zotero/storage/PZFIPXGE/.zotero-ft-cache

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(</J). Its

purpose was to allow for the fact

mass energy of a

system must vary the gravitational

whereas gravitational

mass of the system be independent of potential. a system had

mass m when ...JLJI.....' ... ""...........II.

an

gravitational of potential ¢, then

its gravitational mass Mg was given by

= g(¢)m.

(12)

we now considered a

whose parts lay in regions

of

potential, the gravitational mass of the whole

be given by a g-weighted integral over its volume

M g = g(</J)v

At

expressions for both g (l/J) the source density v re-

mained undetermined. Nordstrom now reversed direction of Einstein's

reasoning. Einstein had shown choosing T as source density enabled

equality of

gravitational mass for complete static systems.

Nordstrom postulated this equality and from it derived Einstein's choice

for source density

1
v = - c-2T

and an expression for g

c2
g(¢) = A+¢'

The constant A could be set
choice A = 0,

as a gauge fre·edom. Under the </J/, Nordstrom's second

theory now provided a very simple relationship between the energy E,

mass m , JI...ll.JI. ...' ... ""..II. ......JI..

gravitational mass M g of a complete stationary system

This dependence of the energy mass of a system on

po-

tential ¢' was closer to familiar classical expressions than the corresponding

(6) of Nordstrom's first theory.

22 John D. Norton

Satisfactory as these results were, they not yet provide an escape

from Einstein's objection to all relativistic scalar theories of gravitation. It

is odd

objection is mentioned nowhere in Nordstrom's paper, even

though a major part of the paper is devoted to developing effects were

able to defeat that objection. These effects emerged from a long series of

analyses of different gravitational systems, including Nordstrom's model

of the electron, stressed rods, light clocks, gravitation clocks, and harmonic

oscillators. Nordstrom found that a very wide range of physical quantities

would depend upon gravitational potential. These included the lengths

of bodies, times of processes, masses, energies, and stresses. When these

dependencies were into account, it

Einstein's

of the law of conservation of energy no longer arose.

A simple thought experiment illustrates most

how

dence arises the 'case of . lengths of bodies

dependence

defeats Einstein's objection. This

our sixth ....,~· Nord- IL-J1.J1."-'\UI.§m...II..JI.IL-

.... It-' ....'J1.AJ1. ..,J1....., ....L.....

strom attributed the thought experiment to HlInC'ltp'1" Lll.JLIL-Jl.ll.'\,,",U;;;;'•.II..lI. HllnCll"Plln

it nowhere himself. Since Nordstrom (1913b) was C1I1l1h11t"n1l1r'ltpril

Zurich,

of both Einstein and von

raises

of precisely

ideas enable escape

obje~tion.

in ,.,.nll"\\C'01l"''\\T.r:l1/"11tr''~n
being stressed. is lowered body yields

Hllnc'ltp'1n now offered is ingenious. If a

being lowered into a gravitational

.Il.\\,I"'lIU.Il..Il.',1~ to expand the body against it ·absorbed . . '7'tr'\\f1I1I"u .... v

themselves?

gr2lvtt:atl,oncll mass of stresses cycle pn~Jlt"O"u_crpnpt""JI'ltll11IO"

blocked. Nordstrom's (1913b, 545-545) account ofEinstein's 1/"1hl.n"llll.nr~!"II1l­

experiment shows us

aC1lustlmellt is easily achieved (see Figure 6).

He wrote:

Herr .Einstein has proved that the dependence in the theory developed here of the length dimensions of a body on the gravitati~nal potential ,must be a general property of matter. He has shown thp.t otherwise

rigid rod stretched between rails
gravitational mass of stresses

Einstein and Nordstrom: Thought Experiments 23

n
Lower

gravitational potential <p'

gravitational mass of
stresses =0

/
tensioning forces

gravitational potential
<pI + d<pl

Escape a net energy gain in cycle by absorbing this extra work 'as work needed to dilate the stressed
rod by l e u s t r e s s S.

work released
in fall
_ Sl d<p' <p'

work absorbed by stress S in dilation
S dl

:. l<pl = constant

Gravitational potential dependence oflength restores energy conservation.
it' 'would be possible to construct an apparatus with which one could pump energy out of the gravitational field. Einstein's example, one considers a non-deformable rod that can be tensioned movably between two vertical rails. One could let the rod fall stressed, then relax it and raise it again. The rod hasa greater weight when stressed than unstressed, and therefore it would provide greater work than would be consumed,in

24 John D. Norton

raising the unstressed rod. However because of the lengthening of the rod in falling, the rails must diverge and the excess work in falling will

be consumed again as the work of the tensioning forces on the ends of the rod.

Let S be the total stress (stress times cross-sectional area) of the rod

and I its length. Because of the stress, the gravitational mass of the rod

is increased by

2- g(l/J) S1 = c2 '

l/J' S1 .

In falling [an infinitesimal distance in which the potential changes by

d</J' and the length of the rod by dl], this gravitational mass provides the

extra work

1, - ¢,Sld¢.

However, at th~ same time at the ends of the rod the work

Sd1

is lost [to forces stressing the rod]. Setting equal these two expressions

provides

1,1

-

-<dp'

"

"'
'Y

=

.-1 d l '

which yields on integration

[<p' = const.

length of a body vary inversely with the

is sufficient-to preserve the conservation of en-

ergy against

of Einstein's earlier thought experiment. Einstein

clearly accepted escape, as acknowledged

his exposition of

Nordstrom's theory 1913, p. 1253) ,.JJJ.-4J1.......U'll-.....JI..IL...

more briefly

addendum to'the

of Einstein Grossmann (1913).

intrusion of these

effects Nordstrom's theory, it

ceased to be a conservative, Lorentz covariant theory of gravitation

b~~amemore ,akin to Einstein'8 'own theory, which· gravitation, space,

were

Just how close it come to

Einstein's theory was

and Adria-an D. Fokker in a

paper the following Febru~ry

Since the times

of all processes and the lengths of bodies

aff~cted equally by

gravitational potential ¢, the times and spaces ofthe background Minkowski

space-time had ceased to be directly measurable by real rods and clocks.

Inst~ad they revealed a non-Minkowskian space-time with the characteristic

Einstein and Nordstrom: Thought Experiments 25

property that there exist preferred coordinate systems (x, y, Z, t) in which the invariant interval is given by

(13)

After postulation of this basi~ property for space-time, the theory de-

veloped a remarkably similar way to Einstein's theory. The trajectory

of a body in free fall in the gravitational field was a geodesic of the space~

time. The law ofconservation ofgravitational and non-gravitational energy-

momentum was given by the vanishing of covariant divergence of the

stress-energy tensor. Finally, the field equation of Nordstrom's second

theory proved to be just
I

R=kT,

where R is the

scalar and k a constant-Einstein was able to in-

troduce generally covariant equations based on the Riemann curvature

tensor his own gravitation theory until November 1915.

In 1914, Einstein could not offer decisive grounds for picking between

his this version of Nordstrom's theory. The strongest argument

he could· muster against Nordstrom's theory was that it failed to satisfy

requirement of the relativity of inertia, a requirement whose essential

content would be transformed into Mach's principle. The presence of the

preferred coordinate systems (x, y, Z, t) in (13) was judged by Einstein as

a residual, absolute

to be jettisoned if principle of

relativity were to be generalized to accelerated motion.

The three soon-to-be classic tests of general relativity could offer no

help in deciding between the two theories. Both Einstein's and Nordstrom's

theory predicted a red shift in light from the sun and of equal magnitude.

Unlike Einstein's theory, Nordstrom's theory predicted no deflection in a

beam of

grazing the sun. However, the world would wait five

years for Eddington's celebrated expeditions. Finally, accounting for the

anomalous motion of Mercury had not yet emerged as a sine qua non of any

new gravitation theory. Einstein's theory of 1913 actually failed to account

for this anomalous motion, a shortcoming that was oddly never mentioned in

Einstein's publications ofthis period. Nordstrom (1914) analyzed planetary

motions according to his theory. He found it predicted changes in

planetary orbits that were very small in comparison with perturbations

due to other planets thus felt justified in concluding that this theory was

"in the best agreement with experience" (p.1I09).

What decisively changed the standards· for evaluation gravitation

theories.vvas a result communicated by Einstein (1915) to the Prussian

Academy on November 15, 1915. He showed his gravitation theory,

26 John D. Norton

now equipped with generally covariant field equations, was to ac-

count almost exactly for the anomalous advance of Mercury's I!-'VA.JUI..llV'A.Il.'U'.Il..lI..

Overnight, the margin of error in astronomical prediction

a gravi-

tation theory dropped by at least an order of magnitude. As von Laue noted

in his sympathetic review (1917, p. 305), Nordstrom's theory was no match

for Einstein's when it came to Mercury, for Nordstrom's theory predicted

a slightretardation of the planet's perihelion. The failure was now deeIp.ed

so complete that von

did not even

to report magnitude of

the retardation.

After the 'excitement of Eddington's eclipse expedition and the

acclaim of Einstein and his 'theory~\ the fate of Nordstrom's theory was

sealed. It could offer competition to the seductive charms of Einstein's

theory. By the time ofPauli's authoritative survey (1921, section 50), in

a paragraph Nordstrom"s theory was dismissed briefly and decisively

as a viable gravitation theory.

NOTES.

1 M. von

to A. Einstein, December 27, 1911, EA 16-008. For further

discussion, see Norton (1985, section 4.1).

2 For philosophical analyses of thought experiments from various perspectives,

see Horowitz and Massey (1991), which contains Norton (1986), and see also Brown

(1991) and Sorensen (1992).

3 Einstein to J.' Stark, September 25, J907, EA 22-333. 4 One of the most informative is Einst~in' (1933, pp. 286-287).

5 Here and henceforth, Greek indices will vary over 1, 2, 3, 4 and Latin indices
over 1, 2~ 3. I will employ the coordinate system (Xl ,X2, X3, X4) = (x, y, z, u = ict)

as was commonin four-dimensional physics in the early 1910s. Summation over

repeated indices win be implied.

6 From the orthogonality of four-velocity UfJ- and four-acceleration dUfJ-Jdr, we

infer from the contraction of (4) with UfJ- that

o = F U = -m a4J dxfJ- = -m d4J ,

fJ- fJ-

aXfJ-' dr

dr

so thatd</JJdr = o.
7 In a lecture given on April 14, 1954,according to notes taken by Wheeler (1979, p~ 188).
8 p?k is tl1e{three-dimensional) stress tensor.
__9 y =lJJl - V2Jc2 •
10 See Norton (1992, section 9), and Janssen (manuscript). 11 Einstein's analysis did not consider the corresponding exchange of momentum associated with the temporary imbalance of external forces, which would lead to the momentum expression in (8). I add this to my analysis below since it is a trivial and 9bvious extension of Einstein's original thought experiment.

Einstein and Nordstrom: Thought Experiments 27

12 I follow Einstein in assuming that we are treating a case in which the forces

between the charges on the body are small compared with the external forces and

can be neglected.

= = = 13 As usual, we have t
J1/ 1 - v2 jc2•

Y(t' + (v / C2 )X' ) and. x

y (x' + vt' ), where y

14 One obvious problem with (9) that Einstein did not mention is that it is ill-

defined for source matter that, unlike dust, has no natural rest frame.

15 Von Laue's (1911a, section 5) definition was unnecessarily restrictive and did

not include bodies rotating uniformly about their axes of symmetry. Nordstrom

(1913b, pp. 534-535) quietly extended the analysis to "complete stationary" sys-

tems, which did include such rotating bodies.

,

16 Under Nordstrom's choice of coordinate system, with X4 = ict, T44 = -(ener-

gy density), whereas under Einstein and Grossmann's (1913) choice of metrical

signature (-, -, -, +),

+(energy density). I have also followed Einstein in

simplifying analysis by ignoring the fact that the total energy of a system must

vary with gravitational potential, whereas its gravitational mass will no,t. Thus

the expression for the proportionality of the inertial and gravitational mass of. a

system must contain a factor is a function ofthe gravitational potential. This

effect is explicitly incorporated into Nordstrom's (1913b) second theory through

the factor g(ifJ), and the proportionality is expressed as relation (12) of Section 6

below. For the analysis of this section and the following, this g factor can be taken as

approximately constant and its effect absorbe~ into other constants in the equations.

17 This conclusion holds for free radiation, and for this reason there is no gravi-

tational bending of Hght in Nordstrom's (1913b) second theory, since it employs T

as its source density.

18 To see this most clearly, imagine that each pair of opposing walls of the box

are held together by a slender rod that carries all the stresses needed to hold the

walls against radiation pressure. One set of opposing' walls and rods forms the set

of baffles. Three identical sets can be fitted together to form the cubical box.

REFERENCES
Abraham, Max (1912). "RelativiUit und Gravitation. Erwiderung auf einer Bemerkung des Hrn. A. Einstein." Annalen derPhysik 38: 1056-1058.
Brown, JaP1es R. (1991). Laboratory of the Mind: Thought Experiments in the Natural Sciences. London: Routledge.
Ehrenfest, Paul (1907). "Die Translation deformierbarer Elektronen und der Flachensatz." Annalen der Physik 23: 204-205.
Einstein, Albert (1907a). "Uber die vom Relativitatsprinzip gefordete Tragheit der Energie." Annalen der Physik 23: 371-384.
- - - (1907b). "Uber das Relativitatsprinzip und die aus demselben gezogenen Folgerungen." lahrbuch der Radioaktivitiit undElektronik 4: 411-462; 5: 98-99.
- - (1912). "Relativitat und Gravitation. Erwiderung auf eine Bemerkung von M. f\braham." Annalen der Physik 38: 1059-1064.

28 John D. Norton

- - (1913). "Zum gegenwartigen Stande des Gravitationsproblems." ,Physika-

lische Zeitschrift 14: 1249-1262.

- - (1915). "ErkHirung del' Perihelbewegung des Merkur aus del' allgemeinen

Relativitatstheorie." Koniglich Preussische Akademie der Wissenschaften

(Berlin). Sitzungsberichte: 831...:.839.

- - ,(1916). "Die Grundlage del' allgemeinen Relativitatstheorie." Annalen der

Physik 49: 769-822; translated without p. 769 as "The Foundation of the

General Theory of Relativity" in The Principle ofRelativity. HendrikA. Lo-

rentz, Albert Einstein, Hermann Minkowski, and Hermann Weyl. New York:

Dover, 1952, pp. 111-164.

- - (1933). "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.

Einstein, Albert and Fokker,Adriaan D. (1914). "Die Nordstromsche Gravitations-

theorie vom Standpunkt'des absoluten Differentialkalktils." Annalen der Physik 44':321~328.

Einstein, Albert and Grossmann, Marcel (1913). Entwurf einer verallgemeinerten

Relativitiitstheorie' und einer Theorie der Grt;lvitation. Leipzig and Berlin:

B.G. Teubner (separatum). Reprinted with added "Bemerkungen" by Ein-

stein in Zeitschriftfiir Mathematik und Physik 63: 225-261.

Einstein, Albert and Infeld, Leopold (1938). The Evolution ofPhysics. Cambridge:

Cambridge University Press.

Horowitz, Tamara and Massey, Gerald, eds. (1991). Thought Experiments in Science

and Philosophy. Savage,'Maryland: Rowman and Littlefield.

Janssen,

(manuscript). "Condensers, Contraction and Confusion: Accounts

of the Trouton-Noble Experiment in"Classical Electrodynamics."

Nordstrom, Gunnar (1912). "Relativitatsprinzip und Gravitation." Physikalische

Zeitschrift 13: 1126-1129. - - (1913a). "Trage schwere Masse in del' Relativit~tsmechanik." Annalen

der Physik 40: 856-878.

- - (1913b). "Zur Theorie der Gravitation vom Standpunkt des Relativitatsprin-

zip." Annalen der Physik 42: 533-554.

- - (1914). "Die Fallgesetze und Planetenbewegungen in del' Relativitatstheo-

rie,"Annalen der Physik 43: 1101-1110.

Norton, John (1985). "What was Einstein's Principle of Equivalence?" Studies

in History and Philosophy of Science 16: 203-246. Reprinted in Einstein

and the History ofGeneral Relativity: Einstein Studies, Vol. 1. Don Howard

and John Stachel, eds. Boston: Birkhauser, 1989, pp. 3-47.

- - (1986) "Thought Experiments in Einstein's Work," presented at the work-

shop "The Place ofThought Experiments in Science and Philosophy," Center

fot Philosophy of Science, University of Pittsburgh, April 17, 1986; pub-

lished in Horowitz and Massey (1991).

- - - (1992). "Einstein, Nordstrom the Early Demise of Scalar, Lorentz Co-

variant Theories of Gravitation." Archive for History of Exact Sciences' 45:

17-94.

Einstein and Nordstrom: Thought Experiments 29
Pauli, Wolfgang (1921). "Relativitatstheorie." In Encyklopadie der mathematischen Wissenschaften, mit Einschluss an ihrer Anwendung. Vol. 5, Physik, Part 2. Arnold Sommerfeld, ed. Leipzig: B.G. Teubner, 1904-1922, pp. 539-775. [Issued November 15,1921]. English translation, Theory ofRelativity. With supplementary notes by the author. G. Field, trans. London: Pergamon, 1958; reprint New York: Dover, 1981.
Sorensen, Roy A. (1992). Thought Experiments. New York: Oxford University Press.
Trouton, Frederick T. and Noble, H.R. (1903). "The Mechanical Forces Acting on a Charged Condensor Moving through Space." Philosophical Transactions of the Royal Society ofLondon 202: 165-181. ,
von Laue, Max (1911a). "Zur Dynamik der RelativiHitstheorie." Annalen der Physik 35: 524-542.
- - (1911b). Das Relativitatsprinzip. Braunschweig: Friedrich Vieweg und Sohn.
- - (1917). "Die Nordstromsche Gravitationstheorie." Jahrbuch der Radioaktivitat und Elektronik 14: 263-313.
Wheeler, John A. (1979). "Einstein's Last Lecture." In Albert Einstein's Theory of General Relativity. Gerald E. Tauber, ed. New York: Crown, pp. 187-190.

In his

became a living oracle. We are told

again of lesser-known scientists grappling

who made pilgrimage to

Einstein, 1lJ ...................... fl."t..J/

ment or endorsement, or

hope

out of their

Our

story of a scientist who had'become hopelessly lost in a

own

as struggled

most

discovery of life.

A correspondent gives

be followed out of

scientist

as a confused dis-

traction, only to discover a

way out a few

makes

our story special is

scientist was not just anyone-it was Einstein

himself-and the discovery was .a.,~.a. f-l.....J1.JI......

was

correspondent was

a

physicist working in Gottingen and taking

the activities ofthe

group centered

The was the so-called 1lJJl.'U'ILJ.Il.....Jl.llJl.

argument,

no physically

acceptable version

be .generally covariant. We conjecture

a serviceable sophisticated escape

__

misunderstood dismissed only to

at a

escape a few months

of point-coincidence. argument.

on basis of an

in wording and

we

will suggest that Einstein may

inspiration for the

forlnulation of his point-coincidence argument another hitherto

unre.cognized source.

Out of the Labyrinth? 31

Our argument for our main conclusion will be somewhat

rest-

ing, as it does, upon our conjectural reconstruction of letters from Hertz

to Einstein on the basis of Einstein's surviving replies to Hertz. Such an

approach raises obvious methodological and historiographical questions

about the use of evidence that is as much conjectured as discovered. How-

ever,

absence of more

evidence, our only alternative is to say

nothing at

is an issue too interesting and important to pass over

in silence.

In the summer of

when our story is set, Einstein's long struggle

toward his .general theory of relativity was drawing to a close. Roughly

two years

he

Grossmann published first outline

of theory, complete in essential details excepting the gravitational

equations offered, w'hich were not generally covariant (Einstein and

lif4JSSmatnn 1913). To

matters worse, Einstein soon suppressed his

concern over

of general covariance by convincing himself that any

generally covariant

one might propose must be physi-

uninteresting. His

argument for this surprising conclusion

argument,"

in its and most complete form in

1066~1067 (see Norton 1987;

1989).

hole argument,

considered a "hole," a region of space-

time devoid of "material processes" (the stress-energy tensor 1ik· = 0), and a

solution gik, a coordinate system x m , of supposedly generally covariant

field equations for metric tensor gik, given a matter distribution

is nonvanishing outside the hole. He then showed the general

covariance ofthe equations allowed him to construct a second solution,

with components g;k' in the same coordinate system x m ,that agreed with the

first solution gik outside the hole came smoothly to differ from it within

the hole. Einstein

the existence of two such solutions in the same

coordinate system unacceptable, for he took it to violate the

of causality,"

seemed here to amount to the

the

field matter distribution outside hole should determine

the

processes or events

the hole. presumption, apparently, was

there is

state of affairs within the hole'(and elsewhere) that is

supposed to described, uniquely, by a theory of gravitation (see Howard

1992).

Einstein constructed these two solutions by means of a transformation from the original coordinate system x m to a new coordinate system

x m / that agreed with the original outside the hole but came smoothly to

32 Don Howard and John D. Norton

differ from it within the hole. Under this transformation first solution gik, in x m , becomesg;k' in x ml, which general covariance guarantees is also

a solution of the field equations. To recover the second solution mentioned

above, Einstein looked upon components g;k as ten functions of arguments. x ml and imagined that these arguments were replaced by numer-

ically identical values of the original x m without changing the functional

form of g;k. The result is two differing solutions of the the same coordinate system x m . (See Figure 1.)

equations.in

It will be important for later discussion to pause here and note these

two solutions have the following characteristic property, although Einstein stress this fact: There exist two coordinate systems x m x ml

agree outside the hole come smoothly to differ

the hole, such

that components of second solution, in the coordinate system x m ,

are precisely same functions of the coordinates as are components

of first solution,

second coordinate sy'stem x m I. 1

[6 in the case v.n..L.lI..Il..D..D.II-J..lI.V,

two-dimensional space-time

if the

of values of second solution is ~2] at (1,

coordinate system, the matrix of values of the first

is also

[~ ~2] at 1) in second coordinate system. Notice, however, if

(1, 1)

coordinates of a p

the hole,

by "

,

,""'.II..II.'1

(1, 1) in second coordinate system be the coordinates

the hole.

range of coordinate

in such a way

any

selected

not use two coordinate systems agreed

to disagree

the

To see how'

theory came

system," 1l.4","",ll.<~IlJ"'''''~ v~>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)

§..

"=~<s.

.S-....;;)

w
U'l

36 Don Howard and John D. Norton

objectively expressed," according to the rules

theory of invariants

of group of

transformations

to

ds Everything we 1111"\\'jrl1l1l"'11 11111""1111"

2•

Lagrange, Gauss,

may be clear itself, says Klein.

It is nevertheless a good idea to explain it further, because there are here, in Einstein's work, imperfections [Unvollkommenheiten], which do not impair the great ideas in his new theory, but hide them from view.
This is connected with the repeatedly mentioned circumstance that Einstein is not innately [von Hause aus] a mathematician, but works rather under the influence of obscure [dunkelen], physical-philosophical impulses. Through his interaction with Grossmann and on the basis of the Zurich tradition he has, to be sure, gradually become 'acquainted with Gauss and Riemann, but he knows nothing of Lagrange and overestimates (par~ntheticany) Christoffel, under the influence of the local Zurich tradition.6

One senses in

words a of jealousy, but they help us

stand how members of the Gottingen group may have regarded Einstein's

mathe:maltlc:al failings with more than a little condescension.

Undeterred by

hole argument, and . . . . . . . . . . . v ..... .lLJu..J1......lLJl...............

.IL,...... - I ! J · u ..

strate how

G6ttingen expertise

ical physics might yield dividends of a kind not yet 'JIl"'nlp;."ut.::II1'1

physic~l-philosophicalimpulses" of Einstein, . . . . .IL.IL............IL ... nilln.selt

task of

generally covarianlJie!~ equations for

stein's theory, a fusion ofEinstein's gravitation theory

matter the-

ory. He communicated the modem grayitational equations of general

relativity to the G6ttingen Gesellschaft der Wissen~chaften on November

20, 1915

1915).

Einstein had 'lost confidence in the

lack of general covariance of theory and returned to the quest for gen-

erally covariant equations. He arrived at the same gravitational field

equations as

they were communicated to the Prussian Acad-

emy on November 25,1915, five days after same equations G6ttingen.7

had communicated

Einstein soon

to the task of informing his correspondents of how

he reconciled his hole argument with· his return to general covariance by

means of a consideration now

as the "point-coincidence argument."g

The

was first published in Einstein's comprehensive 1916 review ar-

ticle, "Die Grundlage allgemeinen Relativitatstheorie" (Einstein 1916,

pp. 117-118). Whereas previously he had argued th~t generally covariant

equations typically can be made to yield different solutions for one and the

same coordinatization of the physical space-time, Einstein now argued that

while the two solutions gik and g;k may be mathematically distinct, they

Out of the Labyrinth? 37

are not physically distinct, for both solutions catalogue the identical set of

space-time coincidences,

exhaust the reality captured by theory.

Thus, Einstein wrote to Ehrenfest on

26, 1915:

The physically real in the world of events (in contrast to that which is dependent upon the choice of a reference system) consists in spatiotemporal coincidences.* Real are,. e.g., the intersections of two different world lines, or the statement that they do not intersect. Those statements that refer to the physically real therefore do not founder on any univocal [eindeutige] coordinate transformation. If two systems of the g/-tv (or in general the variables employed in the description of the world) are so created that one can obtain the second from the first through mere space-time transformation, then they are completely equivalent [gleichbedeutend]. For they have all spatiotemporal point coincidences in common, Le., everything that is observable.
*)and in nothing else! (EA 9-363)

An example of these space-time coincidences would be collision of two

point-masses.

We. illustrate Einstein's point-coincidence argument in away that will

be suggestive below. Let two point-masses originate at a point-event q

outside the

separate, and collide at some point-event within the

hole. See Figure 3. According to the second solution, g;k' the particles will

collide atthe point[-event] with coordinates (1, 1) in the first coordinate

system, xm.According to the first solution, gik, the particles will collide at

the

with coordinates (1, 1) in the second coordinate system, x m'. As

illustrated in Figure 2, Einstein earlier assumed that the two sets of co-

ordinates would represent different point[-event]s, p and p', the physical

space-time. He now understands that, on the contrary, they must repre-

sent the same point[-event] , because the two sets of trajectories agree in all

physically significant quantities and thus cannot pick out physically differ-

ent point[~event]s. For example, measurements of physical time elapsed

along the trajectory qap as determined by the first solution gik would be identical to along qap' as determined by the second solution g;k.9

2. Letters
Einstein later recalled the intense emotions that simmered and boiled within himself through the years of his struggle with general covariance when he wrote of the episode: "But the years of anxious searching in the dark, with their intense longing, their alternations of confidence and exhaustion and final emergence into the light-only those who have experienced it can understand that" (Einstein 1934, pp. 289-290). Into this emotional

World lines of two colliding point-masses
ResolutIon of

w
00

Second

I

I

~oordinat~

system

0 x m,

First solution assigns

o:t::sJ

-, lrOl -021J ~----=.:::::::~:;;;;;..,--------,---. gik =
toxm' =(1,1)

o=t:
a~

§

0.-

~ o
S

~
zo
8
:::s

Second solution assigns

,
gik

-_lorl

toxm = (1,1)

Out of the Labyrinth? 39

and intellectual cauldron around August 1915 was added an exchange in

correspondence with Paul

just a few months before the struggle drew

to its dramatic close that November.

Hertz was born in 1881 in Hamburg. In he was a Privatdozent at

Gottingen and a member of the group clustered around

and Klein.

He taken a degree at Gottingen in 1904 under Max Abraham,

a dissertation on discontinuous movements of an electron (Hertz 1904).10

After publishing a few additional studies on electron theory, he turned his

attention to the foundations of statistical mechanics, an interest cul-

minated his seminal 1916 monograph

Repertorium fur Physik

(Hertz 1916), also led to his acquaintance with Einstein.

ac-

qu,nnt:an(~e was a

result of Hertz's critical remarks (Hertz 1910) on

early papers on subject (Einstein 1902, 1903, 1904), re-

Einstein replied in a short note the Annalen .in 1911

begun corresponding by August 1910 had

acquainted no

than early September 1910, at a

Schweizerische Naturforschende Gesellschaft Basel. 11

Hertz was by

acquainted with several of Einstein's closer friends

colleagues, most importantly Paul Ehrenfest, who had been a student

in Gottingen at same time as· Hertz, 12 and Jakob

another fellow

Gottingen, who was a colleague of Hertz's in Heidelberg from 1. 13 In 1921, Hertz finally received an appointment as Ausseror-

IUl-VJl.JII.v....lI..lI.'b'Jl.Jl.V.lI. Professor in Gottingen, the same year that he and Moritz Schlick

Dut)!lSJnea their

of Helmholtz's epistemological writings

\lllllv.ll.JLltll.ll."--'.II.v...L.i 1921). And in later years, Hertz turned his attention to various

of science, including pioneering studies, very much

in the Gottingen

the formal axiomatics of scientific theories.14

Einstein provided a letter of recommendation for Hertz after emigration

to the United States (EA 12-221). ·He died in

1940.

We not know for certain

was present when Einstein lec-

tured in Gottingen late June early of 1915. Given the nature

of the previous relationship between Hertz Einstein, given Hertz's role

group

and given the character of Hertz's correspon-

dence Einstein later summer, it is more likely, however, that

was present.

\Jle know of the letters that Hertz wrote to Einstein only because Ein-

stein's replies still exist (EA 12-201 and EA 12-203). Einstein's letter

EA 12-203 is dated "22.

(August 22)~ The content is compatible

only with the years 1913~1915. The year must be 1915 because of the

mention in a postscript of a coming visit to Zurich ("Aug. 26 to about Sep~

tember 15"), the address of his friend Heinrich Zanggerbeing given for

40 Don Howard and John D. Norton

correspondence. Einstein made a visit to 1915. 15

description in

Einstein's letter is written in a

encouraging tone. It reflects

on the great problems Einstein faced in finding a way to restrict the

coordinate systems of his theory and sketches the difficulties facing

theory in this area. The letter begins:

One who has himself poked about so much in the chaos of possibilities can understand very well your fate. You haven't the faintest idea what I,· as a mathematical ignoramus, had to go through until I entered this harbor.

about his specific restriction to "adapted" coordinates, he comments:

How can one pick out a coordinate system or a group of such? It appears not to.be possible in any way simpler than th~t which I have chosen. I have groped abo'ut and tried everything possible. . .. The coordinate restriction that was finally introduced deserves particular confidence because it can be broughtinto'connection with the postulate of the complete determination of events.

This

to

first by Einstein 1l1l"'il1l"1t"rllrIllllll".cbrll

U.U~I.IIJ\l.."';U coordinate systems were

o f I"rII1l"'11I"KlI·lIC'llr·.1l"'il

hole

letter's to idea

is to 1l"'oC'·nrll1l",r11 first 1n.4Jl1t"1JI0'1l"1JI1l"'.h

concerns the restriction of coordinate systems. Hertz's idea is presum-

ably also the one

refers to in

opening sentence-"A

surface-theoretical

of

systems

of very

great

sentence of paragraph five- "Perhaps one

could get an overview on

if one succeeded in

geo-

metrical

for

seek"- for such an

is

not given or even mentioned Einstein anywhere else in

Einstein's other

12-201, contains a response to a

by

is cast

language of theory of tW()-a]lmf~nS]lOn;al

Gaussian surfaces. 16

Einstein's EA 12-201 is

clpse similarity of content, it was

at

time asEA12-203. earliest possible date is August since Hertz's

son, Hans, who is mentioned at

of the

was born on Sunday,

August 8. 17 The letter was probably

no later about Saturday,

October 9, since it betrays no doubts on Einstein's part about the restricted

covariance of the Einstein-Grossmann (1913) theory, vyhereas by Octo-

Out of the Labyrinth? 41

ber 12 Einstein is writing to Lorentz that he now realizes something is

amiss with the theory.

The letter responds to another proposal by Hertz, as we shall see, it is

written in a very different tone. The letter is

impatient, discouraging

and almost hostile-Einstein not Hertz's proposal! On the basis of

Einstein's reply in EA 12-201, we reconstruct Hertz's proposal to amount

to an escape the hole argument, coupled with a proposal for setting up

generally covariant gravitational equations. The reconstruction

follows is the only one we found is compatible the entirety

of Einstein's response.

At this point, some readers might like to scan ahead read letter

EA 12-201,

is quoted in full Section 4, order to see the raw

which our reconstruction is based. Readers who like puzzles

might even want to try to

own reconstruction before reviewing

one we offer below in Section 3.

_1lJ'U'U'_'~ Escape

Hertz

to show Einstein he

not be troubled by the dif-

ferences between the two solutions considered in the hole argument. He

considered hole argument for case of a two-dimensional Gaussian

surface. We

now

element of such a surface in the

quadratic differential form ds 2 ~ gll(dx 1)2 + 2g12 dx1 dx 2 +-g22(dx2)2,

!where Hertz used older notation introduced by Gauss, wherein one
writes ds 2 = E du 2 + 2F du dv + G dv 2. In the case of variable curvature,

this geometry seems to. allow the defining of a special coordinate system

(u, v), whose curves are curves of constant curvature and of maximum

curvature

and are adapted to the geometry. We call such

systems

to avoid confusing them with Einstein's "adapted"

coordinate systems. Presumably such coordinates were proposed because

they would be defined in terms of invariant features of surface and be-

cause they might be proved to exist for spaces of both positive negative

curvature, unlike isometric coordinates.

the two solutions of \"IA4.l1,.ll..ll.A.J.lAA\".''-ll.

hqJe .argum{(nt

way

Section 1 above. He considered one solution coefficients

E, F, G in original coordinate system (u, v) and other

coefficients EX,

GX in second coordinate system (UX , VX ) so

that the \ and G are the same functions of the· variables u and v as the functions EX, , and GX are of the variables uX and vx •18 Moreover,

Hertz ensured that the coordinate system (u, v) ·is Hertz-adapted to the

42 Don Howard and John D. Norton

geometry represented by E, F, G, which entails the coordinate system (U X, VX ) is also Hertz-adapted to the geometry represented by EX,

FX ,

and

GX •

He then asked

the nature of

of the ge-

ometry revealed by the admissibility under general covariance of the two

solutions constructed the hole. To do so,' he asked the geometry

within the hole accprding to the two solutions at two points correspond

sense that the coordinates

first point in the first coordinate sys-

(u, v) are numerically equal to the coordinates of the second point in

the

second

coordinate

system

(U X

,

vX ).

To

the points, one must follow

two coordinate'curves corresponding to the coordinate values selected

pursue

they meet in, the hole. Since the two coordinate

systems are Hertz-adapted to superficially different geometries, the coordi-

nate curves diverge entering the hole, according to

system .' was' adapted. to first or second solution of

For coordinate system

to first solution, the curves would

meet at solution,

v) .• .For the coordinate system

second

curves would meet at the

p x (U X

,

vX ).

See

which is our

of

gives in letter

reproduced as Figure 5).

But

the differences between the two solutions

by

no geometrically significant differences.

writes in

EA 12-201, points selected by

have the same coordinate values in of the geometrically significant

systems so --"IU~"IIJIL.V~ \,.IVIl..l.llUJIl.JlJl"II.'V

=u and vX = v.

Moreover, geometries at point

corresponding solutions are

the same. For

G are coefficients assigned by

solution to

GX are coefficients assigned

second solution , then the geometries at two points are the same
so far as EX = E, F X = = and GX G.19

Perhaps

now have said that the two solutions are geo-

metrically the same· in every respect, for these identities would for

correspQridingpoints covering every point of solutions. We can

of each solution as· representing

geometric surface. The con-

structionshows how one of them can be mapped into the other by

that takes point Ptopoint px while preserving all geometric properties. In

modem language, the two are isomorphic.

Curve of constant u, uX

Out of the Labyrinth? 43
First coordinate system (u,v) in which first solution has coefficients E,F, and G

Second coordinate system (UX,V X ) in which second solution has coefficients EX, FX, and GX

Interpretation of figure in Einstein's letter (cf. Figure 5).

rephrase

using only direct

Ein-

stein gives of

Since the solutions

to the same surface

geometrically, we merely recall by the construction, this surface "is

developable [i.e., isomorphically mappable] into itself," a clumsy in-

telligible way of making the

usage of the term "developable"

as

isomorphically mappable was standard at

and was

even

to precisely the case Hertz treats using exactly same set of

equations.

Consider, for example, the discussion of two two-dimensional Gauss-

surfaces embedded in a three-dimensional space that is found in J0-

hannes Knoblauch's Grundlagender Differentialgeometrie. (Knoblauch

1913, pp. 121-124),

regarded as a standard text in Gottingen.2o If

44 Don Howard and John D. Norton

the two surfaces could be upon one another without deformation,

are said to be "developable onto one another." The two surfaces

property

admit two-dimensional coordinate systems (u, v) such

that at corresponding points on the two surfaces, where the coordinate val-

ues are the same, the coefficients E, F, and G of one surface have the same

values as the coefficients E1, F1, Gl of the second surface. Knoblauch

wrote this requirement the now-familiar equations:

E1 = E, Fl = F, G1 = G.

The escape from the hole argument sketched above is obviously very close

in strategy to. the escape Einstein himself would offer shortly as the

coincidence argument, but Einstein's immed1ate response to

posal was just alist ofprotests complaints. Einstein took

coordinates to be the same as the adapted coordinates Einstein hlnllselt

defined (see Section 1 above). The letter from Einstein began

protest Hertz misrepresented Einstein's adapted coordinate sys-

tems, since he failed to retain crucial property stressed Section 1

above, namely two

(Einstein-)adapted coordinate systems

could not come smoothly to agree on the boundary of some region of space-

time.

any case-whetheror-n0t the two coordinate systems were

adapted-they were supposed to have properties in general, could not

obtain. Einstein wrote:

Dear Herr·Hertz,

Berlin, Saturday

If I have understood you~ letter correctly, then you make a completely

erroneous representation of that which I call "adapted coordinate sys-

tems." How do you come to require that a pair. of coordinate systems
= [Figure 5 figure from Einstein's letter] should exist, such that for

ux ·= u

VX = v

one has also

(GX = G) 4Jx = 4J
and overand.abovethis they agree on the boundary of the region? I am rather convinced that (excepting perh.[aps] qIlite special fields)
this is never allowed to be possible. I have never posited the existence of systems ,equivalent in this sense~21

Out of the Labyrinth? 45

50 Diagram in Einstein to Hertz, "Berlin, Samstag" [1915] (EA 12-201).

We can only conjecture about how Einstein came to see the adapted

coordinates of Hertz's proposal as being same as adapted coordi-

nates

for his 1913 theory. Both would use the term

as an appropriate term for coordinate systems they

a way that responds to the geometry of the metric field, but it

to see the use of

alone would be sufficient to lead

that EA 12-203 Einstein had encour-

attempts to a "surface-theoretic interpretation" of

1I"'II1l"'o.iI'"~1l"'1l"'~rlI systems of coordinates of Einstein's theory. If EA 12-203

was

before EA 12-201, we could well imagine Einstein anticipat-

ing such a proposal

when he received EA 12-201. Or, even if

EA 12-201 did predate EA 12-203, Hertz himself might have thought his

adapted coordinates

serve as surface-theoretic interpretation of

- Einstein's adapted coordinates and offered them as such. Finally, a minor

factor that might well be

such circumstances: Einstein complains

later in the letter fie cannot read Hertz's handwriting on page five of

his letter. We

wonder, then, how clearly written the other pages

were.

Einstein's more general complaint about inadmissibility of the two

coordinate systems (UX , VX ) and (u, v) is readily explicable. All he need

assume is

coordinate systems with their components (EX, F X, GX)

(E, G) are coordinate systems components of the same field,

not

fields as is crucial to both the hole

and the

proposal Section 3 above.

this is already assumed in Einstein's

objection

two systems cannot both be adapted coordinate systems.)

As Einstein points only quite special fields can be transformed in

way. indicated. A coordinate transformation in general produces a

quite

set of components

to match in the

way. .IlJl.Jl.U~.Il""U·II-Vu.

46 Don Howard and John D. Norton

Einstein continued in what seems to be an attempt

·to worry

Hertz's proposal. Repointed out

defined special coordinate system

would become degenerate in the case of a space of constant curvature and

then mentioned the problem of extending the definition of these coordinate

systems to the four-dimensional case a way suggested some doubt

about its feasibility. IfEinstein did intend doubt here, he was shortly proven

wrong about the general program of finding four-dimensional coordinate

systems that the natural structure of a region of space-time, for less

two years later Kretschmann showed how a four-dimensional coordinate

system could be constructed in general relativity from curvature invariants

(Kretschmann 1917, pp. 592-599).22 The search for coordinates somehow

"adapted" to the intrinsic geometry of the space was, in any case,

acteristic of Qottingen approach to general' relativity, as retlec·tea

Hilbert's

of what he termed "Gaussian coordinates" , J1.JI.ILJ J1. ...

1916, 58-59),

are now commonly'designated geodesic ....,J1.J ....

coordinates.23 The passage

above continues

Independently of this, I understand how you establish a special coordinate system on a two-dimensional. manifold by curves of constant curvature and those of :maximal curvature gradient. What is problematic [verdiichtig] about this,· however, is that, in regions of constant curvature, the (surfaces) curves (or surfaces) of constant curvature are shifted in~nitelyfar away from one another. The difference, in principle, of the two coordinates that have be~n introduced is also problematic. You could, nevertheless, attempt to s~ewhether such a thing can be done in a four-dimensional manifold.

Hertz generally covariant field V'-lIIl.~"\\.-.Il.'-'.Il..Il.. or not Hertz agreed
again suggests the proposal, as '-'UIL-.Il...Il.J1.JlVlU.
argument. ~.Il.J..Il.U\\.-~~J1..Il..Il.

I have not understood the proposal for the setting-up of a gravitation law, because I cannot read your writing on page 5. After all, I.have said in my work that a usable gravitation law is not allowed to be generally covariant. Are you not in agreement with this consideration?

Einstein systems

objection about two CO()rdllna1te closed these words:

So once again: I would not

of requiring that the world should be

"developable onto itself," and I do not understand how you require such

a dreadful thing of me. In my sense, there is certainly a huge manifold

of adapted systems that do not, however, agree on the boundary.

Out of the Labyrinth? 47

With best regards to you, your wife, and your gentleman son, who is already surprisingly affable and fond of writing, I remain, riveted upon your further communications, yours
A. Einstein

Einstein had understood, in·effect, that Hertz required the transforma-

relcltlI1lg the two coordinate systems to be an isometry of the surface,

so that he could say

surface could be developed onto itself· by

the transformation. As

had pointed out, surfaces admitting such

isometries are exceptional in any case, the transformation could not be

between Einstein's adapted coordinate systems, since such systems would

never agree on the

of region

way He~z required.

Even though Einstein's

. .response· to .Hertz was so prickly and

defensive, he eventually came to appreciate and advocate Hertz's. central

a system is developable onto another, the two represent the same

reality. This· advocacy is nowhere more in evidence than in ·Einstein's

correspondence with Ehrenfest in late December and early January 1916.

Ehrenfest was reluctant to accept the generally covariant form of the theory

of gravitation announced by Einstein in November 1915,and he pressed

his reservations by reminding E~nstein, as had other correspondents, of th.e

earlier hole argument. More specifically, a letter that no longer exists

6. First diagram in Einstein to Ehrenfest, January 5, 1916 (EA 9-372).

48 Don Howard and John D. Norton

from late December 1915, Ehrenfest evidently asked Einstein to consider a

situation in which light from a distant star passes through one of Einstein's

notorious holes and then strikes a screen with a pinhole

directs the

light onto a photographic plate.24 Given generally covariant equations

allow for two different solutions, g~v and g~v, inside the hole, Ehrenfest

asks how we can be sure that light from the distant star following different

paths through the hole determined by the two different solutions can be

guaranteed to strike the same place on the plate.25

We quote the relevant section of Einstein's detailed answer in en-

tirety:

In the following way' you obtain all of the solutions that general covariance brings its train in the above special case. Trace the little figure above [see Figure 6] on completely deformable tracing paper. Then deform the tracing paper arbitrarily in the paper-plane. Then again make a copy on stationery. You obtain then, .e.g., the figure [Figure 7]. If you now refer the figure again to orthogonal stationery-coordinates, then the solution is mathematically a different one from before, naturally also with respect to the gJ-tv' But physically it is exactly the same, because even the stationery-coordinate system is only somethil1g imaginary [eingebildet]. The same points of the plate always receive light. ...
What is essential is this: As long as the drawing paper, i.e., "space," has no reality, the two figures do not differ at all. It is only a matter of '~coincidences,"e.g., whether or not the point on the plate is struck by light. . the difference between your solutions A B becomes a mere difference of repre,sentation,wtthphysical agreement. (EA 9-372)

70 Second diagram in Einstein to Ehrenfest, January 6,

(EA9-372).

Out of the Labyrinth? 49
Aside from the talk of "coincidences," Einstein's point here is exactly Hertz's, namely, that one can have two solutions are mathematically different, while being physically or geometrically (they come to same thing in this context) indistinguishable.

6

.. Escape

Argument

The reconstruction of Hertz wrote to Einstein as conjectured in Sec-

tion 3 above was based on an analysis of Einstein's letters. We then sought

some independent evidence for our conjecture, but the existing documenta-

tion provided none. (fhere is additional correspondence between Einstein

Hertz from early October 1915, concerning whether or not Hertz should

resign his membership some society seemingly concerned with political

matters. And something Einstein wrote this connection so irritated

Hertz that he threatened to break off the correspondence, an eventuality that Einstein earnestly sought to avoid.26 Further communication was no doubt

even more difficult by the fact Hertz soon found himself in the -posted to a flight school in Posen.27

If we could not confirm independently that Hertz suggested such an

escape- from the hole argument, then, we asked ourselves, could we at

determine whether or not such an escape was common knowledge

in Gottingen at the so that Hertz was either initiating or reflecting a

standard response? To our surprise and pleasure we found-after we-had

completed construction of the conjecture of Section 3-that Hilbert

offered almost exactly the escape in second of his· famous . papers

on general relativity

foundations of physics (Hilbert 1916).

The relevant remarks are found in Hilbert's somewhat labored discus-

sion of the "causality problem" in general relativity, the designation Ein-

stein often used for the hole argument (Hilbert 1916, pp.59-63).28 Hilbert

points out the Cauchy problem is not well posed for his own gen-

erally covariant version of general relativity (Hilbert 1915).

theory

has fourteen independent variables-the ten gravitational potentials, g/-tv,

and the four electromagnetic field potentials, qs -but the gravitational field

equations and Maxwell's equations provide only ten il).dependent field equa-

tions. Hilbert illustrates this underdetermination with a pair of solutions,

the first of which represents an electron at rest throughout time, with

the gravitational and electromagnetic fields everywhere time-independent.

In a manipulation reminiscent of the hole argument, the second solution

is obtained by a coordinate transformation that is the identity for the time

coordinate X4 :s; 0, comes to differ for X4 > 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
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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.

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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('T</ +'tu V),

obtained (4)

where

stress-energy tensor for the 4 to' v

tnl"1I"'1Jl"il.TlIi"'Jltll.f""l,nIJlD

is as, riloll,n.clril

r Pl-t~-2,°1<tT'V grafrtp .rlp-ta)/

(5)

~~o' beingthe Christoffel symbols. Differentiatingequation (4) with respect

to x v, Einstein obtained the conservation law for matter plus gravitational

field in the form

a

a.xv(~V +tu V) O.

(6)

It~ust be stressed, however,

1914, noticed ........,J1.JI..IJlU"".....JUI.J1.

Conservation Laws and Gravitational Waves 65

to' v does not transform as a tensor under arbitrary justified transforma-

tions, but only underlinear transformations; nevertheless, we will call

to' v the [stress-]energy tensorS of the gravitational field. Something anal-

ogous holds for the components r~p of the gravitational field strength.

(Einstein 1914, p. 1077)

.

In the spring of 1915, private correspondence with Einstein, Levi-Civita

sharply attacked Einstein's proofs of the covariance of certain. fundamental

quantities of his Entwurf theory (Cattani

1989b); however,

he did not explicitly criticize the pseudotensor character of ta v.

5)
1915, Lorentz published a paper (Lorentz 1915) in which he criticized both the Entwurf theory and the variational formulation Einstein had given to it in 1914. In the second part of his paper, Lorentz proposed a more general variational derivation of gravitational field equations. Lorentz did not specify the form of the Lagrangian; he just assumed it to be a function of the metric tensor and its first-order derivatives. Requiring thatthe action integral be stationary not only for arbitrary infinitesimal variations of the coordinates, as Einstein required, but also for arbitrary infinitesimal variations of the components of metric. tensor~ Lorentz obtained the gravitational field equations in form
aR* (7)
agj1V
V\There R* and M are the Lagrangians for the gravitational field and mat-
ter, respectively. Furthermore, Lorentz showed that equations (7) tum into the Entwurf field. equations when the function chosen by Einstein
is. substituted for R*. As is well known, Einstein himself later realized
that his· choice of a Lagrangian was, in fact, quite arbitrary (Cattani and De Maria 1989b).Unlike Levi-Civita, Lorentz at this point was unaware of the mathematical mistakes Einstein made in his early variational approach, and praised for "his ingenious mode of reasoning" (Lorentz 1915, p. 1089).

a paper, entitled "The Founda- p]reS~~ntf~C1 A A..II.................j..... he discussed a variational princi-
both Einstein (1914, 1915a, 1915b,

66 Carlo Cattani and Michelangelo De Maria

1915c) and Mie (1912), the former for his gravitational field equations,

the latter for his work on nonlinear electrodynamics and his electromag-

netic theory of matter.

restricted his investigation to

situation of an electromagnetic in the presence of a gravitational field.

Hilbert was critical of Einstein's 1914 variational approach as the fol-

lowing quotation·from his paper illustrates:

Einstein gave the fundamental original idea of general invariance a simple expression; however, for Einsteinthe Hamilton principle only plays a subordinate role and his function H is not. at all generally invariant Moreover, the electrical potentials are not included [in his theory]. (Hilbert 1915, I, po 396, footnote)

............'L....,.............. proceeded as follows. He assumed

the "1IUIULJUlll-Jil'II-Jil"-'Ul

acterizing the fields are the' ten gravitational potentials gj1v and the

electromagnetic po~entialsqj1. defined a

world

tion according

following axioms:

Axiom 1 (of Mie about the world function). The law of physical events
is determined through a world function [Lagrangian] 1-[ = A H that
contains fonowing arguments:

and specifically variation of the action integral must vanish for

[changes everyone of the 14 potentials g/-LV, qa 0

Axiom 2 (of general invariance). The

1{ is invari-

ant with respect to arbitrary transformations of the world parameters

[coordinates]

x lX •

I, p. 396)

two ....""".,..,....... functions, one u,"-'Jl,JlJlJl'-"u,

JL1-I.I1.I1.04-.... JL

gravitational field

and one for matter.

used the Riemann .f">1l111l"''\{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<leCl his energy equation the following argument:

Let us consider a spatially bounded material system, whose matter den-
sity and electromagnetic field vanish outside some region. Let S be the boundary surface, at rest, which encloses the entire material system. Then, by integration of the fourth equation over the domain inside S, we get

'£(14 - ~4

4

-f-

4 t4

)dV=

+ + cos(nXt) t42 COS(nx2) t43 COS(nx3») dO'.

Oneis notentitled todefine t44 a~_the energy density of the gravitational field and (t4 1, t42 , t43) as the cOlnllonents of the flux of gravitational energy. But one can certainly maintain, in cases where the integral of t44
is small compared to the integral of the matter energy density 744 , that
the right-hand side represents the material energy loss of the system. It
was only this result that was used in this paper and in my first article on
gravitational waves. (Einstein 1918b, pp. 166-167)

Einstein then considered Levi-Civita's main objection against choice of conservation laws:

Levi-Eivita (and prior to him, although less sharply, H.A. Lorentz) proposed a different formulation ... of the conservation laws. He (as wen as other specialists) is against emphasizing [equations (6)] and against the above interpretation because ta V is not a tensor. (Einstein 1918b, p.166)

A1though Einstein obviously to cluded:

t(J'V is not a tensor, con-

I have to agree 'with this last criticism, but I do not see. why only those
quantities· with the ·transformation properties of the components of a
, tensor should have a physical meaning. (Einstein 1918b, p. 167)

Conservation Laws and Gravitational Waves 81

Finally, Einstein stressed that, even though there is no "logical objection" (Einstein 1918b, p.·167) against Levi-(~ivita's proposal, it has to be dismissed on physical grounds.

I find, on the basis of [equation (20)], that the components of the total energy vanish everywhere. [Equation (20)] , (contrary to [equation (6)]), does not exclude the possibility that a material system disappears completely, leaving no trace of its existence. In fact, the total energy in [equation (20)] (but not in [equation (6)]) is zero from the beginning; the conservation of this value of the energy does not guarantee the persistence of the system in any form. (Einstein 1918b, p. 167)

fact, this result is due to the algebraic form of Levi-Civita's "conser-

(according to

the

stress-energy is equal to zero

everywhere). In Levi-Civita's opinion, the local vanishing of the matter

stress-energy does not allow any energy flux.. From a mathematical point

of view, Levi-Civita's

with a generally covariant gravitational

stress-energy tensor, was ,certainly more general than Einstein's, and ap-

parently more in line the spirit of general relativity. Einstein's choice,

on the other

was more convincing on the basis of physical arguments,

as Levi-Civita himself admitted,. At the time, Einstein stood alone in his de-

fense of a noncovariant definition of gravitational energy. Modern,general

relativists, hov/ever, follow Einstein's rather than Levi-Civita's approach to

conservation laws.

Lorentz l..,evi-Civita were not the only two scientists to criticize Einstein's definition of gravitational stress~energy. In November 1917, Erwin Schrodingershowed, a straightforward calculation, that, given a symmetrical distribution of matter, Einstein's gravitational stress-energy complex ta v can be~ero in a suitable c.oordinate system. Schrodinger evaluated the stress-energy complex, starting from the Schwarzschild metric for the case of an incompressible sphere of matter, and noticed
to determine ta v, we must always specify the co()~dinate system, since their values do not have tensor. character and do not vanish in every system, but only in some of them. The result we get in this particular case, i.e. the possibility of reducing ta v' to be identically zero, is so surprising that I think it will need a deeper analysis..... Our calculation shows that there are some real gravitational fields whose [stress-]energy components vanish; in these fields not only the momentum and'the energy flow but also the energy density and the analogs ofthe Maxwell

82 Carlo Cattani and Michelangelo De Maria

stresses can vanish, in some finite region, asa consequence of a suitable choice of the coordinate system. (Schrodinger 1918, p. 4)

Thus, Schrodinger concluded,

This result seems to have, in this case, some consequences for our ideas about the physical nature of the gravitational field. Since we have to renounce the interpretation of tu V •.•.• as the [stress-]energy components. of the gravitationalfield, the conservation law is lost,and it will be our duty to. somehovyr~place this esselltialpart in. the foundation [of the theorY].,(Schrodinger 1918, pp. 6-7)

Abouttwo andahalfmonths later (on February 5, 1918), Einstein replied to

Schrodinger in the same journal (Einstein 1918a). Oddly enough, Einstein

started by raising further doubts about his choice of the

to

represent gravitationalstress-energy:

Sllrt~Ss-· leIfH:~n~ v C~Jm'DOIlents of

T;, represent a tensor,

for the "[stress-]energy.components" of the

tU v ;
tci (2) .the qUantities ht.==' X:;~.'l7rv gvi aresYII,J1lle c in the
r, while this not true for tUT:. = X:;vtuVgv-c.

For the same reason as mentioned in point (1), Lorentz and Levi-Civita alsoraised doubts about interpreting ta a as the [stress-]energy components of the gravitational field. Even though I can share their doubts, I
am still convinced that it is helpful to give a more convenient expression
for energy components ofthe gravitational field. (Einstein .1918a,

Conservation Laws and Gravitational Waves 83
[t]hese considerations hold mutatis mutandis in all those cases where the field transmits exchange effects between different bodies. But this is not the case for the field considered by Schrodinger. (Einstein 1918a, p. 116)
concluded peremptorily:
Hence, the formal doubts (1) and (2) cannot lead to a rejection of my proposal for the expression of the energy-momentum. It does not seem justifiedto put any further formal demands [on the properties ofa quantity representing gravitational stress-energy]. (Einstein 1918a, p. 116)

one

after Einstein's reply to Schrodinger, Hans Bauer at-

tacked Einstein's choice of to'v (Bauer 1918). discussed an example

complementary to Schrodinger's. ···Schrodinger had shown that·Einstein's

gravitational stress-energy sometimes vanishes despite the presence of a

Bauer now s.howed that it does not always vanish in

absence of a gravitational

He stressed

the partial nonvanishing of the [stress-]energy components has nothing to do with the presence of a gravitational field, but it is due only to the choice of a coordinate system.... This behavior is not surprising, since
is not a tensor. (Bauer 1918, 165)

thrown another stone

physical plausibility

we have to conclude that the "[stress~]energy components" ta v are not related· to presence of a gravitational field as they depend only on the choice of coordinates. They can vanish in presence of a field, as shown . by Schrodinger, and do not always vanish in absence ofa field, as shQwn below. Hence, their physical. significance seems to be very dubious. (Bauer 1918, p. 165)
Einstein replied to Bauer's criticism without delay. In May 1918, published a new reply to Schrodingerand Bauer (Einstein 1918c). once again justified his choice physical arguments. In his opinion,
the. theory of general relativity has been accepted by.most theoretical
physicists and mathematicians, even though almost an colleagues stand
against my formulation of the energy--momentum law. Since I am convinced that lam right, I will in the following present my point of view on these. matters in more detail. (Einstein 1918c, p. 448)

84 Carlo Cattani and Michelangelo De Maria

Einstein reminded his readers how special

combines

conservation laws of energy·and momentum

one (l1tterenl1al i

V\\.IlIl..U,.\l.Il,.JLVJUl

(i.e., the vanishing of the four-divergence of the stress-energy tensor)

is equivalent to the integral form of these conservation laws

in

experience. The generalization ofthis conservation law to general relativity,

he explained, was particularly delicate. Einstein showed how, with his

choice, "the classical concepts of energy and momentum are established as

concisely as we are accustomed to expect classical mechanics" (Einstein

1918c, p. 449). Then he demonstrated the energy and momentum of a

closed system are uniquely determined only when the motion of the system

(considered as a whole)· is expressed "with respect to a given coordinate

system" (Einstein 1918c, pp. 449-450). In particular, he. showed the

stress-energy

closed systems can only be expected to 1t1l"'4Jl1l'''lIC''1t'r,,~

as a tensor

coordinate transformations, viz. those coordinate

transformations that reduce to

at infinity. The

transformationsl.lsedin Schrodinger and Bauer's examples do not

requirement, so they do not'count as counterexamples.

After this article b'yEinstein, the debate on correct

of

conservation. laws. in general relativity· apparently came

U.V~J)V.Il...Il.lU',""u. ...ll.JlJlV_JJ,.a~JLH~';.Il..l~Il..IlJ··Ifl.V' between

Levi-

conservation

general relativity

during the years 1917-1918. Prompted by a mistake

made his

first paper

waves, ·Levi-Civita

the use of non-

covariant

a generally covariant theory.

stimu-

lated Einstein to give a new correct description of gravitational waves.

Meanwhile,

there is no unique definition of the

stress-energy of the gravitational field in general relativity. Following up

on this .insight, Lorentz proposed to interpret field equations as con-

servation· laws. .Levi-Civita independently made same

in a

mathematically more satisfactory way, using the contracted HlI-:111"1\1"'1hl1

tities.

on to

fonnulation of. the ·conservation laws

involving the pseudotensor ta v to represent gravitational stress-energy.

$chrodinger

showed

certain cases, -Einstein's choice of

t(1 v led/to paradoxical results.

This episode makes for interesting case study history of general

relativity for at least two reasons:

clarifies the connections between

variational methods and conservation

general relativity

cross-fertilization; (2) it shows

of Einstein's ""' """

"'" 1'IC""..".Urtl1I"11"..".1n\

Conservation Laws Gravitational Waves 85

in his efforts to complete edifice of general

1916-1918.

Some of most celebrated mathematical physicists, as Lorentz

Levi-Civita, attacked his choice of a pseudotensor to

gravitational

stress-energy on the basis of formal mathematical arguments very

in

spirit of general relativity. Moreover, two young theoretical physicists,

Schrodinger and

came up some

damning counterex~

amples against Einstein's choice. Yet

exploiting the

equivalence principle as a heuristic tool, stubbornly

choice

and justified it

strong physical arguments.

today's UI\,.\l.4Il..!l._II..l\.Jl_U'Il

was right.

ACKNOWLEDGMENTS.

reading

version

useful suggestions and his thorough editing of

b..JI "1I.l~1\"I.!l..!l.~"'.I!. for his critical M. Janssen for many
article.

NOTES

1 See also Cattani's chapter "Levi-Civita's Influence on Palatini's Contribution

to General Relativity" in this volume.

2

his 1914 choice of BJ.L explicitly is

_
BJ.L -

82 8x v (Jx ot

((1 -g/) 2gotfJ gaJ.L

U~8gxJ.fLJV

)

•

3 For a more extensive discussion of these calculations, see Norton (1984). 4 Einstein defined the pseudotensor t~ as (Einstein 1914, p. 1077)

.!. (_ t v ~
a- X

v, g

agar:

a _ v, aH<_g)1/2)

got

aT:

'

got

in order to show explicitly its dependence on H.

5 In this period physicists meant stress-energy tensor when they said energytensor.
6 Einstein to Levi-Civita, August 2, 1917, Einstein Archive, Boston (EA 16-253). English translation by J. Goldstein and E.G. Straus with some modifications.
7 Let us examine Einstein's pendulum clock example a moreclosely. In the reference frame in which there is no gravitational field, the clock is not working since the counterweight that should drive it is not subjected to a gravitational field.

Let us take a concrete example. Suppose our clock is in a spacecraft far from any masses with its engines turned off (frame In this case, the clock is in a situation

of "absence of weight," and consequently cannot work. When the engines are on,the spacecraft accelerates (frame K'). Consequently, objects inside
the spacecraft experience an apparent gravitational field. Our clock will want to start working under the influence of this field. If, in K', we want to prevent this, the clock's gravitational energy be transformed into heat.
8 Here Einstein presumably alludes to the fact that in general relativity gravitational forces are expressed in terms of the Christoffel symbols, which contain first-order derivatives of the metric only.

86 Carlo Cattani and Michelangelo De Maria
9 Levi-Civita to Einstein, August 1917. Only a draft of this letter survives (Levi-Civita Papers, Accademia dei Lincei, Rome). It seems reasonable, though, to assume that the actual letter was not that different from the draft.
10 "Antwort auf einen von Hm. Levi-Civita herrtihrenden Einwand," Einstein 1918b,pp.166-167.

REFERENCES

Bauer, Hans (1918). "Uber die Energiekomponenten des Gravitationsfeldes." Physikalische Zeitschrift XIX: 163-166.

Cattani, Carlo and De· Maria, Michelangelo (1989a). "Gravitational Waves and Conservation Laws in General Relativity: A. Einstein and T. Levi-Civita, 1917 Correspondence." In Proceedings of the Fifth. Grossmann Meeting on General Relativity, D.G. Blair and MJ. Buckingham, eds. Singapore: World Scientific,pp. 1335-1342.

- - - (1989b).• "The 1915 Epistolary Controversy between A. Einstein and T. LeviCivita." Einstein and the History of General Relativity, D. Howard and J. Stachel,eds. Boston: Birkhauser, pp. 175-200.

Einstein, Albert (1914). "Die formale Grundhige der allgemeinen Relativitatstheorie." Koniglich Preussische Akademie der Wissenschaften (Berlin). Sitzungsberichte: 1030-1085.

"Zur allgemeinen RelativiUitstheorie." Koniglich [)reussische Akademieder Wissenschaften (Berlin). Sitzungsberichte: (I) November 4, 778786; (II) November 11, 799-801.

-_.- (1915b). "ErkUirung der Perihelbew~gung des

aus der allgemeinen

Relativitatstheorie." Koniglich Preussische Akademie der Wissenschaften

(Berlin). Sitzungsberichte: November 18, 831-839.

- _ . (1915c). "Feldgleichungen der Gravitation." KoniglichPreussische Akademie der Wissenschaften (Berlin). Sitzungsberichte: November 25, 844-847.

-\- - (1916a). "Naherungsweise Integration der Feldgleichungen der.Gravitation." •Koniglich Preussische Akademie der Wissenschaften (Berlin). Sitzungsberichte: 688-696.

- - (1916b). "Hamiltonsches Prinzip und allgemeine Relativitatstheorie." Koniglich Preussischen Akademie der Wissenschaften (Berlin). Sitzungsberichte: 1111-1116.

- - (1918a)."Notiz.zu E. Schrodingers Arbeit: Die Energiekomponenten des Gravitati.onsfeldes." Physikalische Zeitschrijt XIX: 115-116.

- - (1918b). "Uber Gravitationswellen." Koniglich Prelj,ssische Akademie der Wissenschaften (Berlin). Sitzungsberichte: 154.-167.

_._-- {1918c). "Der Energ·iesatz in der allgemeinen Relativitatstheorie." KonigUch Rreussische Akademie der Wissenschaften (Berlin). Sitzungsberichte: 448459.

Einstein,Albert and Grossmann, Marcel (1913). Entwurfeiner verallgemeinerten Relativitiitstheorie und einer Theorie der Gravitation. l. PhysikalischerTeil

Conservation Laws and Gravitational Waves 87

von AlbertEinstein. II. Mathematischer Teil von Marcel Grossmann. Leipzig and Berlin: B.G. Teubner. Reprinted, with added "Bemerkungen," in Zeitsehriftfiir Mathematik und Physik 62 (1914):225-261.

- - (1914). "Kovarianzeigenschaften der Feldgleichungen der auf die verallgemeinerte Relativitatstheorie gegrtindeten Gravitationstheorie." Zeitschrift flir Mathematik und Physik 63:.215-225.

David (1915). "Die Grundlagen der Physik." Konigliche Gesellschaft der Wissenschaften zu Gottingen, Mathematisch-physikalische Klasse, Nachriehten: (I) (1915): 395-407; (II) (1916): 53-76.

Janssen, Michel (1992). "H.A. Lorentz's Attempt to Give a Coordinate-Free Formulation of the General Theory of Relativity." In Studies in the History of General Relativity, Jean Eisenstaedt and AJ. Kox, eds., Boston: Birkhauser, pp. 344-363.

Levi-Civita, Tullio (1917). "Sulla espressione analitica spettante al tensore gravitazionale nella teoria di Einstein." Rendieonti Aecademia dei Lincei ser. 5, vol. XXVI: 381-391.

Lorentz, Hendrik Antoon (1915). "Het beginsel van Hamilton in Einstein's theorie der Zwaartekracht." Koninklijke Akademie van Wetenschappen te Amsterdam. Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige Afdeeling 23: 1073-1089; English translation: "On Hamilton's Principle in Einstein's Theory of Gravitation." Koninklijke Akademie van Wetenschappen te Amsterdam. Proceedings of the Section of Sciences 19: 751-767.

-~.=. (1916). "Over Einstein's theorie der Zwaartekracht."·Koninklijke Akademie van Wetensehappen te Amsterdam. Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige Afdeeling (I) 24, (1916): 1389-1402; (II) 24, (1916): 1759-1774; 25, (1916): 468-486; (IV) 25, (1917): 1380-1396.

English translation: "On Einstein's Theory of Gravitation," in Lorentz, Collected Papers. Vol. 5. P. Zeeman and A.D. Fokker, eds. The Hague: Martinus Nijhoff, 1937, pp. 246-313.

Mehra, Jagdish (1974). Einstein, Hilbert and the Theory ofGravitation. Dordrecht: D.Reidei.

Mie, Gustav (1912). "Grundlagen einer Theorie der Materie."Annalen der Physik (I) 37, (1912): 511-534; (II) 39, (1912): 1-40; 40, (1913): 1-66.

Norton, John (1984). "How Einstein Found His Field Equations: 1912-15." HistoricalStudies in the Physical Sciences, 14: 253-316. Also printed in Einstein and the History ofGeneral Relativity, D. Howard and J. Stachel, eds. Boston: Birkhauser, 1989, pp. 101-160.

Schrodinger, Erwin (1918). "Die Energiekomponenten des Gravitationsfeldes."

Physikalische Zeitschrift XIX: 4-7.

.

physicist, wrote bUlsn31n

field equations of general theory of

masses at rest. lengthy correspondence

more.acrimonious and

over

ofthis controversy,

both by Einstein flJJ1.'U"VJ1._J1.J1..Il.'l

The two~body problem is

generaltheory. 19ave a talk

history of this ~J1.'U'IIJJ1._.llJl.J1.

1985 conference, of ",hich a slightly extended version is being l!J'I..IlIl..'.llJl.IJIJ1l.llV'lo.ll-

to the Proceedings (Havas 1989). To understand the problem

consider-

ation and put it in its properhistoricaI perspective, it will be necessary,

however, to repeat some 9f the earlier discussion as well as. to elaborate on

part ofit and to provide some technical details.

In his initi~l fonnulation of thegeneral theory, Einstein had assumed

that=-just as inNewtonian mechanics=-th~lawsofmotion are independent

of the force .laws or field equations responsible for the interactions between

bodies, and

postulated

single mass point would move along

a geodesic of the metric. gjtv describing the field. ,For a single body at

rest, this assumption poses no difficulties, and exact solution for such a

body, obtained very early on (Schwarzschild 1916; Droste 1916a), remains

untouched by the subsequent investigations of the

of motion.