat, but rather correspondsto a spacetime of constant curvature. The helium abundance has notbeen studied in any models in the bimetric theory.

Rastall's theoryAs in Rosen's theory, the presence of a flat background metric

rules out closed (k = 1) cosmological models. Rastall (1978) has shownthat the k = 0 models predict a contraction phase, a nonsingular bounce,then an expansion phase. However, the bounce occurs at a radiusflmin/«o � TS> t o ° large to provide an explanation of either the microwavebackground or the helium abundance.

Although the results presented here are very sketchy, they illustrate animportant lesson. For some theories of gravitation, cosmology mayprovide do-or-die tests. This applies particularly to theories whose


predictions for present-day gravitational phenomena (post-Newtonianlimit, neutron stars, gravitational waves, and present cosmologicalobservations) are indistinguishable from those of general relativity, viz.the VMT. For such theories, gravitational effects in the early universemay be sufficiently different from those predicted by general relativitythat the cosmic microwave background and the abundances of the lightelements may help to determine the most viable theory of gravitation.

14

An Update

In this chapter, we present a brief update of the past decade of testingrelativity. Earlier updates to which the reader might refer include "TheConfrontation between General Relativity and Experiment: An Update "(Will, 1984), "Experimental Gravitation from Newton's Principia toEinstein's General Relativity" (Will, 1987), "General Relativity at 75:How Right Was Einstein?" (Will, 1990a), and "The ConfrontationBetween General Relativity and Experiment: a 1992 Update" (Will,1992a). For a popular review of testing general relativity, see "WasEinstein Right?" (Will, 1986).

14.1 The Einstein Equivalence Principle(a) Tests of EEPSeveral recent experiments that constitute tests of the Weak

Equivalence Principle (WEP) were carried out primarily to search for a"fifth-force" (Section 14.5). In the "free-fall Galileo experiment" per-formed at the University of Colorado (Niebauer, McHugh and Faller,1987), the relative free-fall acceleration of two bodies made of uranium andcopper was measured using a laser interferometric technique. The"Eot-Wash" experiment (Heckel et al., 1989; Adelberger, Stubbs et al.,1990) carried out at the University of Washington used a sophisticatedtorsion balance tray to compare the accelerations of beryllium and copper.The resulting upper limits on q [Equation (2.3)] from these and earlier testsof WEP are summarized in Figure 14.1

Dramatically improved " mass isotropy " tests of Local Lorentz Invari-ance (LLI) (Section 2.4(b)) have been carried out recently using laser-cooled trapped atom techniques (Prestage et al., 1985; Lamoreaux et al.,1986; Chupp et al., 1989). By exploiting the narrow resonance lines made


10"r-9 -

10

10"

rio -

10.-12 -

1 1 1

:T-

«i

i i i

1 1 1

Renner

I

1 +«2)

1 1 1

1 1 1

Princeton

1Moscow 1

tl i i

1

1

Boulder

|

LURE

1

i

free-Fall

I -EoMVash

11900 1920 1940 1960 1970 1980 1990

Year of experiment

Figure 14.1. Selected tests of the Weak Equivalence Principle, showingbounds on r/, which measures fractional difference in acceleration ofdifferent materials or bodies. Free-fall and Eot-Wash experimentsoriginally performed to search for the fifth force. Hatched and dashed lineshow current bounds on t] for gravitating bodies (test of the StrongEquivalence Principle) from lunar laser ranging (LURE).

possible by the suppression of atomic collisions in the traps, theseexperiments have all yielded extremely accurate results, quoted as limits onthe parameter 3 [Equation (2.13)] in Figure 14.2. In the THeju framework(Section 2.6), S = � 1 = 2�!], where c0 and ce are re-spectively the limiting speed of test particles and the speed of light. Alsoincluded for comparison is the corresponding limit on 5 obtained fromMichelson-Morley type experiments.

Recent advances in atomic spectroscopy and atomic timekeeping havemade it possible to test LLI by checking the isotropy of the one-waypropagation of light (as opposed to the round-trip speed of light, as testedin the Michelson-Morley experiment). In one experiment, for example(" JPL" in Figure 14.2), the relative phases of two hydrogen maser clocksat two stations of NASA's Deep Space Tracking Network were comparedover five rotations of the Earth by propagating a light signal one-way alongan ultrastable fiberoptic link connecting them (Krisher, Maleki et al.,

An Update 322

10"8 -

X, 10"

10"

10,-20 -

1 1 1 1 1 1 1

Michelson-Morley

« Joos

T

i i i i

JPL

TPAl

Brillet-Hall I

IHughes-Drever

T8 =

e J

l l l l 1 1 1

t

U.

1 1

NIST

lHarvard_

Washington 1

1 * 11980 19901900 1920 1940 1960 1970

Year of experiment

Figure 14.2. Selected tests of local Lorentz invariance showing boundson parameter S, which measures degree of violation of Lorentz invariancein electromagnetism. Michelson-Morley, Joos, and Brillet-Hall experi-ments test isotropy of the round-trip speed of light, the later experimentusing laser technology. Two-photon absorption (TPA) and JPL experi-ments test isotropy of the one-way speed of light. The remaining fourexperiments test isotropy of nuclear energy levels. Limits assume thespeed of Earth is 300 km/s relative to the mean rest frame of the universe.

1990). In another ("TPA"), the isotropy of the Doppler shift was studiedas a function of direction using two-photon absorption in an atomic beam(Riis et al., 1988). Although the bounds from these experiments are not astight as those from mass-isotropy experiments, they probe directly thefundamental postulates of special relativity, and thereby of LLI.

A number of novel tests of the gravitational redshift (Local PositionInvariance) were carried out. The varying gravitational redshift of Earth-bound clocks relative to the highly stable millisecond pulsar PSR 1937 + 21,caused by the Earth's monthly motion in the solar gravitational fieldaround the Earth-Moon center of mass (amplitude 4000 km), has beenmeasured to about 10 % (Taylor, 1987), and the redshift of stable oscillatorclocks on the Voyager spacecraft caused by Saturn's gravitational fieldyielded a one percent test (Krisher, Anderson and Campbell, 1990). The


10"

10,-2

10",-3

Pound-Rebka MillisecondPulsar

IIT -,- Null I' 1 T Redshift Y

Solar

T Redshift t

Y Saturn *

H-Masser

Y

I960 1970 1980

Year of experiment

1990

Figure 14.3. Selected tests of local position invariance via gravitationalredshift experiments, showing bounds on a, which measures degree ofdeviation of redshift from the formula Av/v = AU/c2.

solar gravitational redshift has been tested to about 2 % using infraredoxygen triplet lines at the limb of the Sun (LoPresto, Schrader and Pierce,1991). Figure 14.3 summarizes the bounds on a [Equation (2.21)] thatresult from these and earlier experiments. It is now routine to take redshiftand time-dilation corrections into account in making comparisons betweentimekeeping installations at different altitudes and latitudes, and innavigation systems, such as the NAVSTAR Global Positioning System,which use Earth-orbiting atomic clocks.

(b) The c2 formalismThe THefi formalism (Section 2.6) can be applied to tests of local

Lorentz invariance, but in this context it can be simplified (Haugan andWill, 1987; Gabriel and Haugan 1990). Since most such tests do notconcern themselves with the spatial variation of the functions T, H, e, andfi, but rather with observations made in moving frames, we can treat themas spatial constants. Then by rescaling the time and space coordinates, thecharges and the electromagnetic fields, we can put the THefi action inEquation (2.46) into the form

f (1 - vy2dt +£ea > -c2B2)d4x,

(14.1)

An Update 324

where c2 = //0/roe0iu0 = cl/cl. This amounts to using units in which thelimiting speed c0 of massive test particles is unity, and the speed of light isc. If c # 1, LLI is violated; furthermore, the form of the action above mustbe assumed to be valid only in some preferred universal rest frame. Thenatural candidate for such a frame is the rest frame of the cosmicmicrowave background.

The electrodynamical equations which follow from Equation (14.1)yield the behavior of rods and clocks, just as in the full THsfi formalism.For example, the length of a rod moving through the rest frame withvelocity V in a direction parallel to its length will be observed by a restobserver to be contracted relative to an identical rod perpendicular to themotion by a factor 1 � V2/2 + O(VA). Notice that c does not appear in thisexpression. The energy and momentum of an electromagnetically boundbody which moves with velocity V relative to the rest frame are given by

E = MR + ^MRF2 + ^Mf F'F^, (14.2a)

p> = MR V + 8M\V\ (14.2b)

where MR = Mo�E%s, Mo is the sum of the particle rest masses, E#s is theelectrostatic binding energy of the system, and SMI is the anomalousinertial mass tensor, given by

SMf = - 2 [ ~ llgfif^+JS8*!, (14.3)

where

^ E (14.4a)

(14.4b)ab

Note that (c~2� 1) here corresponds to the parameter S plotted in Figure14.2.

The electromagnetic field dynamics given by Equation (14.1) can also bequantized, so that we may treat the interaction of photons with atoms viaperturbation theory. The energy of a photon is ft times its frequency co,while its momentum is fico/c. Using this approach, one finds that thedifference in round-trip travel times of light along the two arms of theinterferometer in the Michelson-Morley experiment is given byL0(v

2/c)(c~2� 1). The experimental null result then leads to the bound on

(c~2� 1) shown on Figure 14.2. Similarly the anisotropy in energy levels isclearly illustrated by the tensorial term in Equation (14.2a); by evaluating£|Sl> for each nucleus in the various Hughes-Drever-type experiments andcomparing with the experimental limits on energy differences, one obtainsthe extremely tight bounds also shown on Figure 14.2. The behavior ofmoving atomic clocks can also be analysed in detail (Gabriel and Haugan,1990), and bounds on (c~2 � 1) can be placed using results from tests of timedilation and of the propagation of light (Riis et al., 1988; Krisher, Malekiet al., 1990; Will, 1992b). The bound obtained from the " JPL" test of theisotropy of the one-way speed of light (see below) was based on theprediction for the time dilation of hydrogen maser clocks (Gabriel andHaugan, 1990) namely

, (14.5)

where

a = -f ( l -c 2 ) . (14.6)

(c) Kinematical frameworks for studying LLIThere are a number of frameworks for studying tests of special

relativity (or of LLI) that are kinematical in nature, dating back to H. P.Robertson [see Haugan and Will (1987) and Mac Arthur (1986) for recentreviews]. One particularly useful version was developed by Mansouri andSexl (1977a,b,c) (see also Abolghasem, Khajehpour and Mansouri, 1988,1989). It assumes that there exists a preferred universal reference frameE:(7", X) in which the speed of light is isotropic (with unit speed in theappropriate units). The transformation between £ and a moving inertialframe S:{t, x) is given by

T=a-\t-e-x), (14.7a)

X = d-lx-(d~i-b-i)yvxY//w2 + <wT, (14.7b)

where w is the velocity of the moving frame, a, b, and dare functions of w2,and E is a vector determined by the procedure adopted for the globalsynchronization of clocks in S. In special relativity, the functions a, b, anda? have the special forms a'1 = b = y = (1 � w2)'1'2, and d = 1, but E can bearbitrary, depending upon the procedure for synchronization; with eitherEinstein (round-trip light signals) or clock-transport synchronization,£ = � w.

In the low-velocity limit, it will be useful to expand the functions a, b, d,

An Update 326

and E in powers of w2 using arbitrary parameters. Adopting a slightlydifferent convention from Mansouri and Sexl, we write

a(w)x, l+(a-4)w2 + ( a 2 - 4 K + . . . , (14.8a)

b(w)x l+0?+i)w2 + 0?2+f)w4 + ..., (14.8b)

d(w) xl+Sw2 + S2w4 + ..., (14.8c)

.. (14.8d)

In SRT, a, a2, fi /?2, 8 and <S2 all vanish, and with standard synchronization,so do s and e2.

The physics that results from experiments should not depend on thesynchronization procedure, except measurements which depend on adirect, one-time comparison of separated clocks. Thus a measurement ofthe absolute value of the speed of light in S by a time-of-flight techniquebetween two points will depend on the synchronization of the two clocks(a particularly perverse choice of synchronization can make the apparentspeed between those points infinite, for example). However, a study of theisotropy of the speed between the same two clocks as the orientation of theline connecting them varies relative to £ should not depend on how theywere synchronized, as long as they were synchronized by some procedureinitially. Similarly, a measurement of the Doppler shift of an atomicspectral line using a single "clock" as receiver of the signal should notdepend on synchronization, provided that the velocity of the atom isexpressed in terms of observables measured by a single clock. This pointhas been misunderstood by numerous authors who have argued againstthe efficacy of tests of the one-way speed of light. An advantage of theMansouri-Sexl framework is that it allows one to understand explicitly therole of synchronization in a given experiment.

A disadvantage of this and similar kinematical frameworks is that theydo not allow for the dynamical effects revealed by the c2 framework. Thus,the transformation of Equation (14.7) must be understood as being basedon measurements made by a standard rod and a standard atomic clock.Measurements made using different rods or clocks would not yield thesame relationships between the two frames. Nevertheless, for someexperiments, such as the JPL experiment or the two-photon-absorption(TPA) experiment which involved only a single type of atom or atomicclock and the propagation of light, the Mansouri-Sexl formalism can beput to good use (Will, 1992b).

In the JPL experiment, for example, the phases of two hydrogen maseroscillators of frequency v separated by a baseline of L = 21 kilometers were

compared by propagating a laser carrier signal along a fiberoptic linkconnecting them. The phase comparisons could be performed simul-taneously at each end using signals propagated in both directions along thefiber. The phase differences were monitored over a five-day period as thebaseline rotated relative to the Earth's velocity w through the cosmicmicrowave background. The predicted phase differences as a function ofdirection are, to first order in w

A ^ « 2 a ( w n - w n 0 ) , (14.9)

where $ = 2nvL, and where n and n,, are unit vectors along the direction ofpropagation of the light, at a given time, and at the initial time, respectively.The initial phase difference has been set arbitrarily to zero; this istantamount to choosing a convention for synchronization. The observedlimit on a diurnal variation in the relative phase resulted in the bound|a| < 1.8 x 1(T4; this gives a limit on (c~2-l) using Equation (14.6). Thebound from the TPA experiment was |«| < 1.4 x 10~6. The best bound fromsuch isotropy experiments comes from "Mossbauer-rotor" experiments(Champeney, Isaak and Khan, 1963; Isaak, 1970), which test the isotropyof time dilation between a gamma ray emitter on the rim of a rotating diskand an absorber placed at the center; the result is |a| < 9 x 10~8.

(d) Other frameworks for analysing EEPA number of alternative formalisms have been developed to

analyse EEP and Schiff's conjecture in detail. Ni (1977) devised anextension of the THe/i formalism in which the action for test particles andelectromagnetic fields couples minimally to a metric gm, but in which thereis an additional electromagnetic coupling to a scalar field of the form(167T)-1 J \/(-g)</>£"v'"'FllvFpacl4x. EEP is satisfied if and only if <f> s 0. On theother hand, electromagnetically bound test bodies satisfy WEP, butexperience anomalous torques if /dt < 0.1 Ho, where Ho is the Hubble parameter was set by showing thatthis electromagnetic coupling would cause rotations in the plane ofpolarization of radiation from distant radio sources, which are notobserved (Carroll and Field, 1992). Ni (1987) has extended this formalismto incorporate non-abelian gauge fields.

Bekenstein (1982) focussed on a particular model for violation of EEP:a coupling of electromagnetism to a dynamical, dimensionless scalar fieldthat manifests itself as a spacetime variation of the fine structure constant.The dynamics of the scalar field is determined more or less uniquely by a

An Update 328

set of reasonable postulates together with the requirement that thefundamental scale that determines its dynamics be of the order of but nosmaller than the Planck scale (Gh/c3)1'2 x 10~33 cm. He found, however,that the spatial variation of the field is so severely constrained by theEotvos experiment that the length scale must be smaller than 10~3 Plancklengths. This, he argued, rules out any variability of the fine structureconstant.

Coley (1982, 1983a,b,c) studied an extension of the THe/i formalism tonon-metric theories that possess both a metric and an independent affineconnection, retaining the restriction to static, spherically symmetric (SSS)fields. The model contains seven independent functions, whose forms canbe constrained by various experimental tests of EEP.

Horvath et al. (1988) extended the THe/x formalism to include weakinteractions in a "gravitationally modified" standard model. Such aformalism could be used to calculate explicitly the possible WEP-violatingeffects of weak interactions, which were only estimated by Haugan andWill (1976) (see also Fischbach et al., 1985; Lobov, 1990).

(e) Is spacetime symmetric ?Our statement of the metric theory postulates included the

assumption that the metric is symmetric, corresponding to a standardpseudo-Riemannian spacetime. It turns out that a nonsymmetric metric,even if coupled to matter fields in a universal way, does not satisfy thepostulates of EEP (Will, 1989; Mann and Moffat, 1981). Consider a classof theories in which the action for charged test particles and elec-tromagnetic fields coupled to gravity is given by the "minimally coupled"form of Equation (3.20) where now gm =£ gyft and g1" is the inverse of gm

such that gM*gm = g^g^ = S"v. (Mann, Palmer and Moffat (1989) andGabriel et al. (1991a) consider a broader class of electromagnetic actions,but the minimally coupled version illustrates the essential features.) Weconsider nonsymmetric theories having the property that, in an SSSgravitational field, a Cartesian coordinate system can be found in whichthe nonsymmetric g^ takes the form

9W = -T(r), g^H^Sij, gm = -ga = L(r)n, (14.10)

where T, H, and L are functions of r = |x|, «,. = xjr. The inverse of gm isgiven by

gm = g~l

(14.11)

where (-g) = H3T{\~L2/HT). Substituting into Equation (3.20), andidentifying Fm = Et and Ftj = eijkBk, we obtain

/ = - ! % f(f-W2«fc + £ea UtfAa J a J

+ - [{eE2-n-\B2-co{n-Bf]}dix, (14.12)ore J

where

e = (H/T)ll2(\ -L2/HTyw, ft = (/f/T)1/2(l -L2/HT)i/2,

co = L2/HT. (14.13)

Apart from the term co(n � B)2, this action is that of the THefi formalism.The condition for validity of EEP, s = fx = (H/T)v2 for all r is violated bythe action (14.12), if L ^ 0 (nonsymmetric metric).

For example, in the THs/i formalism, the acceleration of an electricallyneutral, composite body of charged particles with total mass M andinternal electrostatic energy £ES is given by Equation (2.117), dropping themagnetostatic terms. In order to apply this directly to nonsymmetrictheories, it suffices to show that the to(n-B)2 term in Equation (14.12)makes no contribution to a, to electrostatic order. This can be shown bydirect calculation, extending the Lightman-Lee procedure appropriately;it can be seen heuristically by noting that, to the required order,0[g(EES/M)], the only part of the vector potential A that results in acontribution to the acceleration of the composite body is that partproduced by the acceleration of each charged particle in the externalgravitational field. This part of A is therefore parallel to g, and thus to n,and hence the relevant part of n � B vanishes; as a consequence, the a>(n � B)2

term will have no effect, to the electrostatic order considered. Higher-ordermagnetostatic effects will result from that term, but, as we saw in Section2.4(a), these are significantly smaller than electrostatic effects. For systemsthat move through the SSS field with velocity V, the co(n � B)2 terms will alsoproduce effects of order V2EEB/M (Gabriel et al., 1991b).

The given forms of s and fi imply that T^H^E^ = 1. Assuming that7"« 1 + 0(m/r), Hx\+ O(m/r), and L2 <4 TH, where m is the mass ofthe external source, we obtain from Equations (14.13) and (2.83),

To as (r2/2m)dL2/dr. (14.14)

Thus nonsymmetric theories in this class violate WEP, and consequently,Eotvos experiments can test their validity. The significance of the resulting

An Update 330

constraints on the nonsymmetric part of the metric will depend on thespecific form of L(r).

In one nonsymmetric theory, Moffat's NGT (Moffat 1979a,b, 1987,1989; Moffat and Woolgar, 1988; for a recent review see Moffat, 1991),L(r) = /2/>2, where /2 is a parameter (which can be negative) defined bye1 = f^/(-g)S°d3x, where S" is a conserved current (W(-g)S")i/1 = 0)of hitherto unspecified microscopic origin, and the integral is over thegravitating source. Thus, in this theory, with the minimal coupling ofEquation (3.20), r o = � 2^/mr3. However, because of an additional mattercoupling in the Lagrangian of NGT, there is an extra WEP-violating termin the gravitational acceleration of a body that depends on the value of its^-parameter, namely, <5a = g(2/2/r3)(/b

2/M), where £ refers to the sourceand <?b refers to the body. Combining the two terms, we obtain for theparameter tj in minimally-coupled NGT,

Thus the constraint placed on NGT will depend on the model adopted forthe f2 parameter. For bulk, electrically neutral, stable matter consisting ofneutrons, protons, and electrons, it is straightforward to show that themost general form of /2 is f2 =fB

2B+fL2L, where B and L are the total

baryon and lepton numbers of the body, and /B2 and fL

2 are arbitrarycoupling parameters (which can be negative) having units of (length)2.Thus tests of WEP will constrain the fB

2 �/L2 plane. Because of the r~3

dependence in Equation (14.15), the most sensitive tests use the Earth asthe gravitating source, and for this purpose, the Eot-Wash III experiment(Adelberger, Stubbs et al., 1990) is the most stringent. We determine EJM,B/M and L/M for each of the test masses in this experiment, and we notethat, for the Earth, L9 « B^/2.05. The experimental limits from E6t-WashIII then provide the constraints on the coupling parameters shown inFigure 14.4. With the coupling parameters constrained by the rough bound2 x 10"44 cm2, we obtain the limit \£9

2\ < (100 m)2.

It should be noted that this result applies only to the minimally coupledelectromagnetic action. Mann, Palmer and Moffat (1989) have presentedan alternative class of couplings of F^ to the nonsymmetric metric, one ofwhich satisfies WEP to electrostatic order, and thus evades the bound givenabove. In this model, e = ju = (H/T)"2, and so the only EEP-violatingeffects come from the o>(n-B)2 term in Equation (14.12).

In fact, this (n � B)2 term is generic to all nonsymmetric theories, and hasimportant observable consequences. It will produce perturbations in the

4 -

-A -

Figure 14.4. Constraints on j \ and J\ of minimally coupled MoffatNGT from the Eot-Wash III experiment. The hatched region is excluded.

energy levels of an atomic system that depend on the orientation of thesystem's wave function relative to the direction n (anisotropies in inertialmass). Such perturbations can be constrained by energy-isotropy experi-ments of the type used to test local Lorentz invariance (Gabriel et al.,1991b). The violation of EEP by this term also produces observable effectsin the propagation of light, such as polarization dependence in thepropagation of light near the Sun (Gabriel et al., 1991c). One consequenceof this is a depolarization of the Zeeman components of spectral linesemitted by extended, magnetically active regions near the limb of the Sun;observations of the residual polarization of such lines place the stringentbound Ko

2| < (535 km)2, substantially smaller than the values preferred byMoffat (1991).

14.2 The PPN Framework and Alternative Metric Theories of GravityThe PPN framework of Chapter 4 is the standard tool for studying

experiments and gravitational theories in the weak-field slow motion limitappropriate to the solar system. Other versions of the PPN formalism havebeen developed to deal with bodies with strong internal gravity (Nordtvedt,1985), and post-post-Newtonian effects (Epstein and Shapiro, 1980;Fischbach and Freeman, 1980; Richter and Matzner, 1982a,b; Nordtvedt,

An Update 332

1987; Benacquista and Nordtvedt, 1988; Benacquista, 1992). A version ofthe formalism with potentials substantially more complicated than thecanonical version has also been proposed (Ciufolini, 1991).

Despite the experimental bound of co > 500 on the coupling constant ofBrans-Dicke theory, variants of the theory became popular again duringthe 1980s, as a result of developments in cosmology and elementary-particle physics. Inflationary models of cosmology involving Brans-Dicke-like scalar fields coupled to gravity have been developed and studied indetail. Scalar fields coupled to gravity or matter are also ubiquitous inparticle-physics-inspired models of unification, such as string theory. Inmany models, the coupling to matter leads to violations of WEP, which canbe tested by Eotvos-type experiments. In many models the scalar field ismassive; if the Compton wavelength is of macroscopic scale, its effects arethose of a "fifth force" (see Section 14.5). Only if the theory can be cast asa metric theory with a scalar field of infinite range or of range longcompared to the scale of the system in question (solar system) can the PPNframework be applied. If the mass of the scalar field is sufficiently large thatits range is microscopic, then, on solar-system scales, the scalar field issuppressed, and the theory is typically equivalent to general relativity. Inany event, the bounds from solar system experiments can provideconstraints on such speculations. The post-Newtonian limit of a class ofmassive scalar-tensor theories, including the Yukawa potentials that resultfrom the massive scalar field, was derived by Helbig (1991) and Zaglauer(1990).

14.3 Tests of Post-Newtonian Gravity(a) The classical testsImprovements in the accuracy of very long baseline interferometry

(VLBI) to the level of hundreds of microarcseconds made new tests of thedeflection of light possible. For example, a series of transcontinental andintercontinental VLBI quasar and radio galaxy observations madeprimarily to monitor the Earth's rotation ("VLBI" in Figure 14.5) wassensitive to the deflection of light over almost the entire celestial sphere (at90° from the Sun, the deflection is still 4 milliarcseconds). The data yieldeda value 5(1+7)= 1.000 + 0.001, comparable to the Viking test of theShapiro time delay (Robertson and Carter, 1984; Robertson, Carter andDillinger, 1991; Shapiro, 1990). A measurement of the deflection of lightby Jupiter using VLBI was recently reported (Truehaft and Lowe, 1991);the predicted deflection of about 300 microarcseconds was seen with about50% accuracy.


1.2

1.1

1.0

0.9

1.1

1.0

0.9

0.8 -

'ft'T 1919

Expedition

Deflection of light� opticalo radio

VLBI VLBI

Expedition

Shapiro time delay4 two-wayD one way

J L _L J_

PSR 1937 + 21Viking IVoyagerJVo

1920 1930 1940 1950 1960 1970Year of experiment

1980 1990

Figure 14.5. Measurements of the coefficient (l + y)/2 from lightdeflection and time delay measurements. The general relativity value isunity. Arrows denote anomalously large values from 1929 and 1936expeditions. Shapiro time-delay measurements using Viking spacecraftand VLBI light deflection measurements yielded agreement with generalrelativity to 0.1 per cent.

Recent" opportunistic " measurements of the Shapiro time delay includea measurement of the one-way time delay of signals from the millisecondpulsar PSR 1937 + 21 (Taylor, 1987), and measurements of the two-waydelay from the Voyager 2 spacecraft (Krisher, Anderson and Taylor,1991). The results for the coefficient 5(1 + y) of all light deflection and time-delay measurements performed to date are shown in Figure 14.5.

Continued radar ranging to Mercury and the other planets has resultedin further improvements in the measured perihelion shift of Mercury. Afterthe perturbing effects of the other planets have been accounted for, theexcess shift is now known to about 0.1 % (Shapiro 1990) with the resultthat & = 42"98 (1.000 + 0.001)^' [see Equation (7.55)]. [For an amusinghistory of how the theoretical value of 42"98 has been misquoted in

An Update 334

numerous books, including the first edition of this book, see Nobili andWill (1986).] In addition, the controversy over the solar quadrupolemoment may be approaching a resolution. Beginning around 1980, theobservation and classification of modes of oscillation of the Sun have madeit possible to obtain information about its internal rotation rate, therebyconstraining the possible centrifugal flattening that leads to an oblateness;current results favor a value J2 as 1.7 x 10~7 (Brown et al., 1989), makingthe correction to Xp [Equation (7.55)] from the solar quadrupole momentsmaller than the experimental error. If further studies of solar oscillationscontinue to support this interpretation, the perihelion shift of Mercury willonce again be a triumph for general relativity.

(b) Parametrized post-Newtonian ephemeridesImprovements in the accuracy of planetary and spacecraft

tracking and in the ability of theorists to model their motions has made ituseful to adopt a slightly different attitude toward tests such as the timedelay and the perihelion shift. As we remarked in Section 7.2, themeasurement of the time delay of light involves a multiparameter least-squares fit of tracking data to a model for the trajectory of the planet orspacecraft and for the propagation of the radar signal. The " time delay "as a distinct phenomenon is never measured directly. Similarly the"perihelion shift" of Mercury is not observed, rather the least-squaresmethod estimates various parameters (ft, y, J2, etc.) that determine part ofthe shift. Although this point of view takes some of the glamour out of thesubject, it is the standard approach in the analysis of relativistic solar-system dynamics.

The goal is to determine the parameters in a model for the relativisticmotion of bodies in the solar system. One might call this model a"parametrized post-Newtonian ephemeris". The current model (Hellings,1984; Reasenberg, 1983) includes such parameters as: (i) the initialpositions and velocities of the nine planets and the Moon; (ii) the massesof the planets, and of the three asteroids Ceres, Pallas and Vesta; (iii) themean densities of 200 of the largest asteroids whose radii are known; (iv)the Earth-Moon mass ratio; (v) the value of the astronomical unit; (vi)PPN parameters, y, fi, a,,...; (vii) J2 of the Sun; (viii) other parametersrelevant to specific data sets, such as station locations, rotation andlibration of bodies, known systematic errors or corrections, etc. The modelalso includes PPN equations of motion for the bodies, and PPN equationsfor the propagation of the tracking signal. In some applications, the modelalso includes equations that tie the coordinate system associated with the


ephemerides to a system tied to distant stars via VLBI. The output of themodel might, for example, be a predicted " range " (round-trip travel time)from a particular station to a planet or spacecraft at a particular epoch, asa function of the parameters. The parameters are then adjusted in the least-squares sense to minimize the difference between the predicted andobserved ranges.

One circumstance that has made it possible to obtain improveddeterminations of the parameters is the ability to combine different datasets in an unambiguous way. In the orbit of Mercury, the effects of /?, y andJ2 are large, but not separable using Mercury radar data alone. In the orbitof Mars, their effects are much smaller (and that of J2 smaller still than thatof fi and y), but the accuracy of Viking lander ranges is so much better thatthe effects can be seen more clearly than with Mercury data. Lunar laser-ranging data has also been incorporated into the data set. In the comingyears, analysis of PPN ephemerides will further improve our knowledge ofPPN parameters, J2, and the dynamics of the solar system (for reviews, seeKovalevsky and Brumberg, 1986; and Soffel, 1989).

(c) Tests of the strong equivalence principleRecent analyses of lunar laser-ranging data continue to find no

evidence, within experimental uncertainty, for the Nordtvedt effect(Section 8.1). Their results for n [Equation (8.9)] are

n = 0.003+0.004, (Dickey et al., 1989)

n = 0.000 ± 0.005, (Shapiro, 1990)

n = 0.0001 ±0.0015, (Muller et al., 1991) (14.16)

where the quoted errors are \a, obtained by estimating the sensitivity of nto possible systematic errors in the data or in the theoretical model.

The third of these results represents a limit on a possible violation ofWEP for massive bodies of 7 parts in 1013 (compare Figure 14.1). ForBrans-Dicke theory, these results force a lower limit on the couplingconstant co of 600. Nordtvedt (1988a) has pointed out that, at this level ofprecision, one cannot regard the results of lunar laser ranging as a cleantest of SEP because the precision exceeds that of laboratory tests of WEP.Because the chemical compositions of the Earth and Moon differ in therelative fractions of iron and silicates, an extrapolation from laboratoryEotvos-type experiments to the Earth-Moon system using various non-metric couplings to matter (Adelberger, Heckel et al., 1990) yields boundson violations of WEP only of the order of 2 x 10"!2. Thus if lunar laser

An Update 336

Table 14.1. Constancy of the gravitational constant

Method

Lunar Laser RangingViking Radar

Binary Pulsar"Pulsar PSR 0655 + 64"

G/G (10-12 yr"1)

0+112 + 4

-2±1011±11

<55

Reference

Miiller et al. (1991)Hellings et al. (1983)Shapiro (1990)Damour and Taylor (1991)Goldman (1990)

" Bounds dependent upon theory of gravity in strong-field regime and onneutron star equation of state.

ranging is to test SEP at higher accuracy, tests of WEP must keep pace; tothis end, a proposed satellite test of the equivalence principle (Section 14.4)will be an important advance.

In general relativity, the Nordtvedt effect vanishes; at the level of severalcentimeters and below, a number of non-null general relativistic effectsshould be present (Mashhoon and Theiss, 1991; Gill et al., 1989;Nordtvedt, 1991).

An improved limit on the "preferred frame" PPN parameter a, of4x 10~4 was reported by Hellings (1984), from analyses of Mercury andMars ranging data. Nordtvedt (1987) has placed an improved bound onthe parameter <x2 of 4 x 10~7 by showing that the failure of conservationof angular momentum in a frame moving relative to the universe whena2 / 0 [Equations (4.104) and (4.114)] would lead to anomalous torqueson the Sun that would cause the angle between its spin axis and the eclipticto be arbitrarily far from its observed value.

Improved observational constraints have recently been placed on G/G,using ranging measurements to Viking (Hellings et al., 1983; Shapiro,1990), lunar laser-ranging measurements (Miiller et al., 1991), and pulsartiming data (Damour, Gibbons and Taylor, 1988; Goldman, 1990;Damour and Taylor, 1991). Recent results are shown in Table 14.1. Thebest limits on G/G come from ranging measurements to Viking. Thecombination of three factors: (i) extremely accurate range measurementsmade possible by anchoring of the landers and orbiters, (ii) the unexpec-tedly long lifetime of the spacecraft (Lander 2 survived for 6 years), and (iii)the ability to combine Viking data consistently with other data sets such asMercury and Venus passive radar, Mariner 9 radar and lunar laser-rangingdata, made it possible to look for G/G at levels below 10~u yr""1. The majorfactors limiting the accuracy of these estimates (and responsible in part for

the difference between the two Viking estimates in Table 14.1, despite beingbased upon similar data sets) are the uncertainty in the masses anddistributions of the asteroids, and the level of correlations among the manyparameters to be estimated in the model. It has been suggested that radarobservations of a Mercury orbiter over a two-year mission (30 cm accuracyin range) could yield A(G/G) ~ l O ^ y r 1 (Bender et al., 1989).

Although bounds on G/G using solar-system measurements can beobtained in a phenomenological manner through the simple expedient ofreplacing G by G0 + G0(t�10) in Newton's equations of motion, the samedoes not hold true for pulsar and binary pulsar timing measurements(Nordtvedt 1990). The reason is that, in theories of gravity that violateSEP, the "mass" and moment of inertia of a gravitationally bound bodymay vary with variation in G. Because neutron stars are highly relativistic,the fractional variation in the mass can be comparable to AG/G, theprecise variation depending both on the equation of state of neutron starmatter and on the theory of gravity in the strong-field regime. Thevariation in the moment of inertia affects the spin rate of the pulsar, whilethe variation in the mass can affect the orbital period in a manner thatcan add to or subtract from the direct effect of a variation in G, given byPJPb = -jG/G. Thus, the bounds quoted in Table 14.1 for the binarypulsar PSR 1913 + 16 and the pulsar PSR 0655 + 64 are theory dependentand must be treated as merely suggestive.

(d) Tests of post-Newtonian conservation lawsOf the five "conservation law" PPN parameters £� f2, £3, £4, and

<x3, only three, C2> C3 and <x3, have been constrained directly with anyprecision. The bound |<x3| < 2 x 10~10 was obtained in Section 9.3 usingpulsar timing measurements.

A remarkable planetary test of Newton's third law was reported byBartlett and van Buren (1986), leading to an improved constraint on £3

(Section 9.2). They noted that current understanding of the structure of theMoon involves an iron-rich, aluminum-poor mantle whose center of massis offset about 10 km from the center of mass of an aluminum-rich, iron-poor crust. The direction of offset is toward the Earth, about 14° to the eastof the Earth-Moon line. Such a model accounts for the basaltic mariawhich face the Earth, and the aluminum-rich highlands on the Moon's farside, and for a 2 km offset between the observed center of mass and centerof figure for the Moon. Because of this asymmetry, a violation of Newton'sthird law for aluminum and iron would result in a momentum non-conserving self-force on the Moon, whose component along the orbital

An Update 338

direction would contribute to the secular acceleration of the lunar orbit.Improved knowledge of the lunar orbit through lunar laser ranging, and abetter understanding of tidal effects in the Earth-Moon system (which alsocontribute to the secular acceleration) through satellite data, severely limitany anomalous secular acceleration, with the resulting limit

<4xlO-12. (14.17)K/«p ) F e

The resulting limit on £3 is |C3| < 1 x 10~8.Data from the binary pulsar PSR 1913 + 16 have finally permitted a

strong test of the post-Newtonian " self-acceleration" effect described inSection 9.3, Equation (9.42). Assuming a theory not too different fromgeneral relativity (but with the possibility of C2 # 0) so that we can use theaccurate values for the pulsar and companion masses obtained from timingdata (Section 14.6(a)), together with the observational bound on variationsin the pulsar period \Pp\ < 4 x 10~30 s~' (Taylor and Weisberg, 1989;J. Taylor, private communication), we obtain from Equation (9.48) thebound \C2\ < 4 x 10~5 (Will, 1992c).

(e) Other tests of post-Newtonian gravityA gyroscope moving through curved spacetime suffers a geodetic

precession of its axis given by dS/dt = Q x S, where £2 = (7 + j)v x V{7,where v is the velocity of the gyroscope and U is the Newtoniangravitational potential of the source [Equation (9.5)]. The Earth-Moonsystem can be considered as a " gyroscope ", with its axis perpendicular tothe orbital plane. The predicted geodetic precession here is about 2arcseconds per century, an effect first calculated by de Sitter. This effect hasnow been measured to about 2 % using lunar laser-ranging data (Bertotti,Ciufolini and Bender, 1987; Shapiro et al., 1988; Dickey, Newhall andWilliams, 1989; Shapiro, 1990).

Current values or bounds for the PPN parameters are summarized inTable 14.2.

14.4 Experimental Gravitation: Is there a Future?Although the golden era of experimental gravitation may be over,

there remains considerable opportunity both for refining our knowledge ofgravity, and for exploring new regimes of gravitational phenomena.Nowhere is the intellectual vigor and continuing excitement of this fieldmore apparent than in the ideas that have been developed for experimentsand observations to push us to the frontiers of knowledge.

Theory and Experiment in Gravitational Physics

Table 14.2. Current limits on the PPN parameters

339

Parameter

7

ia,

a2

a3

Vr4i

C3

Experiment

Time delayLight deflection

Perihelion shiftNordtvedt effect

Earth tides

Orbital preferred-frameeffects

Earth tidesSolar spin precession

Perihelion shiftAcceleration of pulsars

Nordtvedt effect

Self-accelerationNewton's 3rd law

Value or limit

1.000 ±0.0021.000 + 0.002

1.000 + 0.0031.000 ±0.001

< 10~3

< 4 x l O "

<4x lO- 4

<4xlO~7

< 2 x 10~7

<2xlO~'°

< 1.5 xlO"3

<4xlO~5

< 10"8

Remarks

Viking rangingVLBI

J2 = 10~7 assumedrj � 4/?�y �3 assumed

Gravimeter data

Combined solar system data

Gravimeter dataAssumes alignment of solar

equator and ecliptic are notcoincidental

Statistics of dP/dt for pulsars

Lunar laser ranging

Binary pulsarLunar acceleration

" Here rj is a combination of other PPN parameters given by}j = 4/?�y � 3 � y<J � a,+3a2�§£,� |C2. In many theories of gravity, £, = a, = £,. = 0.

(a) GP-B and the search for gravitomagnetismAccording to general relativity, moving or rotating matter should

produce a contribution to the gravitational field that is the analogue of themagnetic field of a moving charge or a magnetic dipole (for reviews of the" gravitoelectromagnetic " analogy for weak-field gravity, see Braginsky,Caves and Thorne, 1977; Ciufolini, 1989). Although gravitomagnetismplays a role in a variety of measured relativistic effects, it has not been seento date, isolated from other post-Newtonian effects [Nordtvedt (1988b) hasdiscussed the extent to which it has been seen indirectly]. The RelativityGyroscope Experiment (Gravity Probe B or GP-B) at Stanford University,in collaboration with NASA and Lockheed Corporation, has reached theadvanced stage of development of a space mission to detect thisphenomenon directly, in addition to the geodetic precession discussed inSection 9.1 (Everitt et al., 1988). A set of four superconducting-niobium-coated, spherical quartz gyroscopes will be flown in a low polar Earthorbit, and the precession of the gyroscopes relative to the distant stars will

An Update 340

be measured. For a polar orbit at about 650 km altitude, the predictedsecular angular precession rate is j(l +y + |a,) 42 x 10"3 arcsec/yr [Equa-tion (9.11)]. The accuracy goal of the experiment is about 0.5 milliarc-seconds per year. A full-size flight prototype of the instrument package hasbeen tested as an integrated unit. Current plans call for a test of the finalflight hardware on the Space Shuttle followed by a Shuttle-launchedexperiment a few years later.

Another proposal to look for an effect of gravitomagnetism is tomeasure the relative precession of the line of nodes-of a pair of laser-rangedgeodynamics satellites (LAGEOS), with supplementary inclination angles;the inclinations must be supplementary in order to cancel the dominantrelative nodal precession caused by the Earth's Newtonian gravitationalmultipole moments (Ciufolini, 1989). This is a generalization of the vanPatten-Everitt proposal involving pairs of polar-orbiting satellites de-scribed in Section 9.1. Current plans involve a joint project of NASA andthe Italian Space Agency. A third proposal envisages orbiting an array ofthree mutually orthogonal, superconducting gravity gradiometers aroundthe Earth, to measure directly the contribution of the gravitomagnetic fieldto the tidal gravitational force (Braginsky and Polnarev, 1980; Mashhoonand Theiss, 1982; Mashhoon, Paik and Will, 1989).

(b) Space tests of the Einstein equivalence principleThe concept of an Eotvos experiment in space has been developed,

with the potential to test WEP to 10"17 (Worden, 1988). Known as theSatellite Test of the Equivalence Principle (STEP), the project is a jointeffort of NASA and the European Space Agency. If approved, it could belaunched in the year 2000.

The gravitational redshift could be improved to the 10~9 level, andsecond-order effects and the effects of J2 of the Sun discerned, by placinga hydrogen maser clock on board Solar Probe, a proposed spacecraftwhich would travel to within four solar radii of the Sun (Vessot, 1989).

(c) Improved PPN parameter valuesA number of advanced space missions have been proposed in

which spacecraft orbiters or landers and improved tracking capabilitiescould lead to significant improvements in values of the PPN parameters(see Table 14.2), of J2 of the Sun, and of G/G. For example, a Mercuryorbiter, in a two-year experiment, with 3 cm range capability, could yieldimprovements in the perihelion shift to a part in 104, in y to 4 x 10~5, in G/Gto 10~14 yr 1 , and in J2 to a few parts in 108 (Bender et al., 1989).


(d) Probing post-post-Newtonian physics in the solar systemIt may be possible to begin to explore the next level of corrections

to Newtonian theory beyond the post-Newtonian limit, into the post-post-Newtonian regime. One proposal is to place an optical interferometer withmicroarcsecond accuracy into Earth orbit. Such a device would improvethe deflection of light to the 10~6 level, and could possibly detect thesecond-order term, which is of order 10 microarcseconds at the limb(Reasenberg et al., 1988). Such a measurement would be sensitive to a new"PPPN" parameter, which has not been measured to date.

(e) Gravitational-wave astronomyA significant part of the field of experimental gravitation is

devoted to designing and building sensitive devices to detect gravitationalradiation and to use gravity waves as a new astronomical tool. Thisimportant topic has been reviewed thoroughly elsewhere (Thorne, 1987).

14.5 The Rise and Fall of the Fifth ForceA clear example of the role of " opportunism" in experimental

gravity since 1980 is the story of the "fifth force". In 1986, as a result of adetailed reanalysis of Eotvos' original data, Fischbach et al. (1986, 1988)suggested the existence of a fifth force of nature, with a strength of abouta percent that of gravity, but with a range (as defined by the range A of aYukawa potential, e~'ix/r) of a few hundred meters. This proposaldovetailed with earlier hints of a deviation from the inverse-square law ofNewtonian gravitation derived from measurements of the gravity profiledown deep mines in Australia [for a review, see Stacey et al. (1987)], andwith ideas from particle physics suggesting the possible presence of verylow-mass particles with gravitational-strength couplings [for reviews, seeGibbons and Whiting (1981), Fujii (1991)]. During the next four yearsnumerous experiments looked for evidence of the fifth force by searchingfor composition-dependent differences in acceleration, with variants of theEotvos experiment or with free-fall Galileo-type experiments. Althoughtwo early experiments reported positive evidence, the others yielded nullresults. Over the range between one and 104 meters, the null experimentsproduced upper limits on the strength of a postulated fifth force of between10~3 and 10~6 the strength of gravity (Table 14.3). Interpreted as tests ofWEP (corresponding to the limit of infinite-range forces), the results of thefree-fall Galileo experiment, and of the Eot-Wash III experiment areshown in Figure 14.1 (Niebauer, McHugh and Faller, 1987; Adelberger,Stubbs et al., 1990). At the same time, tests of the inverse square law of

Table 14.3. Composition-dependent tests of the fifth force

Experimentnameor place

Palisades, NYEot-WashBoulder, COEot-WashIndex, WAMontanaParisBombaySnake RiverEot-Wash IIJapanFlorenceBombay IIIrvine, CAEot-Wash IIIIndex, WA IIFlorence IIJapan II

Year

198619861987198719871988198819881988198819881988198919891989198919891990

Method

FlotationTorsion balanceFree fallTorsion balanceTorsion balanceTorsion balanceBeam balanceTorsion balanceTorsion balanceTorsion balanceFree fallFlotationTorsion balanceTorsion balanceTorsion balanceTorsion balanceFlotationFree fall

Substancecompared

Cu/H2OCu/BeCu/UBe/AlBe/AlCu/CH2

Cu/Pb, C/PbCu/PbC/PbBe/AlAl/Cu, Al/CPlastic/H2OCu/PbCu/PbCu/Be, Al/BeCu/CH2

Plastic/HpAl/Cu,Al/C, Al/Be

Source offorce

CliffHillsideEarthHillsideCliffHillsideLead, brass massesLead massesWater in lockLead massesEarthMountainLead massesLead massesHillsideCliffMountainEarth

Fifthforce?

YesNoNoNoYesNoNoNoNoNoNoNoNoNoNoNoNoNo

gravity were carried out by comparing variations in gravity measurementsup tall towers or down mines or boreholes with gravity variations predictedusing the inverse square law together with Earth models and surfacegravity data mathematically "continued" up the tower or down the hole.Early experiments reported significant differences between predicted andobserved gravity, but these were subsequently explained as resulting fromsystematic errors in the upward continuation results caused by insuf-ficiently controlled biases in the distribution of surface gravity measure-ments, as well as by poorly-accounted-for effects of distant geologicalstructures such as hills and ridges. Independent tower, borehole andseawater measurements now show no evidence of a deviation from theinverse square law (Thomas et al., 1989, Jekeli, Eckhardt and Romaides,1990; Thomas and Vogel, 1990; Speake et al., 1990; Zumberge et al.,1991). The consensus at present is that there is no credible experimentalevidence for a fifth force of nature. For reviews, see Fischbach andTalmadge (1989, 1992), Will (1990b), Adelberger et al. (1991); for acomplete bibliography on the fifth force, see Fischbach et al. (1992).

14.6 Stellar-System Tests of Gravitational Theory(a) The binary pulsar and general relativityThe binary pulsar PSR 1913 + 16 has lived up to, indeed exceeded,

all expectations that it would be an important new testing ground forrelativistic gravity (Chapter 12). Instrumental upgrades at the Areciboradio telescope where the observations are carried out, and improved dataanalysis techniques have resulted in accuracies in measuring times ofarrival (TOA) of pulses at the 15 /us level. Analysis of this TOA data usesa timing model developed by Damour, Deruelle and Taylor (Damour andDeruelle, 1986; Damour and Taylor, 1992) superceding earlier treatmentsby Haugan, Blandford, Teukolsky and Epstein that were described inSection 12.1 [see Haugan (1985) and references therein].

The observational parameters of this model that are obtained from aleast squares solution of the arrival time data fall into three groups: (i) non-orbital parameters, such as the pulsar period and its rate of change, and theposition of the pulsar on the sky; (ii) five "Keplerian" parameters, mostclosely related to those appropriate for standard Newtonian systems, suchas the eccentricity e and the orbital period Ph; and (iii) a set of "post-Keplerian " parameters. The five main post-Keplerian parameters are <<»>,the average rate of periastron advance; y, the amplitude of delays in arrivalof pulses caused by the varying effects of the gravitational redshift and timedilation as the pulsar moves in its elliptical orbit at varying distances from

An Update 344

the companion and with varying speeds [denoted <$ in Section 12.1(d)]; Pb,the rate of change of orbital period, caused predominantly by gravitationalradiation damping; and r and s = sin i, respectively the "range" and"shape" of the Shapiro time delay caused by the companion, where Us theangle of inclination of the orbit relative to the plane of the sky.

In general relativity, these post-Keplerian parameters can be related tothe masses of the two bodies and to measured Keplerian parameters by theequations (Section 12.2)

<«> = 3(27t/Pb)5/3w2/3(l -e2Y\ (14.18a)

y = e(Pb/27iyi3rn2m-m(\ +m2/m), (14.18b)

Pb = -(192K/5)(27rm/Pb)5'3(^/m)(l + ge2 + ||e4)(l -e2)-1'2,

(14.18c)

s^sini, (14.18d)

r = m2, (14.18e)

where m, and m2 denote the pulsar and companion masses, respectively,m = m, + m2 is the total mass, and n = mxm2/m is the reduced mass. Theformula for <a>> ignores possible non-relativistic contributions to theperiastron shift, such as tidally or rotationally induced effects caused by thecompanion [Section 12. l(c)]. The formula for Pb represents the effect ofenergy loss through the emission of gravitational radiation, and makes useof the "quadrupole formula" of general relativity. For a recent survey ofthe quadrupole and other approximations for gravitational radiation, seeDamour (1987). It ignores other sources of energy loss, such as tidaldissipation [Section 12.1(f)].

The values for the Keplerian and post-Keplerian parameters shown inTable 14.4 are from data taken through December 1990 (Taylor et al.,1992).

Plotting the constraints the three post-Keplerian parameters imply forthe two masses w, and m2, via Equations (14.18), we obtain the curvesshown on Figure 14.6. It is useful to note that Figure 12.2 correspondsessentially to the inset in Figure 14.6. From <a>> and y we obtain the valuesm, = 1.4411(7) MQ and w 2 = 1.3873(7) Mo, where the number inparenthesis denotes the error in the last digit. Equation (14.18c) thenpredicts the value Pb = �2.40243(5) x 10~12. In order to compare thepredicted value for Pb with the observed value, it is necessary to take intoaccount the effect of a relative acceleration between the binary pulsarsystem and the solar system caused by the differential rotation of thegalaxy. This effect was previously considered unimportant when Pb was

Table 14.4. Parameters of the binary pulsar PSR 1913 + 16"

Parameter

(i) 'Physical' parametersRight ascensionDeclinationPulsar periodDerivative of period2nd derivative of period

(ii) 'Keplerian' parametersProjected semimajor axisEccentricityOrbital periodLongitude of periastronJulian ephemeris date of periastron

(iii) 'Post-Keplerian' parametersMean rate of periastron advanceGravitational redshift and time dilationOrbital period derivative

Symbol (units)

a<5Pp (ms)

ap sin i (light � sec)e

P*(»)«o(°)Ta (MJD)

<ri>> Cyr 1 )y(ms)Pb (10-12)

Value

19h13m12.s46549(15)16°01'08':i89(3)59.029997929883(7)8.62629(8) x 10"18

< 4 x 10^30

2.341759(3)0.6171309(6)27906.9807807(9)226.57531(9)46443.99588321(5)

4.226628(18)4.294(3)-2.425(10)

3 Numbers in parentheses denote errors in last digit.

An Update 346

com

pani

on

o

Mas

s

1.41

1.40

1.39

1.38

1.37

\ ' \ '

-

i�

�3rovo

1

2

1

(

� ' - .

1

1 1 ^s

-

) 1 2 3 -

- " " -

�

' � � . _ \ . -

1.42 1.43 1.44 1.45Mass of pulsar (M Q )

1.46

Figure 14.6. Constraints on masses of pulsar and companion from dataon PSR 1913 + 16, assuming general relativity to be valid. The width ofeach strip in the plane reflects observational accuracy, shown as apercentage. The inset shows the three constraints on the full mass plane;intersection region (a) has been magnified 400 times for the full figure.

known only to 10% accuracy [Section 12.1(f)(iii)]. Damour and Taylor(1991) carried out a careful estimate of this effect using data on the locationand proper motion of the pulsar, combined with the best informationavailable on galactic rotation, and found

P ° A L ~ -(1 .7 + 0.5) xlO"14. (14.19)

Subtracting this from the observed Pb (Table 14.4) gives the residual

P£BS = -(2.408 ± 0.010[OBS]±0.005[GAL]) x lO"12, (14.20)

which agrees with the prediction, within the errors. In other words,

pGR-5 g= = 1.0023±0.0041(OBS)±0.0021(GAL). (14.21)

The parameters r and J are not yet separately measurable with interesting

accuracy for PSR 1913 + 16 because the 47° inclination of the orbit doesnot lead to a substantial Shapiro time delay.

The internal consistency among the measurements is also displayed inFigure 14.6, in which the regions allowed by the three most preciseconstraints have a single common overlap. This consistency provides a testof the assumption that the two bodies behave as "point" masses, withoutcomplicated tidal effects (conventional wisdom holds that the companionis also a neutron star), obeying the general relativistic equations of motionincluding gravitational radiation. It is also a test of the Strong EquivalencePrinciple (SEP), in that the highly relativistic internal structure of theneutron star does not influence its orbital motion or the gravitationalradiation emission, as predicted by general relativity.

(b) A population of binary pulsars ?In 1990, two new massive binary pulsars similar to PSR 1913 + 16

were discovered, leading to the possibility of new or improved tests ofgeneral relativity.

PSR 2127+11C. This system appears to be a clone of the Hulse-Taylor binary pulsar (Anderson et al., 1990; Prince et al., 1991): Pb �28,968.36935 s, e = 0.68141, < cb > = 4.457° yr"1 (see Table 14.5). Theinferred total mass of the system is 2.706 + 0.011 MQ. Because the systemis in the globular cluster Ml5 (NGC 7078), observed periods Pb and Pp willsuffer Doppler shifts resulting from local accelerations, caused either bythe mean cluster gravitational field or by nearby stars, that are moredifficult to estimate than was the case with the galactic system PSR1913 + 16. This may limit the accuracy of measurement of the relativisticcontribution to Ph to about 2%.

PSR 1534 + 12. This is a binary pulsar system in our galaxy(Wolszczan, 1991). Its pulses are significantly stronger and narrower thanthose ofPSR1913 + 16,so timing measurements have already reached 3 ^saccuracy. Its parameters are listed in Table 14.5 (Taylor et al., 1992).Because of the short data span, Pb has not been measured to date, but it isexpected that in a few years, the accuracy in its determination will exceedthat of PSR 1913+16. The orbital plane appears to be almost edge onrelative to the line of sight (i « 80°); as a result the Shapiro delay issubstantial, and separate values of the parameters r and 5 have alreadybeen obtained with interesting accuracy. This system may ultimatelyprovide broader and more stringent tests of the consistency of generalrelativity than did the original binary pulsar (Taylor et al., 1992).

Table 14.5. Parameters of new binary pulsars"

Parameter

(i) 'Physical' parametersRight ascensionDeclinationPulsar periodDerivative of period

(ii) ' Keplerian' parametersProjected semimajor axisEccentricityOrbital periodLongitude of periastronJulian ephemeris date of periastron

(iii) 'Post-Keplerian' parametersMean rate of periastron advanceGravitational redshift and time dilationOrbital period derivativeRange of Shapiro delay r (jis)Shape of Shapiro delay 5 = sin i

PSR 1534+12

15h34m47.s686(3)12°05'45"23(3)37.9044403665(4)2.43(8) xlO~18

3.729468(9)0.2736779(6)36351.70270(3)264.9721(16)48262.8434966(2)

1.7560(3)2.05(11)-0.1(6)6.2(1.3)0.986(7)

PSR 2127+11C

21h27m36.s188(4)11°57'26!29(7)30.5292951285(9)4.99(5) xlO"18

2.520(3)0.68141(2)28968.3693(5)316.40(7)47632.4672065(20)

4.457(12)****

" Numbers in parentheses denote errors in last digit.* Values not yet available from data.

(c) Binary pulsars and scalar-tensor theoriesIn Section 12.3, we noted that some theories of gravity, such as the

Rosen bimetric theory, are strongly, even fatally, tested by the binarypulsar. Other theories that are in some sense "close" to general relativityin all their predictions, such as the Brans-Dicke theory, are not so stronglytested, because the apparent near equality of the masses of the two neutronstars leads to a suppression of dipole gravitational radiation.

Despite this, two circumstances have made it worthwhile to focus indetail on binary pulsar tests of scalar-tensor theories. The first is theremarkable improvement in accuracy of the measurements of the orbitalparameters of the binary pulsar since 1980, and the continued consistencyof the observations with general relativity, as described above, togetherwith the discovery of new binary pulsars such as PSR 1534+12. Thesecond is the resurrection of scalar-tensor theories in particle physics andcosmology.

With this motivation, Will and Zaglauer (1989) carried out a detailedstudy of the effects of Brans-Dicke theory in the binary pulsar. Making theusual assumption that both members of the system are neutron stars, andusing the methods summarized in Chapters 10-12, one obtains formulasfor the periastron shift, the gravitational redshift/second-order Dopplershift parameter, and the rate of change of orbital period, analogous to Eqs.(14.18c). These formulas depend on the masses of the two neutron stars, ontheir internal structure, represented by "sensitivities" s and K* and on theBrans-Dicke coupling constant a>. First, there is a modification of Kepler'sthird law, given by Pb/2n = (a}/^my'2. Then, the predictions for <a>>, yand Pb are

3, (14.22a)

y = e(Pb/2nyi3m2m-1'^-"3 (a* + #/n2/w+ <>/?), (14.22b)

Pb = -{\92n/5){2nm/Pby»(ji/m)>$-wF{e)

(14.22c)

where, to first order in £, = (2+a)~\ assuming cop 1, we have

0 = 1 -i(sl +52-2^2), (14.23a)

& = 9[l -#+#(Si +s2-2slS2)], (14.23b)

a2* = l - £ s 2 , (14.23c)

n* = (l-2s2)£, (14.23d)

' F(e) = ft} -e2r7/2[/c,(l +y + P)-K2Qe> + p)]t (14.23e)

An Update 350

(14.23f)

- ^ r 2 ) ] , (14.23g)

(14.23h)

T'=l-sx~s2, (14.23i)

G(e) = (1 -e2)"5/2(l -4e2), (14.23J)

^ = s,-s2. (14.23k)

The quantities ,?a and K* are defined by

dJ^>\ ,* = JdJ±m, (14.24)

and measure the " sensitivity " of the mass wa and moment of inertia /a ofeach body to changes in the scalar field (reflected in changes in G) for fixedbaryon number N (see Section 11.3).

The first term in Pb is the effect of quadrupole and monopolegravitational radiation, while the second term is the effect of dipoleradiation (in Section 11.3 we calculated only the dipole contribution).

Estimating the sensitivities ia and K* using an equation of state forneutron stars sufficiently stiff to guarantee neutron stars of sufficient mass,and substituting into Equations (14.23), we find that the lower limit on a>required to give consistency among the constraints on <co>, y and Pb as inFigure 14.6 is 105. The combination of <a>> and y gives a constraint on themasses that is relatively weakly dependent on £,, thus the constraint on <J isdominated by Ph and is directly proportional to the measurement error inPb; in order to achieve a constraint comparable to the solar system value of2 x 10~3, the error in P^BS would have to be reduced by a factor of five.

Damour and Esposito-Farese (1992) have devised a multi-scalar-tensortheory in which two scalar fields are tuned so that their effects in the weak-field slow-motion regime of the solar system are suppressed, with the resultthat the theory is identical to general relativity in the post-Newtonianapproximation. Yet in the regime appropriate to binary pulsars, it predictsstrong-field SEP-violating effects and radiative effects that distinguish itfrom general relativity. It gives formulae for the post-Keplerian parametersof Equations (14.22) as well as for the paramaters r and s that havecorrections dependent upon the sensitivities of the relativistic neutronstars. The theory depends upon two arbitrary parameters /?' and /?";general relativity corresponds to the values fi' = /?" = 0. It turns out(Taylor et al., 1992) that the binary pulsar PSR 1913+16 alone constrainsthe two parameters to a narrow but long strip in the /?'-/?"-plane that


includes the origin (general relativity) but that could include some highlynon-general relativistic theories. The sensitivity of PSR 1534+ 12 to r ands provides an orthogonal constraint that cuts the strip. In this class oftheories, then, both binary pulsars are needed to provide a strong test.

(d) Other stellar-system tests of gravitational theoryThe suppression of dipole gravitational radiation resulting from

the apparent high symmetry of the binary pulsar system suggests that morestringent tests might be found in systems in which the two compact objectsare dissimilar, for example, two very unequal mass neutron stars or aneutron star and a white dwarf. Several candidate systems have beensuggested.

The 11-minute binary 4U1820-30. This system is believed to consist ofa neutron star and a low-mass helium dwarf in a nearly circular orbit witha period of 68 5.008 s. It is not the most" clean " system available for testinggravitational theory, because its evolution is affected by mass transfer fromthe companion low-mass dwarf onto the neutron star, whose X-ray outputcomprises the data from which the binary nature of the system wasestablished (Stella, Priedhorsky and White, 1987; Morgan, Remillard andGarcia, 1988). In fact the rate of mass transfer is believed to be controlledby gravitational-radiation damping of the orbit. Because of this com-plication, the analysis of the implications of Brans-Dicke theory for thissystem is model dependent. Will and Zaglauer (1989) generalized a class ofgeneral relativistic mass-transfer models to the Brans-Dicke theory, andshowed that, if a limit could be placed on \PJPb\ of 2.7 x 10"7 yr"1,corresponding to an early published limit, then bounds on co as large as 600could be placed, depending on the assumed mass of the neutron star andon the assumed equation of state. Unfortunately, recent observations ofthe system using the Ginga X-ray satellite suggest that Ph is opposite in signto that predicted by a gravitational-radiation-driven mass-transfer model(Tan et al., 1991). Evidently, the binary system is undergoing accelerationeither in the mean gravitational field of the globular cluster in which itresides, or in the field of a nearby third body. Whether the effect of suchlocal accelerations on Pb can be sufficiently understood to yield aninteresting bound on co remains to be seen at present.

PSR 1744-24A. This is an eclipsing binary millisecond pulsar, in theglobular cluster Terzan 5 (Lyne et al., 1990), with a very short orbitalperiod of 1.8 hrs, e = 0, and a mass function of 3.215 x 10"4, indicating alow-mass companion of 0.09 MQ. The asymmetry of the system ispromising for dipole gravitational radiation, but the observations are

An Update 352

complicated by the possibility of cluster accelerations as well as by theapparent presence of a substantial wind from the companion (the cause ofthe eclipses), which may complicate the orbital motion. Nevertheless, evenif measurements of Pb can only reach 50 % accuracy relative to the generalrelativistic prediction of Pb/Pbx 1.3 x 10~8yr~', the bound on co couldexceed 1000 (Nice and Thorsett, 1992).

This discussion illustrates both the promise and the problems inherent instellar-system tests of gravitational theory. Dipole gravitational radiationand strong violations of SEP resulting from the presence of neutron starscan lead to potentially large observable effects. Offsetting this are thecomplications of astrophysical effects within the systems, such as masstransfer, and of environmental effects, such as cluster or third-bodyacceleration^. Under the right conditions, however, a significant test mayemerge.

14.7 ConclusionsIn 1992 we find that general relativity has continued to hold up

under extensive experimental scrutiny. The question then arises, whybother to test it further? One reason is that gravity is a fundamentalinteraction of nature, and as such requires the most solid empiricalunderpinning we can provide. Another is that all attempts to quantizegravity and to unify it with the other forces suggest that gravity standsapart from the other interactions in many ways, thus the more deeply weunderstand gravity and its observational implications, the better we maybe able to mesh it with the other forces. Finally, and most importantly, thepredictions of general relativity are fixed; the theory contains no adjustableconstants so nothing can be changed. Thus every test of the theory ispotentially a deadly test. A verified discrepancy between observation andprediction would kill the theory, and another would have to be substitutedin its place. Although it is remarkable that this theory, born 77 years agoout of almost pure thought, has managed to survive every test, thepossibility of suddenly finding a discrepancy will continue to driveexperiments for years to come.

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Index

action principle, 75; n-body, in modifiedEIH formalism, 273; n-body, in PPNformalism, 158-60; n-body, in THejtformalism, 54

active gravitational mass: comparisonwith passive mass via Kreuzer experi-ment, 214; in PPN formalism, 151

advanced time, 228anomalous mass tensors, 40Apollo 11 retroreflectors, 188

Bianchi identities, 76, 128, 230big-bang model, 311binary pulsar, 11, 257, 283; acceleration

of center of mass, 300, 344; arrival-time analysis, 287-92, 343; inBrans-Dicke theory, 306, 349-50;companion, 284; decay of orbit, 297;detection of gravitational radiation,306; determination of masses, 297,344; dipole gravitational radiation,298; formation and evolution, 286; ingeneral relativity, 303-6, 344-6;gravitational radiation emission, 297;gravitational red shift, 290; mass loss,300; measured parameters, 285, 345;periastron shift, 284, 293; post-Newtonian effects, 297; precession ofpulsar spin, 302; pulsar mass ingeneral relativity, 306; in Rosen'stheory, 307; second-order Dopplershift, 290, 296; spin-down rate, 292;test of conservation of momentum,218-20, 338; third body, 301; tidaleffects in, 294, 299; as ultimate test forgravitation theory, 309

binary system: single-line spectroscopic,287; test of conservation of momen-tum, 217-20, 338

black holes, 256; in general relativity,264; motion, see modified EIH formal-ism; motion in Brans-Dicke theory,279; in Rosen's theory, 266; in scalar-tensor theories, 265

boundary conditions for post-Newtonianlimit, 118

Brans-Dicke theory, 125, 182, 190, 203,265, 276, 306, 317, 332, 335, 349; seealso scalar-tensor theories

Cavendish experiment, 82, 153, 191center of mass, 113, 145, 146, 160Christoffel symbols, 70; for PPN metric,

144classical tests, 166; and redshift experi-

ment, 166"comma goes to semicolon" rule, 71completeness of gravitation theory, 18connection coefficient, see Christoffel

symbolsconservation laws: angular momentum,

108; baryon number, 105; breakdownof, for total momentum, 149; center-of-mass motion, 108; and constraints onPPN parameters, 111; energy-momen-tum, 108, 111-12; global, 107-8; local,105-7; rest mass, 106; tests of, for to-tal momentum, 215-20, 337-8

conserved density, 107, 111constants of nature, constancy: gravita-

tional, 202; nongravitational, 36-8;and Oklo natural reactor 36-8

coordinate systems: curvature coordi-

Index 376

nates, 259; local quasi-Cartesian, 92;preferred, 17; standard PPN gauge, 97

coordinate transformation, 69Copernican principle, 312cosmic time function, 80cosmological principle, 312cosmology, 7, 310; in Bekenstein's varia-

ble-mass theory, 317; in Brans-Dicketheory, 317; in general relativity, 313;helium abundance, 315; microwavebackground, 311, 314; in Rastall's the-ory, 318; in Rosen's theory, 318;timescale problem, 311

covariant derivative, 70Cygnus XI, 256

de Sitter effect, 338deceleration parameter, 313deflection of light, 5; derivation in PPN

formalism, 167-70; derivation usingequivalence principle, 170; eclipse ex-pedition, 5, 171; effect of solar corona,172; measurement by radio interferom-etry, 172; optical measurements, 6; ra-dio measurements, 172; VLBI, 332

Dicke, R. H., 10, 16Dicke framework, 10, 16-18Doppler shift in binary pulsar, 290, 293,

296dynamical gravitational fields, 118

E(2) classification, see gravitational radi-ation

Earth-tides, 191eccentric anomaly, 288Einstein Equivalence Principle (EEP),

16-22; and cosmology, 312; implemen-tation, 71; and nonsymmetric metric,328; and speed of gravitational waves,223; and speed of light, 223; and THt;1formalism, 46-50

Einstein- Infeld- Hoffmann (EIH) formal-ism, 267; EIH Lagrangian, 269; see al-so modified EIH formalism

energy conservation: and cyclic ge-danken experiments, 39-43; and Ein-stein Equivalence Principle, 39; inPPN formalism, 158-63; and Scruffsconjecture, 39-43; and Strong Equiva-lence Principle, 82; in THepi formal-ism, 53-8

Eotvb's experiment, 24-7; and Belin-fante-Swihart nonmetric theory, 66;and fifth force, 341; lunar, 185-90;and Nordtvedt effect. 185-90;Princeton and Moscow versions, 25;in space, 340

Eot-Wash experiment, 320equations of motion: charged test parti-

cles, 69; compact objects, see modifiedEIH formalism; Eulerian hydrodynam-ics, 87; n-body, 149, 159; Newtonian,for massive bodies, 145; photons, 143;PPN hydrodynamics, 147; self-gravitat-ing bodies, 144-53; spinning bodies,163-5; in THe/u, formalism, 50

equatorial coordinates, 194Eulerian equations of hydrodynamics, 87

Fermi-Walker transport, 164fifth force, 341-3flat background metric, 79; in Robert-

son-Walker coordinates, 314

gauge transformation, %general covariance, 17; and preferredcoordinate systems, 17; and prior ge-ometry, 17

general relativity, 121-3; black holes,265; derivations of, 83; EIH formalism,267; field equations, 121; locally mea-sured gravitational constant in, 158; lo-cation in PPN theory space, 205; andmodified EIH formalism, 275; motionof compact objects, 267; neutron stars,260; Nordtvedt effect, 152; polarizationof gravitational waves, 234; post-New-tonian limit, 121; PPN parameters,122; quadrupole generation of gravita-tional waves, 246-8; with R2 terms,84-5; speed of gravitational waves,223; standard cosmology, 316

geocentric ecliptic coordinates, 192geodesic equation, 73; for compact ob-

jects, 267geometrical-optics limit: for gravitational

waves, 223; for Maxwell's equations,74-5

gravimeter, 191; superconducting, 198gravitational constant, 120; constancy of,

202-^t, 336; locally measured, 153-8,191; in scalar-tensor theories, 124

gravitational radiation: in binary pulsar,297; detection in binary pulsar, 306,346-7; dipole, 240, 249, 251, 279, 298;dipole parameter, 240, 253; E(2)classification, 226-7; effect on binarysystem, 239; energy flux in generalrelativity, 238; energy loss, 90,238-40; forces in detectors, 237; ingeneral relativity, 223, 234, 246;measurements of polarization, 237;measurement of speed, 226; in

Index 377

modified EIH formalism, 275;negative energy of, 252; PMparameters, 240, 253; polarization,227-38; post-Newtonian formalism,240; quadrupole nature in generalrelativity, 238; in Rastall's theory,225, 236; reaction force, 239; inRosen's theory, 225, 236, 250; inscalar-tensor theories, 224, 234, 248,252, 279; speed, 223-6; speed inRosen's theory, 131; in vector-tensortheories, 224, 235gravitational red shift, 5, 32-6, 322; in

binary pulsar, 290, 296; and cyclicgedanken experiments, 42-3;derivation, 32-3; null experiment, 36;Pound-Rebka-Snider experiment, 33;in TH^ formalism, 62-4; solar, 322;Vessor-Levine rocket experiment, 35

gravitational stress-energy, 109, 241gravitational waveform, 238Gravitational Weak Equivalence Princi-

ple (GWEP), 82; breakdown, 151, 185;see also Nordtvedt effect

gyroscope precession: derivation in PPNformalism, 208-9; dragging of inertialframes, 210; goedetic effect, 209, 338;and LAGEOS, 340; Lense-Thirringeffect, 210; Stanford experiment, 212,

helicity of gravitational waves, 227, 232,252

helium abundance, 315helium main-sequence star, 284, 294Hubble constant, 202, 313Hughes- Drever experiment, 30, 61hydrogenic atom in THe/* formalism,

55-7

inertial mass, 13, 145; anomalous masstensor, 40, 55, 162, 323; dependenceon gravitational fields, 269; inmodified EIH formalism, 273; post-Newtonian, 146

isentropic flow, 106isotropic coordinates, 259

J2, see quadrupole moment

Kerr metric, 256Kreuzer experiment, 214

laboratory experiments as tests of post-Newtonian gravity, 213

Lagrangian-based metric theory, 78-9little group, 233local Lorenz invariance, 23; and

propagation of light, 321;Hughes-Drever experiment, 30;kinematical frameworks, 325;Mansouri-Sexl framework, 325; testsof, 30-1, 320; tests using TH^formalism, 61-2; in TH formalism,48, 323; violations of, 40-1local position invariance, 23; gravitational

red-shift experiments, 32-6, 322; testsusing TH^ formalism, 62-4; in THW

formalism, 49; violations of, 40-1local quasi-Cartesian coordinates, 92local test experiment, 22Lorentz frames, local, 23Lorentz invariance: local, see local

Lorentz invariance; of modified EIHLagrangian, 272

Lorentz transformations of null tetrad,232

Lunar Laser Ranging Experiment(LURE), 188

Mansouri-Sexl framework, 325Mariner 6, 175Mariner 7, 175Mariner 9, 175mass, see active gravitational mass; iner-

tial mass; passive gravitational massmass function, 283Maxwell's equations, 72-3; ambiguity in

curved spacetime, 72-3; geometrical-optics limit, 74-5; in THe/u formalism,50

metric, 22, 68; flat background, 79, 118;nonsymmetric, 328

metric theories of gravity, postulates, 22;see also theories of gravitation

microwave background, 311, 314; Earth'smotion relative to, 197

Minkowski metric, 20, 80, 118modified EIH formalism: in Brans-Dicke

theory, 276; equations of motion forbinary systems, 273; in general relativi-ty, 275; gravitational radiation, 275;Keplerian orbits, 274; Lagrangian, 273;Newtonian limit, 274; periastron shift,274; in Rosen's theory, 280-2; variableinertial mass, 269

moment of inertia of Earth, variation in,195

momentum conservation: breakdown, inPPN formalism, 149; tests of, 215-20,337-8

neutron stars, 255; boundary conditions,259; form of metric, 258; in general rel-ativity, 250; maximum mass, 256; mo-tion, see modified EIH formalism; in

Index 378

Newtonian theory, 257-8; in Ni's the-ory, 263; in Rosen's theory, 261-3; inscalar-tensor theories, 260

Newman-Penrose quantities, 230Newtonian gravitational potential, 87, 88,

151Newtonian limit, 21, 87, 145; conserva-

tion laws, 105; empirical evidence, 21;and fifth force, 341-3; inverse squareforce law, 21, 341-3; in modified EIHformalism, 274

Newton's third law, 152; and Kreuzerexperiment, 214; and lunar motion,337

Nordtvedt, K., Jr., 98Nordtvedt effect, 151; and lunar motion,

185-90; test of, using lunar laser rang-ing, 188-90; 335

null separation, 74null tetrad, 229

oblateness of Sun, 181; Dicke-Golden-berg measurements, 181; Hill measure-ments, 182; and solar oscillations, 334

Oklo natural reactor, 36-8orbit elements, Keplerian, 178, 283, 287;

perturbation equations for, 179osculating orbit, 287

parametrized post-Newtonian formalism,see PPN formalism

particle physics, 20-1passive gravitational mass, 13; anoma-

lous mass tensor, 40, 55, 58, 162; com-parison with active mass, 214; in PPNformalism, 150

perfect fluid, 77-8periastron shift: in binary pulsar, 284,

293; for compact objects, 274perihelion shift: derivation in PPN for-

malism, 177-80; measured, for Mercury,181, 333; Mercury, 4, 176-83; preferred-frame and preferred-location effects,200-1

PM parameters, 240post-Coulombian expansion, 51post-Galilean transformation, 272post-Keplerian parameters, 343-4post-Newtonian limit: for gravitational-

wave generation, 240-6; see also PPNformalism

post-Newtonian potentials, 93, 104PPN formalism, 10, 97; active gravita-

tional mass, 151; for charged particles,214; Christoffel symbols, 144; compari-son of different versions, 104; conser-vation-law parameters, 111; Ed-dington-Robertson-Schiff version, 98;

PPN ephemerides, 334; limits on PPNparameters, 204, 216, 219, 339; metric,99, 104; n-body action principle,158-60; n-body equations of motion,149, 153; passive gravitational mass,150; PPN parameters, 97; PPNparameter values for metric theories,117; post-post-Newtonian extensions,331; preferred-frame parameters, 103;significance of PPN parameters, 115;standard gauge, 97, 102

preferred-frame effects: in Cavendish ex-periments, 148; geophysical tests, 190-9; on gyroscope precession, 210; in lo-cally-measured gravitational constant,190; orbital tests, 200-2, 336; andsolar spin axis, 336; tests from Earthrotation rate, 199; tests usinggravimeters, 199

preferred-frame parameters: in PPN for-malism, 103; in THe/t formalism, 48

preferred-frame PPN parameters, limitson, 199, 202, 336, 339

preferred-location effects: in Cavendishexperiments, 148; geophysical tests,190-9; in locally-measured gravitation-al constant, 190; orbital tests, 200-2;tests using gravimeters, 199

prior geometry, 17, 79, 118projected semi-major axis, 293proper distance, 73, 155proper time, 73, 68PSR 1744-24A, 351PSR 1534+12,347PSR 1913 + 15, see binary pulsarPSR 2127+11C, 347pulsars, 256, 283

quadrupole moment, 145, 177; solar, 180;solar, measurable by Solar Probe, 183;and solar oscillations, 334

quantum systems in THc/x formalism,55-7

quasi-local Lorentz frame, 80

radar: active, 175; passive, 174; and timedelay of light, 174

radio interferometry and deflection oflight, 172

reduced field equations, 241rest frame of universe, 31, 99rest mass, total, 107retarded time, 228Ricci tensor, 73, 230Riemann curvature tensor, 72; electric

components, 227; irreducible parts, 230

Index 379Riemann normal coordinates, 227Robertson-Walker metric, 91, 312Rosen's bimetric theory, 131; absence of

black holes, 266; binary pulsar, 307;cosmological models, 317; field equa-tions, 131; gravitational radiation, 225,236, 250-2; location in PPN theoryspace, 205; and modified EIH formal-ism, 280; neutron stars, 261; post-Newtonian limit, 131; PPN parameters,131

rotation rate of Earth, variation in, 195

scalar-tensor theories, 123-6; Barker'sconstant G theory 125; Bekenstein'svariable-mass theory, 125, 317;Bergmann-Wagoner-Nordtvedt, 123;binary pulsar, 306; black holes, 265;Brans- Dicke, see Brans-Dicke theo-ry; cosmological models, 317; fieldequation, 123; gravitational radiation,224, 234, 248, 50; limits on m, 175,335; location in PPN theory space,205; and modified EIH formalism,276; neutron stars, 260; Nordtvedteffect, 152; post-Newtonian limit, 124;PPN parameters, 125; and stringtheory, 332

Schiff, L. I., 38Schiff s conjecture, 38; proof in THe/u.

formalism, 50-3Schwarzschild coordinates, 259, 265Schwarzschild metric, 256, 265self-acceleration, 149; of binary system,

217, 338; of pulsars, 216self-consistency of gravitation theory, 19semi-latus rectum, 179Shapiro, 1.1., 166Shapiro effect, see time delay of lightsolar corona, 172, 175Solar Probe, 183, 340spacelike separation, 73special relativity, 20-1; agreement of

gravitational theory with, 20-1; andpropagation of light, 325-7; tests inparticle physics, 20-1

specific energy density, 89spin, 163; precession, 165; precession in

binary pulsar, 302static spherical space times, form of met-

ric, 258stress-energy complex, 108stress-energy tensor, 76; in PPN formal-

ism, 104; vanishing divergence of, 77Strong Equivalence Principle (SEP), 79-

83; and dipole gravitational radiation,252; and motion of compact objects,

268; tests of, 184, 335; violations inCavendish experiments, 153;violations of, 102

superpotential, 94

THe/x formalism, 45-66; limitations, 58-9

theories of gravitation: Barker's constantG theory, 125; Bekenstein's variable-mass-theory, 125, 317; Belinfante-Swihart, 64-6; Bergmann-Wagoner-Nordtvedt, 123; bimetric, 130-5;Brans-Dicke, see Brans-Dicke theo-ry; BSLL bimetric theory, 133; confor-mally flat, 141; E(2) classes, 233-7;fully-conservative, 113; general relativ-ity, see general relativity; Hellings-Nordtvedt, 130; Lagrangian-based, 43,78-9, 109; linear fixed-gauge, 139;Moffat, 330; Ni, 137, 263;nonconservative theories, 115;nonviable, 19, 138-41; postulates ofmetric theories, 22; PPN parametersfor, 117; purely dynamical vs. prior ge-ometric, 79; quasilinear, 138; Rastall,132, 225, 236, 318; Rosen, see Rosen'sbimetric theory; scalar-tensor, see sca-lar-tensor theories; semiconservative,114; special relativistic, 7; stratified,135-7; stratified, with time-orthogonalspace slices, 140; vector-tensor, seevector-tensor theories; Whitehead,139; Will-Nordtvedt, 129; withnonsymmetric metric, 328-30

time delay of light: in binary pulsar, 290;as classical test, 166; derivation inPPN formalism, 173-4; effect of solarcorona, 175; measurements of, 174,333; radar measurements, 176

timelike separation, 73torsion, 84transverse-traceless projection, 248

universal coupling, 43, 67-8

vector-tensor theories, 126-30; fieldequations, 127; gravitational radiation,224, 235; Hellings-Nordtvedt, 130;post-Newtonian limit, 129; PPN pa-rameters, 129; Will-Nordtvedt, 129

velocity curve, 283Viking, 175, 336virial relations, 52, 54, 148, 161, 245Voyager 2, 333

Weak Equivalence Principle (WEP), 13,22; and cyclic gedanken experiments,

Index 380

41-2; and electromagnetic interactions, 320; tests using TH£/, formalism, 60;28-9; and Ebtvbs experiment, 24-7; and weak interactions, 29and fifth force, 341; and gravitational Weyl tensor, 230interactions, 29, 82; of Newton, 13; Whitehead PPN parameter, limits on, 199and nonsymmetric metric, 329; and Whitehead's theory, 139strong interactions, 28; tests of, 24-9, Whitehead term, 95, 98

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