QUANTIZED LONGITUDINAL AND TRANSVERSE SHIFTS. . . 3563

states.There is thus no polarization state that can go

through a longitudinal and then a transverse shiftamplifier (or the contrary). In other words, asthese two shifts have no common eigenfunctions,their eigenvalues cannot be simultaneously dis-played.

ACKNOWLEDGMENTS

We express our thanks to Professor A. Kastlerand Professor A. Marechal for their continued in-terest and support in this research, and to Dr. J.Ricard for his always constructive remarks orcx-iticism.

F. Goos and H. H'anchen, Ann. Phys. (Leipzig) 1, 333(1947).

2J. Picht, Ann. Phys. (Leipzig) 3, 433 (1929);F. Noether, ibid. 11, 141 (1931);C. Schaefer and R. Pich,ibid. 30, 245 (1937).

K. Artmann, Ann. Phys. (Leipzig) 2, 87 (1948).4R. H. Renard, J. Opt. Soc. Am. 54, 1190 (1964).5J. Ricard, C. R. Acad. Sci. B 274, 384 (1972);

O. Costa de Beauregard, ibid. 274, 433 (1972). Thesepapers use the energy-flux argument.

F. Goos and H. Hhnchen, Ann. Phys. (Leipzig) 5, 251(1949).

'A. Mazet, C. Imbert, and S. Huard, C. R. Acad. Sci. B273, 592 (1971).

~C. Imbert, C. R. Acad. Sci. B 267, 1401 (1968).C. Imbert, C. R. Acad. Sci. B 269, 1227 (1969);270,

529 (1970).' F. I. Fedorov, Dokl. Adak. Nauk SSSR 105, 465 (1955);O. Costa de Beauregard, Phys. Rev. 139, B1446 (1965);H. Schilling, Ann. Phys. (Leipzig) 16, 122 (1965).' C. Imbert, Phys. Rev. D 5, 787 (1972).

O. Costa de Beauregard, C. R. Acad. Sci. B 274, 433(1972); O. Costa de Beauregard and C. Imbert, Phys.Rev. Lett. 28, 1211 (1972).

C. Imbert, C. R. Acad. Sci. B 274, 1213 (1972).4M. Born and E. W. Wolf, Principles of Optics, 3rd

edition (Pergamon Press, London, 1965), pp. 169-171.See also pp. 127-132 and 215.

~~For an excellent presentation see J. M. Jauch andF. Rohrlich, The Theory of Photons and ELectrons(Addison-Wesley, Reading, Mass. , 1955), pp. 40-47.~~For a codification of this mathematical style using

the Clifford algebra see D. Hestenes, Space-TimeAlgebra (Gordon and Breach, New York, 1966).

Note added in proof. Of course, when using ourequilateral multiplying prism where the light follows ahelical polygonal path, there is a component of the Goos-Hanchen shift parallel to the edges. As previously ex-plained (Ref. 11) and due to the rotation of the incidenceplane at each consecutive reflexion, this shift is commonto the two circular polarization modes and is not re-corded by our apparatus.

PHYSICAL REVIEW D VOLUME 7, NUMBE R 12 15 JUNE 1973

Foundations for a Theory of Gravitation Theories".

Kip S. Thorne, David L. Lee, ' and Alan P. Lightman'California Institute of Technology, Pasadena, California 91109

(Received 10 January 1973)

A foundation is laid for future analyses of gravitation theories. This foundation is applicable to any

theory formulated in terms of geometric objects defined on a 4-dimensional spacetime manifold. Thefoundation consists of (i) a glossary of fundamental concepts, (ii) a theorem that delineates the overlapbetween Lagrangian-based theories and metric theories; (iii) a conjecture (due to Schiff) that the weak

equivalence principle implies the Einstein equivalence principle; and (iv) a plausibility argument

supporting this conjecture for tne special case of relativistic, Lagrangian-based theories.

I. INTRODUCTION

Several years ago our group initiated' a projectof constructing theoretical foundations for experi-mental tests of gravitation theories. The resultsof that project to date (largely due to Will and ¹)and the results of a similar project being carried

out by the group of Nordtvedt at Montana StateUniversity are summarized in several recent re-view articles. ' ' Those results have focused al-most entirely on "metric theories of gravity" (rel-ativistic theories that embody the Einstein equiva-lence principle; see Sec. III below).

By January 1972, metric theories were suffi-

3564 KIP S. THORNE, DAVID L. LEE, AND ALAN P. LIGHTMAN

ciently well understood that we began to broadenour horizons to include nonmetric theories. Themost difficult aspect of this venture has been com-munication. The basic concepts used in discussingnonmetric theories in the past have been defined sovaguely that discussions and "cross-theory analy-ses" have been rather difficult. To remedy thissituation we have been forced, during these lasteleven months, to make more precise a numberof old concepts and to introduce many new ones.By trial and error, we have gradually built up aglossary of concepts that looks promising as afoundation for analyzing nonmetric theories.

Undoubtedly we shall want to change some of ourconcepts, and make others more precise, as weproceed further. But by now our glossary is suf-ficiently stabilized, and we have derived enoughinteresting results using it, that we feel compelledto start publishing.

This paper presents the current version of ourglossary (Secs. II-IV), and uses it to outline somekey ideas and results about gravitation theories,both nonmetric and metric (Secs. V and VI). Sub-sequent papers will explore some of those ideasand results in greater depth.

Central to our current viewpoint on gravitationtheories is the following empirical fact. Only twoways have ever been found to mesh a set of gravi-tational laws with all the classical, special rela-tivistic laws of physics. One way is the route ofthe Einstein equivalence principle (EEP) —(i) De-scribe gravity by one or more gravitational fields,including a metric tensor g ~; and (ii) insist thatin the local Lorentz frames of g & all the nongrav-itational laws take on their standard special rela-tivistic forms. The second way of meshing is theroute of the Lagrangian —(i) Take a special rela-tivistic Lagrangian for particles and nongravita-tional fields, and (ii) insert gravitational fields in-to that Lagrangian in a manner that retains gener-al covariance. The equivalence-principle routealways leads to a metric theory. (Example: gen-eral relativity. } The Lagrangian route alwaysleads to a "Lagrangian-based theory. " [sample:Belinfante-Swihart theory (Table IV, later in thispaper). ] Thus, in the future we expect most of ourattention to focus on metric theories and on La-grangian-based theories; and in the nonmetriccase we might be able to confine attention to theo-ries with Lagrangians.

Since metric theories are so well understood, 4

it would be wonderful if one could prove that allnonmetric, Lagrangian-based theories are defec-tive in some sense. A conjecture due to Schiff'points to a possible defect. Schiff's conjecturesays' that any complete and self-consistent theorythat obeys the creak equivalence Princi Pie (WEP)

must also, unavoidably, obey the Einstein equiva-lence PrinciPle (EEP). (See Sec. III for precisedefinitions. } Since any relativistic, Lagrangian-based theory that obeys EEP is a metric theory,this conjecture suggests that nonmetric, relativ-istic, Lagrangian-based theories should alwaysviolate WEP.

The experiments of Eotvos et al. ' and Dicke etal. ,

' with modifications by Braginsky et al. ' (EDexperiments), are high-precision tools for testingWEP. Hence, the Schiff conjecture suggests that,if one has a nonmetric Lagrangian-based theory,one should test whether it violates the ED experi-ments. (Such tests for the Belinfante-Swihart andNaida-Capella theories reveal violations of ED andWEP. 9)

In this paper, after presenting our glossary ofconcepts (Secs. II-IV), we shall (i) derive a crite-rion for determining whether a Lagrangian-basedtheory is a metric theory ("principle of universalcoupling, " Sec. V), and (ii) discuss and makeplausible Schiff's conjecture (Sec. VI).

II. CONCEPTS RELEVANT TO SPACETIME THEORIES

This section, together with Secs. III and IV, pre-sents our glossary of concepts. To understandthese concepts fully, the reader should be familiarwith the foundations of differential geometry aslaid out, for example, by Trautman. ' He shouldalso be familiar with Chap. 4 of Anderson's text-book" (cited henceforth a,s JLA), from which wehave borrowed many concepts. However, heshould notice that we have modified slightly someof JLA's concepts, and we have reexpressed someof them in the more precise notation and terminol-ogy of Trautman" and of Misner, Thorne, andWheeler (MTW)."

The concepts introduced in this section apply toany "spacetime theory" (see below for definition).In Secs. III and IV we shall specialize to "gravita-tion theories, " which are a particular type ofspacetime theory. To make our concepts clear,we shall illustrate them using four particulargravitation theories: the Newton-Cartan theory(Table I), general relativity (Table II), Ni's theo-ry (Table III),"and the Belinfante-Swihart theory(Table IV)."" Of these theories, general relativ-ity and Ni's theory are metric; the Newton-Cartanand Belinfante-Swihart theories are nonmetric.

Mathematical representations of a theory. Twodifferent mathematical formalisms will be called"different representations of the same theory" ifthey produce identical predictions for the outcomeof every experiment or observation. Here by "out-come of an experiment or observation" we mean

FOUNDATIONS FOR A THEORY OF GRA VITATION THEORIES 3565

TABLE I. Newton-Cartan theory.

1 ~ Reference for this version of the theory:

Chapter 12, and especially Box 12.4 of MTW

2. Gravitational fields.

a.b.c,

Symmetric covariant derivative (affine connection). . . ,

S t lJ. » 1patial metric.U 1niversal time. . . . . . . . . . . ~. . . . . . . . . . . . . . . . . . . . . . . .

~ » ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ » V~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ »

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

3. Gravitational field equations:

a. Vdt=0.b. $(u, n)w=0,

where S. is the curvature operator formed from V; u and n are arbitrary vectors; w is any spa-tial vector ((dt, w} =0).

c. $(v, w) =0for every pair of spatial vectors, v, w. [Note. a, b, c guarantee the existence of the metric, y or".", defined on spatial vectors only, such that

Vg(w ' v) ={View) ' v +w ' (Vgv)

for any u and for any spatial w, v.]d. v Q(u, gnw] = w ~ [CI(u, n) v]

for all spatial v, w and for any u, n, where

J(u, n)g =—2[0(P, n}u +8(P, u) n] .e. Ricci =4mpdt (3 dt,

where Ricci is the Ricci tensor formed from V, and p is mass density.

4. Influence of gravity on matter:

a. Test particles move along geodesics of V, with t an affine parameter.b. Each test particle carries a local inertial frame with orthonormal, parallel-transported spatial

basis vectors (e ~~ e g =6&&, V e J =0) and with e p

= d/d t = (tangent to geodesic world line).c. All the nongravitational laws of physics take on their standard, Newtonian forms in every local

inertial frame.

the raw numerical data, before interpretation interms of theory. Any theory can be given a vari-ety of different mathematical representations.[Example —The Dicke-Brans- Jordan theory hastwo "standard representations: (i) the originalrepresentation, ""in which test particles move

on geodesics but the field equations differ signifi-cantly from those of Einstein; and (ii) the confor-mally transformed representation, " in which thescalar field produces deviations from geodesicmotion but the field equations are nearly the sameas Einstein's. j A theory can be regarded as the

TABLE II. General relativity theory.

1. Reference: Standard textbooks, e.g., MTW.

2. Gravitational field:

The metric of spacetime. . . ~ ~. . . . . . . . . . . . . . . . . .3. Gravitational field equations:

C =87rT,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

where 6 is the Einstein tensor formed from g, and T is the stress-energy tensor.4. Influence of gravity on matter:

a. Test particles move along geodesics of g, with proper time 7' an affine parameter.b. Each test particle carries a local inertial ("local Lorentz") frame with parallel-transported, or-

thonormal basis vectors e-„, and with e p= d/d T = (tangent to geodesic world line) .

c. All the nongravitational laws of physics take on their standard, special-relativistic forms in ev-ery local inertial frame (aside from delicate points associated with "curvature coupling"; seeChap, 16 of MTW~2).

3566 KIP S. THORNE, DAVID L. LEE, AND ALAN P. LIGHTMAN

TABLE III. Ni's "New Theory. "

1. Refe rence: Ni

2. Gravitational fields:

a. Background metric (signatureb. Universal time. . . . . . . . . . . . . .c. Scalar field. . . ~. . . . . . . . . . . . .d. One-form field. . . . . . . . . ~. . . .e. Physical metric. . . . . . . . . ~. . .

3. Gravitational field equations:

a. Background metric is flat,

Riemann(g) =0 .

b. "Meshing" of g, t, Q:

t) 8=0,

+2). . .~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 ~ ~ 0

~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ o ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

t[ati 8~a8

where "~"denotes covariant derivative with respect to th and j tiU)) is the inverse of )( ti&&().

c. g =f2(p)g+(f2(p) -f&(p)) dt (3 dt -g(3dt —dt g.Here f&(y) and f2(y) are arbitrary functions to be determined finally by experiment.

d. Field equations for y and Q follow from the action principle

6 Z d x=0, where 2= G+SG,

a8 6 a8 a8 2

e is a constant to be determined by experiment, SNG —-LNG v'-g, and L NG is the standard Lagran-gian density of special relativity with the metric of special relativity replaced by g.

4. Influence of gravity on matter:

Governed by action principle

Z~ d x=0,

where particle world lines and nongravitational fields are varied.

equivalence class of all its representations. Ta-bles I-IV present particular representations forthe theories described there.

Spacetime theory. A "spacetime theory" isany theory that possesses a mathematical repre-sentation constructed from a 4-dimensional space-time manifold and from geometric objects definedon that manifold. (For the definition of "geometricobject, " see Sec. 4.13 of Trautman. '

) Henceforthwe shall restrict ourselves to spacetime theoriesand to the above type of mathematical representa-tions. The geometric objects of a particular rep-resentation will be called its variables, ' the equa-tions which the variables must satisfy will becalled the physical laces of the representation.[Example —general relativity (Table II}: The

physical laws are the Einstein field equations,Maxwell's equations, the Lorentz force law, etc. ][Example —Belinfante-Swihart theory (Table IV):The physical laws are Hiemann (ti) = 0, and theEuler-Lagrange equations that follow from 5JZd'x=0.)

Manifold maPPing grouP (MMG). The MMG isthe group of all diffeomorphisms of the space-time manifold onto itself. Each diffeomorphismh, together with an initial coordinate systemx"((P}, produces a new coordinate system

x" (d')=x (ft 'iP).

(Events are denoted by capital script letters. }Kinematically possible trajectory (tpt). Consid-

er a given mathematical representation of a given

FOUNDATIONS FOR A THEORY OF GRAVITATION THEORIES 3567

TABLE IV. Belinfante-Swihart theory.

1. References: Summary and analysis of the theory by Lee and Lightman; original paper by Belinfanteand Swihart. ~s

2. Gravitational fields:

a.b.

M tetr1c ~ ~ ~ ~ ~ ~ ~ ~ ~ a ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ o 'g

Symmetric second-rank tensor. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . ... . . . . . . . h

3. Nongravitational variables:

a. Electromagnetic vector potential. . . . . . . . . . . . . . . . . . . . . . . . .. .. . .. ...... . ... ... .. . .. .. .. . .. Ab. Electromagnetic field tensor (second-rank, antisymmetric). . . . . . . . . . . . . . . . . ~ . ~. . . . . . . . ... . Hc. World line of particle J, parametrized in an arbitrary manner. .. . . . . . . . . . . . . . . . . . .. . . . zz (Xz)

[in a given coordinate system, world line is x =zz (Xz)].d. Velocity vector of particle J (defined along world line). .. . . . . . . . . . . . , . . . . . . . . . . .. . . . .. . a&(A. &)e. Momentum vector of particle J (defined along world line). .. . . . . . . . . . . . . . . . . . . . . . . ... .. n &(A, &)

4. Gravitational field equations:

a. Metric is flat: Riemann(g)=0 .b. FieM equation for h follows from varying ha& in 0 f Zd x = 0, where 2 is given below.

5. InQuence of gravity on matter:

Equations for A, H, z z, a z, m z follow from varying these quantities in J ZdS =0.6. Lagrangian density:

a. Z =ZG +ZNG .b. Zo ——-(I/lee)rP t)"" t)~ (ah&&«h&e~&+fbi&«h&o~&)(-t))

where "~ "denotes covariant derivative with respect to g; a and f are constants to be determinedby experiment, and 0 =— det ~j t) &&f .

c. gNG = (1/4+) (&H ~Hv p H ~Ap ([v) ( g)

+eo~ 0 P+Q [-maybe+(7re& ezA&)zz -—vz&az]b [x —zz(hz))dhzJ -eo

~ ' tl "h „+EPI b q R s 'I f * (~ )~did .

d. Here ez and mz are the charge and rest mass of particle J; z&—= dz&/dA, z, bz = (-azar'„); K is

a constant to be determined by experiment; indices are raised and lowered with g~&., and

r~'=- (1/4~)(H ~H '-&q~'H 'H, )

+Q f " "6't —* (X )]de~ .

e. In the action principle one varies h», A&, H», zz(A, &), az(A, &), ~z(A, &) independently; but oneholds rJ» fixed.

spacetime theory. A kpt of that representation isany set of values for the components of all thevariables in any coordinate system. A kpt neednot satisfy the physical laws of the representation.(Example —general relativity (Table II): A kpt isany set of functions (g„e(x) =h"8 (x); F„B(x)

Fa„(x); zP(r~);-. . . ) in any coordinate system,which —if they were to satisfy the physical laws—would represent metric, electromagnetic field,particle world lines, etc. ) (Example —Belinfante-Swihart theory (Table IV): A kpt is any set offunctions ft) z(x) =t)a„(x), h e(x) =h& (x), A (x),H„z(x) =~8„(x), zz(hz), a~(h~), w~a(X~)) in any co-ordinate system. )

Dynamically Possible traj ectory (dPt). A dpt is

any kpt that satisfies all the physical laws of therepresentation.

Covariance group of a representation. A group9 is a covariance group of a representation if (i)9 maps kpt of that representation into kpt; (ii) thekpt constitute "the basis of a faithful realizationof 9" (i.e., no two elements of 9 produce identicalmappings of the kpt)", (iii) 9 maps dpt into dpt.(Example —MMG is a covariance group of each ofthe representations of theories in Tables I-IV. )(Example —Electromagnetic gauge transforma-tions, A„-A „+y „, are a covariance group ofthe representation of Belinfante-Swihart theorygiven in Table IV. ) By complete covariance groupwe shall mean the largest covariance group of the

3568 KIP S. THORNE, DAVID L. LEE, AND ALAN P. LIGHTMAN

yA(tP, [x }), 6' varying and (x"}fixed (3)

constitute a kpt. The diffeomorphismh maps thiskpt into yA((P, [x }), where (x } is the coordinatesystem of Eq. (1). The internal transformation Hconverts y into a new geometric object,

y'=—Hy .

The net effect of G on the kpt (3) is

G: yA(tP, (x })-y„(tP,(x"}).

(4)

(5)

It is often useful to characterize G by the functions

representation. By generally covariant represen-tation of a theory we shall mean any representationfor which MMG is a covariance group. (An argu-ment due to Kretschmann" shows that everyspacetime theory possesses generally covariantrepresentations. } By internal covariance grouPwe shall mean a covariance group that involves nodiffeomorphisms of spacetime onto itself. (Ex-ample —Electromagnetic gauge tr ansformationsare an internal covariance group. ) By externaLcovariance group we shall mean a covariancegroup that is a subgroup of MMG. The completecovariance group of a representation need not bethe direct product of its complete (i.e., largest)internal covariance group with its complete exter-nal covariance group. It may also include trans-formations that are "partially internal" and "par-tially external" and cannot be split up. [Example-%hen one formulates Newton-tartan theory in aGalilean coordinate representation (see the Appen-dix, which should not be read until one has fin-ished this entire section), one obtains a completecovariance group described by Eqs. (A5). Thecomplete external covariance group consists of(A5a) and (ASb). There is no internal covariancegroup. The transformations (A5c) are mixed in-ternal-external transformations that belong to thecomplete covariance group. ]

We shall use the following notation to describea particular element G of the covariance group,and its effect. G consists of a diffeomorphismh[Eq. (1), above] and an internal transformation H:

G =(8tH).

If G is an external transformation (element ofMMG), then H must be the identity operation; if Gis an internal transformation, thenh is the iden-tity mapping; if G is a mixed internal-externaltransformation, then neither h nor H is an identity.Denote the variables of the representation (geo-metric objects) by y, and their components at apoint (P in a coordinate system (x }by y„(6', fx"}).The set of functions

5y.(5, &."'})-=y.(5, ( '})-y.(h '5, ( "})= yA I evaluated at xtv '(p

yA I evaluated at a~=act (tPt ' (6)

Note that these "changes in y" satisfy the relation

5(y, )(tP, ( "})=I5y (+, ( '})]„,where a comma denotes partial derivative, andalso the relation

5y =(Hy) (+, (x"})-(&y)A(6', I:» '}), (8)

5y (6', ( "})= (& y) (+, ( })

+ „—(H, y)„(tP, (x"})d

— 6 =0

(10)

where 2& is the Lie derivative along $ (Sec. 4.15of Trautman'").

Equivalence classes of dpt. Two dpt are mem-bers of the same equivalence class if one of themis mapped into the other by some element of thecomplete covariance group. (Example —WhenMMG is a covariance group, all dpt that are ob-tained from each other by coordinate transforma-tions belong to the same equivalence class. ) If agenerally covariant representation possesses nointernal covariance groups, then there is a one-to-one correspondence between equivalence clas-ses of dpt and the geometric, coordinate-indepen-dent solutions of its geometric, coordinate-inde-pendent physical laws.

Confined, absolute, and dynamical variables.The variables of a generally covariant represen-tation split up into three groups: "confined vari-ables, " "absolute variables, " and "dynamical vari-ables. " The confined variables are those which donot constitute the basis of a faithful realization ofMMG. (Examples —All universal constants, such

where hy is the geometric object obtained by"dragging along with It" (see p. 86 of Trautman").

Of particular interest are the infinitesimal ele-ments of a covariance group. [From them one cangenerate that topologically connected component"of the group which contains the identity. The otherconnected components, if any, are typically ob-tained by bringing into play a discrete set of groupelements (space reflections, time inversions,etc.}.] Let G, =(h„H,) be a one-parameter familyof elements (curve in group space parametrized bye), with G, the identity. Denote by $ the infinites-imal generator of the diffeomorphism A,,:

t'-=[d(h, +)/«], —.. (8)

Then, to first order in e, Eq. (8) reduces to

FOUNDATIONS FOR A THEORY OF GRAVITATION THEORIES 3569

as the charge of the electron, are confined vari-ables. The world line of a particle is not a con-fined variable, as one sees by this procedure: (i)Characterize the world line by the scalar field

0, if 6' is not on world line;T(6') = proper time of particle,

if 6' is on world line.

(ii) ~&erify that an element of MMG can be charac-terized uniquely by the manner in which it mapsthe set of all kinematically possible world lines[all functions 7(x ) that are zero everywhere ex-cept along a curve, and are monotonic along thatcurve] into each other. (iii} Thereby concludethat a particle world line does constitute the basisfor a faithful realization of MMG, and thereforethat it is not a confined variable. } To determinewhether an unconfined variable B is absolute ordynamical, perform the following test: Pick outan arbitrary dpt, and let B„(x")be the functionswhich describe the components of B for that dpt.Then examine each equivalence class of dpt to seewhether these same functions B„appear some-where in it. If they do, for every equivalenceclass and for every choice of the arbitrary initialdpt, then B is an absolute variable. If they do not,for some particular choice of the initial dpt andfor some particular equivalence class, then B isa dynamical variable. Some dynamical variablescontain absolute parts, and some dynamical andabsolute variables contain confined parts. [Ex-ample —Belinfante-Swihart theory (Table IV):is an absolute variable; h „z and all the nongravi-tational variables are dynamical. ] [Example —Ni'stheory (Table III): q and t are absolute variables;P, cp, andy are dynamical. Although P is dynam-ical, it contains an absolute part -the projectionof P on dt (i.e., g t, Bq" ). The remaining, "spa-tial" part of g(/+P t ~~@ dt} is fully dynamical.Although t is absolute, it contains a confined part-its "origin, " or equivalently, its value at somefixed fiducial event (P,. One can remove this con-fined part from t by passing from t to the 1-formfield dt ][Example —. general relativity (Table II):All the unconfined variables are dynamical, andthey contain no absolute parts. It is this featurethat distinguishes general relativity from almostall other theories of gravity {see JLA"; alsoChap. 17 of MTW, where absolute variables arecalled "prior geometry" }.] (Example —Newton-Cartan theory: In the representation of Table I,t and y are absolute variables; V' is dynamical.As in Ni's theory, the origin of t is a confinedvariable and can be split off by passing from t todt. Although the covariant derivative V is dynam-ical, it contains absolute parts. A decompositionof V into its absolute and dynamical parts is per-

formed in the Appendix [Eq. (Ale)]. After that de-composition the theory takes on a new mathemati-cal representation with absolute variables P, y, D,and dynamical variables 4 and V.}

Irrelevant variables. A set of variables of agenerally covariant representation is called irrel-evant if (i) its variables are not coupled by thephysical laws to the remaining variables of therepresentation, and (ii} its variables can be elim-inated from the representation without altering thestructure of the equivalence classes of dpt andwithout destroying general covariance. A variablethat is not irrelevant is called "relevant. " Somevariables contain both relevant and irrelevantparts. (Example —The gauge of the electromagnet-ic vector potential is irrelevant. So is any othervariable that can be forced to take on any desiredset of values by imposing an appropriate internalcovariance transformation. ) [Example —In Ni'stheory (Table IV) and the Newton-Cartan theory(Table I) the origin of universal time t is an irrel-evant variable. ]

Emily reduced, generally covariant representa-tion. A generally covariant representation iscalled "fully reduced" if (i} it contains no irrele-vant variables, (ii) its dynamical variables con-tain no absolute parts, and (iii) its dynamical andabsolute variables contain no confined parts. [Ex-ample —Newton-Cartan theory: The representa-tion of Table I is generally covariant, but not fullyreduced. To reduce it one must follow the proce-dure of the Appendix: (i) Remove the irrelevantorigin of f by passing from t to P =dt; (ii) split V

into its absolute and dynamical parts. The result-ing representation is not quite fully reduced be-cause it possesses the internal covariance trans-formation (A3'a) with an associated, irrelevant"gauge arbitrariness" in D and C. When one re-moves that irrelevance by fixing the "gauge" onceand for all (e.g. , by requiring, for an island uni-verse, that f 8 „]=0 in any Galilean frame wherethe total 3-momentum vanishes), then one obtainsa fully reduced representation. ]

Boundary conditions, Prior geometric con-straints, decomPosition equations, and dynamicallazes. In a given mathematical representation ofa given theory, the physical laws break up intofour sets: (i) boundary conditions —those lawswhich involve only confined variables; (ii) priorgeometric constraints" —those which involve ab-solute variables and possibly also confined vari-ables, but not dynamical variables; (iii) decom-position equations —those which express a dynam-ical variable algebraically in terms of other vari-ables; (iv) dynamical laws —all others. [Ex-ample —Ni's theory (Table III): Equations (3a)and (3b) are prior geometric constraints; Eq. (3c)

3570 KIP S. THORNE, DAVID L. LEE, AND ALAN P. LIGHTMAN

is a decomposition equation; and the equationsthat follow from the variational principle are alldynamical. If one augments the theory by cosmo-logical demands that P and y go to zero at spatialinfinity, those demands are boundary conditions. ][Example —general relativity (Table II): All phys-ical laws are dynamical. ] [Example —Belinfante-Swihart theory (Table IV): Riemann (0}=0 is aprior geometric constraint; the equations obtainedfrom the variational principle are dynamical. ][Example —Newton-Cartan theory (Table I): Inthe mathematical formulation of Table I, Eqs.(3a}-(3d) are all dynamical laws. One has thefeeling, however, that they ought not to be dynam-ical, because they involve only gravitationalfields; they make no reference to any source ofgravity. Only (3e) contains a source, so only it"ought to be" dynamical. The failure of one' s"ought-to" intuition results from one's failure tosplit V up into its absolute and dynamical pieces.Such a split (see Appendix} results in a new math-ematical formulation of the theory, with just onedynamical gravitational law: (Alf), which isequivalent to (3e) of Table I. Of the other gravi-tational equations in the new formulation,(A1a)-(A1d) are prior geometric constraints, and(Ale) is a decomposition equation. ]

Symmetry group. Let G be an element of thecomplete covariance group of a representation.Examine the change produced by G in every vari-able B that (i} is absolute, and (ii}has had all ir-relevant, confined parts removed from itself. If

58„((P,(x j) =0 at all 6' and forall coordinate systems [x

(12}

for every such B, then G is called a symmetrytransformation. Any group of symmetry transfor-mations is called a symmetry group; the largestgroup of symmetry transformations is called thecomplete symmetry group of the representation.[Note: That component of the complete symmetrygroup which is topologically connected to the iden-tity is generated by infinitesimal transformations.One can find all the infinitesimal generators bysolving Eqs. (10) and (12) for f, and for (dH, jdc}, , ] [Another note: If the absolute variablesB are all tensor or affine-connection fields, then~B are all tensor fields, so

(5B„=O for all (P in one coordinate system)

~ (5 B„=O for all (P in every coordinatesystem) . (13)

Hence, in this case one can confine attention to

any desired, special coordinate system when test-ing for symmetry transformations. ] [Example-Belinfante-Swihart theory (Table IV): The com-plete symmetry group consists of the Poincarbgroup (inhomogeneous Lorentz transformations)together with the electromagnetic gauge transfor-mations. One proves this most easily in a globalLorentz frame of g, one can restrict calculationsto this frame because the absolute variable g is atensor. ] [Example —Ni's theory (Table III): Sym-metry transformations are analyzed most easilyin a coordinate system where x' =l =(universaltime), and n, s has the Minkowski form. Any

symmetry transformation must leave 6g 8 =~t=5(q"st g8) =0. Thus, the symmetry transfor-mations are (i) electromagnetic gauge transfor-mations; (ii) spacetime translations, x =x" +a"with a a constant; (iii} time-independent spatialrotations, x' =x' and x' = B'"x' with ([B"(( arotation matrix; (iv} spatial reflections. ] [Ex-ample -general relativity (Table II}: There areno absolute variables, so the complete covariancegroup and the complete symmetry group are iden-tical; they are the MMG plus electromagneticgauge transformations. ] (Example —Newton-Car-tan theory: See Appendix. ) An external symmetrygroup is a symmetry group that is a subgroup ofMMG. An internal symmetry group is a symmetrygroup that involves no diffeomorphisms of space-time onto itself. The complete symmetry groupneed not be the direct product of the external sym-metries and the internal symmetries; it may alsoinclude symmetries that are partially internal andpartially external and cannot be split up. [Ex-ample —Newton-tartan theory in the representa-tion of the Appendix: Transformations (A5c) arepartially internal and partially external. ]

III. GRAVITATION THEORIES ANDEQUIVALENCE PRINCIPLES

We now turn from general spacetime theoriesto the special case of gravitation theories. Wecannot discuss gravitation theories without makingsomewhat precise the distinction between gravita-tional phenomena and nongravitational phenomena.There seem to be a variety of ways in which onemight make this distinction. Somewhat arbitrari-ly, but after considerable thought, we have chosento regard as "gravitational" those phenomenawhich either are absolute or "go away" as theamount of mass-energy in the experimental labo-ratory decreases. In other words, gravitationalphenomena are either prior geometric effects oreffects generated by mass-energy. This meansthat the flat background metric g of Belinfante-Swihart theory is a gravitational field; the metric

FOUNDATIONS FOR A THEORY OF GRA VITATION THEORIES 3571

of general relativity is a gravitational field; but

the torsion of Cartan's modified general relativi-ty,"which is generated by spin rather than bymass-energy, is not a gravitational field.

We try to make the above statements more pre-cise by introducing the following concepts.

Local test experiment. A "local test experi-ment" is any experiment, performed anywhere in

spacetime, in the following manner. A shield isset up around the experimental laboratory. When

analyzed using the concepts and experiments ofspecial relativity, this shield must have arbitrar-ily small mass-energy and must be impermeableto electromagnetic fields, to neutrino fields, andto real (as opposed to virtual) particles. The ex-periment is performed, with freely falling appara-tus, in the center of the shielded laboratory, in aregion so small that inhomogeneities in all exter-nal fields are unimportant. One makes sure thatexternal inhomogeneities are unimportant by per-forming a sequence of experiments of successive-ly smaller size (with size of shield and externalconditions unchanged), until the experimental re-sult approaches a constant value asymptotically.(Examples —The experiment might be a local mea-surement of the electromagnetic fine-structureconstant, or a Cavendish experiment with two leadspheres, or a series of Cavendish experiments in-volving lead spheres and small black holes. )

Local, nongravitational, test experiment. A"local, nongravitational test experiment" is a lo-cal test experiment with these properties: (i}When analyzed in the center-of-mass Galileanframe, using the Newtonian theory of gravity, andusing all forms of special relativistic mass-ener-gy as sources for the Newtonian potential 4, thematter and fields inside the shield must producea 4 with

~4 (at any point inside shield)

-4 (at a,ny point on shield)~

« I .

(ii} When the experiment is repeated, with succes-sively smaller mass-energies inside the shield(as deduced using special relativity theory) —butleaving unchanged the characteristic sizes, intrin-sic angular momenta, velocities, and charges(electric, baryonic, leptonic, etc.} of its variousparts —the experimental result does not change.(Examples: A measurement of the electromagnet-ic fine-structure constant is a local, nongravita-tional test experiment; a Cavendish experiment isnot. )

Gravitation theory. A "gravitation theory, " or"theory of gravity, " is any space-time theorywhich correctly predicts Kepler's laws for a bina-ry star system that (i) is isolated in interstellar

space ("local test experiment"}; (ii) consists oftwo "normal stars" (stars with ~4

~

« I throughouttheir interiors); and (iii) has periastron P largecompared to the stellar radii, P»R. The theory'spredictions must not deviate from Kepler's laws

by fractional amounts exceeding the larger of~4

~and P/R. (Note: To agree with experiment

in the solar system, the theory will have to repro-duce Kepler much more accurately than this. )(Examples —The theories in Tables I-IV are allgravitation theories. )

In the absence of gravity. The phrase "in theabsence of gravity" means "when analyzing any

local, nongravitational test experiment for which

the shield is spherical, has arbitrarily large radi-us, and is surrounded by a spherically symmetricsea of matter. " "To turn off gravity" means "topass from a generic situation to a situation wheregravity is absent. " "To turn on gravity" means"to pass from a situation where gravity is absentto a generic situation. "

Gravitational field In a g.iven representation ofa given gravitation theory, any unconfined, rele-vant variable B is a "gravitational field" if, in theabsence of gravity, it reduces to a constant, or toan absolute variable, or to an irrelevant variable.In particular, every absolute, relevant variableis a gravitational field. [Example —general rela-tivity (Table II}: For local, nongravitational testexperiments, analyzed using Fermi-normal coor-dinates, one gets the same result whether one usesthe correct g or one replaces it by a flat Minkow-ski metric q (absolute variable). Thus g is agravitational field. ] [Example —Newton-Cartantheory (Table I}: t and y are already absolute, sothey are gravitational fields; V can be replacedby the Riemann-flat D of the Appendix without af-fecting local, nongravitational experiments, so itis also a gravitational field. ] (Example —Cartan'smodification of general relativity, with torsion":The torsion is generated by spin. Therefore, itmust remain a dynamical variable in analyses oflocal, nongravitational test experiments. It is nota gravitational field. )

Dicke's weak equivalence principle (WEP).The weak equivalence principle states: If an un-charged test body is placed at an initial event inspacetime, and is given an initial velocity there,then its subsequent soorld line evil/ be independentof its internal structure and composition. Hereby "uncharged test body" is meant an object (i)that is shielded, in the sense used above in defin-ing "local test experiments"; (ii) that has negligi-ble self-gravitational energy, when analyzed usingNewtonian theory; (iii) that is small enough in sizeso its coupling (via spin and multipole moments)to inhomogeneities of external fields can be ig-

3572 KIP S. THORNE, DAVID L. LEE, AND ALAN P. LIGHTMAN

nored. These constraints guarantee that any testof WEP is a local, nongravitational test experi-ment.

WEP is called "universality of free fall" byMTW, "and is called "equality of passive and in-ertial masses" by Bondi."

The experiments of Eotvos et al. ,' Dicke et al. ,'

and Braginsky et al. ' are direct tests of WEP.Braginsky, whose experiment is the most recent,reports that the relative acceleration of an alumi-num test body and a platinum test body placed inthe sun's gravitational field at the location of theearth's orbit is

(relative acceleration) & 0.9 x 10 "(GMe/rorbit )

=0.5xl0 "cm/sec'

(95% confidence) .

If WEP is correct, then the world lines of testbodies are a preferred family of curves (withoutparametrization) filling spacetime —with a singleunique curve passing in each given directionthrough each given event ~ But WEP does not guar-antee that these curves can be regarded as geode-sics of the spacetime manifold; only if thesecurves have certain special properties can theybe geodesics. "

Einstein equivalence PrinciPle (EEP). The Ein-stein equivalence principle states that (i) WEP isvalid, and (ii) the outcome of any local, nongravi-tational test experiment is independent of whereand when in the universe it is performed, and in-dependent of the velocity of the (freely falling) ap-paratus. (Example —Dimensionless ratios of non-gravitational physical constants must be indepen-dent of location, time, and velocity. } The experi-mental evidence supporting EEP is reviewed inSecs. 38.5 and 38.6 of MTW. "

Diche's strong equivalence PrinciPle (S&&).SEP states that (i) WEP is valid, and (ii) the out-come of any local test experiment —gravitationalor nongravitational —is independent of where andwhen in the universe it is performed, and indepen-dent of the velocity of the (freely falling) appara-tus. (Example —The Dicke-Brans- Jordan theory,with its variable "gravitational constant" as mea-sured by Cavendish experiments, satisfies EEPbut violates SEP.)

Two types of effects can lead to a breakdown ofSEP: "preferred-location effects" and "preferred-frame effects. " Perform a local test experiment,gravitational or nongravitational. If the experi-mental result depends on the location of the freelyfalling experimenter, but not on his velocity there,the phenomenon being measured is called a pre-ferred-location effect. If it depends on the velocityof the experimenter, it is called a preferred-

/rame ejfect 2b. [Examples —A cosmological timevariation in the "gravitational constant" (as mea-sured by Cavendish experiments) is a preferred-location effect. Anomalies in the earth's tidesand rotation rate due to the orbital motion of theearth around the sun and the sun through the gal-axy" are preferred-frame effects. ]

A theory of gravity obeys SEP if and only if itobeys EEP, and it possesses no preferred-frameor preferred-location effects.

Any theory for which the complete external sym-metry group excludes boosts will presumably ex-hibit preferred-frame effects. But preferred-frame effects can also show up when boosts are inthe symmetry group. (Example —The vector-ten-sor theory of Nordtvedt, Hellings, and Will" ex-hibits preferred-frame effects but possesses MMGas a symmetry group. ) For further discussionsee "metric theory of gravity, "below.

IV. PROPERTIES AND CLASSESOF GRAVITATION THEORIES

Completeness of a theory. A gravitation theoryis "complete" if it makes a definite prediction(not necessarily the correct prediction} for theoutcome of any experiment that current technologyis capable of performing. (Standard quantum-mechanical limitations on the definiteness of theprediction are allowed. ) To be complete, the the-ory must predict results for nongravitational ex-periments as well as for gravitational experiments.Of course, it can do so only if it meshes with andincorporates (perhaps in modified form) all thenongravitational laws of physics. If a theory iscomplete so far as all "classical" experimentsare concerned, but has not yet been meshed withthe quantum-mechanical laws of physics, we shallcall it classically complete.

Self-consistency of a theory. A gravitation theo-ry is "self-consistent" if its prediction for the out-come of every experiment is unique —i.e., if, whenone calculates the prediction by different methods,one always gets the same result.

Reference 2 discusses completeness and self-consistency in greater detail, and gives examplesof incomplete theories and self-inconsistent theo-ries.

&elativistic theory of gravity. A theory of grav-ity is "relativistic" if it possesses a representa-tion ("relativistic representation") in which, inthe absence of gravity, the physical laws reduceto the standard laws of special relativity. (Ex-amples —General relativity, Ni's theory, and theBelinfante-Swihart theory are relativistic; theNewton-Cartan theory is not, nor is Cartan's tor-sion-endowed modification of general relativity. '

)

FOUNDATIONS FOR A THEORY OF GRAVITATION THEORIES 35'73

Metric theory of gravity. By "metric theory"we mean any theory that possesses a mathematicalrepresentation ("metric representation") in which

(i) spacetime is endowed with a metric; (ii) theworld lines of test bodies are the geodesics ofthat metric; and (iii) EEP is satisfied, with thenongravitational laws in any freely falling framereducing to the laws of special relativity. Any

theory or representation that is not metric willbe called nonmetric. [Examples —General relativ-ity and Ni's theory are metric theories, and therepresentations given in Tables II and III are met-ric; the Belinfante-Swihart theory is nonmetric, "but can be made metric by suitable modifica-tions. "'" The Newton-Cartan theory is nonmet-ric. The Dicke-Brans-Jordan theory is metric;the representation of Ref. 16 is a metric repre-sentation; the representation of Ref. 18 ("confor-mally transformed representation"; "rubber metersticks" ) is nonmetric. ]

In any metric theory, the metric that enters in-to EEP is called the "Physical metric. " All othergravitational fields are called "auxiliary gravita-tional fields. " Relevant auxiliary scalar fieldstypically produce preferred-location effects; otherrelevant auxiliary gravitational fields (vector,tensor, etc. ) typically produce preferred-frameeffects. This is true independently of whether ornot the auxiliary fields are absolute variables orare dynamical —i.e., independently of whether thecomplete external symmetry group is MMG or ismore restrictive.

Clearly, every metric theory is relativistic,but relativistic theories need not be metric [ex-ample: the Belinfante-Swihart theory] ¹"ha. sgiven a partial catalog of metric theories. Willand Nordtvedt" have developed a "parametrizedpost-Newtonian formalism" for comparing metrictheories with each other and with experiment.

Prior geometric theories. Any gravitation theo-ry will be called a "prior geometric theory" if itpossesses a fully reduced, generally covariantrepresentation that contains absolute variables.(Examples —The Newton-Cartan theory, Ni's the-ory, and the Belinfante-Swihart theory are priorgeometric; general relativity and the Dicke-Brans-Jordan theory are not. )

Lorentz-symmetric rePresentations and theo-ries. A generally covariant representation iscalled "Lorentz symmetric" if its complete exter-nal symmetry group is the Poincare group —withor without inversions and time reversal. We sus-pect that, for any theory, all fully reduced, gen-erally covariant representations must have thesame complete external symmetry group. As-suming so, we define a theory to be "Lorentz sym-metric" if its fully reduced, generally covariant

representations are Lorentz symmetric. (Ex-ample —General relativity is not Lorentz symmet-ric; the complete external symmetry group of itsfully reduced, standard representation is too big-it is MMG rather than Poincare. ) (Example —Ni'stheory is not Lorentz symmetric; as with the New-ton-Cartan theory, the complete external symme-try group is too small. ) (Example —Belinfante-Swihart theory is Lorentz symmetric. )

Elsewhere in the literature one sometimes findsLorentz-symmetric theories called "Lorentz-in-variant theories" or "Qat-space theories. "

Lagrangian-based rePresentations and theories.A generally covariant representation of a space-time theory is called Lagrangian-based if (i) thereexists an action principle that is extremized withrespect to variations of all dynamical variablesbut not with respect to variations of absolute orconfined variables, and (ii) from the action prin-ciple follow all the dynamical laws but none of theother physical laws. The issue of whether theother physical laws (boundary conditions, decom-position equations, and prior geometric con-straints) are imposed before the variation orafterwards does not affect the issue of whetherthe representation is Lagrangian-based. A theoryis called Lagrangian-based if it possesses a gen-erally covariant, Lagrangian-based representa-tion. (Examples —General relativity, Ni's theory,and the Belinfante-Swihart theory are all Lagran-gian-ba, sed. )

The Lagrangian density 2 of a Lagrangian-basedrepresentation (which appears in the action prin-ciple in the form 5JZd'x =0) can be split up intotwo parts: Z =KG+X„G. The gravitational partZG is the largest part that contains only gravita-tional fields. The nongravitational part gNG is therest.

U. UNIUERSAL COUPLING

We turn attention, now, from our glossary ofconcepts to some applications. We begin in thissection by analyzing the overlap between metrictheories and relativistic, Lagrangian-based theo-ries.

As motivation for the analysis, consider anyrelativistic representation of a relativistic theoryof gravity. In the absence of gravity that repre-sentation reduces to special relativity —so, inparticular, it possesses a flat Minkowski metric

By continuity one expects the representationto possess, in the presence of gravity, at leastone second-rank, symmetric tensor gravitationalfield g 8 that reduces to q 8 as gravity is turnedoff. Indeed, this is the case for all relativistictheories with which we are familiar. (Example-

3574 KIP S. THOBNE, DAVID L. LEE, AND ALAN P. LIGHTMAN

general relativity: The curved-space metric g ~

reduces to q 8 when gravity is turned off. ) (Ex-ample —Ni's theory: There are a variety of sec-ond-rank, symmetric tensor gravitational fieldsthat reduce to g &. They include the Qat back-ground metric q 8, the physical metric g 8, anytensor field of the form [1+f(qr)]q„s, where f(rp)is an arbitrary function with f (0) = 0, etc.) [Ex-ample —Belinfante-Swihart theory: q 6, g 8+h 8,rl s(1+3k„"}—17h "h all reduce to ti„s when

gravity is turned off.Next consider any Lagrangian-based, relativis-

tic theory. Being relativistic, it must possess agenerally covariant, Lagrangian-based represen-tation in which, as gravity is turned off, the non-gravitational part of the Lagrangian Z„G ap-proaches the total Lagrangian of special relativity.Adopt that representation. Then, in the presenceof gravity Z„G will presumably contain at least onesecond-rank, symmetric, tensor gravitationalfield g 8 that reduces to q 8 as gravity is turnedoff. Roughly speaking, if Z„G contains preciselyone such g 8 and contains no other gravitationalfields, then the theory is said to be "universallycoupled. '"'

More precisely, we say that a Lagrangian-based,relativistic theory is universally coupled if it pos-sesses a representation ("universally coupled rep-resentation") with the following properties: (i)The representation is generally covariant and La-grangian-based. (ii) Z„o contains precisely onegravitational field, and that field is a second-rank,symmetric tensor g &

with signature +2 through-out spacetime. (iii) In the limit as gravity isturned off g s becomes a Riemann-flat second-rank, symmetric tensor field q &,

' and whenever

$~8 is replaced by such an g &, J„G becomes thetotal Lagrangian of special relativity. (iv} Theprediction for the result of any local, nongravita-tional experiment anywhere in the universe is un-changed when, throughout the laboratory, one re-places $„8by a Riemann-Qat second-rank, sym-metric tensor.

The following theorem reveals the key role ofuniversal coupling as a link between Lagrangian-based theories and metric theories: Conside~ allLagrangian-based, relativistic theories of gravity.Every such theory that is universally coupled is ametric theory; and, conversely, every metrictheory in this class is universally coupled.

Proof: Let 5 be a Lagrangian-based, relativis-tic, universally coupled theory. Adopt a univer-sally coupled representation. Use that represen-tation to analyze any local, nongravitational testexperiment anywhere in spacetime. Use themathematical tools of Riemannian geometry,treating the unique gravitational field g 8 that ap-

pears in gNG as a metric tensor. In particular,introduce a Fermi-normal coordinate system (P„8=q 8, I &„=0 at the center of mass of the labora-tory). Condition (iv) for universal coupling guar-antees that the predictions of the representationwill be unchanged if we replace g 8 by g„zthroughout the laboratory. Do so. Then condition(iii) for universal coupling guarantees that Z„ isthe total Lagrangian of special relativity. The dy-namical laws that follow from

gG+gNG d g=0

by varying all nongravitational variables also fol-low from

in this representation and coordinate system theyare the laws of special relativity. Thus, the out-come of the local, nongravitational test experi-ment is governed by the standard laws of specialrelativity, irrespective of the location and velocityof the apparatus. This guarantees that theory 5 isa metric theory.

Proof of converse: Let g& be a Lagrangian-based,metric theory. Adopt a Lagrangian-based, metricrepresentation. Since all unconfined, nongravita-tional variables are dynamical, they must all bevaried in 5fZd'x=0. Moreover, since they appearin Z„G but not in ZG, their Euler-Lagrange equa-tions are obtained equally well from

ZNG d4x = 0.

Call those Euler-Lagrange equations (obtained byvarying all unconfined, nongravitational variablesin 0 fZ„od x =0) the "nongravitational laws. " Leta freely falling observer anywhere in spacetime,with any velocity, perform a local, nongravitation-al test experiment. Analyze that experiment in alocal Lorentz frame of the physical metric g 8 us-ing the above nongravitational laws. Because thetheory is metric, the predictions must be the sameas those of special relativity. Hence, the nongrav-itational laws —in any local Lorentz frame of g 8any&chere in the universe —must reduce to the lawsof special relativity. This is possible only if (i)those laws —and hence also gNG —contain no refer-ence to any gravitational field except g z,

"and(ii) g„o is some version of the total special rela-tivistic Lagrangian, with q 8 replaced by g ~.These properties of Z„G, plus the definition of"metric theory, " guarantee directly that the fourconditions for universal coupling are satisfied.Hence, theory @ is universally coupled. QED.

FOUNDATIONS FOR A THEORY OF GRAVITATION THEORIES 3575

VI. SCHIFF'S CONJECTURE

Schiff's conjecture states that any comPleteand self con-sistent gravitation theory that obeys8'EP must also, unavoidably, obey EEP.

General relativity is an example. It endows

spacetime with a metric; it obeys WEP by pre-dicting that all uncharged test bodies fall alonggeodesics of that metric, with each geodesic worldline determined uniquely by an initial event and aninitial velocity; it achieves completeness by de-manding that in every local, freely falling framethe nongravitational laws of physics take on theirstandard special relativistic forms; and by thismethod of achieving completeness, it obeys EEP.

The Newton-Cartan theory is another example.It was complete and self-consistent within theframework of nineteenth century technology. Itendows spacetime with an affine connection; itobeys WEP by predicting that all uncharged testbodies fall along geodesics of that affine connec-tion, with each geodesic world line determineduniquely by an initial event and an initial velocity;it achieves completeness by demanding that inevery local, freely falling frame the laws of phys-ics take on their standard nongravitational New-tonian form; and by this method of achieving com-pleteness, it obeys EEP.

Before accepting Schiff's conjecture as plausible,one should search the literature for a counterex-ample —i.e., for a theory of gravity which some-how achieves completeness, and somehow obeysWEP, but fails to obey EEP. Several Lagrangian-based theories which one finds in the literaturemight conceivably be counterexamples, but theyhave not been analyzed with sufficient care to al-low any firm conclusion. Subsequent papers"will show that the most likely counterexample,Belinfante-Swihart theory, actually fails to satisfyWEP, violates the ED experimental results, andis thus not a counterexample at all.

One can make Schiff's conjecture seem veryplausible within the framework of relativistic, La-grangian-based theories (the case of greatest in-terest; see Sec. I) by the following line of argu-ment. "

Consider a Lagrangian-based, relativistic theo-ry, and ask what constraints WEP places on theLagrangian. WEP probably forces ZNG to involveone and only one gravitational field (and that fieldmust, of course, be a second-rank symmetric ten-sor g 8 which reduces to g z far from all gravi-tating bodies). If g„o were to involve, in addition,some other gravitational field y, then to satisfyWEP g 8 and (It} would have to conspire to produceidentically the same gravitational accelerationson a test body made largely of rest mass, as on a

body made largely of electromagnetic energy, ason a body made largely of internal kinetic energy,as on a body made largely of nuclear binding ener-gy, as on a body made largely of. . . . This seemsimplausible, unless g 8 and y appear everywherein J„G in the same "mutually coupled" formf ((p) g ((

—in which case one can absorb f ((p) into

g 8 and end up with just one gravitational field inThus, it seems likely that WEP forces Z„G

to involve only g 8. This means that the theory isuniversally coupled- and, hence, by the theoremof Sec. V, it is a metric theory.

This argument convinces us that Schiff's conjec-ture is probably correct, when one restricts atten-tion to Lagrangian-based, relativistic theories.And it is hard to see how the conjecture could failin other types of theories.

A formal proof of Schiff's conjecture for a morelimited class of theories will be given in a subse-quent paper. '

APPENDIX: ABSOLUTE AND DYNAMICAL FIELDSIN NEWTON-CARTAN THEORY

In order to separate the absolute gravitationalfields of Newton-Cartan theory from the dynamicalfields, one must change mathematical representa-tions. In place of the representation given in Ta-ble I, one can adopt the following.

1. Gravitational fields.a. Symmetric covariant derivatives (two of

them): D and V.b. Scalar gravitational field: 4.c. Spatial metric [defined on vectors w such that

(B,w&=0]: y.d. Universal 1-form: P.

(Note: t has been replaced by P in order to removefrom the theory the "irrelevant" choice of originof universal time; see "irrelevant variables" inSec. II A. D and 4 will turn out to be absolute anddynamical parts of V; see below. }

Z. Gravitational field equations.

a. P is perfect: d /=0.b. p is covariantly constant: Dp=0.c. D is flat: Riemann (D}=0.d. Compatibility of D and y'.

(Ala)

(A1b)

(Alc}

D„(v w) =(D„v) w+v ~ (D„w} for any vector n, andfor any spatial vectors v, w. (A Id)

e. Decomposition of V:

V = D +A(3) P(3 P, where A is the spatial vector"dual" to dC: (dC, w) =A w for all spatialw.

(Ale)

3576 KIP S. THOBNE, DAVID L. LEE, AND ALAN P. LIGHTMAN

f. Field equation for 4:

D ~ A -=(divergence of A) = 4vp. (Alf)

=(dx ",D„es) =0, (A2a)

the condition Ddt = 0 becomes s't/sx ex 8 = 0; inparticular, s't/sx'ax'=0, so t =ax'+b for someconstants a and b. Renormalize x' so I =x'. (v)In the resulting coordinate frame P, y, and A havecomponents

3. Influence of gravity on matter. Same as in

part 4 of Table I where t is any scalar field suchthat P = dt.

To prove that this and the formalism given inTable I are different mathematical representationsof the same theory, we can show that they becomeidentical in Galilean coordinate frames. The re-duction of the formalism of Table I to a Galileanframe is performed in Exercise 12.6 of MTW. "The reduction of the above formalism proceeds asfollows: (i) Let t be any particular scalar fieldsuch that P =df (ii) A. t some particular event inspacetime pick a set of basis vectors [e } suchthat (a) e„~e, e~ are spatial, (~p e,) = 0, and or-thonormal, e, e, =b»,' (b) e, is not spatial, (P, e, )s0. (iii} From each vector e construct a vectorfield on all of spacetime by parallel transportwith D. The resulting field is unique because D isflat; and it has De =0. Hence, the commutatorsvanish:

[e, e8]=D e8 —Dse

This guarantees the existence of a coordinate sys-tem fx "} in which e =s/sx . (iv) The condition(valid in any coordinate frame} (dx', e&) =0, when

compared with (dt, ei) =0, guarantees that the sur-faces of constant x' and constant t are identical,i.e., t = f (x'). Moreover, because the connectioncoefficients of D vanish in this coordinate frame,

4 - 4 ' = 4 —a'(t)x ' + constant,

all other variables, including I' 8,(A3)

left unchanged.

In coordinate-free form the internal transforma-tions are

D D'= D+aP(3P,

4-4 '=4 —b,(A3'a}

where a is any vector field which is covariantlyconstant in the surfaces of P,

D a =V a =0 for all spatial vectors m;

and where b is any scalar field such that

(A3'b)

Chap. 12 of MTW. " Thus, the two formalisms aredifferent mathematical representations of the sametheory.

In the above formalism it is easy to verify thatD, P, and y are absolute gravitational fields,while 4 is a dynamical gravitational field. In fact,D, P, and y are the absolute parts of V; 4 is itsdynamical part; Eqs. (Ala)-(Ald) are the priorgeometric constraints of the theory; Eq. (Ale} isthe decomposition of V into its absolute and dynam-ical parts; and Eci. (Alf) is the dynamical fieldequation for 4.

The covariance group for the above mathematicalrepresentation of Newton-Cartan theory is slightlylarger than that for the representation of Table I.For Table I the covariance group is MMG. For theabove representation it is the direct product ofMMG with a group of internal covsriance transformations. In a Galilean frame the internal trans-formations are

Pp——1, P) =0, y)~ =5j)

A'=0, A =84/sx'; (A2b) (db, w) =a w for all spatial vectors w .

(A3'c)

g2@=4m

ex&ax& (A2c)

and the connection coefficients of V are I'z~

=A "t Bt „, i.e. ,8@F pp j p all other F 8„vanish ~ax (A2d)

so the field equation for 4 is Poisson's equationThe complete symmetry group for the above

mathematical representation of Nevrton-Cartantheory is best analyzed in a Galilean coordinatesystem. [Because the absolute objects are all ten-sors or affine connections, one can restrict atten-tion to a single coordinate system; see Eq. (13)and associated discussion in the text. ] The sym-metry transformations are those which leave

This Galilean coordinate version of the aboveformalism is identical to the Galilean coordinateversion of the formalism of Table I, as given in

5y)„——5P = 6™Py

=0 (A4)

FOU NDA TIONS FOR A THEORY OF QRA VITATION THEORIES 3577

Clearly, the symmetry transformations include(i) spacetime translations

X ~X =X +C (A5a}

(A5b)

~~R~~[

a constant rotation matrix. They also in-clude (iii) the combination of an arbitrary time-dependent spatial translation with a carefullymatched internal covariance transformation

where c are constants, and (ii) spatial rotations

"f00 00 00 + () d2

where c' =dt2 '

4- 4 =4 —c'(t}x~

Note that these symmetry transformations areprecisely the transformations that lead from oneGalilean coordinate system to another (cf. Sec.12.3 of MTW1').

x'-x' =x'+c'(t), where c' are arbitraryfunctions of t,

(A5c}

*Supported in part by the National Aeronautics andSpace Administration under Grant No. NGR 05-002-256and the National Science Foundation under Grant No.GP-27304, and No. GP-28027.

)Imperial Qil Predoctoral Fellow.)NSF Predoctoral Fellow during part of the period of

this research.K. S. Thorne and C. M. Will, Astrophys. J. 163, 595

(1971).K. S. Thorne, W. -T. Ni, and C. M. Will, in Pro-

ceedings of the Conference on Experimental Tests ofGravitation Theories, edited by R. W. Davies, NASA-JPL Tech. Memo 33-499, 1971 (unpublished).

C. M. Will, Physics Today 25 (No. 10), 23 (1972).4C. M. Will, Proceedings of Course 56 of the Inter-

national School of Physics "Enrico Fermi", edited byB. Bertotti (Academic, New York, to be published)[also distributed as Caltech Report No. OAP-289,1972 (unpublished)].

50ur form of Schiff's conjecture is a classical analogof Schiff's original quantum mechanical conjecture.Schiff briefly outlined his version of the conjecture onpage 343 of his article in Am. J. Phys. 28, 340 (1960).So far as we know, he never pursued it in any detailuntil November 1970, when his interest in the issue wasrekindled by a vigorous argument with one of us (KST)at the Caltech-JPL Conference on Experimental Testsof Gravitation Theories. Unfortunately, his sudden death2 months later took him from us before he had a chanceto bring his analysis of the conjecture to fruition.

~R. V. Eotvos, D. Pekar, and E. Fekete, Ann, Phys.(Leipzig) 68, 11 (1922); also R. V. Eotvos, Math. u.Nature. Ber. Aus. Ungam. 8, 65 (1890)~

7P. G. Roll, R. Krotkov, and R. H. Dicke, Ann. Phys.(N.Y.) 26, 442 (1964).

V. B. Braginsky and V. I. Panov, Zh. Eksp. Teor.Fiz. 61, 875 (1971) [Sov. Phys. -JETP 34, 3463 (1972)].

A. Lightman and D. Lee, Caltech Report No. OAP-314,1973 (unpublished).'OA. Trautman, lectures in A. Trautman, F. A. E.

Pirani, H. Bondi, I.ectures on General Relativity(Prentice-Hall, Englewood Cliffs, N. J., 1965).

~~J. L. Anderson, Principles of Relativity Physics(Academic, New York, 1967); cited in text as JLA.' C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation (Freeman, San Francisco, 1973); cited in

text as MTW.' W. -p. Ni, this issue, Phys. Rev. D 7, 2880 (1973).

Our normalizations and signature differ from those of Ni.~4D. Lee and A. Lightman, following paper, Phys. Rev.

D 7, 3578 (1973)~

'+F. J. Belinfante and J. C. Swihart, Ann. Phys. (N. Y.)1, 168 (1957); 1, 196 (1957); 2, 81 (1957).

C. Brans and R. H. Dicke, Phys. Rev. 124, 925(1961).

P. Jordan, Z. Physik 157, 112 (1959).R. H. Dicke, Phys. Rev. 125, 2163 (1962).

' This definition of "faithful realization" differs fromthat given on page 26 of JLA; we think that this is whatJLA intended to say and should have said.

E. Kretschmann, Ann. Phys. (Leipzig) 53, 575 (1917).For the topological properties of groups see, e.g. ,

L. S. Pontrjagin, Topological Groups (Qxford Univ.Press, Oxford, 1946).

This concept is due to C. W. Misner; see Chap. 17 ofMTW (Ref. 12).

Note added in proof. A. Trautman (private communi-cation) argues that we should define as "gravitational"those phenomena which are absolute or "go away" as theamounts of mass-energy and spin in the experimentallaboratory decrease. Such a definition, he argues, wouldbe "in accordance with our knowledge, based on specialrelativity, where mass and spin are equally fundamental. "He goes on to say, "I feel that it is a little misleadingto arrange the definitions (by omitting the phrase 'andspin' above) so as to exclude the Einstein-Cartan theoryfrom the framework of relativistic theories of gravity(Sec. IV of this paper). Clearly, the Einstein-Cartantheory reduces to special relativity when G- 0. More-over, in the real world, spin, charge, baryon number,etc. are always accompanied by energy and momentum.Therefore, if the amount of mass-energy in the labora-tory decreases, then not only gravitational phenomena goaway: so do electromagnetic and nuclear effects as wellas the (so far hypothetical) effects of torsion. "

We (the authors) find Trautman's argument cogent.Nevertheless, we adhere to our set of definitions —fora very pragmatic reason. By refusing to allow torsioninto "relativistic theories of gravity" we make it possiblefor ourselves to prove certain theorems about relativistictheories which otherwise we would not know how to proveand which might well be false. An example is the theorem

35'78 KIP S. THORNE, DAVID L. LEE, AND ALAN P. LIGHTMAN

on universal coupling proved in Sec. V of this paper.For details and references on Cartan's theory with

torsion, see A. Trautman, Bull. Acad. Pol. Sci. Ser. Sci.Math. Astron. Phys. 20, 185 (1972), and references citedtherein.

24R. H. Dicke, lectures in Relativity, Groups, and

Topology, edited by C. DeWitt and B. DeWitt (Gordonand Breach, New York, 1964).

For a more detailed discussion of WEP see Chap. 38of MTW-where WEP goes under the name "universalityof free fall".

~H. Bondi, Rev. Mod. Phys. 29, 423 (1957).See, e.g. , Chap. 10 of MTW.

28K. Nordtvedt, Jr. and C. M. Will, Astrophys. J.177, 775 (1972).

29This definition of "metric theory" originated in Chap.39 of MTW. Here and henceforth we shall adhere to it,even though earlier work by our group (e.g. , Ref. 1)used a slightly less restrictive definition. (Any theorythat is "metric" according to the present definition isalso "metric" according to the old definition. )

3 A. Lightman and D. Lee (unpublished).W. -T. Ni, Astrophys. J. 176, 769 (1972).

32C. M. Will and K. Nordtvedt, Jr. , Astrophys. J.

177, 757 (1972); also earlier references by Nordtvedtand by Will cited therein. For reviews see Refs. 2, 3,4, and 12.

We have adapted the principle of universal couplingfrom R. V. Wagoner, Phys. Rev. D 1, 3209 (1970).Wagoner enunciated this principle only for the specialcase of scalar-tensor theories, and he gave it the morerestrictive name "principle of mutual coupling. " How-ever, our concept is a straightforward generalizationof his.

34To see more clearly why no gravitational fields otherthan g~8 can enter the Euler-Lagrange equations,argue as follows: The only gravitational effects whichvanish as the size of the frame vanishes arise fromterms of a Taylor series type expansion of somegravitational field(s) B. But if there are any B otherthan g„s, then somewhere in spacetime there willbe local Lorentz frames of g~& in which the lowestorder Taylor series term of some of the B does notvanish, thus violating the local validity of specialrelativity.

35This is a classical version of Schiff's originalquantum-mechanical line of reasoning (Ref. 5).

PHYSICAL REVIEW D VOLUME 7, NUMBER 12 15 JUNE 1973

Analysis of the Belinfante-Svvihart Theory of Gravity"'

David L. Lee" and Alan P. LightmanCalifornia Institute of Technology, Pasadena, California 91109

(Received 8 January 1973)

We show that the Belinfante-Swihart (BS) theory can be reformulated in a representation in whichuncharged matter responds to gravity in the same way as in metric theories. The BS gravitationallymodified Maxwell equations can also be put into metric form to first order in the deviations of thephysical metric from flat space, but not to second order; consequently, the theory is nonmetric exceptin first order. We also show that the theory violates the high-precision Eotvos-Dicke experiment, butcannot be ruled out by the gravitational precession of gyroscopes.

I. INTRODUCTION AND SUMMARY

This paper analyzes the most complete and ex-tensively developed nonmetric theory that exists:the 1957 theory of Belinfante and Swihart. ' 'Belinfante and Swihart (BS}constructed their the-ory as a Lorentz-symmetric~ linear field theorywhich would be easily quantized. However, as weshall show, in terms of measurable quantities thetheory has all the nonlinearities of typical "curved-spacetime" theories. Moreover, it is nearly ametric' theory: We construct a new mathematicalrepresentation which has metric form to first

order in deviations of the physical metric fromflatness, but does not have metric form to higherorders.

Section II gives a brief summary of the originalBS representation. Included are discussions ofnonlinearities and the behavior of rods and clocks.Section III presents our new mathematical repre-sentation of the theory. Section IV gives a pre-scription for obtaining the post-Newtonian limit'of the theory, and Sec. V considers various exper-imental tests. Contrary to previous calculations'it is found that both the geodetic and the Lens-Thirring precessions of gyroscopes cannot dis-