Editors.· of the Center University ..a...f''-JfL:,ll,.'-JfA1Il. 2: "'flIH1JD1VlVlJ1O VOltlme 3: ! John Eannan Department of Fistory and Philosophy of Science University of Pltltsblur2:h Pittsburgh, PA 15260 John Norton Department of History and Philosophy of Science University of Pittsburgh Pittsburgh, PA 15260 and of Science University of Pittsburgh Plt1tsbur~~h9 PA 15260 l,;ofneres:s-l;al:aloe:il1l2 In-Publication Data The Attraction of gravitation : new studies in the history .of general relativity I edited by Earman, Michel Janssen, John D. Norton. p. em. -- (Einstein studies : v. 5) Includes bibliographical references and index. ISBN 0-8176-3624-2 (alk. paper). -- ISBN 3-7643-3624-2 (alk. paper) 1. General relativity (physics)-- History. 1. Earman, John. II. Janssen, Michel, 1953- III. Norton, John D., 1960- IV. Series. QCI73.~.A85 1993 93-30748 530.1 '1--dc20 CIP Printed on acid-free paper. © The Center for Einstein Studies 1993. The Einstein Studies series is published under the sponsorship of the Center for Einstein Studies, Boston University. Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. ISBN 0-8176-3624-2 ISBN 3-7643-3624-2 Typeset in TEX by TEXniques, Inc., Newton, MA. Printed and bound by Quinn-Woodbine, Woodbine, NJ. Printed in the U.S.A. 987654321 I Preface Acknowledgments A Note on Sources vii . . . . . . . . . . . . . . xi ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Part I Disputes with Einstein Nordstrom: Some Lesser-Known Thought Experiments in Gravitation 3 JOHN D. NORTON Out of the Labyrinth? Einstein, Hertz, and Gottingen Answer to the Hole Argument 30 DON HOWARD AND JOHN D. NORTON Conservation Laws and Gravitational Waves in General Relativity (1915-1918) . . . . . . . . . . . . . . . . . . . . . . . .. 63 CARLO CATTANI AND MICHELANGELO DE MARIA The General-Relativistic Two-Body Problem and the Einstein-Silberstein Controversy . . . . . . . . . . . . . . . . . . . . 88 PETER HAVAS Part II The Empirical Basis of General Relativity Einstein's Explanation of the Motion of Mercury's Perihelion JOHN EARMAN AND MICHEL JANSSEN Pieter Zeeman's Experiments on the Equality of Inertial and Gravitational Mass A.I Kox '. . .. 129 173 I I vi Contents Part III Variational Principles in General Relativity Variational Derivations of Einstein's Equations 185 S. KICHENASSAMY Levi-Civita's Influence on Palatini's Contribution to General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 206 CARLO CATTANI Part N The Reception and Development of General Relativity The American Contribution to the Theory of Differential Invariants, 1900-1916 225 KARIN REICH The Reaction to Relativity Theory Germany, III: "A Hundred Authors against Einstein" ..... . . . . . . . . . . . . . . . .. 248 HUBERT GOENNER Attempts at Unified Theories (1919-1955). Alleged Failure and Intrinsic Validation/Refutation _ ...... __.. "u...... ••••>.0 274 SILVIO BERGIA . . . . . """ Fock: of Gravity Il-JI'hllUnC,nnlrl'T Gravity ofPhilosophy 308 GENNADY GORELIK So Chandrasekhar's Contributions to General Relativity 0 •••• 0 •••• 0 ••• 0 ••• 0 • • • •• KAMESHWAR C. WALl • 0• • • • • • • •• 332 Part V Cosmology and General Relativity Lemaitre and the Schwarzschild Solution JEAN EISENSTAEDT 0 •••••••••••••••••• 0 353 E.A. Milne and the Origins of Modem Cosmology: An Essential Presence .0 0 0 0 •••••••••••••• " ••••••• 0 • • • • • •• JOHN URANI AND GEORGE GALE 390 Contributors 421 Index 423 I I The attraction of gravitation is universaL Over the last few decades it has to a resurgence of interest in Einstein's general theory of relativity, our best theory of gravitation. In the mid-1980s, this interest began to extend to the history of general relativity, which is now enjoying international at- tention ·of unprecedented vigor and intensity. This volume represents the latest outcome of this new interest. Most of the papers began as presenta- tions at the Third International' Conference on the History Philosophy of General Relativity and, after considerable development and revision, have been brought to their present form. The conference was held at the University of Pittsburgh at Johnstown, Pennsylvania (U.S.A.), June 27-30, 1991. Members of the local organizing committee were John Earman, Al Janis, Michel Janssen, Ted Newman, Norton, Alan Walstad (Uni- versity of-Pittsburgh) and Clark Glymour (Camegie~Mellon University, Pittsburgh). Members of the National and International Committee were Jean Eisenstaedt (Institut Henri Poincare, Paris), Hubert Goenner (Univer- sity of Gottingen), Joshua Goldberg (Syracuse University), Don Howard (University of Kentucky), A.I Kox (University of Amsterdam Einstein Papers, Boston), Jiirgen Renn (Einstein Papers,:B~ston), Stachel (Boston University). This is the volume in the Einstein Studies series to be devoted to the history of general relativity. There are now sufficiently many scholars working in the. area to support a series of conferences volumes of research articles explicitly devoted to the history of general relativity. John Stachel w'as the first to tap into this interest when he organized the first viii The Attraction of Gravitation international conference on the history of general at Osgood Massachusetts (U.S.A.), May 8-11, 1986. He and Don Howard founded the series Einstein Studies and edited its first volume, Einstein and the History of General Relativity (Birkhauser Boston, 1989), which contained papers from the Osgood conference elsewhere. Following the success of the first conference, Jean Eisenstaedt organized the Second Rn'lt,01l''1l''l1lJl'ltllnnlJlH Conference on the History of General Relativity, which was at the International Center of Mathematical Research (CIRM) at Luminy, France, September 6-8, 1988. He and A.I Kox edited a proceedings volume, Studies in the History of General Relativity, which appeared as· Einstein Studies, Volume Three (Birkhauser Boston, 1992). The and diversity of papers in this volume demonstrate the ever growing vitality of research in the history of general relativity. We have divided the volume into five sections. The first group of papers deals with disputes between· Einstein and other figures in the history of general ity. These papers remind us that science is a collaborative enterprise, even in the case of general relativity? whose genesis is celebrated almost exclu- sively as' the work of just one person. The papers show us how Ols:putes might sometimes further the interests of science other not. Norton's paper recounts how prospects of a covariant gravita- tion theory were explored an extended exchange between Einstein Nordstrom at Einstein was laying down foundations of general Howard and Norton's paper recalls months of Einstein's struggle-With general relativity, when he still remained convinced through hole argument general covariance was physically uninteresting. They conjecture Hertz at Gottingen communicated a serviceable escape from the hole argument to Einstein- which he misunderstood brusquely rejected. main focus of C.arlo Cattani and Michelangelo De Maria's paper is the debate over the correct formulation of conservation laws general relativity. They show Einstein tenaciously defended formulation against criticism ous authors, foremost among them Levi-Civita. Peter Havas' portrays an accommodating Einstein entering a dispute with berstein over the two-body in general relativity. We dispute as it grovvs from a disagreement into an U'VJl...ll.Jl..ll..lILV.II..II..ll.V'-'I!.O -D'.............................. that surfaced in press. relativity is not F,'VJl.Jl.'l,;il.II.4.ll. for its IntImate ,("lln1t"ll1l"IJlA....1I" an empirical base, second group of papers examines some episodes related to the empirical evidence .tohn Earman Michel Janssen analyze Einstein's paper of November which was the work of only one week. They ask if blnlste]ln a(~nlC~ve:a Preface to Volume Five ix speed by sacrificing mathematical rigor. A.I Kox discusses Pieter Zeeman's little-known experiments on the equality of gravitational mass, dra'lVing on the recently discovered Zeeman Nachlass. The mathematical complexity of general stimulated consid- erable research into the development of new useful mathematical per- spectives on general relativity. .This is by two papers in the third section, "Variational Principles in General Relativity~" In the first, S. Kichenassamy gives an overview of the early use of variational princi- ples in general relativity, carefully distinguishing the different notions of variation employed. Carlo Cattani's paper on reveals Pala- tini's contribution to gen~ral relativity is not exhausted by the celebrated principle to VUJl.B.U\L.l!..VJI..IlI;..ll..Il which his name is attached. The reader may find it read these two papers conjunction with Cattani and De Maria's paper first section. The largest group of papers in the volume addresses the reception and development 'of general relativity. Karin Reich investigates the Ameri- can reception and development of the theory of differential invariants, the of mathematics essential to the historical foundation of general rela- tivity to its further development. Hubert Goenner dissects·· a less happy episode in the reception of Einstein's work, the malicious 1931 denun- ciation .«4 Hundred Authors against Einstein. Goenner exposes the often murky background motivations of the volume's contributors. Silvio Bergia gives an extensive survey of attempts to formulate unified field the- ories along the lines suggested by general relativity. Bergia evaluates these attempts with a carefully chosen set of criteria, articulated at the time of the attempts, thus minimizing the danger of anachronism in his survey. Gennady Gorelik recounts the life of one of the foremost Russian rel- ativists, Fock, revealing a fascinating and complex figure who negotiated controversy within his home country and internationally with dignity and principle. Kameshwar Wali explains why Chandrasekhar's en- try into active research in general relativity was delayed until the 1960s. He then reviews Chandra's substantial contributions from the 1960s to the 1990s, starting with relativistic instabilities and post-Newtonian approxi- mations and continuing through rotating stars and black holes. In the final section, papers by Jean Eisensta~dt and by George Gale and John Urani explore the ever fertile interaction ot cosmology and general relativity. Eisenstaedt shows how Lemaitre's interest in cosmology was crucial for his important contribution to the modern interpretation of the Schwarzschild solution. Gale and Urani maintain that E.A. Milne's "kine- matic relativity" was not merely a dead-end curiosity to be relegated to a footnote in the history of 20th century philosophy. They argue that Milne's x The Attraction of Gravitation program not only helped shape the debate about nature of cosmology but also played a direct role in the development of the Robertson-Walker metric. John Earman Michel Janssen John Norton 1993 The editors gratefully acknowledge the support, assistance, and encouragement of many people and organizations, their officers, and staff: the Center for Einstein Studies, Boston University; the Center for Philosophy of Science, University of Pittsburgh; the Collected Papers of Albert Einstein Project, Boston University; the Department of History and Philosophy of Science, University of Pittsburgh; the Department of Philosophy, Carnegie-Mellon University; the Franklin 1. Matchette Foundation, New York; the University Center for International Studies, University of Pittsburgh; Adam Bryant, Suzanne Durkacs, Einstein Studies Series Editors Don Howard and John Stachel; Sara Fleming for preparing the index; and the staff of Birkhauser Boston. A NOTE ON SOURCES In view of the frequent citations of correspondence or other items in the Einstein Archive, we have adopted a standard format for such citations. For example, the designation "EA 26-107" refers to item number 26-107 in the Control Index to the Einstein Archive. Copies of the Con- trol Index can be consulted at the Jewish National and University Library Hebrew University), Jerusalem, where the Archive is housed; and at Manuscript Library, Princeton Universify, and Mugar Memorial Library, Boston University, where copies of the Archive are available for consultation by scholars. Late in 1907, Einstein his attention to the question of gravitation in new theory of relativity. It was obvious to his contemporaries that Newton's theory of gravitation required only minor adjustments to bring it into agreement with relativity theory. Einstein's first published words on question (Einstein 1907b, part V), however, completely ignore the possibility of such simple adjustments. Instead he looked upon gravita- as the vehicle for extending the principle of relativity to accelerated motion. He proposed a new gravitation theory violated his fledgling light postulate related the gravitational potential to now variable speed of light. Over the next eight years, Einstein developed these earliest ideas into his greatest scientific success, the general theory of relativity, and gravitation theory was changed forever. Gravitational fields were no longer pictured as just another of space and time, like electric and magnetic fields. They were part of the very fabric of space and time itself. In light of this dazzling success, it is easy to forget just how precarious were Einstein's early steps toward his general theory of relativity. These steps were not based on novel experimental result~. Indeed, the empirical result Einstein deemed decisive-the equality of inertial and gravitational mass-was known in some preliminary form as far back as Galileo. Again, there were no compelling theoretical grounds for striking out along the path Einstein took. In .1907~ it seemed that any number of minor modifications could make Newtonian gravitation theory compatible with Einstein's new special theory of relativity. One not have to look for relativistic 4 John D. Norton salvation of gravitation theory in an extension of the !l-'JLAJIl.JI.....,JL!I-'A_ of relativity. Einstein himself would label motivations for new approach "epistemological" (Einstein 1916, section 2). Through the years of his struggle to develop and disseminate gen~ral relativity, one of Einstein's greatest strengths was his celebrated mastery of thought experiments. If you doubted that merely uniformly accelerating your coordinates could create a gravitational field, Einstein would have yqu visualize drugged physicists awakening trapped in a box as it was uniformly accelerated through gravitation-free space (Einstein 1913, pp. 1254-1255). Would not objects in the box fall just as though the box were unaccel- erated but under the influence ofa gravitational field? Was not a state of U.l.JI..!I...Il..'U'.IlJIlJl.JI. acceleration. fully equivalent to the presence of a homogeneous gravitational field? As vivid and compelling as Einstein's thought experiments proved to be, they still could not mask the early difficplties of Einstein's precarious speculations. Even a loyal supporter, Max von Laue, author of the earliest textbook~ on special general relativity, had objected to Einstein's idea that acceleration could produce a gravitational field. How this be possible, he complained, since this gravitational field would have no source masses. 1 Einstein's evolving theory had to compete with a range of far more conservative more plausible approaches to gravitation, it was to these physicists such as von Laue looked for· a relativistic treatment of gravitation. We must ask, therefore, about Einstein's own toward these al- ternatives. In particular, of the possibility of a small modification to Newtonian gravitation theory order to render it Lorentz covariant and thus compatible with special relativity? Einstein considered this possi- bility? What reasons could he give for turning away from this conservative but natural path? It turns out that Einstein considered and rejected this conservative path in the months immediately prior to his first publication of 1907 on relativity and gravitation. He felt such a theory must violate equality of inertial and gravitational mass. He was forced to revisit these considerations in 1912 with the explosion of interest relativistic gravitation theories. He first continued to insist that a simple Lorentz co- variant gravitation theory was not viable. In the course of following year, however, he came to see he was wrong and that there were ways of constructing Lorentz covariant gravitation theories compatible with the equality of inertial and gravitational mass. After an initial enchantment and subsequent disillusionment with Abra- ham's theory of gravitation, Einstein found himself greatly impressed by a Lore~tz covariant gravitation theory due to the Finnish physicist Gunnar Einstein and Nordstrom: Thought Experiments 5 Nordstrom. In fact,by late 1913, Einstein nominated Nordstrom's theory as the only viable competitor to his own emerging general theory of relativity (Einstein 1913). This selection came, however, only after a series of exchanges between Einstein and Nordstrom that led Nordstrom to significant modifications of his theory. Einstein's concession to the conservative approach proved to have a silver lining; under continued pressure from Einstein, Nordstrom made his theory compatible with the equality of inertial and gravitational mass by assuming that rods altered length and clocks their rate upon falling into a gravitational field so that the background Minkowski space-time had become inaccessible to direct measurement. As Einstein and Fokker showed early 1914 (Einstein and Fokker 1914), the space-time actually revealed by direct clock rod measurement had become curved, much like space-times of Einstein's own theory. Moreover, Nordstrom's gravitational equation was equivalent to a geometrical equation in the Riemann-Christoffel curvature tensor played the central role. In contraction, the curvature scalar, is set proportional to the trace of the stress-energy tensor. is remarkable about this field equation is that it comes almost two years before Einstein recognized the importance of the curvature tensor in constructing field equations for his own general theory of relativity! In this regard, the conservative approach actually anticipated Einstein's more daring approach. Einstein now an answer to the objection that general relativity troduced an unnecessarily complicated mechanism for treating gravitation, the curvature of space-time. He had shown that the conservative path led to this same basic result: Gravitational fields come hand-in-hand with the curvature of space-time. Elsewhere, I have given a more detailed account of Einstein's response to the conservative approach to gravitation and his entanglement with Nord- strom's theory of gravitation (Norton, 1992). My purpose in this chapter is to concentrate on one exceptionally interesting aspect of the episode. As in Einstein's better-known work on his general theory of relativity, the episode was dominated by a sequence of compelling thought experiments.2 These experiments concentrate the key issues into their simplest forms and present them a way that makes the conclusions emerge ~onvincingly and effort- lessly" In this chapter I will review this sequence of thought experiments as it carries us through the highlights of the episode. In particular, we will see how one of the more arcane areas of. spe- cial relativistic physics proved decisive to the development of relativistic gravitation theory. It emerged from the work of Einstein, von Laue, and others that stressed bodies behave in strikingly nonclassical ways in rela- 6 John D. Norton tivity theory. For example, a moving body can acquire energy simply by being subjected to stress,even though it may not be deformed elastically by the stress. Nonclassical energies such as these provided Einstein with the key for incorporating the equality and gravitational mass into relativistic physics. 10 First Thought Experiment: Masses a Tower from The bare facts of Einstein's initiation into the problem of relativizing grav- itation theory are known. late September 1907, Einstein accepted a commission from Johannes Stark, editor of Jahrbuch der Radioaktivitiit und Elektronik, to write'a review article on the principle of relativity.3 That review (Einstein 1907b) was submitted a little over two months later," on December 4, 1907. Its concluding part contained the earliest statement of what came to be the principle of equivalence and of conjectures about gravitation that followed we know only from reminiscences by Einstein is that, in this brief period between September and December, he considered and rejected a conservative Lorentz covariant theory of gravitation.4 that .Jl-JJl.Jl.JLU"'......JILJI..IiL 1I"t:ll1"4JI111t:ll1il knew how one could take Newton's theory of gravitation render it Lorentz covariant with modifications to its equations. Newton's theory· is given most conveniently in the usual Cartesian coordinates (x, y, z) by the fleW equation V 24J = 82 + aay22 + aaz22 ) 4J = 4rrGp (1) for the gravitational field 4> generated by a mass density p, where G is the gravitational,constant, and by the force equation = -mV4> (2) for the gravitational force f on a body of mass m. The adaptation to special relativity of the to alluded was obvious. simply replaces covariant operator \/2 of(1) with the manifestly Lorentz 0 2 to recover 2 (2 2 o .4> = at V - - c12 - ( 2 ) 4> = 4rrGv, (3) where v is' an invariant mass density and t the time coordinate. An analo- gous modification of (2) would also be required. Einstein (1933, pp. 286- 287) continued to explain that outcome of his investigations was not satisfactory. Einstein and Nordstrom: .Thought Experiments 7 These investigations, however, led to a result which raised my strong suspicions. According to classical mechanics, the vertical acceleration of a body in the vertical gravitationalfield is independent of the horizontal component of its velocity. Hence in such a gravitational field the vertical acceleration of a mechanical systern or of its center of gravity works out independently of its internal kinetic energy. But in the theory I advanced, the acceleration of a falling body was not independent of its horizontal velocity or the internal energy of the system. This did'not fit with the old experimental fact that all bodies have the same acceleration in a gravitational field. This law, which may also be formulated as the law of the equality of inertial and gravitational mass, was now brought home to me in all its significance. I was in the highest degree amazed at its existence and, guessed that in it must lie the key to a deeper understanding of inertia and gravitation. I had no serious doubts about its strict validity even without knovving the results of the admirable experiments of Eotvos, which-if my memory is right-I only came to know later. I now abandoned as inadequate the attempt to treat the problem of gravitation, in the manner outlined above, within the framework of the .special theory of relativity. It clearly failed to do justice to the most fundamental property of gravitation. result troubled Einstein in the theory he advanced came from the relativistic adaptation of the force law (2). As Einstein pointed out in his reminiscences, this adaptation could not be specified so unequivocally. We can proceed directly to result, however, if we use four-dimensional methods of representation not, available to Einstein in 19070 The natural adaptation of (2) is FJt = mdU-J-t dr = -ma-l/-J , dXJt (4) where FJt is the gravitational four-force acting on a body of rest mass m with four-velocity UJt; r is the proper time.5 We can now apply (4) to the special case of a body whose three-velocity v has, at some instant of time, no vertical. comppnent in a static gravitational field. If the gravitational field at that instant at the mass acts along the z-axis of coordinates, so that the z-axis is the vertical direction in space, then it follows from (4) that the vertical acceleration of the mass is given by dvz dt =-(1- V c 2 2 )da4>z . (5) We see immediately that this vertical acceleration is reduced as. the horizontal speed v is increased, illustrating Einstein's claimed dependence of the rate of fallon horizontal velocity. 8 John D. Norton The "old experimental fact," which this result contradicts, surely be- longs to the famous fable which Galileo drops various objects of different weights from a tower. Einstein and (1938, 37-38) , iden- tify this ,story when they ,wrote: What experiments prove convincingly that the two masses [inertial and gravitational] are the same? The answer lies in Galileo's old experiment in which he dropped different masses from a tower. He noticed that the time required for the fall was always the same, that the motion of a falling body does not depend on the mass. We can combine these ingredients to make explicit the thought experiment suggested by Einstein's analysis. Masses are dropped from a ,high tower, some with various horizontal velocities and some with none. According to (5),the masses with greater horizontal velocity fall slower, contradicting Einstein's expect~tioilandthe classical result that they should all fall alike. See Figure 10 Trajectories after equal times Vertical fall slowed by horizontal velocity in a Lorentz covariant' theory of gravitation. It is not so obvious why Einstein the outcome of ,this first thought experiment to be so troubling that he felt justified in abandoning the search for a Lorentz covariant theory of gravitation. The dependence is effect, ~econd order vic. one might well wonder hovv even Einstein and Nordstrom: Thought Experiments 9 most ingenious experimentalist could compare rate offall of a mass with that of another whizzing past at a horizontal velocity close to the speed of light. Even if this were possible, the experiment surely notbeen done in 1907. How could Einstein reject this effect as incompatible with an "old experimental fact" whose origins lay with Galileo? answer resides in the.fact that Einstein derived the dependence of vertical acceleration on the "horizontal velocity or the internal energy of the system." Einstein meant by this was made clear in 1912 when the Hl111lnl1C'1I1l physicist Gunnar Nordstrom published the first of a series of papers on a Lorentz covariant, scalar theory of gravitatio~(Nordstrom 1912). The essential assumptions and content of Nordstrom's theory were contained in equations (3) and (4) above. Nordstrom did correct, however, a problem (4). It turns out that this force law can only hold for a mass moving so the. rate of change of the gravitational potential along its world line is zero.6 (This holds instantaneously for the special case used to derive [5].) force law (4) requires modification if it is to apply to masses along whose trajectories 4J is not constant~ Nordstrom found two SUl1taO,Le modifications. He favored the one in which the rest mass m of the body is assumed to vary with the gravitational potential ¢. In particular, he readily derived the dependence m = moexp(~), (6) where mo is the value of m when ¢ = o. By October 1912, when Nordstrom sent his paper to Physikalische ZeitschriJt, Einstein's novel ideas on gravitation had become a matter of public controversy.. In July, Einstein found himself immersed in a vitriolic dispute Abraham, who saw Einstein's admission of a variable speed of light a "death blow" to relativity theory (Abraham 1912). In his response, Einstein (1912, pp. 1062-1063) published his 1907 grounds for abandoning Lorentz covariance the most general form he could manage. In any Lorentz covariant gravitation theory, he argued, be it a four-vector or six-vector theory, gravitation would act on a moving body· with a strength vvould vary with velocity. Any such theory was unacceptable, since it violated the requirement of the equality of and gravitational mass. Therefore it is not at all surprising that Nordstrom attracted Einstein's when he published just such a theory. "\L.\l..\1o.,lJl..lI.\L.,II.V'JI..lI. Einstein's reaction was so swift Nordstrom was able to mention it in·an addendum to his original paper! The addendum began (Nordstrom 1912, p.1129): Addendum to proofs. From a letter from Herr Prof. Dr. A. Einstein I learn that he had already earlier concerned himself with the possibility 10 John D. Norton used above by me for treating gravitational phenomena in a simple way. He however came to the conviction that the consequences of such' a theory cannot correspond with reality. In a simple example h~ shows that, 'according to this theory, a rotating system in a gravitational field will acquire a smaller acceleration than a non-rotating system. Einstein's reflection on the acceleration of fall of a spinning system is actually only a slight elaboration of the situation considered in the first thought experiment above. Each element of a suitably oriented spinning body ·in a gravitational field has a horizontal velocity. Thus, according to (5), which obtains Nordstrom's theory, each element will fall slower than the corresponding element without that velocity. What is true for each part holds for the whole., A spinning body falls slower than the same body without rotation. This example now makes clear Einstein's remark the body. is se~ into rotation, its parts gain overall el1ergyand its. inertia are increased. However, 1t1h1l"'jnlllll'1l1'lh a decrease the gravitational force acting on so fall is decreased. is, its rate, of decreases as and inertia increases. Presumably the spinning one example of a general effect of type. In reminiscences, Einstein used the example. of a gas.? As the gas is each molecule moves faster and more slowly. the aggregate of molecules', the more a colder gas. two examples COfl1prise experiment. See Figure 2. Einstein's is a far greater to Lorentz covariant theories of gravitation such as Nordstrom's, to effects might testable. transcend detection by of the tops or hot gases, it escape an to that of the Eotvos experiment? Nordstrom seemed to so, continued his appendix by dismissing Einstein's argumenton basis of the effect being "too small to yield a contradiction with experience." This disrrrissal depended on a assumption: no common systems of matter in which a great part,of energy, and thus is due to the kinetic energy of systems, if they existed, would slower t4an6thers· according to Nordstrom's theory. have-been right that no measurable effect would arise from spinning of a body, but could he energy of commonpl~cematter not already have a significant kinetic component? theory of matter was then in a state of scarcely able to assure him either way. Amore prudent Einstein was unwillingto take it turn Einstein and Nordstrom: Thought Experiments 11 not spinning gravitational acceleration gravitational acceleration cold gas hot gas gravitational acceleration gravitational acceleration molecule has greater horizontal velocity and falls slower 2. Spinning bodies fall slower than when not spinning. Hot gases fall slower than cold gases, in Nordstrom's theory. out that a significant total energy of various types. of ordinary matter was due, different proportion, to an internal kinetic energy, then Nordstrom's theory might well be by simple.observations of the of diff~erent substances from a tower. By the time of submission of his next paper on the theory in January 1913, Nordstrom become more wary (Nordstrom 1913a). While still insisting (p..878) that no observable effect would arise in the case ofspinning bodies, he was prepared to raise the question of whether the "molecular motions of a falling body" would influence rate of fall. He did not state 12 John D. Norton directly since theory. be measurable, effect spe:cm.meonaw~of1n~n~\nr~lh~,n Energy Nordstrom's paper of January 1913 was devoted to a question that would ultimately completely direction of development of his· theory. The paper asked which represented the inertial mass of ~ body. The question was Recent work in relativistic theory of shown there were inertial effects arose a was stressed for which there were no classical analogs. Nordstromob- served (1913a, p: ·856) that it had proved possible to ignore question develop a complete mechanics of extended bodies introduci~g concept of inertial mass. could no longer afforded, he continued, when one worked a relativistic nr~'T1I1r.'li"lIr'll1l"\l ory, because very close connection between.inertial masses. to represent mass of a in a v.!ay that al- lowed effects in stressed bodies cannot be ~i"i"'II'"1l hlllli"Oril rff1l1l"":::ll~1tH;" to an mass. of results to which Nordstrom referred reached its mature work of von Laue There von Laue essentially presented the theory of relativistic continua, no- tion of the general stress-energy tensor of matter. to· which Nordstrom took following. form. Ifone a stress to a body without deforming it or setting it into motion, then both the energy and momentum of body unchanged its rest frame. However, if one viewed this same process from a frame of reference in which the body was in motion, the energy and momentum body might change. For if body was influenced by a shear stress8 P~y its rest viewed from a frame of reference moving at vel()city v in x direction, in that frame the body would a momentum in the y momentum density gy to stress is given by9 v0 gy = y-c Pxy · (7) stress was a normal stress P~x in the rest frame, then, when viewed in the relatively moving frame, body would have acquired both energy an x-directed momentum. The energy density Wand momentum density Einstein and Nordstrom: ThoughtExperiments 13 gx acquired is given by W = Y 2 V2 2c'" 0 Pxx' = 2V 0 gx Y 2c" Pxx · (8) are effects for are no classical analogs. They proved decisive relativistic analysis of a of celebrated thought ex- periments most notably Lewis Tolman bent lever and the capacitor. 10 One of clearest earliest analyses of these nonclassical effects is due to a experiment of Einstein (1907a, section 1; 1907b, sec- tion 12) was given in the context of his discussion of inertia of energy. He imagined an extended body at rest carrying a charge distribu- He imagined at some definite instant its rest frame, the comes influence of an external electromagnetic field. The """~...'..,......,"""'.. forces are assumed to balance so the body remains at rest. effect of the continued action of the forces, however, is to induce a state of stress the body. Einstein now redescribed this process from a frame the body moved uniformly. Because of the relativity of simultaneity, does come under the influence of the external field at one .ll.1l..ll.LJ\\,.Q..ll.1l.JLIl-. For a brief period, some charge elements are under the influence of field and some are not. this period, the external forces exerted by do not balance, so that there is a net external force· exerted on body. is done on or'by the force as the body moves, and there a net transfer of energy. This energy is the energy described (8) and associated the of a stressed state the body. 11 The beauty· of this thought experiment is it derives the effects of equations (8) directly from the most fundamental, nonclassical effect ofspe- relativity, of simultaneity. Forces applied simultaneously one of reference need not be seen as simultaneously in another. The resulting temporary imbalance leads to an energy and momen- tum transfer in the latter frame only these transferred quantities emerge as those of (8). Einstein's analysis is mathematically quite complicated, however, since he considers a body of arbitrary shape and charge distribu- Ke1caJ>ltullatlng Einstein's analysis for a case is to essential physics. case is a of cross section charges at end. is the thought experiment. See Figure 3. has rest I, cross-sectional area A, extends from x' = 0 to x' = 1in its rest (x', t ' ). At a specific instant t ' = 0 in its rest frame, rod comes influence of a field that applies equal 14 John D. Norton Rest frame (x~t') of rod F F F F Area A Both forces tumedon att'=O No change in energy or momentum of rod. Rod moves . at v in (x, t) Instant at which forces are ttl"" ttl"" F tumedon ". t'=O t=Y.!c.2-l .... } F ............ ==~e----3~ t=O ............. x 2 Energy F1y.cL2 and v momentum F1'YCI lost from rod. 3. Stressing a moving rod changes its energy and momentum. oppositely forces F to the charges. For concreteness, assume forces are directed away along forces a tensile stress on the rod P~x = -FlA. If we redescribe stressing of rod moves at velocity v +x 'i\,.lI..ll..Il.~""'''''\L..Il.''-'.lI..Il'l not activated simultaneously because of force F on the trailing end is activated ata time y '2rI Einstein and Nordstrom: Thought Experiments 15 on the leading end. For this short time period external force F on the trailing end is not balanced by the other external force. As a result, work is done by the motion of the rod against the force. resulting loss of energy from the rod is Fly ~~ the loss of momentum Fly ~ . Recalling the above expression for p~x and . the volume of the rod in the frame (x, t) is V = Al/y'~.we recover expressions for the energy E and x-momentum Gx gained by rod in the process of being stressed: 2 E = Y2 2Vc : Px0x V and Gx = Y2 2V 0 PXXV . ,C Division of these expressions by the volume V yields (8). In his paper (1913a), Nordstrom had asked the right question. What quantity represents total mass of a body, including contributions to its inertial properties that arose from stresses? He sought his answer in the form ofthe source density v for equation (3), and he looked in the right place his answer. He expected density to be a quantity derived from the stress-energy tensor Tj-tv, recently introduced by von Laue. After extensive discussion, he settled upon '1 / c2 times the rest energy density of the source as his source densityv. ·The rest frame required for this choice was instantaneous local rest frame of a continuous matter distribution- "dust"-which Nordstrom assumed contributed to the source matter. We \tvould now express Nordstrom's choice in'manifestly covariant form as (9) where Bj-t is the four-velocity vector field of the continuous of matter. Nordstrom's answer was close to correct answer-but not close enough, as was pointed out by Einstein, in section 7 of his physical part of Grossmann (1913).14 He reported that von Laue himself, also at the University of Zurich, had p-ointed to Einstein the only viable choice, trace of the stress-energy tensor Einstein proposed to call this scalar "Laue's scalar." What was distinctive aboutthis choice wasthat it enabled a gravitation theory thatemployed 16 John D. Norton it to satisfy the requirement of the equality ofinertial gravitational mass, at least "up to a certain degree," as Einstein it. This degree included examples such as those in second thought experiment above', as we now see. The key result that enabled satisfaction of this equality was due to von Laue. Von Laue (1911a) found a single general solution to a range of problematic examples within relativity theory. They all involved systems whose properties appeared to violate the principle of relativity. For exam- ple, on the basis of classical electromagnetic theory, Trouton and Noble (1903) believed a charged, parallel-plate capacitor would experience a net turning couple it was set m.otion with its plates oblique to the ~..II..Il..'V""'l\,.JIl..,",1L1L of motion~althoughtheir experiment yielded a celebrated result. Again, Ehrenfest (1907) had raised the possibility a nonspher- ical or nonellips~idal.electron could not persist translational motion unless forces are applied to it. In both cases the projected behavior would provide an in.dicator of motion of the system, violating the principle of relativity. What these exan1pleshad in common was presence of stresses the systems with the proper of these stresses, threat to the principle of evaporated. Von Laue noticed systems were static systems," is, they ffi2L1nt:m.Iled a static frames· of reference interacting with systems. The basic result characterizing these systems was in rest frames, (10) where the integral over the rest volume VO of whole body. It follows from (10) that the energy momentum of a complete static system transforms Lorentz transformation exactly like energy and momentum of a point-mass. Since the dynamics of a point-mass was compatible the principle of . so was the dynamics of a com- plete static system"and not expect a violation of the of relativity in the dynamics of these systems. Von Laue's analysis very general and powerful because it needed to ask very of the systems. All one needed to know was whether the system static system. If it'was, one could ignore the details simply a box, drawn around system. Its overall dynamics was now _. . . .·.......JL.......................,...... In effect, what Einstein was to report (1913,. section 7}was von Laue's machinery could Einstein and Nordstrom: Thought Experiments 17 to the problem of selecting a gravitational mass density. If one chose T as the gravitational mass density, von Laue's result (10) entailed that the total gravitational mass of a complete stationary system its rest frame was equal to its inertial mass. For,using (10), for suc~ a system we have16 = = TO dV O = total 44 energy total inertial mass' (11) where I follow Einstein in simplifying the analysi~ by neglecting factors of c2, so that energy and inertial mass become numerically equal. The power subtlety of this rather beautiful result stood out clearly in the example Einstein employed in his discussion. This example is our thought experiment. The trace T for electromagnetic radiation vanishes. it seem electromagnetic radiation can have no gravitational mass. I7 But what of a system of electromagnetic radiation enclosed within a massless box with mirrored walls? Would such a system have any gravitational mass? The radiation itself would not, although.that ll."U~ll.""JIlV.ll..ll. would exert a pressure on the walls of the box. These walls would become stressed and, simply because of this stress, the walls would acquire a gravitational mass. Since it is a complete static system, we need do no direct computation of the distribution of stresses in the walls. The result (11) tells us immediately the total gravitational mass of the system in its rest frame is given by the system's total inertial mass. See Figure 4. The same reasoning can essentially be applied to the spinning bodies heated gases of the second thought experiment, if they are set in a gravitation theory that uses T as its source density. Molecules of gas with horizontal motion slower than those without this motion, thus they do have a smaller effective gravitational mass. They exert a pressure on the walls of containing vessel, however,which becomes stressed. These stresses alter the value of T and thereby contribute to gravitational mass. Since (11) applies here, we read immediately from it that the gravitational mass of a gas enclosed in a·vessel in its rest frame is given by inertial mass of the whole system. Similarly, the individual masses comprising ~ spinning.body have a smaller effective gravitational mass because of their motion, the spin- ning body is stressed by centrifugal forces. We know from (11),without calculation, the contribution of the stresses to the total gravitational mass exactly compensates for the reduction due the motion of the individ- masses. As before, the total gravitational mass is given by the total mass. Jll1.Jl ................ " ........ 18 John D. Norton Box with mirrored, massless walls encloses electromagnetic radiation. Tensile stresses in "'"5~)·"P>-."·r~~---L_ walls in reaction to radiation pressure Electromagnetic radiation has no gravitational mass. Walls acquire gravitational mass due to stresses. I _-------11\...- - - - - - Moving mass Motion-induced\ elements have stresses reduced contribute to gravitational gravitational mass. mass,. ~pressure-\ molecules have reduced gravitational mass. induced stresses in walls contribute to gravitational 4. Equality of inertial and gravitational mass for complete stationary systems in a gravitation theory with source density At -this point, one might would have to capitulate and cease his opposition to Lorentz gravitation theories. .s ob- jection to these theories had been to satisfy the requirement ofequality of inertial and gravitational mass. Most damaging was his con- clusion this equality would in the type of cases ~ealt with Einstein and Nordstrom: Thought Experiments 19 second thought experiment above. But now his analysis of the choice of T as source density showed how a Lorentz covariant, scalar theory of grav- itation could escape Einstein's objection in exactly most damaging cases. Einstein was no mood for retraction, and good reason. Having presented T as only viable choice of gravitational source density, he proceeded to argue that the choice was a disaster. A theory that employed T as the gravitational source density must violate the law of conservation of energy. Einstein's argument was presented within a thought experiment- our thought experiment-and it was beguilingly simple. See Figure 5. He imagined electromagnetic radiation trapped a'mirrored, massless box. We assume it shape for simplicity. The system is lowered into a field. Since it has gravitational mass, an amount of energy proportional to mass is extracted. Einstein now introduced another apparatus to raise radiation. He a 1I1tn'S!l0'1I1I11PI"1I shaft extending out of gravitational field. are two mirrored, massless baffles, firmly fixed together. The is .Il.UU'.Il.U"JI.Vll.J1. space between the·baffles is raised out gravitational as baffles are raised. We shall.again assume the space between baffles is cubic. have already seen that the gravitational mass of the mirrored'box to lower is entirely to the stresses its walls. It now follows immediately the system of radiation baffles has only «Hlt~-R (III nu the gravitational mass of the radiationlbox system, for in elevating between 1I"0IrlI1l0l1/"·1Itn.n lI.-.I!.I!.4IIJIlJ'lo"/_ baffles, one need move only one-third as many stressed members. 18 Only as energy need therefore be supplied to raise in the apparatus as is released when the .Il.U'-l~Jl.U"JIl.V1UI. is box. Since no energy is involved in raising and lowering massless box when devoid of radiation, a complete cycle of raising lowering yields a gain of energy. violates the of conservation of energy. Einstein must have been very pleased outcome. In a single blow, it out not Lorentz covariant, scalar theories of gravita- any relativistic gravitation theory employed a scalar potential. the complexity" (Einstein and Grossmann 1913, 1, section 7) of Einstein's second-rank tensor theory seemed UJl.JlQ;..u.V'-'JI.~Cl.A.lIJJI.'V. Einstein's 1hl'''1l1l111n111l''\h was short lived. In 1913, Nordstrom (1913b) sub- 20 John D. Norton mirrored, massless box ~-==-+-i\l!>,. mirrored shaft Transfer radiation. Lower into gravitational field. massless, mirrored baffles gravitational mass = E gravitational E mass 3 5. Trace T as source density violates energy conservation. his so-called "second" theory to Annalen der Physik. trace T as its gravitational source opportunities!t for equality o f .ll............'.llI;l,..a..Q,.Q,..i4. ,.~itwas •. " .... I • ....., ... I"' ..... to an escape attack on ............................ '" ....LI"",....... scalar theories of gravitation. basic Q.nllll1lJ11l"ll,,",,1l"'lCO remained (3) (4), except four-force FI1 was .IlVIl--'.Il""~Vu. a four-force density K/1-: Einstein and Nordstrom: Thought Experiments 21 KJL = -g(¢)vaa-x¢-JL, where u = ict. The major alteration was factor g(JV.ll.~.II.Jl...ll"\l.V -;an-::llD'P7i311 VV,J.II.~JlI..II..II."!\I."'"system to a given was ~V.D..D..Il.ll.""~ so contrived that it selected a single !t-'.II. ....JL.II....,.... !t-'......., VV'J.ll..~JlI.A.II."Il-"'" system those came smoothly to agree on the of any given region of space-time. entails a will become IlJ_~l ~J1 below: ..lI..II......... ... ........ any region ofspace-time, it is impossible for there to be ,two different coordinate systems that come smoothly to agree at the also show his 1913 were covariant between these adapted coordinate systems, so these were generally covariant, at least the maximum covariance by hole ,argument.2 Einstein's failure/to offer generally equations was a great worry< and embarrassment to frequent protestations of unac- cepta~ility of generally covariant equations,however, such as Out of the Labyrinth? 33 stein 1914a, and his publication in October 1914 of a lengthy review article (Einstein 1914b) of the theory suggested the theory had achieved some stability its then non-generally covariant irn1l"1l1l1I1I116fJlf"-ann In late June and early July of 1915, visited Gottingen and gave six lectures on his theory to a group including David Hilbert;l Felix and, more likely than not, Emmy Noether and Paul Hertz. Einstein described this visit to several correspondents. Thus, on August16, he wrote to Berta and Wander Johannes de Haas: "To my great delight, I succeeded in convincing completely" (EA 70-420).3 one earlier, on reported enthusia~tically to Sommerfeld: In Gottingen I had the great pleasure of seeing everything understood, down to the details. I am quite enthusiastic about Hilbert. A man of consequence. (EA21-381; reprinted in Hermann 1968, p. 30)4 That report to Sommerfeld, however, also showed that Einstein was not yet to .ll"".....'V.ll..Il.~..,.ll.llI\,,''l..IJ. new theory. He wrote Sommerfeld he would prefer not to one or two papers on his new theory (Einstein 1914b) in the collection Das Relativitiitsprinzip, since none of the current presentations were "complete." As it Einstein been understood in Gottingen even better realized. was particularly excited, writing to Schwarz- schild on July 17, 1915: "During the summer we had here as guests following: Sommerfeld, Born, Einstein. Especially lectures of the last on gravitational theory were an event" (quoted Pyenson 1979a, p. 193, n. 83). The excitement in Gottingen was tempered, however, by a widely shared belief Einstein's mathematical abilities not be up to the task of perfecting the new theory of gravitation. Typical of this attitude are a couple of remarks in Felix Klein's lecture notes on general from the summer of on the first day of lectures, July 15, 1916, to his audience in relativity theory was by a "fog of mystery" [Nebel derMystik], adding: Einstein's own way of thinking is partly to blame for this mystery, for it starts out again from the most general philosophical speculations and is guided, above all, more by strong physical instinct than by clear mathematical insight.5 More to the point, however, is a remark later in same in the of a section entitled "On the Choice of Coordinates Encountered in Einstein." In Einstein's new theory, tells his students, we enter upon the terrain of arbitrary coordinates, "familiar" to us from work of Lagrange, Gauss, and Riemann;l where the g/-tv and the ds2 must be treated according to the rules of Ricci's absolute differential calculus, or "more I I I, Secon~ coordi~atesystem x m . ~ ~ First assigns Coordinate transformation takes (2,2) to (1,1) w ~ :ot:si o::r:: a~ § 0.. Use coefficients g'ik to construct ::~::sr second solution ~ Z §. :o:s Space-Time Second solution assigns g'ik= tox m = (1,1) First coordinate system x m I I I ~ru,,~+-rn£'t,ll"'\n of the two solutions of the hole ar.Q:ument. Second II coordina~e system8 x m, in each coordinate system coordinated with different events in space-time First coordinate system x m I I B 2. Properties of the two solutions of the hole argument. First solution assigns gik = to x m' = (1,1) o C f""1' Second solution assigns o , g ik - - fLlO-021J' :~ ;. (l) ~ to x m = (1,1) §.. "=~ O. In the second solution, 50 Don Howard and John D. Norton the electron adopts a nonvanishing velocity. and fields become dependent after x~ = O. possibility of such at first seems to threaten of causality, however, proposes to rescue it by offering a of it means for an object, a law, or an.expression to be "physically " According to some- thing· should be regarded as physically meaningful only if it is invariant with respect to arbitrary transformations of coordinate system. in this sense, the causality principle is .satisfied, since, he asserts, physi- cally meaningful expressions, which is to say, expressions, are unambiguously by the generally covariant equations.29 It is at this point in Hilbert's exposition that his argument converges upon we believe proposed to Einstein. Hertz, we believe, exploited a geometrically coordinate system to display essential agreement between two s-olutions E, F, G and EX, F X , GX • summarized his basic anq promised to prove the by exploiting geometrically Gaussian coordinate system: The causality principle holds in this sense: From a knowledge· of the 14 physical potentials, g/LV, qs' follow all assertions about them for the future necessarily and uniquely, insofar as they have physical significance. In order to prove this claim, we employ the Gaussian space-time coordinate system. 1916, p. Hilbert's emphasis) by noting selection ofGaussian coordinates provides extra constraints needed to ensure the are determined fourteen equations. Gaussian coordinate sys- tem is defined, most the assertions then made· about the Gaussian cQordinate system are·of invari- ant character. present can the and therefore physically content of the no contradiction the causality remains. proceeded to indicate three ways in assertions can be given expression. o f our 1l._'bo'''U'J!.J!.IUlll.lI..\I,,.ll._ll.Jl'U'J!.lI. of Hertz's proposal, two of ways resorted to specially adapted coordinate systems.30 first recapitulated the use of "111I"II'{I'1'lI1I"'1I1'lI1I"II1I" coordinate systems, as termed Gaussian (geodetic "' I coordinates, elaborated on its application to the example of the electron at rest. second allowed character for an assertion there exists a coordinate system in which some nominated relation holds. As an illustration, he . resorted again to the case of the electron and roR6JI"U1l1f'l\t:llrll invariant character for the assertion that there exists·a. coordinate system ac~ording to whose X4 time coordinate the electron is at rest. Out of the Labyrinth? 51 That Hertz, as we reconstruct and working in Gottin- gen, should rely so heavily on specially adapted coordinate systems to reveal the physically significant elements of a theory provides strong evidence for our reconstruction. It also raises the further question of the origin of these ideas. Were they Hertz's own? Or was he acting, in effect, as a spokesperson for and Gottingen group? Hertz's proposal to Einstein-as reconstructed by us-would have pro- vided a serviceable escape the hole argument. The escape route ac- "...".""'",,,. by Einstein, however, his point-coincidence argument, dif- fered in crucial ways from Gottingen group. The latter was escape, principally on the mathe:maltlc;al was physicist's escape, relying prin- physical reality. Was the point-coincidence argument outpouring of Einstein's genius? Q..ll.~LJlIl-'.lLJII...Il.Jl..ll. Q..ll..Il...Il.'I\,JIl-..Il..lI.'V.IL ....' __ can we We believe that are at least two first of these,chronologically, is Joseph Petzoldt, a Privatdozent at Technische Berlin-Charlottenburg, founder 1912 of Gesellschaft fur positivistische Philosophie (of Einstein was a of numerous books and articles promoting JLQ..ll.V'_.lL~'__ "relativistic positivism," a melange of chief notion of substance. i-1I"'n,rlIl1i-"'I11r'o1l"'llnU 1l1("A't":llnh'uC'lIf'~g was a critique of the most 1l'1l"'tf'll"nr'o1I"'t"nll"'lli- contribution for the purposes .of our discussion was in 1895 of "Das Gesetz ("The Law of Uniqueness" or "Univocalness") (Petzoldt 1895), according one of its forms, a theory would be acceptable determined a rnodel describe. Petzoldt's "law of uniqueness" major discussion stimulated by it form an essential part of background to Einstein's hole this very methodological point-coincidence arguments, since it is that lies at the root ofboth.31 By 1915, Einstein Petzoldt were in personal' contact one an- other. l"here ·is evidence was attending Einstein's lectures on relativity! in Berlin in either the winter semester of 1914-1915 or the summer semester of 1915. A postcard from Einstein to Petzoldt in late 1914 or early ·1915 makes it clear Einstein had been reading Petzoldt's work and approved of its general tendency: "Today I have read with great 52 Don Howard and··John D. Norton interest your book its entirety, and I from it have for a long time been your companion your way of (EA 19-067); the book was most likely Petzoldt's Vas Weltproblem vom Standpunkte des relativistischen ·Positivismus aus, historisch-kritisch dargestellt 1912b).32 Against this background, one may wonder had ab- sorbed the point of view exemplified by a remark in Petzoldt's "Die Rela- tivitatstheorie im erkenntnistheoretischer Zusammenhang des relativistis- chen Positivismus" (Petzoldt 1912a), which would have appeared early in 1913 in the proceedings· of Deutsche Physikalische Gesellschaft. The relevant remark concerns way Petzoldt's epistemological perspectival- ism is· allegedly embodied special relativity. Petzoldt writes, The task of physics becomes, thereby, the unique [eindeutige] general representation'ofevents from different standpoints moving relative to one another with constant velocities, and the unique setting-into-relationship of these representations. Every such representation of whatever totality of events must be uniquely mappable onto every other one of these representations·of the samel) events. The theory of relativity is one such mapping theory~ What is· essential is that unique connection. Physical concepts must be bent to fit for its sake. We have theoretical and technical command only, of that which is represented uniquely by means of concepts. 1)'Better: representations· of events in arbitrarily many of those systems of reference that are uniquely mappable onto one another are representations of "the same" event. Identity must be defined, since it is not given from the outset. (Petzoldt 1912a, p. 1059) It.is the footnote grabs one's attention, for it expresses a rUIlaa.mf~ntGll presupposition of Einstein's point-coincidence argument. ing about this way of talking identity mapping, especially of what are clearly, from context, Minkowskian events,vvas not commonplace the pre-1915 on To appreciate role of second figure possibly Inrluencllng stein's point-,coincidence argument, recall that Einstein's struggle to find generally covariant equations came to a close November 25, 1915 to Prussian Academy \JL.I.Il..ll..ll.l.JlIL.'Il"I.Il..Il..ll. !915b). Already in his preceding communication of Novem- ber 18,1915, he general covariance, "time space have been robbed of trace of objective reality" (Einstein p. 831), by.whichhe "the relativity postulate its most tnr'.on.o-r"JlU formulation 0 •• turns the space-time coordinates into physically meanIng- less parameter&" (Einstein p. 847). This makes it clear of the Labyrinth? 53 time, late November, Einstein was in possession of an answer to the hole argument involving essentially coordinatizations are not sufficient for individuation of points in the space-time. Curi- ously, however, when begins infomling correspondents about these developments in late time, the talk of co- incidences so characteristic of point-coincidence argument. It seems likely to us coincidence talk came from work of Erich es- say, "Uber die Bestimmbarkeit der berechtigten Bezugssysteme beliebigerRelativitatstheorien," is a lengthy and labored discussion of ~Ilo.IQ~Ilo.I.Il..I.l.Il..II..Il..II..Il.f.\I.~JI.""'.lI..II. of coordinate systems in which of spatiotemporal coincidence plays role. The paper clearly anticipates essen- eleme~nts of point-coincidence argumient, as Kretschmann himself a he cited his own paper "for (Kretschmann 1917, p. 576) on the point-coincidence argument, citing Einstein's version of the argument solely for the introduc- of German "Koinzidenzen," replacing Kretschmann's 1915 "Zusammenfallen" (see below).33 In paper, Kretschmann argues that only what he calls "topolog- ical" relations the form ofcoincidences have empirical significance, since all observation requires we bring a of the measuring ... .Il.ll.ll~ .Il.llU'''''..... ....,• ... .Il.ll"" contact the measured object: What is observed.here-ifwe neglect, at first, all direct metrical determinations-is only the completely or partially achieved spatiotemporal co... incidence [Zusammenfallen] or non-coincidence [Nichtzusammenfallen] of parts of the measuring instrument with parts of the measured object. Or more generally: topological relations between spatiotemporally extended objects. (Kretschmann 1915, p. 914) A similar insistence on observability .of coincidences figures promi-, nently in the best-known of Einstein's statements of the point-coincidence argument,where Einstein writes: All ·our space-time verifications invariably amount to· a determination of space-time coincidences· [Koinzidenzen].... Moreover, the results of our measurings are nothing but verifications ()f such mee~ings of the material points of our measuring instruments with other material points, coincidences [Koinzidenzen] between the hands of a clock and points on the clock dial, and observed point-events happening at the same place at the same time. (Einstein 1916, p. 117)34 is, to be sure, the 0l1e difference noted later by Kretschmann, which is that Einstein uses the "Koinzidenzen," not Kretschmann's "Zusam- 54 Don Howard and John D. Norton menfallen." The former term is more suggestive of the topologist's notion of the intersections of lines at extensionless points, whereas. latter is more suggestive of macroscopic congruences of bodies at the level of ob- servational practice. Thus, Kretschmann can talk more comfortably of "completely or partially achieved coincidences [Zusammenfallen]." The similarity is nonetheless striking. Kretschmann proceeds the 1915 paper to develop now-familiar ideas concerning coordinate systems. In particular, he urges on the basis of his earlier assertions on coincidences that, "in no case can a soundly based decision be made, through mere observations, between two quantitatively different but. topologically' equivalent, mappings of. the world of appear- ance onto a space-time reference system" (Kretschmann 1915, p. 916). An immediate application of Kretschmann's remark not offered by Kretschmann) is the case of two solutions, gik g;k the same coordinatesystemxm).of hole argument. 'They are "two qu,lntlltat]Lvejly different ... mappings of the world of appearance onto a [single] space- time coordinate system." Nonetheless, they are "topologically equivalent," since they 'agree on point-coincidences, hence observation no soundly decision between But if observation' reveals no difference, does there any difference . ~\\.,~ IOcJ •• W'I1 .... •• we development of Kretschmann's ideas, we everd~fferences between two solutions, 'gik merely matters of convention: "Insofar as assertions of a system of physical laws cannot be reduced to purely' topological relations, they are to be considered as mere-at most methodologically grounded-conventions" (Kretschmann 1915, p. 924).35 Of course, is reason to discussion to be to Einstein's argument. However, the J1 •• _ ity between expositions of point-coincidence argument Kretschmann's discussion is so striking it cannot be (dare we say!) a mere coincidence have resulted from some sort of be- tween Einstein The only question to be resolved is the na- ture is extremely suggestive is that KJet:scJlmanl['l' paper appeared in an issue of Annalen der Physik was JL _ ....... on December 11,,1915, five days before earliest of the surviving let- ters in which Einstein articulates point-coincidence argument, his to Ehrenfest of December 26 (EA 9-363). We are unaware of any invocation of point-coincidences the corpus of Einstein's writings- both and unpublished-prior to letter. is more, when, in a letter of December (EA21-610),'Einstein 1I1"'a1t",n1l"'nr"llorll '''6.1Hn11l'''11'11"''7}'' Schlick about the' exciting developments of November Out of the Labyrinth? 55 only on space time having lost the last vestige of physical reality, with no mention of point-coincidences. These facts make almost irresistible the conclusion that Einstein read Kretschmann'spaper of its content -when it appeared, found the ideas on coincidences extremely congenial, and turned to refine and exploit to explain to correspondent Ehrenfest where his hole argument failed. Other paths oftransmission ofthese ideas between Einstein and Kretsch-IUann are possi~le, but seem less Ii~ely.. Kretsch~ann completed his Ph.D. in .1914. underMa~ Planck and Heinrich Rubensin '. standing for the Promotionspriifung on February 5 of that year..... B,ut Kretschmann reports he finished his studies in Berlin in 1912 (see the Lebenslauf at the end the manuscript of his 1915 paper was submitted from Konigsberg, where he had finished Gymnasiunl in 1906 and where he became aPrivatdozent in 1920. Were he present after Einstein's had some· contact the ideas C'1I1l1l""l,nRlIt:.l&r8 they may however, cannot have been or engaging to Kretschmann as far as Einstein's still incomplete general theory relativity was concerned. While he was elsewhere rather long-winded, Kretschmann's··1915 paper contains only a·brief discussion of Einstein's 977-978), citing just two ofthe earlier joint publications by Ein- stein Grossmann (Einstein and Grossmann 1913, 1914),andomitting ~eIUajorreyiew~icly?f,]'\Toveill~~U 914 (Bi~~tein 1914b). T~ediscu~­ lllalc rttOll is.sk~tchy alld fails to: e any seri0l1sc()~tact with the idea of adapted coordinates,.an.ideathat·was a major focus ofEinstein's Berlin work on the theory at that time and very relevant to the subject of Kretschmann's paper. Finally, of course, the possibility of such earlier transmission completely fails to explain the extraordinary' fact that the point-coincidence argument and mention of space-time coincidences in this general context appear for , the first time in a letter of Einstein'sof December 26, 1915, only days after issue of the Annalen containing Kretschmann's paper was distributed.36 are· due to Rudolf Hertz, Paul Inv'alulabJLe a:~SH;taIlce in our research. Thanks are Havas, for having drawn our -attention to the notes from general relativity during the summer of 1916. The rewas ., supporfediri'partby grants from the National ...,....,JL....,.It.lt...~...., ~-()UnlC1atl0n (no.• 5E5-8421040,. DH), the American Philosophical the Deutscher Akademischer Austauschdienst (DH), and Kentucky Research Foundation. (DH). We would like to 56 Don Howard and John D. Norton thank the Hebrew University of Jerusalem, which holds the copyright, for permission to quote from Einstein's letters, thank Niedersachsische Staats- Universitatsbibliotek, Gottingen, for permis- sion to quote from Felix Klein's lectures. Items in the Einstein Archive are cited by nunlbers in the Control Index. NOTES 1 To see this, note that the first solution transformed from x m to x ml has the functional form g;k of the coordinates x ml , which is the same functional form as·the components of the second solution in the coordinate system x m . 2 For a summary of the· mathematical machinery Einstein used to analyze his adapted coordinates, see Norton (1984, section 6). 3 This letter is dated on the basis of its place in a sequence of letters discussing the shipment of the de Haas's furniture from Berlin to the Netherlands, the shipment being overseen 'by Einstein. 4 For more on this 'visit, see the discussion in Pais 1982, pp. 250 and 259. 5 Cod. Ms. Klein 21L, p. 63, Niedersachsische Staats- und Landesbibliothek Gottingen. ' 6 Cod. Ms. Klein 21L, p. 69, Niedersachsische Staats- Landesbibliothek Gottingen. 7 This timing, the fact that Einstein and· Hilbert engaged in an intense corre~ spondence through November 1915 and then had a brief falling out after that cor- respoJ;1denC,e, has raised the possibility that Einstein stole the field equations from Hilbert: We do not this possibility seriously for the reasons given in Norton (1984, pp. 314-315). 8 See, for example, Einstein to Paul Ehrenfest, December 26, 1915 (EA 9-363), December 29, 1915 (EA 9-365), and January 5, 1916 (EA 9-372), as well as Einstein to Michele Besso, January 3,1916 (EA7-272; reprinted in Speziali 1972, pp. 63- 64). 9 Notice that such magnitudes as "time elapsed" are in tum reducible to space- time coincidences. A crude physical time could be measured by an idealized light clock, which is a small rigidly co-moving rod along whose length a light pulse is repeatedly reflected. The time elapsed is measured by the number of collisions of the light pulse with the mirrored ends of the rod. 10 Hilbert was the titular director of Hertz's dissertation, but Hertz actually did the work under Abraham, who was then Privatdozent; see Pyenson 1979b, p. 76. 11 See Einstein to Hertz, August 14, 1910 (EA 12-195) and August 26, 1910 (EA 12-198).·· For more on the beginning of their acquaintance, see Stachel et al. 1989,.p. 44, and Klein et al. 1993, p. 315. 12 See the Hertz-Ehrenfest correspondence in the Ehrenfest scientific correspon- dence in the Archive for the History of Quantum Physics. 13 SeePyenson 1990, as well as Laub to Einstein, May' 16, 1909 (EA 15-465), Einstein to Laub, May 19, 1909 (EA 15-480), and Einstein to Laub, OctOber 11, 1910 (EA 15-489), November 4, 1910 (EA 15-491). 1,4 See, for example, Hertz 1923, 1929a, 1929b, 1930, 1936~, 1936b. Out of the Labyrinth? 57 15 See Clark 1971, p. 184. The chief purpose of Einstein's trip was to meet the novelist Romain Rolland at Vevey, this as part of Einstein's efforts to promote international intellectual cooperation in spite of the barriers raise by World War I. For more on the meeting with Rolland and Einstein's related activities, see Nathan and Norden 1968, pp. 12-18. The year could not be 1913, because Einstein was then still in Zurich, and such a trip would not likely have been undertaken in late August 1914, immediately-after the outbreak of the war. 16 See below. In particular, Hertz uses the older"E, F, and G" notation for what we would now call the components of the metric tensor. 17 Rudolf Hertz, (Paul's son), private communication. 18 To see the correspondence between our account of the hole argument in Section 1 and Hertz's construction, notice that our second solution, gik' in the first coordinate system, x m , corresponds to Hertz's E, F, G in (u, v), while our first solution, gik, in the second coordinate system, x m', corresponds to Hertz's EX, F X, GX in (U X , VX ). Of course, there is the inconsequential change of context. Einstein's argument is formulated in a space-time with an indefinite metric, whereas Hertz's argument is formulated for the space of a two-dimensional Gaussian surface. 19 Obviously, this construction and the point-coincidence argument have the following in common: They pick out a point in the physical space by the intersection of curves with invariant geometrical properties. In Hertz's case, the curves are curves of constant curvature and maximal curvature gradient; in the case of the point-coincidence argument, they' are geodesics. 20 In his Vorlesungen aber die Entwicklung der Mathematik im 19. Jarhundert (Klein 1927, pp. 147-148), Felix Klein lists Knoblauch 1913 as one of the "great textbooks" appearing around the tum of the century, along with Darboux's Lerons sur la theorie generqle des suifaces (Darboux 1914-1915) and Bianchi's Vorlesungen iiber Differentialgeometrie (Bianchi 1910). Although first published in 1927, Klein's lectures were delivered in the years 1915 through 1917. 21 Einstein's replacing of G, the g22 component of the metric, by ¢ is explicable in terms of his 1913 theory. In Einstein's 1913 theory, the g"time" "time" component of the metric in a static field in a suitably adapted coordinate system represents the single gravitational potential of the field, commonly represented by ¢. Note that the angle brackets indicate a strikeout in Einstein's original. 22 In a footnote, Kretschmann comments thatthe possibility of finding "absolute" coordinates, meaning coordinates picked out uniquely by the geometry of the space being thus coordinatized, had been pointed out to him already in a letter from Gustav Mie in February 1916; see Kretschmann 1917, p. 592, n. 1. 23 For more on- Hilbert's introduction of "Gaussian coordinates," see Stachel 1992, pp. 410-412. 24 The alJproximate date of Ehrenfest's letter to Einstein can be determined from his remark, in a letter to Lorentz of December 23,1915, that he had invited Einstein to spend the holidays in Leiden. Einstein's reply to Ehrenfest's thought experiment is contained in the same letter of January 5,1916 (EA 9-372), in which he explains that the border's being blocked was the reason why he could not have come to Holland at that time. We thank A.J. Kox for making available transcriptions of 58 Don Howard and John D. Norton the Ehrenfest-Lorentz correspondence, these from his forthcoming edition of the scientific correspondence of Lorentz. 25 The reconstruction of Ehfenfest's thought experiment is based upon Einstein's reply of January 5 (EA 9-372) and on the description found in Ehrenfest's letter to Lorentz of January 9, in which he enclosed Einstein's letter, asking for Lorentz's opinion. 26 See Einstein to Hertz, undated 1915 (EA 12-205), October 1915 (EA 12-206), Hertz to Einstein, October 8, 1915 (EA 12-207), and Einstein to Hertz, October 9, 1915 (EA12-20S). Though the dating of some of these letters is problematic, they seem clearly to form a sequence written over a short period. It should be not.ed t.hat. most of Hertz~s are missing, the letter of October S having survived because Hertz retained a copy in his files. 27 See Hertz to Hilbert,February 17, 1916(Cod.Ms. Hilbert 150, Handschriftenabteilung, Niedersachsische Staats- und Universitatsbibliothek Gottingen). 28 Hilbert's only footnoteinthis section of the paper (Hilbert 1916, p. 61) cites Einstein's most complete version (1914b,p. 1067) of the hole argument. on 29 For more Hilbert and the causality principle in general relativity, see Stachel 1992, 410-412. 30 The third merely allowed invariant character to a fully covariant law, such as the law of conservation of energy-momentum expressed as the vanishing covariant divergence ofthe stress-energy tensor.. 31 For more on Petzoldt and a more detailed bibliography of his vvritings, see Howard 1992. 32 For the dating of Einstein's postcard to Petzoldt and other details about their relationship, see Howard 1992'- 33 For more on Kretschmann's papers, see Norton 1992, 295-301. 34 See Howard 1992~ n. 25" for a critical discussion of Friedman's (1983, pp. 2225) interpretation ofthis passage as anticipating the verificationist theory of meaning that later became popular among the logical positivists. 35 In a footnote to the word "convention," Kretschmanhcarefully indicates the precise sense of the word intended. It is to mean that which is not demonstrable through observation, rather than something arrived at by some kind of free agreement. 36 We might also conjecture that Einstein was· asked to review .the paper by Planck, the editor of Annalen. Kretschmann's paper is dated October 15 and was received onOctober21. Ifit was sent out for review, Einstein would have been the obvious reviewer. The shorttime between submission and publication, October 21 to December 21, suggests that, even though Kretschmann was a first-time author in the Annalen, the manuscript was not sent out for review, since a two-month period between submissi"on and publication was more or less normal for established authors (see Pyenson 1983). This would not be surprising,. since Planck had supervised Kfetschmamfs Ph.D., was presumably confident of Kretschniann'sscholarship, and possibly already familiar with the work submitted. Out of the Labyrinth? 59 REFERENCES Bianchi, Luigi (1910). Vorlesungeniiber Differentialgeometrie, 2nd ed. Max Lukat, trans. Leipzig and Berlin: B.G. Teubner. Clark, Ronald W. (1971). Einstein: The Life and Times. New York and Cleveland: World. Darboux, Gaston (1914-1915). Lerons sur la theorie generale des surfaces et les applications geometriques du calcul lnfinitesimal, 2nd ed. Paris: GauthierVillars. Einstein, Albert (1902). "Kirietische Theorie des Warmegleichgewichtes und des zweiten Hauptsatzes der Thermodynamik." Annalen der Physik 9: 417-433. -,_._- (1903). "Eine Theorie der Grundlagen der Thermodynamik." Annalen der Physik 170-187. - - (1904). "Zur allgemeinen molekularen Theorie der Warme." Annalen der Physik 14: 354-362. - - (1911a). "Bemerkungen zudenP. HertzschenArbeiten: 'Uberdiemechanischen Grundlagen der Thermodynamik' ,," Annalen der Physik34: 175-176. - - (1911b). "Uberden ,Ein~uB der Schwerkraft auf die Ausbreitung des Lichtes." Annalen der Physik 35: 898-908. - - (1914a). "Prinzipielles zur veraHgemeinerten RelativiHitstheorie undGravitationstheorie.',' Physikalische Zeitschrift 15: 176-180. -_.- (1914b). "Die formale Grundlage def allgemeinen RelativiUitstheorie." Koniglich Preussische Akademie der Wissenschaften (Berlin). Sitzungsberichte: .1030-1085. - - (1915a). "Erldarung der Perihelbewegung, des Merkuraus der.allgemeinen RelativiHitstheorie." .Koniglich ,Preussische Akademie der Wissenschaften (Berlin). Sitzungsberichte: 831-839. - - - (1915b).. "Der Feldgleichungen· der Gravitation."· Koniglich·· Preussische Akademie der Wissenschaften (Berlin). Sitzungsberichte: 844-847. --.- (1916). "Die Grundlage der allgemeinen RelativiHitstheorie." Annalen der Physik 49: 769-822. Reprinted as a separatum Leipzig: Johann Ambrosius Barth" 1916. Page numbers are cited from the English translation: "The"' Foundations of the General Theory of Relativity." In Hendrik A. Lorentz, Albert. Einstein,. Hermann Minkowski, and Hermann Weyl, The Principle of Relativity. W. Perrett and G.B. Jeffrey, trans. London: Methuen, 1923; reprint New York:.Dover, 1952. (1934). "Einiges· tiber die Entstehung der allgemeinen RelativiUitstheorie." In Mein Weltbild.Alpsterdam: .Querido, pp.248--256. Qu'otations are from the.\English translation:. "Notes on the Origin of the General Theory of Relativity." In Ideas and Opinions. Carl Seelig, ed.; Sonja Bargmann, trans. New York: Crown, 1954, pp. 285-290. ~UIStt~ln, Albert and Grossmann, Marfel (1913). Entwurf einer verallgemeinerten Relativitiitstheorie und einerTheorie der Gravitation. I. Physikalischer Teil vonA.lbert Einstein. Mathematischer Teil von Marcel Grossmann. Leipzig 60 Don Howard and John D. Norton and Berlin: B.G. Teubner. Reprinted with added "Bemerkungen,"-Zeitschrift fUr Mathematik und Physik 62 (1914): 225-261. - - - ' (1914). "Kovarianzeigenschaften der Feldgleichungen der die verallgemeinerten RelativiHitstheorie gegrlindeten Gravitationstheorie." Zeitschrift fUr Mathematikund Physik 63: 215-225. Eisenstaedt, Jean and Kox, AJ., eds. (1992). Historical Studies in General Relativity. Einstein Studies, Vol. 3. Boston: Birkhauser. Friedman, Michael (1983).iFoundations ofSpace-Time Theories: Relativistic Physics and Philosophy ofScience. Princeton, N-ew Jersey: Princeton University Helmholtz, Schriften zur Erkenntnistheorie. Paul Hertz and Moritz Schlick, eds.Berlin: Julius Springer. Hermann,'.Armin, ed. (1968). Albert Einstein/Arnold Sommerfeld. Briefwechsel. Basel and Stuttgart: Schwabe. Hertz, Paul (1904). '·'Untersuchungen tiber unstetige Bewegungen eines Elektrons." Ph.D. dissertation. Gottingen. - - - .(1910). "Uber die mechanischen Grundlagen der Thermodynamik." Annalen der Physik 33: 225-274~ 537-552. ---. '(1916). "Statistische Mechanik.'.' Repertorium der Physik. rich Weber and Richard Gans, eds. Vol. 1, Mechanik und Wiirme. Part 2, Kapillaritiit, Wiirme, Wiirmeleitung, kinetische· Gastheorie und statistische Mechanik. Rudolf Heinrich Weber and Paul Hertz, eds. Leipzig Berlin: B.G. Teubner, 436-600. ----,(1923).UberdasDenken·und seine·BeZiehungzur Anschauung. Part 1, Ober den funktionalen Zusammenhang zwischenauslosendem Erlebnis und Enderlebnis bet elementaren prozessen. Berlin: Julius Springer. (1929a). "UberAxiomensysteme beliebiger Satzsystemet Annalen der Phi- losophie undphilosophischen Kritik. 8: 179-204. - ,- (1929b).."Uber Axiomensysteme flir beliebiger Satzsysteme." Mathematis- cheAnnalen 457-514. -".- (1930). "Uberden Kausalbegriff imMakroskopischen, besonders in der klassischen Physik." Erkenntnis 1: 211~227. - - - (1936a). "Kritische Bemerkungen zu Reichenbachs Behandlung des Hume- schen Problems." Erkenntnis 6: 25-31. - - (1936b). "RegelmaBigkeit, Kausalitat und Zeitrichtung." Erkenntnis 6: 412- 421. Hilbert, David (1915). "'Die Grundlagen derPhysik. (Erste Mitteilung)." Konigliche Gesellsphaft der Wissenschaften zu Gottingen. Mathematisch-physikalische Klasse. Nachrichten: 395-407. - - -.•' (1916). "Die Giundiagell def Physik:zweite Mitteilung." Konigliche Ge- sellschaft ·der Wissenschaften zu Gottingen.· Mathematisch-physikalische Klasse. Nachrichten: 55-76. Howard, Don (1992)0 "Einstein and Eindeutigkeit: A Neglected Theme in sophical Background to General Relativity." In Eisenstaedt and Kox 1992, pp. 154-243. Out of the Labyrinth? 61 Howard, Don and Stachel, John, eds. (1989). Einstein and the History of General Relativity. Based on the Proceedings of the 1986 Osgood Conference, North Andover, Massachusetts, May 8-11, 1986. Einstein Studies, Vol.. 1. Boston: Birkhauser. Klein, Felix (1927). Vorlesungen iiber die Entwicklungder Mathematik im 19. Jahrhundert.Part 2, Die Gru~dbegriffe der lnvariantentheorie und ihr Eindringen in die mathematische Physik. Richard Courant and S. Cohn-Vossen, eds. Berlin: Julius Springer. Klein, Martin, Kox, AJ., Renn, Juergen, and Schulmann, Robert, eds. (1993). The Collected Papers ofAlbertEinstein. Vol. 3, The Swiss Years: Writings, 19091911. Princeton: Princeton University Press. Knoblauch, Johannes (1913). Grundlagen der Differentialgeometrie. Leipzig and Berlin: B.G. Teubner. Kretschmann, Erich (1914). "Eine Theorie der Schwerkraft Rahmen der ursprtinglichen Einsteinschen Relativitatstheorie.".Ph.D. dissertation. Berlin. - - (1915). "Uber die prinzipielle Bestimmbarkeit der berechtigten Bezugssysteme beliebiger RelativiHitstheorien." AnnalenderPhysik48: 907-942,943982. - - (1917). "Uber den physikalischen Sinn der RelativiUitspostulate~ A. Einsteins neue und seine ursprtingliche Relativitatstheorie." Annalen der Physik 53: 575-614. Nathan, Otto and Norden, Heinz, eds. (1968). Einstein on Peace. New York: Schocken. Norton, John (1984). "How Einstein Found his Field Equations: 1912-1915." Historical Studies in the Physical Sciences 14: 253-316. Reprinted in Howard and Stachel1989, pp. 101-159. - - - (1987). "Einstein, the Hole Argument and the Reality of Space." In Measurement, Realism and Objectivity. J. Forge, ed. Dordrecht and Boston: D. Reidel, pp. 153-188. - - (1992). "The Physical Content of General Covariance." In Eisenstaedt and Kox 1992, pp. 281-315. Pais, Abraham (1982). 6Subtle is the Lord . ... ': The Science and the Life ofAlbert Einstein. Oxford: Clarendon; New York: Oxford University Press. Petzoldt,·· Joseph (1895). "Das Gesetz der Eindeutigkeit." Vierteljahrsschrift fUr wissenschaftliche Philosophie und Soziologie 19: 146-203. - - - (1912a). "Die Relativitatstheorie im erkenntnistheoretischer Zusammen- hang des relativistischen Positivismus." Del!:tsche PhY$ikalische Gesell- schaft. Verhandlungen 14: 1055-1064. . - - - (1912b). Das Weltproblem vom Standpunkte des relativistischenPositivismus aus, historisch-kritisch dargestellt, 2nd ed. Wissenschaft nnd Hypothese, vol. 14. Leipzig and Berlin: B.G. Teubner. Pyenson, Lewis (1979a). "Mathematics, Education, and the Gottingen Approach to Physical Reality, 1890-1914." Europa: A Journal of Interdisciplinary 62 Don Howard and John D. Norton Studies 2, no. 2. Quotations are taken from the reprinting in Pyenson 1985, pp. 158-193. - - (1979b): "Physics in the Shadow of Mathematics: The Gottingen Electrontheory Seminar of 1905." Archive for History of Exact Sciences 21: 55-89. - - - (1983). "Physical Sense in Relativity: Max Planck Edits the Annalen der Physik, 1906-1918." In Proceedings of the Ninth International Conference on General Relativity·and Gravitation. Ernst Schmutzer, ed. Berlin: Akademie-Verlag, pp. 285-302. Reprinted in Pyenson 1985, pp. 194...;214. - - (1985). The Young Einstein: The Advent of Relativity. Bristol and Boston: Adam Hilger. _ . - (1990). "Eaub, Jakob Johann." In Dictionary of Scientific Biography, vol. 17, suppl.2; Frederic L. Holmes, ed. New York: Charles·Scribner's Sons, pp. 528-529. Speziali, Pierre, ed. (1972)~ Albert Einstein-Michele Besso. Correspondance 19031955. Paris:-Hermann. Stachel, John (i989}.. "Einstein's Search for General Covariance, 1912-1915." In Howard and Stachel1989, pp. 63-100. - - . (1992). "The Cauchy Problem in General Relativity: The Early Years." In Eisenstaedtand Kox 1992, pp.. 407-418. Stachel, John,· Cassidy, David Renn, Juergen, .and Schulmann, l~obert, eds. (1989). The Collected Papers olAlbert Einstein. Vol. 2, The Swiss Years: Writings, 1900-1909. Princeton: Princeton University Press. ( This chapter deals two closely debates general relativity 1916-1918, one on gravitational waves, the other on the correct formulation of conservation laws. Both issues involve-the definition of a quantity rep- resenting the stress-energy of the gravitational field. Such definitions were typically proposed in the context of deriving the gravitational field equa- tions from a.variational principle. A proper understanding of the debates on gravitational waves and conservation laws therefore requires some dis- cussion of the rather complicated history of attempts to derive gravitational field equations from a/variational principle. 1 We will trace Einstein's work on gravitational waves and work on conservation laws during the years 1916-1918 in this more complex network. 'We objections to Einstein's approach Levi- Civita, Schrodinger, at alternative approaches suggested by Lorentz Levi-Civita; and at Einstein's response'to of them. In particular, we examine 1917 correspondence between Einstein and Levi-Civita. We will see how Levi-Civita's criticism of Einstein's formulation'of conservation laws strengthened Einstein his· conviction physical considerations force one to adopt a noncovariant .II. '-'.II..II..II..II.'-'L.II.\l.4.\I,..a.'-J.Il...Il. of conservation laws for matter plus gravitational field. Einstein and Grossmann 1914 and Einstein 1914, Einstein used a variational method to derive field equations of limited covariance of his 64 Carlo Cattani and Michelangelo De Maria so-called Entwurf theory (Einstein and Grossmann 1913). He used conservation of energy-momentum of matter plus gravitational field~the stressenergy of the latter'being represented by apseudotensor rather a tensor-to define the Lagrangian for the gravitational field to restrict the covariance of his theory. Einstein believed he had found a very general argument to fix the Lagrangian for the gravitational field. This Lagrangian leads to the field ~quationsof the Entwurf theory. By substituting the gravitational tensor into the law of conservation of energy-momentum of matter (with stress-energy tensor ~ V), Einstein was able to derive certain constraints on H that he thought uniquely fixed its form. Imposing conservation of energy-momentum of matter and unaware of the contracted Bianchi identities, he obtained a set of equations to be satisfied by the gravitational field: -8,8.x-VS'av - ' B0'=0, (a, v, .' .. = 0, 1,2,3) (1) Einstein Cllhr"'Il'lrr.clril So' v.. = (3) and used these conditions to define the form of Entwurf field equations in form3 aC:a(~_ggafJr~fJ) = -X('T1l111l"''\{Tn1l"1I111l'''O scalar R. For a function As long as gravitational no derivatives of gj1V higher than of second order, 1t must be sum of these two functions: (8) By ev~luating "Lagrangian derivatives" I, p. 397) of /H respect to various obtained the evolution tions for both gravitational electromagnetic potentials. next step was to show that Axiom 2 allows one to give explicit proof of the cavan- anceof these evolution equations. Splitting the Lagran.gian into the scalar curvatureinvariant for the gravitational field and a Lagrangian Conservation Laws and Gravitational Waves 67 the electromagnetic field, ...... ~L'-'_JELlI,. arrived at correct gravitational field equations: -x Gil-V = r1::::;;~v, v-g (9) where (10) Finally, the evolution equations for electrodynamics in a curved space-time by generalizing Mie's for space-time. In conclusion, we want to stress the of method: (1) derived equations in the context of electro- magnetic of matter. As a consequence, his variational method not be generalized to other matter. To accomplish have to specify how matter Lagrangian depends on the potentials '-J'JLJLI~ .... lI,.lI.41l.-,.... .... gJ1,v' (2) generally covariant field equations, he made use of Lagrangian derivatives were not generally covariant. (3) was unaware of contracted Bianchi identities, so he arrived at the explicit form ofthe gravitational tensor in a rather clumsy way. In 1916, DutHISJl1ea .L.J'-J'JI.'''''JLllll.-1L.J ity (Lorentz field equations gravitational gravitational As ~pposed' to the unspecified Lagrangian of his 1915 article, Lorentz now chose curvature scalar n as the Lagrangian for grav- itational field. come to realize the Lagrangian to be a generally covariantscalar (Lorentz 1916, I, p. 248,p. 251; see also Janssen 1992). Lorentz n the variation of the action into two parts. The first part, which is no longer a scalar leads to gravitational field equations; the second vanishes identically on account of the boundary conditions. Moreover, he showed that the form of his gravitational tensor coincided with Einstein's "onlyfor one special choice of coordinates" (Lorentz 1916, 68 Carlo Cattani and Michelangelo De Maria p. 281, italics in the original). Lorentz the correct gravitational field equations (Lorentz 1916, p. 285). We want to stress, however, that Lorentz made some assumptions in deriving mGlth~~m~atH;alJlV ll1l'lnl'll'll:,rfJI1l'"1l"'fJI'lnI1t"arll his results. He assumed that variations of the components of the metric tensor have tensor character. Moreover, he to make a special choice of coordinates. Lorentz also discussed the conservation of energy-momentum of matter plus gravitational field, and arrived at the equations (6) obtained by Einstein in 1914 (Lorentz 1916, 292). Lorentz too was aware of the fact the complex'ta V is nota (Lorentz 1916, p.294). Whereas this was p-erfectly acceptable.to Einstein, Lorentz wrote that [e]vidently. it would be more satisfactory if we could ascribe a stress-energy-tensor to the gravitation field. Now this can really be done. (Lorentz 1916" III,p. 295~ italics in the original) A "natural" candidate for this tensor, according to Lorentz, was gravitational tensor GJlvof Einstei~'s generally covariant field equations. Therefore ,he suggested one interpret these equations as conservation laws. In Lorentz's opinion this interpretation of the field \,.1\..11 Qo.l1U.lI.-ll.VlI.JlO and the conception to which they have led, may look some"what starAccording to it-we should have to imagine behind the directly obseryallie world with its· stresses, energy etc. ,', the gravitation field is hidden with stress~s, energy etc. that are everywhere equal and opposite to the former; evidently this is in agreement with the interchange of momentum and energy which accompanies the action of gravitation. On the way of a lightbeam, e.g.,· there would be 'everywhere in the gravitation field an energy current equal and opposite to the one t:?xisting in the beam. If we remember that this hidden'energy-current·can be fully described mathematically by the quantities gab and that only the interchange just mentioned makes it perceptible to us, this mode of viewing the phenomena does not seem unacceptable. At all events we are forcibly led to it if we want to preserve the advantage of a stress-energy-tensor also for the gravitation field. (Lorentz 1916, III, p. 296, italics in the original) In part IV' of his paper, Lorentz compared'his definition of the stress~nergy components of the gravitational field with the definition given by Einstein. While expression contained first and second order derivatives ofthemetric, "Einstein on the contrary has given valuesfor the stress-energy componefltswhichcontainthe derivatives only and which therefore are in many respects much more fit for application" (Lore,ntz 1916, IV, p. 297). Thus Lorentz defin,ed a stress-energy complex withcomponents to' v' are homogeneous·and'quadratic functions of the first-order derivatives of the me~ricanddo not contain any higher-order derivatives. The divergence of Conservation Laws and Gravitational Waves 69 Lorentz's complex coincides the divergence of Einstein's ta- v. Lorentz -H showed when = 1 and gOlfJ = DOlfJ his complex is the same as Einstein's. He added that "it seems very agreement will exist in general" (Lorentz 1916, IV, p. 299). In conclusion, we want stress Lorentz showed, for the first time, the quantity representing gravitational stress~energy was not uniquely defined. In 1916, Einstein- returned to a variational approach to derive his gravi- field equations. 1I"01l"11r'hndJln remarked that both Lorentz and Hilbert had succeeded giving general relativity a clear form by deriving the field equations from a single variational principle. His aim now was to present the basic relations of the theory as clearly as possible and a more general way. In fact, he considered his new approach more general and "in contrast especially with Hilbert's treatment" (Einstein 1916b, p. 1111), since he rejected some of restrictive hypotheses' on the nature of matter. H , starting point was the universal function 1t ~ H assumed to be a function of the metric tensor and its first-order derivatives and a linear function of its second-order ~erivatives. Furthermore, he generalized the variat~onal principle to any physical phenomenon by assuming 1-l to be dependent on matter variables qp (not necessarily ofelectromagnetic origin) and their first-order derivatives. Thus, he replaced his 1914 Lagrangian by (11) Integrating a Lagrangian of this form one arrives at variational principle the usual boundary conditions, D 1t*dr = 0, (12) where 1{* no longer depends on the second-order derivatives of the metric. Einstein had to start from a function of the form of (11) because, according to his principle of general relativity, the Lagrangian 1{ must be invariant under arbitrary coordinate transformations. However, the reduction of 1{ to 1t* (i.e., the reduction to a quadratic function of the metric's first-order derivatives) enabled Einstein to make use of the mathematical machinery developed in his 1914 paper. Meanwhile, the problems he had struggled 70 Carlo Cattani and Michelangelo De Maria in 1914 been overcome: the theory was now generally covariant and his choice ofa Lagrangian was no longer (Norton 1984; Cattani 1989b). Einstein's next step was to the Lagrangian into a gravitational and a matter part (see equation (8) above). Einstein concluded that in order to satisfy his principle of general relativity, gravitational part of the Lagrangian "(up to a constant factor) must be the scalar of the Riemann curvature tensor; since there is no other invariant required properties" (Einstein 1916b, p. 1113). Clos~ly following variational approach, Einstein showed, using an infinitesimal- coordinate transformationx~/-= x~+!:ix~, conditionBI-t = o(see equation (3) above) still holds. fact, Einstein proved that this condition--could be obtained by showing that li.J Rdr = 1.5. JR* dr where Theref9re, the relation BJt='O now every coordinate system, to the invariance·of R and to the principle of general Bit played a fundamental role Einstein's new derivation of conservation laws. In fact,; according to Einstein, v...I\.U"~>"/"llULv...I\.\I,.JI.'U'.lI..lI.U' (7). ·These equations ,allowed way, conservation laws. ,.,... a + axa (aagR~f*L g V~) . = v X (.ler v ter ), (13) where conditions (2)-(3) are JI..Il..Il..Il.llJ'U'U'~__q it follows (R*8~ - a~:g~a)'. aga (15) "'Whenequation (13) is with respect to xv, the left-hand,side tumsinto Bf-l.Since B~ vanishes, obtained in this way is equation (6), expressing conservation of t(}talenergy-momentum. Conservation Laws and Gravitational Waves 71 As in his previous theory, Einstein ~ v as· representing stress-energy density for matter and t(j v as representing the stress-energy density of the gravitational (Einstein 1916b, p. 1116). He concluded that although· t(j v was not a tensor, the equations expressing the conserva- tion of total energy-momentum are generally covariant, since they were directly from the 'U'VQ.l\.ll..ll.ll..Jl.""'-' of general relativity (Einstein 1916b, p. 1116). As we see,this claim led Levi-Civita, in 1917, to dispute not only the tensor character of t(j v also equations used as his conservation laws for matter gravitational field De 1989a). on In paper from 1916, Einstein tried to compute components of t(j v for special case of a weak field, doing so discovered the existence of waves. The metric for the weak is written, as in form (16) Minkowski metric YJLV (and its first-order derivatives) are Inl1nlteS:imcal ~U".Il..II.\L..ll.\L.jl..""'0. weak-field approximation the equa- tions reduce to (17) where Y '. JLV .= YJLV - 21:y8JLv , JL Y YJL· (18) The Y~v are defined only up to a gauge transformation. Einstein therefore imposed gauge condition way, found solutions of the weak-field equations,vanishing are the analogs of retarded potentials in electrodynamics.. There- fore, according to Einstein, "gravitational fields propagate as waves speed of light" (Einstein 1916a,p. 692). Multiplying equation (17) by aY~v / 8x(j , Einstein obtained the conservation law for the total energy- mome:ntu:m in the usual (6), where aY~f3 a.Y~f3 _ 1.8 v·",·.(aY~f3.). 2 (19) axJL 8x V 2JL LJ ax r. a{3r 72 Carlo Cattani and Michelangelo De Maria deriving the conservation law, however, Einstein made a trivial math- ematical error used y/Ol/3 instead of yOl/3 in conservation law for matter). As we shall see, two years elapsed before discovered this "regrettable error in computation" (Einstein 1918b, p. 154). The error caused some "strange results" (Einstein 1916a, p. 696). Einstein obtained three different types of gravitational waves compatible with (17): not just longitudinal and transversal 'ones but also a "new type" of wave (Einstein 1916a, p. 693). Using equation (19) to compute the energy carried by these waves, he found the paradoxical-result that no energy transport was associated with either the longitudinal or the transversal waves. tried to explain this absurdity by'treating these waves as fictitious: The strange result that _there should exist gravitational waves without energy transport ... can easily be explained. They are not "real" waves, but "apparent" ones, because we have chosen as the coordinate system the one vibrating ~sthe waves. (Einstein 1916a, p. 696) Einstein found only the kind of waves transport energy. He concluded, however, that the mean value of the energy radiated by this new type of waves was very small, because of a damping factor Ijc4 and because of the small value of the gravitational constant X 1.87 · 10-27) entered into its expression. the possibility of gr2lvlt:atlOtlcll l JL\\-I1o-.J1.\L..Q.lL.J1.,",,'.II..B. was bothersome. As Einstein.stated in his paper: Nevertheless, due to .the motion of the electrons in the atom, the atoms should radiate not only electromagnetic energy, but also gravitational energy, though in a little quantity. Since, this does not happen in nature, it seems that the quantum theory should modify not only the electrodynamics of Maxwell, but also the new theory of gravitation. (Einstein 1916a,p.696) 80 Einstein's choice of a noncovariant stress~energy complex (Einstein 1916b) and strange results on waves (Einstein 1916a) motivated Leyi-Civita to try a satisfactory definition of a gravitational stress- energy theory (Levi-Civita 1917). In opinion, it was Einstein's use of pseudotensor quantities physically unacceptable results on gravitational waves. He wrote: The idea of a gravitational [stress-energy] tensor belongs to the majestic construction of Einstein. But the definition proposed by the author is unsatisfactory. Firstof all, from the mathematical pointof,view, it lacks ~he invariant character it should have in the spirit of general relativity. Conservation Laws and Gravitational Waves 73 More serious is the fact, noticed also by Einstein, that it leads to a clearly unacceptable physical result regarding gravitational waves. He thought that the way out of this last problem was through the quantum theory.... Indeed, the explanation is closer at hand: everything depends on the correct form of the gravitational [stress-energy] tensor. (Levi-Civita 1917,p.381) In Levi-Civita's opinion, general relativity called for a generally covariant gravitational stress-en1ergy tensor. Since no. differential invariants of the first order exist, one cannot have a stress-energy tensor containing only first- order derivatives of the metric; since the definition of ta v in (Einstein 1916b) only contains first-order derivatives, Levi-Civita concluded that "Einstein's choice the gravitational tensor is not justified" (Levi-Civita 1917, p. 391). Levi-Civita, in fact, showed that Einstein's stress-energy complex was covariant under linear transformations only. He proposed a new for the gravitational stress-energy tensor, and, consequently, a new for the conservation law. Starting from the Ricci tensor RJ-lv, Levi-Civita, like Hilbert in 1915, = GJ-lV ril.a.lI"1n.alril RJtv - ~ gJ-lV R and wrote the gravitational field equations in of (9). Using, for the first time, the contracted Bianchi iden- tities, Levi-Civita showed that the covariant divergence of GJ-l v vanishes: VvGJ-lv = O. Consequently, Vv~v = O. This conservation law for matter will Levi-Civita pointed out, since "~v includes the complete con- tribution of all phenomena (but gravitation) which take place at the point in consideration" (Levi-Civita 1917, p. 389). Levi-Civita now made·a move similar to the one we saw Lorentz make earlier: proposed to interpret equation (9) both as field equations and as conservation laws. Defining the stress-energy tensor for the gravitational field as'-- = = + def 1 Ajtv -Yjtv -~v =} AJ-lv ~v = 0, X (20) he identified A/lV as the components of a [stress-]energy tensor of the space-time domain, Le., depending only on the coefficients of ds2 • Such a tensor can be called both gravitational and inertial, since gravity and inertia shnultaneously depend on ds 2• (Levi-Civita 1917, p. 389) Acco~?ingtRLevi-Civita, A/Lv completely characterizes the contribution of gravityto the local mechanical behavior. With this interpretation, it follows from equfltion (20) that no net flux of energy can exist. This equilibrium is guaranteed by the "real" existence of both quantities which, being tensors, are independent of the choice of coordinates. Hence, 74 Carlo·Cattani and Michelangelo De Maria [n]ot only the total force applied to every single element vanishes" but also (taking into account the inertia of the Aj.tv) the total stress, the flux, and the energy density. (Levi-Civita 1917, p. 389) So, for Levi-Civita, gravitational stress-energy is characterized by the only element independent of the coordinates, the Riemann tensor. In Levi-Civita's approach, the problems Einstein ran into are avoided. Einstein to the possibility that gravitational waves transporting energy are generated the absence of sources. Einstein's weak-field equations h~ve solutions for ~v = 0 representing such spon- taneous gravitational waves. Moreover, the energy flux, computed on the basis of equation (17), could be zero in one coordinate system and nonzero in another. Einstein invoked the of theory to solve these problems. Levi-Civita ,claimed that it was enough to define the gravitational stress-energy. tensor the way sugg~sted to reinterpret field equations accordingly.. This precludes situations of the sort Einstein encountered, for, according to (20), stress-energy tensor ,AJLv vanishes whenever the stress-energy tensor ~v for vanishes. the summer of 1917, the Great a vacation to country, gave him a copy ofLev~-Civita's paper (Levi- published in Rendiconti dell'Accademia o n August 2, JJ...4 .....·..........JLlLlL...... '1 Einstein wrote a long was very close to war front), in order to rebut criticism of his theory, especially use of a pseudotensor to represent gravitational stress-energy. Einstein gave physical considerations to show stress-energy of the field cannot be represented by a generally covariant tensor. Einstein began letter··expressing his for work": I admire the elegance of your of calculation. It must be nice toride throughthese fields upon the horse of true mathematics, while people like me have to make their way laboriously on foot. . .. I still don't understand your objections to my view of the gravitationalfield. I would like to tellyou again'what causes me to persist· in my view. , (Einstein to Levi-Civita, August 2, 1917,p. 1) Conservation Laws and Gravitational Waves 75 He proceeded to discuss the example of a counterweight pell0UUUlTI clock to show that Levi-Civita's choice of a tensor to represent the stress- energy of the gravitational field is problematic from a physical of view: I start with a Galilean space, i.e~, one with constant g/-tv. Merely by changing the reference system [i.e., by introducing an accelerated reference system], I obtain a gravitational field. If in K' a pendulum clock driven by a weight is set up a state in which it is not working, gravitational energy is transformed into heat, while relative to the original system K, certainly no gravitational field and thereby no energy of this field is present.7 Since, in K, all components of the energy "tensor" in question vanish identically, all components would also have to vanish in K', if the energy of gravitation could actually be expressed by a tensor. (Einstein to Levi-Civita, August 2, 1917, p. 1) stress-energy could be expressed by a tensor, no gravita- j:;".Il.f..lI.'If.ll.\\,U\\...Il."-JJl.Jlll.4.1l. occur in , in which case, contrary to experience, gravitational energy be transformed into heat. In short, the pen- example shows that it should be possible for the components of gravitational stress-energy to be zero in one reference frame nonzero in U.D.J1.'-,\\...D..ll\",1.1l.. Therefore, gravitational stress-energy cannot be represented by a generally covariant tensor. Notice how Einstein's reasoning here is deeply rooted in conception of equivalence principle. To the physical argument of the pendulum clock, Einstein adds an ar- gument against the tensor character of gravitational stress-energy of a more mathematical In general, it seems to me that the energy components of the gravitational field should only depend upon the first-order derivatives ofg/-tv, because this is also valid for the forces exerted by the fields. 8 Tensors of the first order (depending only on Bg/-tv/8xa = g~V), however, do not exist. (Einstein to Levi-Civita, August 2, 1917, pp. 1-2) In his letter, went on to criticize Levi-Civita's interpretation of the gravitational field equations (20) as conservation laws. .Einstein gave some examples. showing such conservation laws would have strange and undesired consequences. He wrote to Levi-Civita, You think that the field equations ... should be conceived of as energy equations, so that [Q;:] would be the [stress-]energy components of the gravitational field. However, with this conception it is quite incomprehensible how something like the energy law could hold in spaces where gravity can be disregarded. Why, for example, should it not be possible on your view for a body to cool off without giving off heat to the outside? (Einstein to Levi-Civita, August 2, 1917, p. 2) 76 Carlo Cattani and Michelangelo De Maria On Levi-Civita's proposed of the conservation laws, the for matter to lose energy, it seems, is to transfer it to It does not seem to allow for possibility of energy ...m. .......~lJlU.m.'''''.m. one place to another. At the same time, Levi-Civita's proposal did seem to allow for processes one would like to rule out. Einstein wrote: The equation gt + 7;.4 = 0 (21) allows~4 to decrease everywhere, in which case this change is com- pensated for by. a decrease of the, physically not perceived, absolute 91.... value of the quantity I maintain, therefore, that what you [Levi- Civita]. call the ep.ergy law has nothing to do with what is otherwise so designated in physics. (Einstein to Levi-Civita, August 2, 1917, p. 2) these grounds, Einstein rejected Levi-Civita's .Il.ll..ll.IL,.~""'.IlfIJ.a.VII,.'4\L..Il'-".ll..Il of equations as conservation laws, and on to tion of the conservation layvs (6).He argued this was .Il"-".Il..ll..IlJl.Ul.ll.tl.4\L..lI."-".Il.lL perfectly sensible from a physical point of view, even though it a pseudotensor representing gravitational stress-energy: [My] conclusions are correct, whether or not one admits that the t~ are "really" the components of the gravitational [stress-]energy. That is to ~ay, relation d dx~ =0 holds true with the vanishing of 4r v and ta v at [spatial] infinity, where the integral is extended over the whole three-dimensional space. For my conclusions, it is only necessary 144 be the· energy density of matter, which neither one of us doubts. (Einstein to Levi-Civita, August 2, 1917, p. 2) Finally, Einstein lIJ"-".l!..Jl..8.II,.,-'-, out that, in his definition, the· gravitational stress-energy exhibits desired behavior at spatial infinity: ... (in the static case) the field at infinity must be completely determined by the energy of matter and of the gravitational field (taken together). This is the case with my interpretation.. .. (Einstein to Levi-Civita, August2, 1917, p. 2) Levi-Civita's At the end of August 1917, Einstein received Levi-Civita's answer,9 flattery as well as criticism: Conservation Laws and Gravitational Waves 77 I am very grateful that you kindly appreciate the mathematics of my last articles but the credit of having discovered these nevv fields of research goes to you. (Levi-Civita to Einstein, August 1917, draft, p. 1) letter, Levi-Civita criticized Einstein's the gravitational energy, wondering why a of first-order derivatives of tensor be taken as stress-energy (pseudo)tensor, and asking for a more convincing motivation of choice. the other granted Einstein his interpretation of field equations as conservation laws was not very fecund: I recognize the importance of your objection that, in doing so, the energy principle would lose all its heuristic vC:\lue, because no physical process (or almost none) could be excluded a priori. In fact, [in order to get any physical process] one only has to associate it a suitable change of the ds2 • (Levi-Civita to Einstein,August 1917, draft, p. 1) tensor seems to be referring to Einstein's example of a stress-energy whose energy component decreases everywhere. Ein- stein's conservation laws (4) such a stress-energy tensor. It looks as if Levi-Civita's conservation laws, I.e., the gravitational field equations, do It looks as would be possible for almost any matter stress-energy tensor to a metric field such the field equations are satisfied. The conservation laws thus seem to lose "heuristic value" of restricting the range of acceptable matter stress-energy tensors. Of course, through the contracted Bianchi identities" the field equations do, in fact, restrict the range of acceptable stress-energy tensors. In his letter, Levi-Civitastressed having no prejudice against a definition gravitational stress-energy dependent on the choice of coordinates, or, as he it, dependent on the expression of ds2, in analogy with what happens for the notion of force of the field. . . . In the case of the equations of motion, written in the forf!l x v}. 2 d v _ { dx'" dx v & ' ds 2 - - (f {t ds one can explicitly connect the right-hand side (which does not define either a covariant or a contravariant system) with the ordinary notion of force. According to you, the same should happe~ for your ta v (which do not constitute a tensor). I am not in principle opposed to your point of view. On the contrary, I am inclined to presume that it is right as are aU intuitions of geniuses. But I would like to see each conceptual step [canceled: logical element] to be clearly explained and described, as is done (or, at least, as is known can be done) in the case of the equation above, where we know how to recover the ordinary notion of force. (Levi-Civita to Einstein, August 1917, draft, pp. 1-2) 78 Carlo Cattani arid Michelangelo De Maria At the same time, Levi-Civitainsisted that, at least from a logicalpoint of view, there 'Has wrong his own choice of a generally covariant tensor to represent gravitational stress-energy: [canceled: Let me add some opinions for a logical defense]. While I maintain an attitude of prudent reserve and wait, I still want to defend the logical flawlessness of my tensor 9JLV. (Levi-Civita to Einstein, August 1917, draft, p. 2) Next, Levi-Civita attacked the· counterweight pendulum-clock example: I want to'.stress that, contrary to. whatyou claim,'thereis no contradiction between the accounts of the pendulum-clock in the two systems K and K', the first one fixed (in the Newtonian sense),the second one moving with constant acceleration. You say that: (a) K, the- energy·tensor zero because the gJLV are constant; (b) in K", thisis not the case; instead, there.is a physical phenomenon with·an observable transformation of energy into heat; (c) due to the .invariant. character of a tensor, the simultaneous validity of (a) and (b) implies that there is something wrong with the premises'. contest (a), since we can assume .... gJLv. constant outside of the ponderable bodies, but [not] in the space taken by your pendulumclock. (Levi-Civita to Einstein, August 1917, draft, p. 2) to Einstein's comment on 1!"'£:l!IC''lI'''Ilr\\1'l''lIrU£:l!IrfI behavior regard to the last consideration of your letter (point 4), if I am not wrong, it [the behavior of the gravitational field at infinity] is not a consequence of the special form of your ta v, is equally valid for my AJLv. It.seems to me that the behavior at infinity can be obtained from [our equation (20)] by using the circumstance that the divergence of the tensor A JLV is identically zero; therefore, the divergence of ~v also. vanishes, it red~ces asympto.tically to. ~a7xVirv =0, because the gJLV tend to the values EJLv the constant Minkowski values of the metric tensor]. (Levi-Civita to Einstein, August 1917~draft, p. 2) So, Levi-Civita invoked the contracted Bianchi identities to show his conservation laws, like Einstein's, exhibit the desired 'behavior at infinity. Conservation Laws and Gravitational Waves 79 In an addendum, Levi-Civita finally remarked: An indication in favor [of our equation (20)] is the negative value of the energy density of the gravitational field Aoo (assuming 100 > 0). This is in agreement with the old att¥mpts to localize the potential ellergy of a Newtonian body, and explains the minus sign as due to the exceptional role of gravity compared to all other physical phenomena. (Levi-Civita to Einstein, August 1917, draft, p.2) on Waves .a. ..._ ....a.'"'JII..lL_A 18) After Levi-Civita's August 1917 letter, the polemic between two scien- tists stopped Einstein in 1918 published a new paper on gravitational waves (Einstein 1918b). In introduction, he recognized earlier approach to gravitational waves (in Einstein 1916a) was not transparent enough, and it was lIlarred by a regrettable error in computation. ,Therefore, I have to tum back to the same argument. (Einstein 1918b, 154) Because of this error, he had obtained wrong expression for his stress- energy complex. Correcting the error, Einstein could easily derive the correct expression for the stress-energy complex. As a consequence, he only two n.hllrlJlll1l''IIal"1l of waves, thereby resolving physical para- doxes of his previous results. ~instein could now assert with confidence [aJ mechanical system which always maintains its spherical symmetry cannot radiate, contrary to the result of my previous paper, which was obtained· on the basis of an erroneous calculation. (Einstein 191 ~b, p. 164) the last section of (Einstein 1918b), "Answer to an objection advanced by Mr. Levi-Civita,"lO Einstein publicly gave his reply to Levi-Civita's objections. Einstein gave improved versions of some of arguments. already given in his August 1917 letter to Levi-Civita. He (6) must be looked upon '-""1lUlU\I...A.1iIo...I.1l..II. as tVa cannot be considered components of tensor. In this section of his paper, Einstein gave ample credit to Levi-Civita his contributions to general relativity: In a recent series of highly interesting· studies, Levi-Civita has contributed significantly to. the clarification of some problems in general relativity. In one of these papers [Levi-Civita 1917], he defends a point 80 Carlo Cattani and Michelangelo De Maria of view regarding the conservation laws different from mine, and disputes my conclusions about the radiation of energy through gra"itational waves. Although we have already settled the issue to the satisfaction of both of us in private correspondence, I think it is fitting, because of the importance of the problem, to add some further considerations concerning conservation laws.... There are different opinions on the question whether or not tVa should be considered as the components of the [stress-]energy of the gravitational field. I consider this disagreement to be irrelevant and merely a matter of words. But I have to stress that [our equation (6)], about which there are no doubts, implies a simplification of views that,is important for the signific'ance of the conservation laws. This has to be underscored for the fourth equation (a = 4), which I want to define as the energy equation. (Einstein 1918b, p.166) Without entering into the· mathematical details of ta v, Einstein oelt'en