Studies in of General Relativity · 2019-04-20 · Studies, Vol. 1), the history of general...

Jean Eisenstaedt A. J. KoxEditors

Studies in the History of General Relativity

Based on theProceedings of the2nd International Conference on the History of General Relativity, Luminy, France, 1988

BirkhäuserBoston Basel Berlin

Jean EisenstaedtCentre National de laRecherche ScientifiqueUniversité Pierre et Marie Curie 75231 Paris Cédex 05France

A. J. KoxInstituut voor Theoretische FysicaUniversiteit van Amsterdam1018 XE AmsterdamThe NetherlandsandThe Collected Papers of Albert EinsteinBoston UniversityBoston, MAU.S.A.

Library of Congress Cataloging-in-Publication Data

International Conference on the History of General Relativity (2nd :1988 : Luminy, Marseille, France)

Studies in the history of general relativity : proceedings of theSecond International Conference on the History of General Relativity, Luminy, Marseille, France, 6-9 September 1988 / Jean Eisenstaedt, A. J. Kox, editors.

p. cm. — (Einstein studies : v. 3) Summaries in French.Includes bibliographical references and index.ISBN 0-8176-3479-7 (alk. paper)1. General Relativity (Physics) -History-Congresses.

I. Eisenstaedt, Jean, 1940-. II. Kox, Anne J. III. Title.IV. Series.QC173.6 1572 1988530.1'1''9-dc20

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Preface to Volume Three

In May 1988, the Osgood Hill Conference Center in North Andover, Massachusetts, housed a conference that was unique in its kind: the first international conference exclusively devoted to the history of general relativity. As John Stachel, who organized the meeting, wrote in the introduction to its proceedings (Einstein and the History of General Relativity, Don Howard and John Stachel, eds., Boston: Birkhäuser, 1989; Einstein Studies, Vol. 1), the history of general relativity—as well as general relativity itself—has in the past been a much neglected field of research. In recent years, however, a new interest in general relativity and its history has arisen, among historians as well as philosophers of science. Inspired by the success of the Osgood Hill Conference, its participants decided to try to make it the first in a regular series of meetings. Thus, the Second International Conference on the History of General Relativity was held September 6-9, 1988, at the International Center of Mathematical Research (CIRM) at Lummy, France. It was organized by Jean Eisenstaedt (Institut Henry Poincaré, Paris), with the assistance of an organizing committee consisting of S. Goldberg (Smithsonian Institution, Washington) A. J. Kox (University of Amsterdam and The Collected Papers of Albert Einstein), Μ. de Maria (University of Rome), J. Renn (The Collected Papers of Albert Einstein), and J. Stachel (The Collected Papers of Albert Einstein). The organizing committee was advised by a scientific committee, which consisted of the following persons: P. G. Bergmann (New York University), J. Goldberg (Syracuse University), P. Havas (Temple University), W. Israel (University of Alberta), Μ. Jammer (The Hebrew University of Jerusalem), A. Lichnerowicz (Collège de France), Μ. A. H. Mac- Callum (Queen Mary College), J. Merleau-Ponty (Université Paris XI) J. D. North (University of Groningen and International Academy of the History of Science), A. Papapetrou (Institut Henri Poincaré), J. C. Pecker (Collège de France), E. Schucking (New York University), J. Stachel

vi Preface to Volume Three

(The Collected Papers of Albert Einstein), and V. P. Vizgin (Academy of Sciences of the USSR).

At the end of the Luminy conference, it was decided to continue the series with a third meeting, to be organized by Professor John Norton and to be held June 27-30, 1991, in Pittsburgh, Pennsylvania. The proceedings of this conference will appear in a future volume of Einstein Studies.

The names of the participants at the Luminy meeting are listed in the back matter of this volume, along with the addresses of the contributors and French summaries of their papers.

viii Contents

Contents

Preface to Volume Three ............................. V

Acknowledgments ............................... .. viiA Note on Sources .................... .. vii

Introduction ........................................... xi

Part I The Reception and Development of General Relativity

General Relativity and Portugal: A Few PointersTowards Peripheral Reception Studies ...................... David Lopes Gagean and Manuel da Costa Leite

3

The Reception of the Theory of Relativity in Germany as Reflected by Books Published Between 1908 and 1945 .....Hubert e Μ. Goenner

15

General Relativity in the Netherlands, 1915-1920 ............A. J. Kox

39

The Reception of General Relativity Among British Physicists and Mathematicians (1915-1930) ............. ..José Μ. Sanchez-Ron

57

US Air Force Support of General Relativity: 1956-1972 .......Joshua N. Goldberg

89

Mathematics and General Relativity: A Recollection ..........André Lichnerowicz

103

General Relativity at the Turning Point of Its Renewal ......... ..André Mercier

109

Part II Conceptual Issues in General Relativity

Einstein and Mach’s Principle ............................ Julian B. Barbour

. 125

Einstein and Eindeutigkeit: A Neglected Themein the Philosophical Background to General Relativity ........ Don Howard

. 154

The Correspondence of Albert Einsteinand Gustav Mie, 1917-1918 ...................... ....................... ..JôzsEF Illy

, 244

An Outline of the History of Einstein’sRelativistic Ether Conception .............................LUDWIK KOSTRO

. 260

The Physical Content of General Covariance ................ John Norton

281

Part III Technical Aspects of General Relativity

Albert Einstein’s 1907 Jahrbuch Paper: The First Step from SRT to GRT ..................................................................Arthur I. Miller

319

On the Origin of the Concept of an Einstein Space ........... Gunnar Berg

336

H.A. Lorentz’s Attempt to Give a Coordinate-Free Formulation of the General Theory of Relativity ........................ Michel Janssen

344

Quantization of the Gravitational Field, 1930-1988 ...........Peter G. Bergmann

364

The First Steps of Quantum Gravity and the Planck Values .... Gennady E. Gorelik

367

Dirac Equations in Curved Space-Time .....................S. Kichenassamy

383

Levi-Civita and the General-Relativistic Problem of Motion ....Thibault Damour and Gerhard Schäfer

393

Myron Mathisson’s and Jan Weyssenhoff’s Work on the Problem of Motion in General Relativity ........................... Bronislaw Sredniawa

400

x Contents

The Cauchy Problem in General Relativity—The Early Years .. John Stachel

. 407

Spherical Coordinates in General Relativity from 1915 to 1960: A Physical Interpretation ............................... Christiane Vilain

. 419

Part IV Relativistic Cosmology

Contributions of Lemaître to General Relativity (1922-1934) Odon Godart

. 437

Schrôdinger and Cosmology ............................. Helmuth Urbantke

. 453

French Summaries ..................................... . 460List of Participants .................................... . 466Contributors’ Addresses ................................ . 467

Introduction xi

Introduction

Jean Eisenstaedt and A. J. Kox

This volume of the Einstein Studies series presents the proceedings of the Second International Conference on the History of General Relativity, which was held in Lummy, France, in the fall of 1988. In spite of our efforts, we are not able to include in this volume all of the papers presented at the meeting: personal circumstances prevented Peter Havas from finishing his paper dealing with the history of the controversy between Einstein and Ludwik Silberstein before this volume went to press. It is planned to include his paper in a future volume of Einstein Studies. However, not all of the papers in this volume were delivered at Luminy. Joshua Goldberg’s paper on the history of the US Air Force support for research in general relativity was actually presented at the Osgood Hill conference on the history of general relativity (see Preface), and the paper by Thibault Damour and Gerhard Schäfer on Levi-Civita and the problem of motion in general relativity was delivered at the Fifth Marcel Grossmann Conference at the University of Western Australia. Gennady Gorelik’s history of the first stages of quantum gravity is included, even though the author was unable to attend the meeting.

Several of the papers presented here are devoted to the reception and development of the general theory of relativity. David Lopes Gagean and Manuel da Costa Leite concentrate on the situation in Portugal, Hubert F. Μ. Goenner offers a survey of the reception of the theory of relativity in Germany through the books published between 1908 and 1945, and in addition, he provides an extensive bibliography of this period. One of us (AJK) presents an account of the reception and further development of general relativity in the Netherlands. Finally, José Μ. Sânchez-Ron gives a detailed description of the reception of Einstein’s theory of gravity among British physicists and mathematicians between 1915 and 1930.

On a more personal level, Joshua N. Goldberg tells the intriguing story of the relativity research group at Wright-Patterson Air Force Base,

xii Jean Eisenstaedt and A. J. Kox

Ohio, of which he was a leading member; André Lichnerowicz analyzes mathematical research in the field during the years 1937-1945 and discusses his own work in this context; and André Mercier offers us a description and a personal testimony about the origins of the international meetings on general relativity (GRG), in particular of the first one, which was held in Bern in 1955.

The discussion of conceptual issues in general relativity constituted a second theme of the conference. Julian B. Barbour discusses in a historical perspective the question of the “Machian” character of Einstein’s theory of gravitation, while Don Howard explores the role in Einstein’s thinking of what he calls the Eindeutigkeit principle. Jôzsef Illy gives an account of the discussion on general covariance as it took place in the correspondence between Einstein and Gustav Mie, whereas Ludwik Kostro treats another fundamental topic in Einstein’s thinking through his concise history of Einstein’s relativistic ether conception. Finally, John Norton analyzes the question of Einstein’s attitude to the physical content of general covariance and reviews Kretschmann’s objection to Einstein’s use of covariance principles.

A third set of papers deals with the more technical aspects of general relativity. Arthur I. Miller treats the prehistory of general relativity through the analysis of Einstein’s 1907 Jahrbuch paper, and Gunnar Berg sketches the origin of the Einstein space. Michel Janssen analyzes Lorentz’s 1916 geometrical approach to general relativity. There are two papers on quantum gravity: Peter G. Bergmann presents a short review of the quantization of the gravitational field from 1930 to the present, and Gennady E. Gorelik concentrates on the first steps toward quantum gravity, in particular on Matvey P. Bronsteyn’s work in this field. S. Kichenassamy’s study focuses on the histoiy of the description of the coupling of an electron with a gravitational field. Thibault Damour and Gerhard Schafer give a technical analysis of Levi-Civita’s approach to the problem of motion in general relativity, while the work of the Polish physicist Myron Mathisson on the problem of motion is analyzed by Bronislaw Sredniawa. In addition, John Stachel retraces the early history of the Cauchy problem in general relativity, and Christiane Vilain gives an overview of the question of spherical coordinates in general relativity.

Finally, we should mention two papers on cosmology: Odon Godart gives a review of the work of Lemaître, and Helmuth Urbantke surveys Schrodinger’s contributions to cosmology.

The First Steps of Quantum Gravity and the Planck Values

Gennady E. Gorelik

Every contemporary discussion of quantum gravity inevitably includes the so-called Planck values, which are combinations of three ftmdamental constants: c (the velocity of light), G (the gravitational constant), and h (Planck’s constant):

XPl = cxGyhz,

where the exponents x, y, z are rational numbers. Depending upon the choice of x, y, z the Planck values can have any dimensionality—length, time, density, and so forth.

Today, the Planck values are connected with the quantum limits of general relativity (GR). Such an interpretation of these values is reached in different ways—from arguments within the framework of tentative variants of quantum-gravitational theory to simple dimensional considerations. Because the latter approach does not require any complex theoretical constructions, one can surmise that the quantum-gravitational role of the Planck values may have been known to Planck himself.

In reality, however, Planck introduced his cGh values in 1899, without any connection to quantum gravity. Quantum limits to the applicability of general relativity (and, implicitly, their Planck scale) were first discovered in 1935 by the Soviet theorist Matvey P. Bronsteyn (1906— 1938). It was not until the 1950s that the explicitly quantum-gravitational significance of the Planck values was pointed out almost simultaneously by several physicists. In the following, the history of the Planck values will be reviewed. The topic is an important one, because the Planck scale is almost the only element certain to be a part of a future synthesis of general relativity and quantum theory (Misner, Thome, Wheeler 1973).

368 Gennady E. Gorelik

1. The Birth of the Planck Values

In the fifth installment of his continuing study of irreversible radiation processes (Planck 1899), Max Planck introduced two new universal physical constants, a and b, and calculated their values from experimental data. The following year, he redesignated the constant b by the famous letter h (and in place of a, he introduced k = b/a, the Boltzmann constant).

In 1899, the constant b (that is, h) did not yet have any quantum theoretical significance, having been introduced merely in order to derive Wien’s formula for the energy distribution in the black-body spectrum. However, Planck had previously described this constant as universal. During the six years of his efforts to solve the problem of the equilibrium between matter and radiation, he clearly understood the fundamental, universal character of the sought-for spectral distribution.

It was perhaps this universal character of the new constant that stimulated Planck, in that same paper of 1899, to consider a question that was not directly connected with the paper’s main theme. The last section of the paper is entitled “Natural Units of Measure” [“Natürliche Maassein- heiten”]. Planck noted that in all ordinary systems of units, the choice of the basic units is made not from a general point of view “necessary for all places and times,” but is determined solely by “the special needs of our terrestrial culture” (Planck 1899, p. 479). Then, basing himself upon the new constant h and also upon c and G, Planck suggested the establishment of

units of length, mass, time, and temperature that would, independently of special bodies and substances, necessarily retain their significance for all times and all cultures, even extraterrestrial and extrahuman ones, and which may therefore be designated as natural units of measure. (Planck 1899, pp. 479-480)

These new units are chosen in such a way that in the new system of units every mentioned constant (c, G, h) is equal to one. Thus, Planck introduced the following units:

lPl = (hG/c)1/2 = 10-33 cm

mPl = (hc/G)1/2 = 10-5g

tPl = (hG/c5)1/2 = 10-43s

(where modern notation and values are used). In cosmology, a “natural unit” of density is also now very popular:

ρPl = c5/hG2 = 1094g/cm3.

Quantization of the Gravitational Field 369

While this last section of the paper bears only a slight connection to the rest, it is strongly connected with Planck’s broader philosophy of physics, in particular, with his attention to the absolute elements of the scientific world picture and his opposition to anthropomorphism in physics (Goldberg 1976). Eddington (1918) seems to be the first who attempted to uncover a theoretical significance in the Planck length, a significance that may be the clue to some essential structure of gravitation. But, both Planck’s and Eddington’s suppositions were ignored or even ridiculed (see, for example, Bridgman 1922).

2. Gravitation and Quanta

The quantum-gravitational meaning of the Planck values could be revealed only after a relativistic theory of gravitation had been developed. As soon as that was done, Einstein pointed out the necessity of unifying the new theory of gravitation with quantum theory. In 1916, having obtained the formula for the intensity of gravitational waves, he remarked:

Because of the intra-atomic movement of electrons, the atom must radiate not only electromagnetic but also gravitational energy, if only in minute amounts. Since, in reality, this cannot be the case in nature, then it appears that the quantum theory must modify not only Maxwell’s electrodynamics but also the new theory of gravitation. (Einstein 1916, p. 696).

(For a similar comment, see Einstein 1918.) Einstein obviously had in mind the problem of the instability of the atom. But his conclusion was hardly based on quantitative estimates. Atomic radiation, calculated according to classical electrodynamics, results in the collapse of the atom in a characteristic time interval on the order of 10-10 s (in stark contradiction with reality), whereas atomic gravitational radiation, calculated according to Einstein’s formula, is characterized by an enormous time collapse on the order of 1037 s. Thus, there was no reason to worry that the relativistic theory of gravitation would contradict empirical data, but the analogy with electrodynamics determined the direction of Einstein’s thinking.

The “cosmological” value of the time of atomic gravitational out- radiation reminds one that during these years Einstein was also thinking about cosmological problems. Even though the effect of gravitational radiation is small, Einstein had to reject its possibility, apparently because of his prerequisites for a cosmology. In a static, eternal universe, any instability of the atom is inadmissible, wholly independently of its


magnitude. It is interesting to compare this attitude, quite natural for that time, with the current, more tolerant attitude toward the instability of the proton (Salam 1979). The change is explained by the acceptance of an evolutionary picture of the universe.

For two decades after Einstein pointed out the necessity of a quantum-gravitational theory in 1916, only a few remarks about this subject appeared. There were too many other more pressing theoretical problems (including quantum mechanics, quantum electrodynamics, and nuclear theory). And, the remarks that were made were too superficial, which is to say that they assumed too strong an analogy between gravity and electromagnetism. For example, after discussing a general scheme for field quantization in their famous 1929 paper, Heisenberg and Pauli wrote:

One should mention that a quantization of the gravitational field, which appears to be necessary for physical reasons, may be carried out without any new difficulties by means of a formalism wholly analogous to that applied here. (Heisenberg and Pauli 1929, p. 3)

They grounded the necessity of a quantum theory of gravitation on Einstein’s mentioned remark of 1916 and on Oskar Klein’s remarks in an article of 1927 in which he pointed out the necessity of a unified description of gravitational and electromagnetic waves, one taking into account Planck’s constant h.

Heisenberg and Pauli obviously intended that quantization techniques be applied to the linearized equations of the (weak) gravitational field (obtained by Einstein in 1916). Being clearly approximative, this approach allows one to hope for an analogy with electromagnetism, but it also allows one to disregard some of the distinguishing properties of gravitation—its geometrical essence and its nonlinearity. Just such an approach was employed by Léon Rosenfeld, who considered a system of quantized electromagnetic and weak gravitational fields (Rosenfeld 1930), studying the mutual transformations of light and “gravitational quanta” (a term that he was the first to use).

The first really profound investigation of the quantization of the gravitational field was undertaken by Matvey P. Bronsteyn. The essential results of his 1935 dissertation, entitled “The Quantization of Gravitational Waves,” were contained in two papers published in 1936. The dissertation was mainly devoted to the case of the weak gravitational field, where it is possible to ignore the geometrical character of gravitation, that is, the curvature of space-time. However, Bronsteyn’s work also contained an important analysis revealing the essential difference between quantum electrodynamics and a quantum theory of gravity not thus restricted to


weak fields and “nongeometricness.” This analysis demonstrated that the ordinary scheme of. quantum field theory and the ordinary concepts of Riemannian geometry are not sufficient for the formulation of a consistent theory of quantum gravity. At the same time, Bronsteyn’s analysis led to the limits of quantum-gravitational physics (and to Planck’s cGh-values).

3. Bronsteyn and His Way to Quantum Gravity

In the 1930s, the name of the Leningrad theorist Matvey Petrovich Bronsteyn (1906—1938) was well known in Soviet physics, and this in spite of Bronsteyn’s youth. He was a friend and colleague of the bright young physicists George Gamow, Lev Landau, and Dmitriy Ivanenko, as well as the astronomers Victor Ambarzumian and Nikolai Kosyrev. Bronsteyn’s papers were published in the main physical journals. In 1932, Dirac’s Principles of Quantum Mechanics was published in his translation and with many of his footnotes. At Leningrad University and the Leningrad Physical-Technical Institute, the students and faculty appreciated Bronsteyn’s participation in the journal Physikalische Dummheiten, which circulated in a single copy. Many more readers knew Bronsteyn’s popular scientific articles and books and his three wonderful books for children. The latter are being reprinted today, fifty years after the death of their author. In 1937, Bronsteyn was arrested and then killed by Stalinism. For two decades, his name was banned as the name of an “enemy of the people.”

Bronsteyn had broad scientific interests—his papers dealt with astrophysics, semiconductors, cosmology, and nuclear physics. But, the investigation of quantized gravity became his most important achievement (see Gorelik 1983; Gorelik and Frenkel 1985, 1990).

One of his first papers of 1926 contained evidence of his acquaintance with general relativity, but the full measure of his knowledge of the topic was in an extensive and profound survey article on relativistic cosmology that he published in 1931. The appearance of this review, which contains sections on the foundations of general relativity, on the formulation of the cosmological problem, and on all of the then-known cosmological models, was stimulated by Hubble’s discovery of 1929.

In this article, one finds no doubts about the possibility of constructing a cosmological theory based solely on general relativity. In Bronsteyn’s subsequent papers concerning cosmology, such doubts are characteristic, and they are based on his general conception of physics. Bronsteyn looked at physics through the “magic cGh cube.” He drew a chart describing the structure of theoretical physics based on the role


played by the three fundamental constants c, G, and h, and on the corresponding limits of applicability of different theories (ignoring some of these constants). His next paper on the topic (Bronsteyn 1933b) contains a chapter entitled “Relations of Physical Theories to Each Other and to Cosmological Theory” the basic analysis of which was illustrated by the following scheme:

Here, as Bronsteyn wrote, “solid lines correspond to existing theories, dotted ones to problems not yet solved” (Bronsteyn 1933b, p. 15). Relativistic quantum theory, that is, ch theory, was then the center of attention. But, as a rule, gravity was ignored.

Bronsteyn also wrote about the quantum limits of general relativity in other papers, in the context of astrophysics and cosmology (see, for example, Bronsteyn 1933a). One phrase from Bronsteyn’s popular article of 1929 (which concerned Einstein’s attempt to unify gravitation and electromagnetism) reveals his general attitude to the problem of the quantum generalization of general relativity: “The construction of a space-time geometry that could result not only in laws of gravitation and electromagnetism but also in quantum laws is the greatest task ever to confront physics” (Bronsteyn 1929, p. 25). Thus, it is no surprise that Bronsteyn chose the quantization of gravitation as the topic of his dissertation (when the system of scientific degrees was introduced in the USSR). However, this choice was rather surprising in the contemporary scientific environment, because in the 1930s, fundamental theorists were


concentrating on quantum electrodynamics and nuclear physics.Bronsteyn defended his thesis in November 1935. His examiners,

the prominent Soviet theorists Vladimir Fock and Igor Tamm, praised the thesis highly as “the first work on the quantization of gravitation resulting in a physical outcome” (Gorelik and Frenkel 1985, p. 317). Bronsteyn’s dissertation and two corresponding publications (Bronsteyn 1936a, 1936b) were mainly devoted to the quantization of weak gravitational fields, but one of the most important results is an analysis of the compatibility of quantum concepts and classical general relativity in the general case (this latter topic will be examined seperately).

Bronsteyn considered gravitation in the weak-field approximation (where it is described by a tensor field in Minkowski space) in accordance with Heisenberg and Pauli’s general scheme of field quantization (Heisenberg and Pauli 1929). From quantized weak gravitation, he deduced two consequences: (1) a quantum formula for the intensity of gravitational radiation coinciding in the classical limit with Einstein’s formula, and (2) the Newtonian law of gravitation as a consequence of quantum-gravitational interactions.

One might think that what we see here is merely an expression of the simple requirement of the correspondence principle. In fact, Bronsteyn’s results had real significance, because the peculiar character of the gravitational field (namely, its identification with the space-time metric) gave rise to doubts about the possibility of synthesizing quantum concepts and general relativity. For example, Yakov I. Frenkel (the head of the theoretical department at the Leningrad Physical-Technical Institute, where Bronsteyn worked) was very sceptical about the possibility of quantizing gravitation, because he, like some other physicists, considered gravitation as a macroscopic, effective property of matter. It should be mentioned that even in the 1960s, Léon Rosenfeld supposed that the quantization of gravitation might be meaningless since the gravitational field probably has only a classical, macroscopic nature (Rosenfeld 1963)—and Rosenfeld was the first to consider a formula for the expression of quantized gravitation. On the other hand, Einstein’s attitude was also well known. He believed that the correct full theory was separated from general relativity by a much smaller distance, so to speak, than from quantum theory.

Bronsteyn’s attitude was near the golden mean, since he was sure that one should look for the synthesis of both great theories—general relativity and the quantum theory. His investigation brought to light deep connections between the classical and quantum ways of describing gravitation and so gave testimony to the effect that the quantum generalization of general relativity is both possible and necessary.


4. “... an essential difference between quantum electrodynamics and the quantum theory of the gravitational field.” Quantum Limits of General Relativity

Before obtaining the two mentioned results and just after deducing commutation relations, Bronsteyn analyzed the measurability of the gravitational field, taking into account quantum restrictions.

The question of measurability had occupied an important place in fundamental physics since Heisenberg had discovered the restrictions on measurability resulting from the indeterminacy relations (restrictions on the measurability of conjugate parameters). As physicists looked toward the development of ch-theory (relativistic quantum theory), the question of measurability attracted great attention, especially after Landau and Peierls rejected in their 1931 paper the concept of an “electromagnetic field at a point,” based upon its immeasurability within the ch-framework. This paper led to the detailed and careful analysis of the situation that was undertaken in 1933 by Bohr and Rosenfeld. They saved a local field description in quantum electrodynamics, but at the high price of assuming arbitrarily high densities of mass and electrical charge.

Bronsteyn had an eye for this subject, and just after Bohr and Rosenfeld’s paper, he summed up the situation in a short but clear note (Bronsteyn 1934). So it was quite natural that in considering quantum gravity Bronsteyn decided to analyze the problem of the measurability of the gravitational field within the cGh-ffamework.

He proceeded from the weak-field approximation, where the metric gik = δik + hik, and δik is the Minkowski metric, with hik << 1. In this case, the Einstein field equations have the form

Ohik = K(Tik - ^6ikT), (1)

where Tik is the energy-momentum tensor, and κ = IôtG/c2. In order to analyze the measurability of the gravitational field, Bronsteyn considered the following thought experiment for measuring the value of the Christof- fel symbol r’-fc (gravitational field strength) by observing the movement of a test body.

In the weak-field approximation, the equation of a geodesic

dïx1 _ τι1 dx* dxk n

can be written as (x1 = x)

d2x _ dhm x dhmc2 dt2 1,00 cdt 2 dx

(2)


To measure Γι,οο averaged over the volume V and the time interval T, one should measure the component of momentum px of a test body with volume V at the beginning and at the end of the interval T:

_ d2x _ px(t + T) - px(t)1,m~c2dt2~ c2pVT

where p is the density of the test body. If the measurement of the momentum has indeterminacy ~ Δρχ, then

ΔΓι,οο ~ APx/(?pVT. (3)

The indeterminacy Δρχ is the sum of two parts: the ordinary quantum mechanical term (Δρ^)] = h/Ax, where Δχ is the indeterminacy in the coordinate x, and “the term connected with the gravitational field that is created by the measurer due to the recoil in the measurement of momentum.” The latter term can be estimated with help of Eq. (1):

Dhoi = kToi = Kpvx/c.

If a separate measurement of momentum requires a time interval Δί(Χ T), then the indeterminacy in hoi, connected with the indeterminacy in the recoil velocity vx ~ Ax/At, is of the order

Δ/ίοι ~ up ~-(cAi)2,οΔί

and, in accordance with Eq. (2), ΔΓχ,οο ~ up Ax. So the indeterminacy in the momentum that is connected with the gravitational field of the test body would be

(Apx)2 ~ 62ρνΔΓι;οο · Δί = c2Kp2VAxAt.

The total indeterminacy

(Apx) = (Δρτ)ι + (ΔΡχ)2 ~h/Ax + c2Kp2VAxAt (4)

is minimized byAx = (h/KC2p2V At)1/2. . (5)

Then,

(Δρ^πύη ~ (hnc2p2VAt)1/2,

1 thKC2At\ V2(ΔΓ1ι00)πύη ~ J (6)

376 Gennady B. Gorelik

Two conditions determine the lower bound on the duration Δί of the measurement of momentum. First, it is necessary that Δί > Δχ/c in order for the recoil velocity to be less than c. Hence, from Eq. (5), we get

Δί> n =(/i/KcyV)1/3· (7)

Second, from the very meaning of the measurement of the field in volume V, it follows that the indeterminacy Δχ < V1/3. Hence,

At > T2=h/(?Kp2V5^3. (8)

Having obtained these two lower limits, Bronsteyn remarked that the ratio

“depends on the mass of the test body, being quite insignificant in the case of the electron and becoming of order 1 in the case of a dust particle weighing one hundredth of a milligram” (Bronsteyn 1936b, p. 216). There are then two limits for ΔΓ, respectively

(AOnrim £1 //t2Kcÿ/3

(10)

(ΔΓ)πώ12 Z hc2TpV4/3 ' (11)

As we have seen, to measure Γ in a certain volume V as precisely as possible, one should use a test body with as large a mass (density) as possible. Hence, only the first limit is essential.

Bronsteyn wrote that the preceding considerations are analogous to those in quantum electrodynamics. But there arises here the essential difference between quantum electrodynamics and quantum gravity, because

in formal quantum electrodynamics, which does not take into consideration the structure of the elementary charge, there is no consideration limiting the increase of density p. With sufficiently high charge density in the test body, the measurement of the electrical field may be arbitrarily precise. In nature, there are probably limits to the density of the electrical charge... but formal quantum electrodynamics does not take these limits into account.... The quantum. theory of gravitation represents a quite different case: it has to take into account the fact that the gravitational radius of the test body (wpV) must be less than


its linear dimensionsnpV < V1/3.

(Bronsteyn 1936b, p. 217)

(12)

It follows that Eq. (10) gives “the absolute minimum for the indeterminacy”:

(13)

Bronsteyn understood that “the absolute limit is calculated roughly” (in the weak-field framework), but he believed that “an analogous result will be valid also in a more exact theory.” He formulated the fundamental conclusion as follows:

The elimination of the logical inconsistencies connected with this requires a radical reconstruction of the theory, and in particular, the rejection of a Riemannian geometry dealing, as we see here, with values unobservable in principle, and perhaps also the rejection of our ordinary concepts of space and time, modifying them by some much deeper and nonevident concepts. Wer’s nicht glaubt, bezahlt einen Taler. (Bronsteyn 1936b, p. 218)

(The same German phrase concludes one of the Grimm brothers’ very improbable fairy tales).

In such a way, the quantum limits of general relativity were revealed for the first time. The only thing lacking for the modem theorist is the Planck scale of these limits. Of course the Planck values were implicitly present, because all three constants c, G, and h, were involved in Bronsteyn’s analysis. One such value, the Planck mass (hc/G)1/2, appeared in Bronsteyn’s text, “a dust-particle weighing one hundredth of a milligram.”

It is not difficult to supplement Bronsteyn’s reasoning in order to demonstrate the Planck scale of quantum gravity, if only one tries to measure the gravitational field as precisely as possible in a volume as small as possible (Gorelik 1983). Then, one can obtain, for example, the indeterminacy of the metric

Ag = lpi/cT.

The true value of Bronsteyn’s fundamental result concerning the quantum limits of general relativity was not recognized by his contemporaries. For example, Vladimir Fock had, on the whole, a very high opinion of Bronsteyn’s dissertation, but in a summary written for an abstract journal, Fock gave only the following vague characterization of


this important result: “a few observations on the measurability of the field quantities are appended” (Fock 1936, p. 88).

In the 1930s, it was much easier to unite the constants c and h with e, me, and mp than with G (Heisenberg 1930). Only a deep understanding of the structure of theoretical physics allowed one to single out just the three constants c, G, and h. It was clear to Bronsteyn that a synthesis of general relativity and quantum theory would be inevitable for astrophysics, cosmology, and fundamental physics in general (Gorelik and Frenkel 1985, 1990). Today, when such a synthesis is eagerly sought, Bronsteyn’s analysis of measurability does not lose its significance.

5. The Planck Values and Quantum Gravity

For two decades after Bronsteyn’s work, there was calm in the field of quantum gravity. Only in the mid-1950s did the length lo = (Gh/c3)1/2 appear almost simultaneously in a few different forms in a few papers. For example, in 1954, Landau pointed out that the length I = G1//2h/ce(= a-(1/2)lOt very near to the Planck length) is the limit of the region outside of which quantum electrodynamics cannot be considered as a self- consistent theory because of the necessity of taking into account gravitational interactions (Gm2/r ~ e2/r, when m ~ p/c ~ h/lc) (Abrikosov, Landau, Khalatnikov 1954).

In a report at the Bern relativity jubilee conference in 1955, Oskar Klein remarked that lo corresponds to the gravitational limit of the region of applicability of special relativistic quantum theory. He reasoned in the following way. A relativistic particle represented by a wave packet in volume λ3 has energy he/λ and mass h/ολ. A difference between the gravitational potentials at the center and the edge of the wave packet Δφ ~ Gh/c)? would not change the metric significantly if Δ92 C c2 or λ 3> lo (Klein 1954, 1956, p. 61). In his discussion of Klein’s report, Wolfgang Pauli also referred to Landau’s remark and conjectured that there is a connection between the problem of the quantization of gravity and the problem of divergences in quantum field theory (Pauli 1956, p. 69). Such a possibility continues to attract the attention of theorists.

The most famous quantum-gravitational interpretation of the length lo was developed in John Archibald Wheeler’s 1955 paper entitled “Geons.” The main subjects of the paper are hypothetical gravitational- electromagnetic pure field configurations (geons). In addition, Wheeler evaluated the manner in which quantum fluctuations of the metric Agik depend on a characteristic space scale £. Using Feynman quantization, he showed that Agtk ~ lo/that is, the fluctuations become significant


when £ ~ 1$. Thus, the length Zo = (Gh/c3)1/2 corresponds to the quantum limits of general relativity. This is the approach that became the most popular.

The term “Planck values,” which is now generally accepted, was introduced later (Misner and Wheeler 1957). According to Wheeler, he did not know in 1955 about Planck’s “natural units” (private communication).

6. Conclusion

A history of the Planck values provides interesting material for reflections on timely and premature discoveries in the history of science. Today, the Planck values are more a part of physics itself than of its history. They are mentioned in connection with the cosmology of the early universe as well as in connection with particle physics. In considering certain problems associated with a unified theory (including the question of the stability of the proton), theorists discovered a characteristic mass ~ 1016mp(mp is the proton mass). To ground such a great value, one first refers to the still greater mass 1019mp. In the words of Steven Weinberg:

This is known as the Planck mass, after Max Planck, who noted in 1900 that some such mass would appear naturally in any attempt to combine his quantum theory with the theory of gravitation. The Planck mass is roughly the energy at which the gravitational force between particles becomes stronger than the electroweak or the strong forces. In order to avoid an inconsistency between quantum mechanics and general relativity, some new features must enter physics at some energy at or below 1019 proton masses. (Weinberg 1981, p. 71).

The fact that Weinberg takes such liberties with history in this quotation is evidence of the need to describe the real historical circumstances in which the Planck mass arose. As we saw, when Planck introduced the mass (ch/Gf/2(~ 1019mp) in 1899, he did not intend to combine the theory of gravitation with quantum theory; he did not even suppose that his new constant would result in a new physical theory. The first “attempt to combine the quantum theory with the theory of gravitation,” which demonstrated that “in order to avoid an inconsistency between quantum mechanics and general relativity, some new features must enter physics,” was made by Bronsteyn in 1935. That the Planck mass may be regarded as a quantum-gravitational scale was pointed out explicitly by Klein and Wheeler twenty years later. At the same time, Landau also noted that the Planck energy (mass) corresponds to an equality of gravitational and electromagnetic interactions.


Theoretical physicists are now confident that the role of the Planck values in quantum gravity, cosmology, and elementary particle theory will emerge from a unified theory of all fundamental interactions and that the Planck scales characterize the region in which the intensities of all fundamental interactions become comparable. If these expectations come true, the present report might become useful as the historical introduction for the book that it is currently impossible to write, The Small-Scale Structure of Space-Time.

Acknowledgment·. The author is thankful to Jean Eisenstaedt and Don Howard for their help in preparing this report.

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List of Participants

Julian B. BarbourGunnar BergMargot Bergmann Peter G. BergmannMichel BiezunskiManuel da Costa Leite François de Gandt Nathalie DeruelleCharles DuvalJohn EarmanJean EisenstaedtOdon GodartHubert F. Μ. GoennerJoshua N. GoldbergPeter HavasDon HowardJôzsef IllyMichel JanssenS. KichenassamyLudwik Kostro

A. J. KoxAndré LichnerowiczAndré MercierArthur I. MillerJohn NorthJohn NortonAchille PapapetrouΜ. PeninJürgen RennJim RitterJean-François RoberSolange Rutz da FonsecaJosé Μ. Sânchez-RonSkuli SigurdssonBronislaw SredniawaJohn StachelG. TuynmanHelmuth UrbantkeChristiane Vilain

Contributors 467

Contributors

Julian B. Barbour, College Farm, South Newington, Banbury, Oxon 0X15 J4G, Great Britain

Gunnar Berg, Department of Mathematics, Uppsala University, Thunbergsvagen 3, S-75238 Uppsala, Sweden

Peter G. Bergmann, Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA

Manuel da Costa Leite, Universidad Nova de Lisboa, Faculdad de Ciências e Tecnologia, 2825 Monte da Caparica, Portugal

Thibault Damour, Institut des Hautes Études Scientifiques, 91149 Bures-sur- Yvette, France and Départaient d’Astrophysique Relativiste et de Cosmologie, CNRS, Observatoire de Paris, 92195 Meudon Cédex, France

Odon Godart, Institut d’Astrophysique et de Géophysique Georges Lemaître, Université Catholique de Louvain, Chemin du Cyclotron 2, B-1348 Lou- vain-la-Neuve, Belgium

Hubert F. M. Goenner, Institut für Theoretische Physik, Universität Göttingen, Bunsenstrasse 9, 3400 Göttingen, Germany

Joshua N. Goldberg, Department of Physics, Syracuse University, 201 Physics Building, Syracuse, NY 13244, USA

Gennady E. Gorelik, Institute of History of Science, Academy of Sciences USSR, Staropansky per. 1/5, Moscow 103012, USSR

Don Howard, Department of Philosophy, University of Kentucky, Lexington, KY 40506, USA

Jozsef Illy, Institute of Isotopes, The Hungarian Academy of Sciences, H-1525 Budapest, Hungary

Michel Janssen, Department of History and Philosophy of Science, 1017 Cathedral of Learning, University of Pittsburgh, Pittsburgh, PA 15260, USA

S. Kichenassamy, Laboratoire de Physique Théorique, Institut Henri Poincaré, 11, Rue Pierre et Marie Curie, 75231 Paris Cédex 05, France

468 Contributors

Ludwik Kostro, Institute of Experimental Physics, University of Gdansk, Ul. Wita Stowosza 57, 80-952 Gdansk, Poland

A. J. Kox, Instituut voor Theoretische Fysica, Universiteit van Amsterdam, Val- ckenierstraat 65, 1018 XE Amsterdam, The Netherlands and The Collected Papers of Albert Einstein, Boston University, 745 Commonwealth Avenue, Boston, MA 02215, USA

André Lichnerowicz, Collège de France, 6, Place Marcelin Berthelot, 75231 Paris Cédex 05, France

David Lopes Gagean, Gabinete de Filosofia do Conhecimento, Rua Virginia Vitorino n.9 5 Dto, 1600 Lisbon, Portugal

André Mercier, Institut für Theoretische Physik, Sidlerstrasse 5, 3012 Bem, Switzerland

Arthur L Miller, Department of History and Philosophy of Science, University College London, Gower Street, London WC1E 6BT, Great Britain

John Norton, Department of History and Philosophy of Science, 1017 Cathedral of Learning, University of Pittsburgh, Pittsburgh, PA 15260, USA

José Μ. Smchez-Ron, Departamento de Fisica Teorica C-XI, Universidad Autonoma de Madrid, Canto Blanco, 28049 Madrid, Spain

Gerhard Schäfer, Max Planck Institut für Physik und Astrophysik, Institut für Astrophysik, Karl Schwarzschild Strasse 1, 8046 Garching bei München, Germany

Bronislaw Sredniawa, Institute of Physics, Jagellonian University, Reymonta 4, 30-318 Cracow, Poland

John Stachel, Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA

Helmuth Urbantke, Institut für Theoretische Physik, Universität Wien, Boltz- manngasse 5, A-1090 Wien, Austria

Christiane Vilain, Département d’Astrophysique Relativiste et de Cosmologie, CNRS, Observatoire de Paris, 92195 Meudon Cédex, France

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