The definition of rigidity in the special theory of relativity and the genesis of the general theory of relativity

The Definition of Rigidity in the Special Theory of Relativity and the Genesis of the General Theory of Relativity Giulio Maltese* and Lucia 0rlando”f 1. Introduction The general theory of relativity (GR), probably Einstein’s greatest achievement in theoretical physics, was constructed during the years 1907-1915-a fecund and intensive period which began 2 years after Einstein’s statement of the special theory of relativity (SR). While the latter proved from the outset a thoroughly complete theory, the path toward GR was strewn with difficulties, certainly not free from errors and sometimes requiring revision of the validity of principles initially deemed generally true. The mathematics needed to set out the new theory was thoroughly new and substantially more complex than that involved in SR; Einstein tackled the major difficulties of the task during the period 1912-1915, which culminated in the final version of his gravitational theory. The path from SR to GR is well known, at least in outline. In 1907 Einstein wrote a paper reviewing the state of the theory of relativity; the paper was published in the Jahrbuch der Radioaktivitiit und Elektronik’ and included the very first statement concerning the equivalence between an accelerated frame of reference and a frame in a gravitational field. This paper marks the beginning of a long, hard path from the particular to the general case. In 1911 Einstein was forced to abandon one of the cornerstones of SR, i.e. the principle of the constancy of the velocity of light, to construct a broader theory of relativity able to describe the gravitational field.2 The outlined approach was developed in two papers, written at the beginning of 1912, the content of which is known *Group for the History of Physics of the National Research Council, sections of Bologna and Rome. Seminar for the History of Science, University ‘La Sapienza’, Rome, Italy. tDepartment of Physics and Seminar for the History of Science, University ‘La Sapienza’, Rome, Italy. Received 15 February 1995; injinalform

24 July 1995.

‘Einstein (1907~). ‘Einstein (191 lb). Pergamon

Stud. Hist. Phil. Mod. Phys., Vol. 26, No. 3, pp. 263-306, 1995

Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 1355-2198/95 %9.50+0.00 263

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as ‘the scalar theory of the gravitational field’.3 Einstein’s goal was initially to construct a theory where the speed of light varied with respect to position in a gravitational field and could actually be regarded as the gravitational potential. However, he soon had to recognize that he was up against a big snag: the equations of his theory were no longer linear since, according to the hypotheses, the energy of a gravitational field does in fact have inertia and therefore acts as a source of the field itself. This implied that the principle of equivalence held only locally and that, to generalize the theory, it was necessary to abandon one of its cornerstones. In the late spring of 1912, Einstein realized he had to replace the scalar theory of gravitation with a tensorial theory, where the gravitational field was to be described through quantities that behaved like the components of a tensor. In August 1912, he entered into fruitful collaboration with Marcel Grossmann, who helped him to develop the mathematical part of the new theory, later known as the Entwurf theory.4 However, contrary to Einstein’s initial expectations, the field equations he employed in the first version of his theory were not generally covariant. Even though Einstein initially thought he had discovered the proof that general covariance could not be attained, he later realized that his argument against general covariance was fallacious5 and eventually, in November 19 15, succeeded in setting out the correct equations of the gravitational field.6 Reconstructing Einstein’s path from SR to GR is one of the most attractive and complex tasks in the history of physics; in the past it has been tackled from different points of view and with different methodologies. One of the problems that have teased historians and prompted several papers, is to ascertain what made Einstein realize in 1912 that he needed a tensorial theory to describe the gravitational field.7 We well know how difficult it is to reconstruct how Einstein 3Einstein (1912a,b). ?See Einstein and Grossmann (19 13). ‘See Norton (1984), pp. 298-303. Cattani and De Maria (1989) showed how Levi-Civita’s criticisms played a role in Einstein’s growing dissatisfaction with his Entwurftheory during the year 1915. %ee Einstein (19 I 5). ‘See Earman and Glymour (1978), Lanczos (1972), Norton (1984), Pais (1982), Stachel (1979, 1980), Vizgin and Smorodinskii (1979). Another important point about the genesis of GR concerns the reason why Einstein abandoned the right choice he had initially made when he rejected the Ricci tensor in the equations of the gravitational field and ended up by constructing a theory that was not generally covariant. Interpretation of the limited covariance of the equations is still a moot point among historians of physics. Some believe that Einstein’s difficulties were of an essentially mathematical nature, since, as they argue, he was not familiar enough with the problems and methods of tensorial analysis [see Vizgin and Smorodinskii (1979), Earman and Glymour (1978) Hoffmann (1982) Pais (1982) pp. 2162231. By contrast, others stress their physical nature, originating from the difficulty of getting rid of the influence of the old theory while constantly striving to generalize it. According to this point of view, Einstein failed to accomplish a ‘break’ with tradition as sharp as was required and chose an erroneous Newtonian limit for the new theory, i.e. he failed to select the correct limiting case to which the new and broader theory was to be reduced [see Norton (1984), Stachel(1986), Maltese (1991)]. According to this account, Einstein’s theory of gravitation during the years 1907-1912 is the result of a continuous endeavour to generalize the footnote 7 continued on p. 265

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got to the idea of a tensorial theory to describe the gravitational field. We have no conclusive evidence to shed light upon the crucial point in Einstein’s path, which he arrived at in the summer of 1912. Accepted historical reconstructions have had to ground their explanatory hypotheses on the scant correspondence available for that period* and on much later reminiscences.Q We know that Einstein solved the impasse represented by the failure of his scalar theory of the gravitational field by assuming, correctly, that the new gravitational theory had to be grounded on a tensorial formalism. Thus, the gravitational potential would be represented by 10 quantities-the components of a symmetric tensor of the second rank. However, we can only conjecture on what determined the transition to the tensorial theory of gravitation. Stachel suggested that analysis of the stationary gravitational field, intended as a completion of the static case, might have put Einstein on the right track;‘0 according to this hypothesis Einstein realized that in a rotating disk-an obvious example of an accelerated frame of reference equivalent to a stationary gravitational field-Euclidean geometry did not hold any longer, and it is very likely that this helped him to recollect Gauss’ theory of surfaces, where Euclidean geometry holds locally, as was the case for the principle of equivalence. Rather than pursuing the certitude of what was the actual cause, given this state of affairs it seems more feasible and probably more significant for the history of science to contribute to a more detailed understanding of the context in which one or more points set Einstein on the right track to create GR. We feel that in this matter there is an issue which has not yet been examined with due attention: the meaning that terms like ‘rigid body’ and ‘rigid motion’ have within SR and how such meanings-and the resulting definitions-might have influenced the creation of GR. In particular, these points may well have influenced Einstein while searching for accelerated motions equivalent to given gravitational fields. footnote

7 continued from p. 264

existing theory. The path from the particular to the general case clearly compelled Einstein to tackle the problem of whether the generalized theory did indeed cover the special case he had started from. Manv of the difficulties faced bv Einstein in building GR can be explained bv taking into account all the generalizations of the previous theory, where the process of recovering the initial particular case was actually ruled out by the very way the theory had been developed. Thus, rejection of the Ricci tensor might well have been caused by an erroneous choice of the metric. The one factor that led Einstein to reject the Ricci tensor was therefore his misunderstanding of the proper metric to which the gravitational field described by his new theory was to reduce in the Newtonian limit. It has been suggested that Einstein’s method reveals the systematic usage of a guideline, namely the generalized principle of correspondence, according to which the new and more general theory must include as a limiting case the theory that is to be replaced [see D’Agostino and Orlando (1992, 1994)]. 8See Klein et al. (1993), Pais (1982), pp. 2lC216 and Stachel (1980). ‘See, for example, the passage of the so-called ‘Gibson Lecture’ (which Einstein held in Glasgow on 20 June 1933) quoted in Viigin and Smorodinskii (1979), p. 495. See also Schilpp (1949) pp. 63-71, and Einstein (1956a), pp. 285-290. “Stachel (1980).

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The problem of the definition of the rigidity of a body immediately appears in SR, as a direct consequence of the fact that signals propagate with finite velocity. As we shall see, this problem came up in 1907, even before Einstein formulated the principle of equivalence in the context of his research on the ‘completeness’ of the system including the postulates of SR and the consequences of the hypothesis of equivalence between mass and energy. From the outset, Einstein’s concern in respect to the problem of the rigid body went beyond the scope of SR and proved a sort of lens to bring into focus some thorny questions regarding the nascent gravitational theory. The first of the two papers we refer to had been inspired by a note, published by Ehrenfest, concerning the problem of the rigidity of the electron: this issue was of great interest at that time in the context of attempts to demonstrate the electromagnetic nature of the mass of the electron.” Again in the context of the theory of the electron and, in particular, of the efforts to evaluate the expression of its electromagnetic mass, Max Born published in 1909 a long paper, in the Ann&n der Physik, concerning the definition of rigidity and the theory of the electron in the kinematics of the principle of relativity or, as we say today, in SR.i2 Born presented his work at the annual meeting of the German Society of Scientists and Physicians, which took place in Salzburg on 21-25 September 1909. Einstein, too, attended this meeting and, as Born himself later recalled, they discussed a puzzling implication of Born’s definition of rigidity, according to which it was impossible to set into rotation a rigid body at rest. l3 In the following years, Einstein often demonstrated that he continued to bear in mind Born’s definition of rigid body and the importance of this issue for the development of GR. We have evidence that Einstein was considering this problem in the late spring of 1912, a period of intense meditation that would eventually generate the idea of the tensorial theory of gravitation. This induced Pais to wonder, in his well-known biography of Einstein, whether it might be argued that the formalism employed by Born, which ‘manifestly has Riemannian traits’ according to Pais, gave Einstein the inspiration for general covariance. This paper purports to articulate the hypothesis foreshadowed by Pais by investigating, on the one hand, the reason that made Born suggest his definition of rigidity on the ground of the principle of relativity and, on the other hand, the influence this reason held over Einstein on his path towards GR. More precisely, we intend to investigate whether it is possible to speak of an influence exerted by Born’s definition of rigidity on the genesis of GR and what this influence consisted in. In the first place, it must be said that no documentary “Einstein (1907a,b), Ehrenfest (1907). “Born (1909a). 13Bom (1910), p. 233. 14Pais (1982), p. 214.

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evidence exists on this issue but that there are some clues which make it conceivable that something more than a merely formal link exists between Born’s theoretical work and the genesis of GR. For example, we know for certain that Einstein had in mind the kind of motion defined by Born in the period that was to be crucial for the genesis of GR. We also know that, in the same period, Paul Ehrenfest was interested in Einstein’s efforts, and was absorbed in the most difficult problem faced by Einstein in the scalar theory of the gravitational field: the reason for the non-validity of the principle of equivalence in finite domains. Ehrenfest succeeded in proving that the class of relativistic motion introduced by Born, the so-called hyperbolic motion, represented the only kind of motion compatible with the principle of equivalence. Finally, we know that Ehrenfest informed Einstein of his attempts.15 We feel that, necessary as it is to pursue reconstruction of what actually happened relying on the available material, it is still insufficient for the task we have taken on. As we have seen, the available documents cannot remove all uncertainties. On the other hand, plain historical reconstruction would not allow us to grasp all the analogies existing at various levels between the two theories-no merely formal analogies, but related to their ultimate essence. Our interpretative approach rests on the assumption that Born’s work played a sort of maieutic role, pointing out to Einstein possible lines of development for his gravitational theory. This role would have been evident, for example, in defining the relationship between the ‘particular’ and the ‘general’ case, since it showed as necessary that, while building up the new gravitational theory, properties that were generally valid in the old theory had to hold locally in the new. Moreover, Born’s ideas appear to have led Einstein to complete reconsideration of the connection between physics and geometry. This thesis is supported by the following observation: GR was not only a new theory. but also represented a new kind of theory,‘6 since it was a metric theory of the gravitational field, i.e. a theory where metrics, considered a premise of the physical theory until then (we can speak of laws if we already know how to measure quantities), becomes contextual with the phenomena that the laws should explain and, in the case under examination, is deeply related to the structure of the gravitational field. It now seems quite likely and, according to Einstein’s own words,*7 even necessary that, in the process of founding a metric theory of the gravitational field, a fundamental role was played by the notion

“See Klein rt al. (1993), Dots 393, 394, 409, 411. See also the letter Ehrenfest wrote to Sommerfeld on 23 June 1912 (Archive for the History of Quantum Physics, Accademia Nazionale delle Scienze detta dei XL (Rome), Microfilm 30. Sect. 7) and Ehrenfest (1913). 16D’Agostino and Orlando (1992), p. 199 17Einstein (1914). pp. 1079-1080.

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of rigidity, i.e. by that sort of bridge by means of which ‘geometry becomes a physical theory’.‘*

2. The Idea of Rigidity in the 1907 Theory of Relativity Between the end of the 19th and the beginning of the 20th century the achievements of the electromagnetic theory, mostly developed by Lorentz, induced Wien to propose that research should proceed toward an electromagnetic world picture in which the laws of mechanics would be deduced from the laws describing the electromagnetic field. ‘9 In particular, the lack of results in experiments aiming at revealing the motion of the earth relative to the ether had been ascribed to interactions between the ether and the ‘ions’ (the ultimate components of matter bearing an electric charge) in Lorentz’s electromagnetic theory of 1895. It was therefore necessary to build up a theory of such particles that could explain their mechanical properties and, in particular, their mass starting from purely electromagnetic assumptions. Such a need gave rise to the ‘theories of the electron’ of Abraham and Lorentz, published in 1903 and in 1904 respectively. According to the former, the electron was a small rigid charged sphere, whose rigidity was, however, conceived as a kinematic constraint without considering which forces had to be present to hold the electron together. In the latter theory, on the contrary, the electron was taken to be deformable and subject to Lorentz-Fitzgerald contraction. The hypothesis that the electron was deformable was indeed hazardous, since it was necessary to postulate the existence of internal forces preventing the electron from exploding under the electrostatic repulsive force; obviously it was not possible to postulate that such forces were electromagnetic without violating the foundations of the theory itself. However, Lorentz’s theory was the first to provide an explanation for the second-order experiment performed by Michelson and Morley, as also for the more recent experiment (1903) by Trouton and Noble and for Lord Rayleigh’s research (1902) on the possible double refraction exhibited by an isotropic substance at rest on the moving earth owing to the strain originating in the contraction.20 “Einstein (1914), p. 1079. “See Miller (1991) Chap. 1, and Miller (1977). Lorentz’s theory had succeeded in explaining the results of all the experiments accurate to first order in v/c (where v is the speed of the earth with respect to the ether and c is the speed of light) aiming at evidencing the motion of the earth with respect to the ether. The only weakness in the theory had been the introduction of the ad hoc hypothesis of contraction (which was in fact named after Lorentz and Fitzgerald) to account for the negative result of the experiment performed by Michelson and Morley in 1887, accurate to second order in vlc. “Another theory grounded on the assumption of a deformable electron was put forward by Bucherer and Langevin. In its state of rest the electron was assumed to be spherical, while it became ellipsoidal when moving. Unlike Lorentz’s theory, according to Bucherer and Langevin’s scheme the longitudinal and transversal factors of contraction were different, since the authors also assumed that the volume of the electron had to be the same both in a state of rest and in motion.

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In 1905 Einstein published his celebrated paper on the electrodynamics of moving bodies.21 Here all the experimental results that had baffled previous theories fitted in perfectly with Einstein’s scheme, and those hypotheses that had previously been added ad hoc (such as the contraction) appeared as necessary corollaries of the postulates of the theory. In 1907, Ehrenfest published a short paper in the Annalen der Physik on the following problem: in his 1903 theory, Abraham had shown that a rigid ellipsoidal uniform distribution of charge could move in stable force-free translational motion only in the direction of its major axis. Ehrenfest wondered what would happen if, in the problem described by Abraham, the rigid electron was replaced with a deformable one: this problem would be a test for Einstein’s electrodynamical theory. If the deformable electron was unable to perform force-free motion even in the direction parallel to its major axis, then the theory of relativity would be facing a crisis, since it would be possible to define a state of absolute rest; a new hypothesis would be required to deny the existence of such an electron and save the theory. But if the electron was actually able to move in force-free motion, this result would have been reached ‘in a purely deductive way’*2 from the electrodynamical theory according to Einstein’s formulation, since this was, according to Ehrenfest, a ‘closed system’.23 In reply Einstein pointed out that in his opinion the postulates of SR did not represent a closed system, and were not even to be regarded as a ‘system’, but ‘rather only as a heuristic principle, which, considered by itself alone, contains assertions about rigid bodies, clocks and light signals’.*4 Einstein went on asserting that the problem of determining the laws of motion of electrons considered as extended particles was to be tackled by applying an electrodynamical approach, guided by the principle of relativity, This in turn required specification of the distribution of the charge on the surface of the electron: Einstein proposed that the charge be distributed uniformly over a rigid frame with appropriate forces maintaining stability in the frame. This assumption demanded a treatment of the rigid body in terms of the principle of relativity: If we regard the system as a rigid body [...I then the problem of the motion of the electron can be solved in a deductive manner without ambiguity if and only if the dynamics of a rigid body is adequately known. While the theory of relativity is true, we are still far from the latter goal. We have, first of all, a kinematics of parallel translation and a formula for the kinetic energy of a body in parallel translation that is not interacting

with other bodies; both the dynamics

and the kinematics

of the rigid

“Einstein (1905). “Ehrenfest (1907), p. 204. 231&d. 24‘[,,,] sondern

lediglich als ein heuristisches Prinzip. welches fiir sich allein betrachtet Aussagen iiber starre Kiirper, Uhren und Lichtsignale enthllt’, Einstein (1907a). p. 206.

nur

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bodies required by the problem under examination are to be considered still unknown.25 This first explicit reference to the problem of the motion of a rigid body did not remain in isolation, since it was followed by Einstein’s remarks in another paper received by the Annalen der Physik less than a month after the paper quoted above,26 in which Einstein planned to investigate the consequences of mass-energy equivalence and, in particular, to inquire into possible special cases in which such a principle leads to incompatible consequences. Here Einstein was extending a work he had undertaken the previous year, which demonstrated (at least up to first order in v/c) that the equivalence between mass and energy did not contradict the principle of constancy of motion of the centre of mass.27 His standpoint was clear from the outset: no general reply to such a question was possible, ‘since we do not yet possess a world-picture according to the principle of relativity’.z8 Thus attention had to be confined to particular cases, and Einstein decided to investigate two such cases, one of which concerned the motion of an electrically charged rigid body. He succeeded in demonstrating that such a body has an inertial mass which exceeds that of the uncharged body by the electrostatic energy divided by c2. However, this result did not imply that the dynamics of the parallel motion of a rigid body was available in the framework of the principle of relativity. To make this clear, in the paragraph entitled ‘Remarks concerning the dynamics of rigid bodies’, Einstein considered the example of a rigid rod of length 1(as measured in its rest frame k) moving along the x axis of a reference frame K with speed v. Let two equal and opposite impulses act simultaneously (to an observer in reference frame k) on the two ends of the rod: the rod is clearly in equilibrium. Relative to K, however, the impulses are not simultaneous, being separated by a lapse of time At = yZvlc*, where y = l/4(1 - v2/c2). D uring this time the impulses do not compensate each other and the first one to act should make the energy of the rod increase during the time At. However, to the observer in reference frame K the rod’s velocity remains unchanged: this means a violation of the principle of conservation of energy. Einstein conjectured that a straining force propagating ‘5‘Wennwir das Geriist als einen starren [...I Kiirper ansehen, so kann das Problem der Bewegung des Elektrons dann und nur dann auf deduktivem Wege ohne Willkiir gellist werden, wenn die Dynamik des starren Kiirpers hinreichend genau bekannt ist. Falls die Relativitltstheorie zutrifft, sind wir von letzterem Ziele noch weit entfernt. Wir besitzen erst eine Kinematik der Paralleltranslation und einen Ausdruck fiir die kinetische Energie eines in Paralleltranslation begriffenen Korpers, falls letzterer mit anderen Kiirpern nicht in Wechselwirkung steht; im tibrigen ist sowohl die Dynamik als such die Kinematik des starren Kiirpers fur den vorliegenden Fall noch als unbekannt zu betrachten’, Einstein (1907a), pp. 207-208. 26Einstein (1907b). The paper was received on 14 May 1907; the preceding paper had been received on 16 April. *‘Einstein (1906). See also Miller (1991), pp. 236237 and 354357, and Pais (1982) Chap. 7, Sect. 7d. *s‘[...] weil wir ein vollstlndiges, dem Relativitlsprinzip entsprechendes Weltbild einstweilen nicht besitzen’, Einstein (1907b), pp. 371-372. Translation as in Miller (1991), p. 354.

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with an infinite velocity instantaneously arose within the rod, thereby compensating for the force at the other end of the rod. However, immediately after this he added: Such an assumption does not agree with the principle of relativity, as I will demonstrate later. We are clearly compelled in our case [...I to suppose that the body undergoes a state modification of an unknown property which propagates with a finite velocity and in a short time accelerates the body until another force acts on the body and restores the equilibrium. Thus, if the relativistic electrodynamics is correct, we are still far from possessing a dynamics of parallel translation of rigid bodiesz9

This remark was followed by an argument against the hypothesis of a velocity higher than the speed of light, which would ultimately have overthrown the concept of causality. The argument was later resumed in the review paper that Einstein wrote in the same year for the Jahrbuch der Radioaktivittit und Elektronik;so in this paper, too, Einstein showed he was aware of the inadequacy of the available

theory with respect to the rigid body concept.

This may

be inferred from the ‘kinematical part’ of the paper when the need is stated for a Cartesian frame of reference, i.e. a set of three mutually perpendicular and rigidly

connected

rigid

rods

together

with

a rigid

unit

measuring

rod,

to

determine the spatial labelling of a process. At this point Einstein remarks: ‘instead of “rigid” bodies one could just as well speak here and in the sequel of solid bodies account

free of deforming

the hypothesis

forces’.31 Later,

of extending

when Einstein

the principle

of relativity

first takes into to accelerated

frames of reference, the issue is presented differently. The problem is now to evaluate the possible influence of acceleration on the shape of a body; the answer,

however,

whose

individual

velocity

relative

S. Einstein

is very much like the previous material

points

to the nonaccelerated

possess

one. Einstein

a certain

reference

considers

acceleration

a body

y but

not

frame S at a fixed instant

t of

wonders:

What influence does this acceleration y have on the shape of the body with respect to S? If such an influence exists, it will consist in a dilatation of constant ratio in the direction of the acceleration, and possibly in the two directions perpendicular to this direction. [...I Those dilatations [...I must be even functions of y; and they can be thus disregarded when one restricts oneself to the case when y is so small that terms of the *9‘Eine derartige Annahme ist, wie nachher gezeigt wird, mit dem Relativitatsprinzip nicht vereinbar. Wir sind also in unserem Falle offenbar genotigt, [...I eine Zustandslnderung unbekannter Qualitat im K&per anzunehmen, welche sich mit endlicher Geschwindigkeit in demselben ausbreitet und in kerzer Zeit eine Beschleunigung des Kijrpers bewirkt, falls innerhalb dieser Zeit nicht noch andere Krlfte auf den K&per wirken, deren Wirkungen die der erstgenannten kompensieren. Wenn also die Relativitatselektrodynamik richtig ist, sind wir noch weit davon entfemt, eine Dynamik der Paralleltranslation des starren Kijrpers zu besitzen’, Einstein (1907b), p. 381. “‘Einstein (1907c), p. 423. 3“Statt von ‘starren’ Kiirpem, kijnnte hier sowie im folgenden ebenso gut von deformierenden Kraften nicht unterworfenen festen Kijrpern gesprochen werden’, Einstein (1907c), p. 415. Translation as in Schwartz (1977), p. 517.

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second and higher powers in y may be neglected. Since we wish to confine ourselves in the sequel to this case, we do not have therefore to assume any influence of the acceleration on the shape of a body.32

The paper published in the Jahrbuch represented the first step on the path that eventually led Einstein to GR, through the statement of the principle of equivalence. We may infer from this point that Einstein took the path unaware of what the definition of a rigid body was to be in the framework of the theory developed so far. On the other hand, he was perfectly aware of the major role played by such a concept, the absence of which he himself believed significantly restricted the scope of the theoretical scheme of SR, which did not include the dynamics of extended bodies except for a limited number of special cases. Between the years 1908 and 1911, Einstein’s interest shifted to the quantum theory, leaving relativity aside; however, in those years this theory was being enriched by some important contributions, like the Minkowskian space-time formulation of SR and the definition of rigidity put forward by Max Born in the framework of such a formulation. We will now turn to Born’s work. 3. Born and the Definition of Rigidity ‘in the Kinematics of the Principle of Relativity’33 The paper by Max Born, where the definition of rigidity was first expressed in the theory of relativity, was his Habilitationsschrzji at the University of Gottingen, where he graduated in 1906, and was developed between the winter of 1908 and the spring of 1909.J4 Born’s interest in the theory of relativity can be traced back to the beginning of 1908, when he was still at the University of Breslau. Since he had encountered some difficulties in this new theme of research, he had turned to Minkowski for advice. Clearly impressed by Born’s work, the famous mathematician replied inviting him to become his assistant and asked him to attend the annual meeting of the German Society of Scientists 32‘Was fur einen Einflu5 hat diese Beschleunigung y auf die Gestalt des Korpers in bezug auf S? Falls ein derartiger Einflu5 vorhanden ist, wird er in einer Dilatation nach konstantem Verhlltnis in der Beschleunigungsrichtung sowie eventuell in den beiden dazu senkrechten Richtungen bestehen. [...I Jene [...I Dilatationen miissen [...I gerade Funktionen von y sein; sie kiinnen also vemachllssigt werden, wenn man sich auf den Fall beschrlnkt, da5 y so klein ist, da5 Glieder zweiten und hdheren Grades in y vemachliissigt werden diirfen. Da wir uns im folgenden auf diesen Fall beschranken wollen, haben wir also einen EinfluB der Beschleunigung auf die Gestalt eines Kiirpers nicht anzunehmen’, Einstein (1907c), pp. 454455. Translation as in Schwartz (1977) p. 899. 33From the titles of Born’s papers. See Born (1909a,b, 1910). ?See Born (1978), pp. 132-138, for an autobiographical recollection of the context where the research matured. From a letter written by Einstein on 20 December 1907 we know that Einstein sent three copies of a paper, which most likely regarded SR, to R. Ladenburg, then Privatassistent in physics at the University of Breslau. One of the two other addressees-not explicitly mentioned by Einstein-was probably Born, who later wrote that Ladenburg was so enthusiastic about relativity that he visited Einstein (probably in the summer of 1909) to persuade him to attend the Eighty-First Meeting of the German Society of Scientists and Physicians (see below). See Klein et al. (1993), Dot. 68.

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and Physicians, which was to be held in Cologne, where, on 21 September 1908, Minkowski presented his famous paper ‘Raum und ZeiF proposing his spacetime formulation of SR to express the results of the still young theory of relativity. Interest in such a topic thoroughly overcame Born’s residual doubts, and he decided to move to Gottingen as an assistant of Minkowski. The untimely death of his master (12 January 1909) did not divert Born from the path he had undertaken and from the subject of his research: evaluation of the self-energy of the electron. As we recalled in the previous section, one of the hypotheses under investigation at that time was the electromagnetic nature of the mass of the electron. According to Abraham’s theory of the electron, in the formula for the mass of a rigid electron there had to be some dependence of mass on velocity. Lorentz had derived an analogous formula for a deformable electron which undergoes contraction in the direction of motion: he had found, too, that the mass should depend on velocity, but his formula was simpler than Abraham’s36 This, then, was the state of knowledge when Born set about to begin his research. He did not accept as correct the procedure adopted by both Abraham and Lorentz, which consisted in calculating the self-energy of an electrically charged rigid body and in later using this expression as a Hamiltonian to derive the equations of motion. This technique implied that the energy at a constant velocity had to be the same as in an accelerated motion. Since this was just the point Born was sceptical about, he decided to start his work writing the equations of motion for an electron based on the principle of relativity; in this task he had to face the problem of finding a definition for rigidity within the framework of Einsteinian theory. He obviously had read Einstein’s 1907 paper published in the Jahrbuch and was in particular aware of Einstein’s attempts to clarify the behaviour of rods and clocks in accelerated frames of reference. In fact, Born pointed out that the classical definition of a rigid body was still viable in SR, since in many papers on this subject the rigid body concept had been employed without giving it a new definition. The difficulties arose when accelerations came under focus. According to Born ‘in this direction there is only an attempt carried out by Einstein who, however, did not fully clarify the j5Minkowski (1909). See also Born (1978), pp. 130-131. 36Abraham (1903), Lorentz (1904). Abraham’s formulas for the electron’s longitudinal mass (m,) and transverse mass (+) are m,=F

28

[

-1_8’

““1; mT=&++jY)log+]

- log1 _B

where b = v/c, m, = e*/8nrc2 is the electron’s rest mass and r is the electron’s radius. mL. and )nT represent the inertia of electromagnetic origin opposed by the electron to variations in its state of motion along and normal to its own trajectory, respectively. Lorentz’s predictions for the electron’s longitudinal and transverse mass are mL = 3~1

See also Miller (1977), p. 1041.

4% _

p2)3/2;

mT

= ~(1

4mo _

g2~1/2

Studies in History and Philosophy of Modern Physics

274

t

X

Fig. 1. Adaptedfrom Born (1909b). p. 814, Figure 1.

state of affairs’ (Einstein had supposed that rods and clocks were unaffected by accelerations).37 In SR, the classical definition of rigidity is broken. This is simply described in Fig. 1, where the rigid motion of a segment is represented according to classical mechanics: the distance between those world-lines describing the motion of the ends of the segment is always equal to r, i.e. to the length of the segment. In SR this no longer holds, since the length of the segment, that is the quantity: r=x

2

--x

1

(where xi and x2 are the coordinates of the ends) is not invariant under Lorentz transformations. Another definition for rigidity is therefore required. According to Born: The possibility of this [a new definition of rigid body] is likely from the very beginning, since, from all points of view, Newtonian kinematics represents an asymptotic case of the new [kinematics], i.e. a case in which the velocity of light c is supposed to be infinitely great. The method I am about to explain consists in defining rigidity through a differential law, instead of [using] an integral law.38 37‘Es liegt in dieser Richtung nur ein Versuch vor, den Einstein gemacht hat, ohne die Sachlage ganz zu kllren’, Born (1909a), p. 3. Born quotes paragraph 18, ‘Space and time in a uniformly accelerated system’, of Einstein (1907~). (See note 32.) “‘Die Mijglichkeit davon ist von vomherein wahrscheinlich, weil in jeder Beziehung die Newtonsche Kinematik einen Grenzfall der neuen darstellt, nlmlich den, bei dem die Lichtgeschwindigkeit c als unendlich grog angesehen wird. Die Methode, die ich einschlage, besteht darin, die Starrheit statt durch ein Integralgesetz durch ein Differentialgesetz zu definieren’, Born (1909a), p. 3.

275

Dejinition of Rigidity in the Special Theory of Relativity

Since he considered Newtonian mechanics as the limiting case of a broader theory, namely the theory of relativity, Born was led to think that, in general, the condition of rigidity to be sought should not be expressed by an integral relationship, i.e. a relationship between finite differences in the values of the coordinates, but rather by a differential equation:

Born’s approach consists in trying to express a condition like Eq. (2) in Newtonian mechanics and to apply it later to the kinematics of the principle of relativity. He considers the flow of a deformable medium in the manner of Lagrange, expressing the coordinates of a point x,y,z by means of its initial coordinates &q,[ and of the time t:

x = x(S,rl,C0, Y =

Y(blLJ),

(3)

Then he remarks that the line element ds2 = dx2 + dv2 + dz2

(4)

can be represented through a quadratic form containing the differentials dt, dg, dc. To do so, Born employs the formulas:

(5)

obtaining:

d =wX2 +P22dv2 +p3+K2 + @,,Wv

+ ‘&d&X + ‘@,,dvK,

where pij is the element of the matrix of the ‘deformation given by: P = ATA,

quantities’

(6) P

(7)

Studies in History and Philosophy of Modern Physics

276 where:

(8)

A=

Thus, according to Newtonian mechanics, the differential condition of rigidity can be expressed by requiring that the elements of the matrix P do not change with time:

&lo

(9)

*

dt-

The problem is now to write Eq. (9) according to the kinematics of the principle of relativity, where reference frames are connected by Lorentz instead of Galilean transformations. In such a case, the problem is that a subspace where t = const does not in general transform into a subspace where t’ = const, and where t and t’ are the proper times of two reference frames related by Lorentz transformations. It follows that in Eq. (6) there will also be terms containing the proper time dz of the reference c, g, [, r and that it will not in general be possible to write a differential condition of rigidity like Eq. (9). The problem can be described geometrically by saying that the distance I between two world-lines (Fig. 2), describing two points moving in the xt plane, cannot in general be represented through a segment belonging to a straight line that remains parallel to the x axis; the problem is therefore to devise an expression, associated with the two moving points, that is invariant under Lorentz transformations. Born considers the four-velocity of a point moving along a world-line: it is the time-like vector (i.e. directed from the origin 0 towards the hyperbola c2t2 - x2 = 1) of unit magnitude tangent to the worldline. He then goes on to consider the condition of normality between two vectors xi, t, and x2, t,: 2 c t2t,

-

x2x,

=

0,

tw

which is clearly Lorentz-invariant. Geometrically, this condition means that one of the two vectors is directed from 0 towards one of the hyperbolas Ztl, while the other is tangent to the hyperbola in its point of c2t2 -x2=

Definition of Rigidity in the Special Theory of Relativity

Fig. 2. Adaptedfrom Born (1909b),

271

p. 815, Figure 3.

contact with the first vector. Since the condition of normality is Lorentzinvariant, it still holds under a Lorentz transformation of coordinates. Thus, we can choose a set of coordinate transformations such that the new time axis coincides with the direction of the velocity: the new x axis will be the locus of the points which are considered simultaneous by an observer moving at the above-mentioned four-velocity. Along this direction let us take a segment of length r times the semi-axis of the hyperbola c*t* - x2 = 1: if we repeat such a construction for all the points on the world-line describing the motion of the point under examination we obtain the distance r between two points rigidly connected. Let us see what it means analytically to consider the normal to the velocity or, according to Born’s own words, the ‘normal section’ to the world-line. Taking a four-dimensional space and again considering the flow of a deformable medium in the manner of Lagrange by specifying the coordinates x, , . . , x4 (x4 = ict) as functions of the initial coordinates and proper time is,,..., &, (& = icr) of a moving point, the world-line element: ds* = x dX;

(11)

is in general a quadratic form of the differentials d5, i = 1, .... 4: ds= = x Aikd<,d& i,k

(12)

Studies in History and Philosophy of Modern Physics

278

If we consider the four-velocity ui, i = 1, . ... 4 and a segment dxi along the direction normal to it, we have: C

ui

dxi =

z Uizd
(13)

i

Using Eq. (13) in Eq. (12), we can eliminate the terms containing d& and write, as required, ds2 as a quadratic form of the three spatial differentials only:

I

i,k=

The new definition of rigidity follows immediately from Eq. (14). In fact, the deviations of thep,, from their initial values give the deformation of the volume element. For a rigid body they must vanish, so that: ‘pik

_ 0

(15)

’

at,

which is a definition analogous to Eq. (9) that holds in Newtonian mechanics. As before, Born considers the quantity pik as the element of a matrix P whose expression, now, is the following:39 P=

A=SA,

(16)

where (reverting to the earlier notation):

(17)

where, for example, x5 = &z/a&O, 0,0,r) and: YPX,

-7

Y: 1+-z C

s= Y?, 7

Y,4 ic 39Bom (1909a), p. 15.

w

y

ic ‘95 ic

? z2

1+;

C

VT ic

VT ic

1 - t;

(18)

Dejinition of Rigidity in the Special Theory of Relativity

279

As we have seen, in addition to the paper published in the Annalen der Physik, Born presented the results of his work at the above-mentioned meeting of the German Society of Scientists and Physicians (Salzburg, 21-25 September 1909). On that occasion, he expounded a less quantitative version of his work, focusing on its geometrical aspects; he took into account the above-mentioned motion of a rigid segment in the plane xt. We have already pointed out that the fundamental feature of such motion is that the subsequent positions assumed by the line joining the two points are not parallel. This fact has an important consequence, which Born did not fail to grasp: not only are the extensions of the various lines joining the two points not parallel, but they have an envelope and, by definition, the envelope is something they cannot get out of, i.e. they cannot ‘penetrate’ it. Born infers that this new kind of ‘rigid body’ must have a finite extent: ‘Such a body must therefore have a limited extent, and, since the more the curvature the smaller the distance between the envelope and the generating curve, it follows that the greater the acceleration, the smaller the [rigid] body’.lO If a body has the shape of a sphere of radius R, the following equation holds: aR < c2,

(19)

where a is the acceleration of the body. When a ---fco, R + 0, and the size of the rigid body becomes infinitesimal. Thus, in general, the condition of rigidity holds locally. The definition of rigidity given by Born aroused great interest and in turn prompted other papers4i which, among other points, examined the problem of the number of degrees of freedom of a rigid body like Born’s. It was von Laue who finally clarified the problem in 1911, showing that such a number cannot be limited, according to the theory of relativity. In fact, since no action can propagate with a velocity greater than the speed of light, an impulse simultaneously acting on n different points of a body will necessarily result in at least n degrees of freedom.42 However, while the rigid body concept had lost its place in relativistic mechanics, the concept of rigid motion related to Eq. (15) was still valid: it is precisely in that kind of motion that the body moves in such a way that none of its individual parts is subjected to stress. As Herglotz pointed out, the body must move in such a way that each volume element undergoes a Lorentz contraction.43 40‘Demnach kann em solcher Kiirper nur eine endliche Ausdehnung haben, und da die Enveloppe urn so nlher an der Ausgangkurve liegt, je gr6Ber deren Krtimmung ist, so folgt, daB der K&per urn so kleiner sein muD, je gr6Ber seine Beschleunigung ist’, Born (1909b), p. 815. 4’See for example the papers quoted in Pauli (1921), p. 131 (footnote). ‘?on Laue (1911); see also Pauli (1921) pp. 131-132. 43Herglotz (1910), p. 397.

280

Studies in History and Philosophyof Modern Physics

4. Einstein, the Genesis of the General Theory of Relativity and the Concepts

of Rigidity and of Hyperbolic Motion 4.1. The Years 1909-1911 Einstein attended the above-mentioned meeting in Salzburg, where he presented a paper, later published in the Physikalische Zeitschrift immediately after Born’s paper.44 He was undoubtedly greatly interested by Born’s paper on the definition of rigidity in relativistic mechanics: Born later recalled that during the meeting he had a private discussion with Einstein about a puzzling implication of his definition of rigidity, according to which it was impossible to set into rotation a rigid body at rest .45 The point was first made in print by Ehrenfest and soon became known as ‘Ehrenfest’s paradox’.46 Ehrenfest’s paper was submitted to Physikalische Zeitschrift on 29 September 1909, on the same day that Einstein wrote a letter to Sommerfeld, just a few days before leaving Bern to take the chair of associate professor of theoretical physics at the University of Ziirich. In the letter, Einstein pointed out the importance of the problem of the rigid body for generalizing the existing theory of relativity: The treatment of the uniformly rotating rigid body seems to me to be of great importance on account of an extension of the relativity principle to uniformly rotating systems along analogous lines of thought to those that I tried to carry out for uniformly accelerated translation [...I.47

Some months later, Einstein’s words reveal he was rather discouraged by the difficulties he had encountered, as we can read in another letter to Sommerfeld: Now on the other critical point, the rigid body. I am not working too much on it now. In fact, it seems to me that the empirical evidence is insufficient to expound a theory for an arbitrarily accelerated body. If Fizeau’s work and the measurements of the velocity of light in vacuum had not given evidence, the ground to base the theory of relativity would have not been firm enough; analogously, in my opinion, we face the “Einstein (1910). 45See note 13. 46Ehrenfest (1909). Ehrenfest’s argument is as follows: consider a cylinder of radius R (as measured when it is at rest) set into motion at a constant angular velocity about a fixed axis normal to its endfaces. Let R’ be its radius as measured by an observer at relative rest. Since the individual parts of the periphery of an endface move with the linear velocity R’w tangent to the direction of motion, they undergo a Lorentz contraction: it follows that 2nR’ < 2xR. But any radius of an endface of the cylinder is normal to the direction of rotation: they are not affected by the contraction and, therefore, R’ = R. Thus, the radiuses obey two mutually contradictory conditions. On Ehrenfest’s paradox see for example Pauli (1921), pp. 131-132 and Miller (1991), pp. 245-253. On this subject Einstein published a paper in 1911 (Einstein, 191la) to underline the reality status of the contraction of moving bodies, which had been questioned by one of the participants in the debate (VariEak, 1911). See also the letter Einstein wrote to Ehrenfest on 12 April 1911, Klein et al. (1993), Dot. 264. 47‘Die Behandhmg des gleichfiinnig rotierenden starren Kijrpers scheint mir von grosser Wichtigkeit wegen einer Ausdehnung des Relativitatsprinzips auf gleichfiirmige rotierende Systeme nach analogen Gedankenglngen, wie ich sie [...I fiir gleichfiirmig beschleunigte Translation durchzufiihren versucht habe’, letter to Sommerfeld of 29 September 1909, Klein et al. (1993), Dot. 179, p. 210. Translation as in Stachel (1980). Einstein started his new job on 15 October 1909.

Definition

of Rigidity

in the Special Theory of Relativity

281

problem of acceleration [of a rigid body]. According to my opinion, for the time being, we can assert something only about a system with an infinitely small acceleration. However, we must try to devise hypotheses on the behaviour of a rigid body allowing uniform rotation. I have little time at the moment, since my new job demands more than I had expected.48 Again, in March 1910, Einstein wrote to Laub: ‘I am very interested in the latest research by Born and Herglotz. Indeed, in relativity a “rigid” body with 6

degrees of freedom seems not to exist’.4y We wish to emphasize two fundamental points. On the one hand, Einstein realized from the outset the whole complexity of defining a rigid body in the theory of relativity and the importance of the problem. On the other hand, interest in this matter was not confined to Born and Einstein: besides the numerous papers we have mentioned so far,50 it is worth recalling that Ehrenfest’s paper was submitted just four days after the end of the Salzburg conference. It seems reasonable to assume that Ehrenfest, too, knew about the discussion between Born and Einstein or, at least that he deemed the issue of defining rigidity in SR a crucial subject. We shall later resume discussion on the role played by Ehrenfest in the matters described in this paper. 4.2. The Scalar Theory of the Gravitational Field Einstein did not resume his work on gravitation until he settled in Prague, where he arrived in March 19 11. His stay in Prague lasted 16 months, since he went back to Zurich in August 1912, to take on a position of full professor at the Eidgeniissische Technische Hochschule. After the paper of June 1911 on the bending of light rays and the equivalence between mass and energy,51 Einstein constructed the so-called ‘scalar theory of the gravitational field’ between February and March 1912:s2 we have already pointed out that it was a theory where the gravitational field was described by a variable velocity of light c, considered as a function of the position in the presence of a gravitational field. 48‘Jetzt zu dem anderen Schmerzenskind, dem starren k&per. Mit diesem beschiiftige ich mich wenig. Es scheint mir namlich, dass die Erfahrungsdaten nicht hinreichen, urn die Theorie beliebig beschleunigter K&per aufzustellen. Wenn der Fizeausche Versuch und die Messungen iiber die Vakuum-Lichtgeschwindigkeit nicht vorllgen, hatte das Material fur die Aufstellung der Relativitatstheorie gefehlt; so Ihnlich aber stehen wir nach meiner Meinung da der Beschleunigung gegentiber. Nur tiber unendlich langsam beschleunigte Systeme llsst sich nach meiner Meinung jetzt etwas aussagen. Immerhin sollte man versuchen Hypothesen tiber das Verhalten starrer Korper zu ersinnen, welche eine gleichmassige Rotation erm8glichen. Ich habe jetzt wenig Zeit, weil mich mein neues Amt mehr in Anspruch nimmt als ich dachte’, letter to Sommerfeld of 19 January 1910, Klein et al. (1993), Dot. 197, p. 229. 49‘Die letzten relativitlstheoretischen Untersuchungen von Born und von Herglotz interessieren mich sehr. Es scheint wirklich dass es einen ‘starren’ K&per mit 6 Bewegungsfreiheiten in der Relativitiitstheorie nicht gibt’, letter to Laub of 16 March 1910, Klein et al. (1993), Dot. 199, p. 232. “See notes 41, 42, 43. “Einstein (1911b). 52The theory is contained in the papers listed as Einstein (1912a,b), which were submitted to the Annalen der Pfiysik on 24 February and on 20 March, respectively, and were both published on 23 May. See Klein et al. (1993), Dots 365 and 375.

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Studies in History and Philosophy of Modern Physics

In both papers Einstein showed he was taking into account Born’s definition of uniformly accelerated motion since he needed such a definition in the method adopted in his early papers on gravitation. Following his own approach, Einstein compared the descriptions of a phenomenon obtained in different frames of reference: an inertial frame, a frame uniformly accelerated with respect to the first and a frame situated in a static gravitational field (i.e. a field generated by a mass distribution at rest). In fact, in both papers, Einstein takes pains to clarify that the acceleration should be considered uniform ‘according to Born’ (‘im Bornschen Sinne’). 53 As a matter of fact, what Einstein was referring to was the kind of motion Born called ‘hyperbolic motion’, i.e. uniformly accelerated motion according to the principle of relativity. While in fact the motion of a rigid body with constant acceleration [.,.I in a coordinate system xt can be represented in the old mechanics via an array of parabolas, the representation of such a hyperbolic motion is provided by an array of hyperbolas, whose asymptotes are the straight lines x f ct. This motion, too, is to be considered, in a certain sense, as uniformly accelerated since the [...I ‘value of the acceleration’ a is constant for every point. The parabolas of the old mechanics are evidently the limit of such hyperbolas when c = 00.~~

It should be noted that Einstein’s allusion to hyperbolic motion seems to be citing something that the reader must have been very familiar with at that time (even though we cannot be sure that Einstein was aware of all the implications deriving from such a kind of motion, see below). Born had been the first to undertake the task of constructing a theory to describe the behaviour of an accelerated body; it seems quite reasonable that, after 1909, the concept of hyperbolic motion became a landmark for all those who were striving to extend SR in a dynamic sense. Apart from the above-mentioned explicit references to Born’s work, there are in the 1912 papers additional references to the condition of rigidity, a concept that proves to be fundamental for the measurability of physical quantities. These references reveal a situation far from being devoid of difficulties and discrepancies, as we can see from the following passage: Such a system K [uniformly accelerated], according to the equivalence principle, is strictly equivalent to a system at rest in which a matter-free static gravitational field [i.e. generated by masses situated at infinity] of a certain kind exists. Let spatial measurements in K be made with measuring rods which-when compared with each other at rest at some point of K-have the same length; assume that the theorems of 53Einstein (1912a), p. 356; Einstein (1912b), p. 443. “‘Wihrend Gmlich in der alten Mechanik die Bewegung eines starren Korpers mit konstanter Beschleunigung [...I in einem xt Koordinatensystem durch eine Schar von parallelen Parabeln dargestellt wird, ist das Bild dieser Hyperbelbewegung eine Schar von Hyperbeln, deren Asymptoten die Geraden x f ct sind. Auch diese Bewegung ist in einen gewissen Sinne als gleichfijrmig beschleunigt zu bezeichnen, da bei ihr der [...I “Betrag der Beschleunigung” a fur jeden Punkt konstant ist. Die Parabeln der alten Mechanik sind offenbar der Grenzfall c = cc von solchen Hyperbeln’, Born (1909b), pp. 815-816.

Dejinition of Rigidity in the Special Theory of Relativity

283

[Euclidean] geometry are valid for lengths measured in this way, and thus also for the relationship between the coordinates x,y,z and other lengths. This stipulation is not automatically permissible, but contains physical assumptions which ultimately could prove to be invalid. For example, they most probably do not hold in a uniformly rotating system in which, on account of the Lorentz contraction, the ratio of the circumference of a circle to its diameter would have to differ from n using our definition of length. The measuring rod, as well as the coordinate axes, are to be treated as rigid bodies. This is permissible in spite of the fact that, according to [special] relativity theory, rigid bodies cannot really exist. For one can imagine the rigid measuring body replaced by a large number of small non-rigid bodies so aligned alongside each other that they do not exert any pressure forces on each other since each is separately held in place.55

If, on the one hand, Einstein gives the impression of acknowledging the result explicitly achieved by Born, i.e. the non-existence of the relativistic rigid body, on the other hand he was nevertheless forced to use such a concept (otherwise he would not have been able to make measurements within the new theoretical context) and, for want of a better solution, he simply proposed using the rigid measuring body he had already considered in 1907 (a solid body made of many non-rigid bodies close to each other but not interacting). Moreover, in the 1912 theory Einstein was still working in the context of Euclidean geometry, even though the accepted definition of rigid body implied that Euclidean geometry was no longer valid; indeed, he was now becoming aware of this. We agree with Stachel, according to whom: ‘The tentative nature of his conclusions reflects Einstein’s puzzlement during this period over the problem of the relationship between coordinates and measurements with rods and clocks’.56 Pais is inclined to emphasize the importance of the problem of the rigid body: ‘The sequence of these remarks may lead one to surmise that the celebrated problem of the rigid body in the special theory of relativity stimulated Einstein’s step to curved space, later in 1912’.57 We must also take into account another severe drawback of the 1912 theory which is to be considered fundamental, especially in the light of later ‘5‘Ein solches System K ist nach der Aquivalenzhypothese streng gleichwertig einem ruhenden System, in welchem ein massenfreies statisches Gravitationsfeld bestimmter Art sich befindet. Die raumliche Ausmessung von K geschieht durch MaBstPbe, welche-im Ruhezustande an der namlichen Stelle von K miteinander verglichen--die gleiche Lange besitzen; es sollen die Sitze der Geometrie gelten fur so gemessene Llngen, also such fiir die Beziehungen zwischen den Koordinaten x,y,z und anderen Langen. Diese Festsetzung ist nicht selbstverstandlich erlaubt, sondern enthllt physikalische Annahmen, die sich eventuell als unrichtig erweisen kiinnen; sie gelten z.B. hochst wahrscheinlich nicht in einem gleichformig rotierenden Systeme, in welchem wegen der Lorentzkontraktion das Verhaltnis des Kreisumfanges zum Durchmesser bei Anwendung unserer Definition fur die Langen von n verschieden sein mtil3te. Der MaDstab sowie die Koordinatenachsen sind als starre K&per aufzufassen. Dies ist erlaubt, trotzdem der starre Korper nach der Relativitatstheorie keine reale Existenz besitzen kann. Denn mann kann den starren MeBkiirper durch eine groDe Anzahl kleiner nicht starrer Kiirper ersetzt denken, die so aneinander gereiht werden, da8 sie aufeinander keine Druckkrafte austiben, indem jeder besonders gehalten wird’, Einstein (1912a), p, 356. Translation as in Stachel (1980). p. 3. s%tachel (1980). p. 3. 57Pais (1982) p. 202.

284

Studies in History and Philosophyof Modern Physics

development of the theory and of the role played by Ehrenfest in this matter. Einstein, who had set out to obtain equations of the gravitational field such that: v2c = 0

(20)

in the absence of matter, had to recognize that the gravitational field itself possesses an energy which, in turn, corresponds to an inertia, acting, of course, as a source of the gravitational field; this implied that Eq. (20), in the absence of matter, had to be replaced by:

pc -

y2

(21)

In turn, this implied that the principle of equivalence could no longer hold generally (i.e. not limited to an infinitely small region of space) and we can understand Einstein’s unwillingness to state it: ‘I am reluctant to take this step, because I am now abandoning the ground of the absolute validity of the principle of equivalence. It seems that this principle holds only in infinitely small regions of space’.58 So, while attempting to generalize his theory, Einstein arrived at a point where the new theory no longer covered the specific case he had started from. In particular, the principle of equivalence, the discovery of which Einstein was later to acknowledge as ‘the happiest thought of my life’,59 held only locally in the theory of the static gravitational field, i.e. held only in infinitely small regions of space. 4.3. The Transition to the Tensorial Theory of Gravitation Einstein completed his scalar theory of the gravitational field in March 1912. We can follow his feelings about the theory through his correspondence of that period, since his initial and cautious satisfaction gradually gave way to concern as he became aware of the difficulties he was coming up against. In a letter to Ehrenfest of 10 March, Einstein seemed quite satisfied: ‘The investigations on the statics of gravitation (mechanics of mass-points, electromagnetism, statics of gravitation) are ready and satisfy me very much. I really believe I have found a piece of truth. I am now thinking about the dynamic case, once again going from the special to the general’.60 However, just a few days later, we find that Einstein was fully aware that giving up the constancy of the velocity of light Ss‘Zudiesem Schritt entschliel3e ich mich deshalb schwer, weil ich mit ihm den Boden des unbedingten iiquivalenzprinzips verlasse. Es scheint, daI3 sich letzteres nur fur unendlich kleine Felder aufrecht erhalten IIDt’, Einstein (1912b), pp. 455456. 59Pais (1982), p. 178. 60‘Die Untersuchungen tiber die Statik der Gravitation (Punktmechanik Elektromagnetik Gravitostatik) sind fertig und befriedigen mich sehr. Ich glaube wirklich, ein Stuck Wahrheit gefunden zu haben. Nun-denke ich iiber den dynamischen Fall nach, such wieder vom Speziellen

footnote 60 continued on p. 28.5

Dejkition of Rigidity in the Special Theory of Relativity

285

meant in practice breaking the symmetry between spatial coordinates and time. This symmetry had been a cornerstone of SR and, in particular, of Minkowski’s formulation: ‘However, I have not yet managed to find the dynamical laws of the gravitational field. The simple schema of the equivalence of the four dimensions does not hold here in the way it does with Minkowski’.cl Two days later, Einstein wrote to Besso: ‘Recently, I have been working furiously on the gravitation problem. It has now reached a stage in which I am ready with the statics. I know nothing as yet about the dynamic field, that must follow next [...] I am still very far from being able to interpret rotation as rest! Every step is devilishly difficult, and what has been derived so far is surely still the easiest [part]‘.62 Nearly 2 months later, on 20 May, Einstein wrote to Zangger: ‘The investigations on gravitation have led to some satisfactory results, although until now 1 have been unable to penetrate beyond the statics of the gravitation’.63 At the beginning of June 1912, Einstein’s uneasiness seemed to be increasing: ‘The further development of the theory of gravitation meets with great obstacles’;@ ‘the research on the statics of the gravitation has been completed and I greatly trust the results. However, the generalization [of the static case] appears very difficult’.65 From these fragments we can infer that Einstein was confident enough that his treatment of the static gravitational field was correct, whereas he deemed that further generalization was very difficult. However, in a letter Ehrenfest received from Einstein on 20 June, there is some evidence of uneasiness regarding also the static field and originating from comparison between such a field and those ‘Bornian fields’ used as a test case for the theory of the static field: footnote 60 continued from p. 284 zum Allgemeineren iibergehend’, letter to Ehrenfest of 10 March 1912, Klein et al. (1993). Dot. 369. p, 428. Translation as in Pais (1982) p. 210. 6“Es ist mir aber noch nicht gelungen, die dynamischen Gesetze des Gravitationfeldes aufzufinden. Das einfache Schema de;gleichberechtigten 4 Dimensionen gilt hier nicht in dieser Weise wie bei Minkowski’. letter to Smolukowski of 24 March 1912, Klein et al. (1993). Dot. 376, P. 434. Part of the translation as in Stachel (1980), p. 11. 62‘In der letzten Zeit arbeitete ich rasend am Gravitationsproblem. Nun ist es soweit, dass ich mit der Statik fertig bin. Von dem dynamischen Feld Weiss ich noch gar nichts, das sol1 erst jetzt folgen. [...I ich [bin] noch weit davon entfemt, die Drehung als Ruhe auffassen zu konnen! Jeder Schritt ist verteufelt schwierig, und das bis jetzt abgeleitete gewiss noch das einfachste’, letter to Besso of 26 March 1912, Klein et al. (1993), Dot. 337, pp. 435 and 436. Part of the translation as in Pais (1982), p. 210. This letter is not included in the correspondence edited by Speziali (1972). 63‘Die Untersuchungen iiber Gravitation haben zu einigen befriedigenden Resultaten gefiihrt, wenn ich such iiber die Statik der Gravitation hinaus bis jetzt nicht habe vordringen kbnnen’, letter to Zangger of 20 May 1912, Klein et al. (1993) Dot. 398, p. 467, Einstein’s italic. Translation as in Pais (1982), pp. 210-211. @‘Die Weiterentwicklung der Theorie der Gravitation stosst auf grosse Hindemisse’, letter to Zangger, 1912, probably written after 5 June, Klein et al. (1993), Dot. 406, p. 480. Translation as in Pais (1982), p. 211. 65‘Die Untersuchungen fiber die Statik der Gravitation habe ich nun fertig und setze grosses Vertrauen in die Resultate. Aber die Verallgemeinerung scheint sehr schwierig zu sein’, letter to Hopf of 6 June 1912, Klein et al. (1993), Dot. 408, p. 483. Part of the translation as in Pais (1982). p. 211.

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It seems that the equivalence [principle] can hold only for injinitely smaN systems [and] that therefore Born’s accelerated jnife system cannot be considered as a static

gravitational field, that is, it cannot be generated by masses at rest. A rotating ring does not generate a static field in this sense, although it is a field that does not change with time. [...I In such a field, the reversibility of light paths does not hold. In the theory of electricity, my case corresponds to the electrostatic field; on the other hand, the general static case would, in addition, include the analog of a static magnetic field. I am not that far yet. The equations found by me must refer only to the static case of masses at rest. Born’s field of finite extension does not fall in this category. It has not yet become clear to me why the equivalence principle fails for jinite fields (Bom).66 This letter shows us that Einstein was puzzled that the principle of equivalence did not hold in finite regions of space even for a static gravitational field, i.e. a field generated in answer

by masses at rest. We find this perplexity

to some of Abraham’s

Physik on 4 July

criticisms

again in a paper written

and received

by the Annalen der

1912:

In the near future our task will be to devise a relativistic scheme where we will be able to express the equivalence of gravitational and inertial mass. I made an initial, modest attempt towards this goal in my papers on the static gravitational field. [...I I must admit that I could maintain such a view only in infinitely small regions of space and that I cannot devise any plausible explanation for such a state of affairs.67 There is another

aspect to consider

the kind of gravitational

about Einstein’s

field equivalent

scalar theory. It concerns

to a uniformly

accelerated

frame of

reference ‘in the Bornian sense’: it can be demonstrated that such a field is uniform only locally. A frame of reference like this, which we shall call K,, can be formed

considering

the bundle X2 -

of hyperbolas

c2t2=d,

O
(in the plane x - t): co,

(22)

66’[Es scheint], dass der Aequivalenzsatz nurfiir unendlich kleine Felder gelten kann, dass also das Bornsche beschl. endliche System nicht als statisches Gravitationsfeld aufgefasst werden kann, d.h. sich nicht durch ruhende Massen erzeugen llsst. Ein sich drehender Ring erzeugt nicht ein statisches Feld in diesem Sinne, obwohl es ein zeitlich unvednderliches Feld ist. In einem solchen Felde wird die Reversibilitiit der Lichtwege nicht gelten. Mein Fall entspricht in der Elektrizitatstheorie dem elektrostatischen Felde, wogegen der allgemeinere statische Fall noch das Analogon des statischen Magnetfeldes mit einschliessen wiirde. So weit bin ich noch nicht. Die von mir gefimdenen Gleichungen sollen sich nur auf den statischen Fall ruhender Massen beziehen. Das Bornsche Feld endlicher Ausdehnung f%llt nicht darunter. Warum das Aequivalenzprinzip fur endliche Felder (Born) versagt, ist mir noch nicht klar geworden’, letter to Ehrenfest, written before 20 June 1912, Einstein’s italic, Klein et al. (1993), Dot. 409, p. 486. Part of translation as in Pais (1982), pp. 213-214. 67‘Aufgabe der nachsten Zukunft mul3 es sein, ein relativitatstheoretisches Schema zu schaffen, in welchem die Aquivalenz zwischen trager und schwerer Masse ihren Ausdruck findet. Einen ersten, recht bescheidenen Beitrag zur Erreichung dieses Zieles habe ich in meinen Arbeiten tlber das statische Gravitationsfeld zu geben gesucht. [...]Es ist zuzugestehen, daB ich diese Auffassung nur fur unendlich kleine RBume widerspruchsfrei durchfilhren konnte, und da8 ich hierfiir keinen befriedigenden Grund anzugeben weiD’, Einstein (1912c), p. 1063.

Dejnition of Rigidity in the Special Theory of Relativity

287

where ?/a is the value of the acceleration. Observers located in different points of KB have different values for the acceleration, which can be considered constant only in an infinitely small region around a given point. It is with such a limitation that the principle of equivalence can be placed within the framework of the Minkowskian formulation of space-time. Thus, a uniform gravitational field cannot be built employing a reference system accelerated ‘in the Bomian sense’. However, we have no evidence that Einstein did actually realize this limitation when he started constructing his scalar theory of the gravitational field: the influence of Born’s paper had probably only led Einstein to understand that the classical concept of rigid body cannot really exist in SR and that the procedure to make measurements would somewhat be affected by this.68 Einstein had at any rate been able to grasp that the principle of equivalence held only locally even though, at the beginning of the summer of 1912, he was still wondering why this occurred, as we can read in the letter to Ehrenfest quoted above. We must now consider in more detail the role played by Ehrenfest as Einstein’s privileged interlocutor in those months. They had already worked together on problems concerning gravitational theory: in February 1912 Ehrenfest visited Einstein in Prague and was his guest from 23 to 29 February. The two men discussed at length many topics regarding physics and, among them, the new theory Einstein was constructing in those very days.69 At the end of April, Einstein sent Ehrenfest the proofs of his papers on the scalar theory of the gravitational field. 70 Ehrenfest immediately started investigating a problem that was certainly of great interest to Einstein: ‘find the most general field of world-lines which is equivalent to a stationary gravitational field’? He briefly described the results of his efforts to Einstein in the letter quoted above (of 14 May) and in another letter, written after 16 May.72 Einstein’s reply was received by Ehrenfest on 20 June. We have already quoted a passage from this letter, which showed Einstein’s difficulties in understanding why the principle of equivalence seemed to hold only locally. In the same letter, Einstein commented on Ehrenfest’s remarks contained in his two previous letters which, evidently, he had not understood: ‘I could not yet understand the principle concerning the gravitational field you stated. You ought to explain it better to me. [...I Please explain it to me again and precisely what you have demonstrated. As I said 68See note 5.5. 6gIn Ehrenfest’s diaries we can find evidence of his stay; it seems that a deep friendship had arisen between him and Einstein. See Klein (1970), p. 176. “Einstein (1912a,b). See Einstein’s letters to Ehrenfest of 25 and 26 April, Klein et al. (1993), Dots 384 and 387. “‘Zu bestimmen das allgemeimte Weltlinien-Feld, das einem stationiiren Gravitationsfeld aequivalent ist’, letter of Ehrenfest to Einstein of 14 May, Klein et al. (1993), Dot. 393, p. 460. Ehrenfest’s italic. “Klein et al. (1993), Dots 393 and 394.

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before, I do not even understand your statement’.73 Ehrenfest replied to Einstein’s request-received on 20 June-in a letter probably written on 29 June.T4 It is quite a long letter consisting of several partly overlapping drafts: it seems as if Ehrenfest actually started writing his letter again and again, probably dissatisfied with his own explanation. The core of his argument was later included in a paper published in 191375 and we shall refer to it for details. At this point it is worth focusing on an aspect that clearly emerges from this letter and provides us with unique information about Ehrenfest’s view. He dwelt upon Einstein’s statement, according to which the principle of equivalence holds locally, and ‘therefore Born’s accelerated finite system cannot be considered as a static gravitational field’.76 Ehrenfest planned to demonstrate that, even if in general the principle of equivalence holds locally, there may however exist a special case such that it holds in a finite region of space: ‘However, I do not see in your [...I paper that you point out that, also in a special case, Born’s hyperbolic motion does not fulfil the macro-equivalence. For the time being I can assert: it is not proved that a finite field of hyperbolic motion cannot be considered as a static gravitational field’.77 Thus, it seems that Ehrenfest aimed at demonstrating that one of the special cases in which the principle of equivalence holds in a finite region of space is represented by Born’s hyperbolic motion, which is to be regarded as an exception with respect to what Einstein had discovered and which therefore confirms the substance of Einstein’s difficulties, as Ehrenfest himself wrote to Sommerfeld on 23 June: On the occasion of the publication of Einstein’s papers on gravitation, I considered

the following problem: which is the most general field of world-lines that in the space x,y,z, t is equivalent to a stationary gravitational field with respect to geometricaloptical behaviour? I could demonstrate through purely geometrical arguments that, besides the fields of world-lines of the uniform translation and of Born’s hyperbolic motion, this requirement can be fulfilled only by degenerate and physically meaningless fields. This result clarifies (as I believe) why Einstein meets such special difficulties in the dynamical part of his work.78 73‘Der Satz iiber Gravitationsfelder, von dem Sie mir erzahlen, ist mir noch nicht verstiindlich geworden. Sie mtissen mir ihn deutlicher erkllren. [...I Ich bitte Sie mir nochmals genau zu sagen, was Sie bewiesen haben. Ich verstehe-wie schon gesagt,-nicht einmal die Behauptung’, letter to Ehrenfest written before 20 June; Klein et al. (1993), Dot. 409, pp. 485486. ?Clein ef al. (1993), Dot. 411. “Ehrenfest (1913). 76‘[...]also [kann] das Bomsche beschleunigte endliche System nicht als statisches Gravitationsfeld aufgefasst werden’, Ehrenfest’s letter to Einstein of 29 June, Ehrenfest’s italic, Klein et al. (1993), Dot. 411, p. 488. The expression ‘therefore’ (‘also’) is underscored three times in the original, and the underscoring is set off on both sides by a pair of question marks (Klein et al., 1993, p. 496, note 8). “‘Ich sehe aber nicht, dass Ihre f...] Arbeit zeigen wiirde, dass such im speziellen Fall der Bomschen Hyperbelbewegung die Makro-Aequival&z fehlt’, Ehrenfest’s letter-to Einstein of 29 June, Ehrenfest’s italic, Klein et a/. (1993). Dot. 411. D. 488. 78‘[...]Anllsslich des Erscheinens her Gravitation&eiten von Einstein habe ich mir folgende Aufgabe iiberlegt. Was ist das allgemeinste Weltlinienfeld im x,y,z,t-Raum, das in Bezug auf footnote 78 continued on p. 289

Definition

of Rigidity

in the Special Theory qf Relativity

289

This passage is followed by a concise explanation on how to verify that what Ehrenfest calls a ‘stationary gravitational field’ is equivalent to the field described by the world-lines of Born’s hyperbolic motion. Demonstration of the point was resumed in a paper published shortly after, in February 1913,79 a few months before the publication of Einstein and Grossmann’s tensorial theory. Before continuing to analyze and comment on Ehrenfest’s work, it must be remarked that both in his letter to Einstein and, as we shall see, in his subsequent paper, Ehrenfest means ‘static gravitational field’ when he writes ‘stationary gravitational field’. A static field is a field generated by masses at rest; a frame of reference exists where the elements of the metrics, as we say today, are independent of time and there are no time-space cross-terms. A stationary field is a field generated by masses whose distribution of density does not change with time; a reference frame exists where the elements of the metrics are independent of time but there are nonzero time-space cross-terms. For instance, in a disk rotating with angular velocity CO,the nonzero cross-term is g,,, the value of which is W/C, where r is the radial distance. If a certain path is possible for a light ray in a stationary field, then this path must always be a possible path for light. Moreover, in a static field every ‘light path’ is reversible: this means that if a certain path is a possible path for light, then the reverse path must always be a possible path for light. The latter property does not hold in a stationary field. Returning to Ehrenfest’s statements, we remark that a certain confusion between the terms ‘static’ and ‘stationary’, when speaking of a gravitational field, was quite common in the years preceding the birth of GR; however, Ehrenfest’s arguments, published in the 1913 paper, do not leave any doubt about this. Ehrenfest began referring to the local validity of the footnote 78 continued from p. 288 geometrisch-optisches Verhalten einem stationliren Gravitationsfeld aequivalent ist? Durch rein geometrische Betrachtungen konnte ich zeigen, dass ausser den Weltlinienfeldem der gleichformigen Translation und der Bomschen Hyperbelbewegung nur noch physikalbedeutungslose degenerierte Felder diese Aequivalenzforderung erfiillen kiinnen. Dieses Resultat klart auf (wie mir scheint) warum Einstein im dynamischen Teil seiner Arbeiten schliesslich auf eigenthiimliche Schwierigkeiten stiisst’, Ehrenfest’s letter to Sommerfeld, 23 June 1912, Archive for the History of Quantum Physics, Accademia delle Scienze detta dei XL, Microfilm 30, Sect. 7, Ehrenfest’s underlining. As far as we know, the content of this important letter has not yet been considered in existing accounts of the genesis of GR. It is also worth noting that in the previously quoted letter to Einstein, Ehrenfest remarks (and he does so repeatedly) that the postulates on which he grounds his concept of equivalence are geometrical-optical rather than dynamical [Ehrenfest’s letter to Einstein of 29 June, Klein et al. (1993), Dot. 411, pp. 488, 489,491,492, 4931. His standpoint is, in his own words: ‘If we ascertain the most general set of motions that satisfies this m postulate of macro-equivalence, it is much easier to extract from it that particular subclass which is now macro-equivalent to a stationary gravitational field also with respect to dynamical, electrodynamical, thermodynamical experiments etc.’ (‘1st die allgemeinste Bewegungsclasse festgestellt, die dieser opt&hen Makro-Aequiv. Forder geniigt so ist dann schon vie1 leichter aus ihr weiter diejenige Unterclasse herauszuschllen, die nun such noch in Bezug auf dynamische, elektromagnetische, thermodynamische etc. Experimente mit einem station&en Gravitationsfeld makroaequivalent ist’, Ehrenfest’s letter to Einstein of 29 June, Ehrenfest’s underlining, Klein et al. (1993) Dot. 411, pp. 492493). 79Ehrenfest (1913).

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equivalence principle as a sort of ‘difficulty’ Einstein had run across. In his attempt to clarify the issue, Ehrenfest wondered where such a difficulty had arisen from: whether from the hypothesis of equivalence itself or from other hypotheses put forward by Einstein in his attempt to explain the dynamical phenomena, like those ‘concerning the dynamic actions of rigid kinematic connections’.80 The conviction Ehrenfest arrived at was, in his own words: ‘All the statical$elds of attraction with the exception of a very particular class, are in contradiction with Einstein’s hypothesis of equivalence. Already the statical field of attraction brought about by several centres of attraction which are stationary with respect to each other, is not compatible with the hypothesis of equivalence’.81

Ehrenfest considered a frame of reference in a static gravitational field, i.e. a reference where the postulates of the ‘constancy paths of light’ (once a light ray passes through the points A, B, .. ., F, G this path must always be a possible path for light) and of the ‘reversibility of the light paths’ (if the path through the points A,B, . . . . F,G is a possible path for light, then also the reverse path GJ, ...) B, A must always be a possible path) held. He constructed his reference frame in order to answer the following question: ‘How must the points of this (accelerated) laboratory in which there is no gravitation move, so that the observers in it shall observe constancy and reversibility paths of light in the sense of the hypothesis of equivalence? (i.e. which accelerated frames are equivalent to

static gravitational fields?). 82 The answer he obtained was that Born’s hyperbolic motion was the only special and physically meaningful case of an accelerated motion compatible with the principle of equivalence and such that the velocity of light is independent of time.s3 s’Ehrenfest (1913) p. 1188. “Ehrenfest (1913), pp. 1188-l 189, Ehrenfest’s italic. “Ehrenfest (19 13) p. 1189, Ehrenfest’s italic. 83To demonstrate his assertions Ehrenfest considers a two-dimensional (for the sake of simplicity) and gravitation-free frame of reference (we can call it S(x,y,r)) and a laboratory L with an acceleration y with respect to the positive direction of the x axis of S. According to SR, within the frame S the rays of light lie on the light cone forming an angle of 45” with respect to the t axis. Again following SR, if we arbitrarily select some points in the laboratory A, B, . . . their motions can be represented in S through curved world-lines, which we call u,b, .. .. respectively. Ehrenfest now supposes that an observer in L sends a light ray from A to B; if we call sr the path of the light ray, s, contains by definition the points A and B. According to the postulate of constancy of light paths, given any s, there must be infinitely many possible paths and the curves a,& will be intersected by this bundle of paths or, in other words, the curves, a.6, lie on the surface given by those infinitely many paths. Ehrenfest now applies the second postulate for static fields, i.e. the reversibility of light paths: given path s, from A to B, it must be possible to build the reverse path s’, from B to A. It follows that the curves 4 b, ... are also intersected by s’r, .... s’,, i.e. they lie on the surface given by two systems of infinitely many curves. What kind of surface are we talking about? Since, as we pointed out at the beginning, the light paths lie on a light cone forming an angle of 45” with respect to the t axis, the surface is an equilateral hyperboloid of revolution (light hyperboloid), the axis of which is parallel to the t axis and is described by the equation A(g+y’+z*)+Bx+Cy+Dt+E=O.

The motion of a generic point A will be described through the curve a lying on Hub (the light hyperboloid to which the curves a and b belong). Ehrenfest then supposes that the observer sends a light signal from A to another point B’ of the laboratory. Curve a must therefore also lie on the footnote

83 continued on p. 291

Definition

of Rigidity in the Special Theory of Relativity

291

The paper we have just examined was published in February 1913 but, as we have seen, the core of its arguments had already been developed by Ehrenfest in June 1912 and presented to Einstein in a set of three letters, very briefly in the first two and in more detail in the third.g4 We know for certain that Einstein received the first two letters-and he did not understand Ehrenfest’s point-whereas we cannot be sure that he ever received the third and more detailed letter. We do not possess any acknowledgement from Einstein: this phase of his correspondence with Ehrenfest broke off to resume later, when Einstein had already started with the tensorial theory of gravitations5 However if Einstein, as seems the case, read Ehrenfest’s arguments (and in any case he had the opportunity to meditate on the first two letters), it is likely that such arguments may have led him to realize that the compatibility of hyperbolic motion with the equivalence principle was to be considered as an exception and that, in general, it was not possible to construct arbitrary gravitational fields through accelerated motions, however complex they may be. Thus, even in the very initial and elementary aspects of the theory, there occurred oddities which suggested that SR could not be generalized to cover gravitation, but ought to be regarded as the limiting case of a broader theory. As to the starting point in such a quest, Einstein may, as Stachel conjectured, have been inspired by the case of the rotating disk, the simplest and most immediate example of a stationary gravitational field and the natural continuation of the path Einstein had started on when tackling the static field.86 In the above-mentioned letter, which Ehrenfest received on 20 June, Einstein also considered the problem of how to complete his theory of the static field with discussion of the case of the stationary field. As we pointed out before, the most natural example of such a field is the one produced by a rotating disk. Pais remarks that, in those weeks of intense meditation in the late spring of 1912, Einstein was focusing on Born’s work on rigid bodies of three years before, and wonders whether Born’s formalism, which ‘manifestly has Riemannian traits’ could possibly ‘give Einstein the inspiration for general covariance’.87 At any rate, it seems quite reasonable to agree here with Stachel, according to whom the physical example, the ‘missing link’ footnote 83 continued from p. 290 corresponding hyperboloid Hub’, i.e. must belong to the intersection between Hub and Hab’, which

is a plane. It can be shown that a is either a hyperbola or a straight line. In conclusion, given the requirement that the field of the world-lines of motion satisfy the conditions for the static field, the motion of an arbitrary point A in the most general case will in fact be precisely Born’s hyperbolic motion. Finally, due to the principle of equivalence, the hyperbolic motion is equivalent to a static gravitational field. (See Ehrenfest, 1913, pp. 1189-l 191.) s4Klein et al. (1993), Dots 393, 394, 411. “Einstein’s next letter to Ehrenfest was written on 20-24 December 1912 (Klein et al., 1993, Dot. 425). %tachel (1980). “Pais (1982) pp. 214 and 216.

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between the scalar and the tensorial theories of gravitation, was the field generated by a rotating disk, which Einstein probably took into account in generalizing his theory for static fields. Einstein knew that, in a rotating frame of reference, Euclidean geometry no longer holds; together with the fact that the principle of equivalence held only locally, this may well have reminded him of Gauss’ theory of surfaces (which states that Euclidean geometry holds only in infinitely small regions of space) to describe the metric laws of the space-time continuum in a gravitational field. His line of reasoning may well have been as he himself was to describe later on: in a rotating reference frame Euclidean geometry no longer holds. Due to the principle of equivalence, this reference frame may be considered as a system at rest equivalent to a gravitational field. ‘We therefore arrive at the result: the gravitational field influences and even determines the metrical laws of the space-time continuum. If the laws of configuration of ideal rigid bodies are to be expressed geometrically, then in the presence of a gravitational field the geometry is not Euclidean’.88 As a matter of fact, the paper that originated from the collaboration between Einstein and Grossmann, which began in August 1912, opens with the statement that the equivalence between an accelerated body and a gravitational field can hold only in an infinitesimal region of space: what had been an inexplicable point in the scalar theory of the gravitational field became one of the basic points of the new theory.89 To summarize thus far: from his very first steps on the path from SR to GR, Einstein had intended to extend SR not only because it did not cover gravitational phenomena-as traditionally recognized so far by historiographical research-but also because it did not possess extended bodies dynamics. How to deal with the dynamics of a rigid body is one of the problems left unresolved by SR, and it was a question that absolutely needed to be solved within a theoretical framework in order to develop a theory capable of describing the dynamics of extended bodies. This clearly emerges from the 1906-1907 papers we mentioned and quoted in the second section of this paper. It should be pointed out that it was Ehrenfest who first realized the contradiction encountered by the classical definition of rigidity within a theory where revision of the concept of simultaneity had led to the introduction of Lorentz’s hypothesis of contraction.gO It was Born who, in his 1909 paper, demonstrated how the attempt to build a relativistic dynamics for the extended bodies raised serious difficulties for the very concept of rigid body. We have examined Ehrenfest’s contribution to the theory of the static gravitational field and the role his contribution played in Einstein’s decision to abandon the theoretical scheme of the special theory of relativity. We shall now 88Einstein (1922), p. 59 (page numbers refer to the 6th ed., 1956). 89Einstein and Grossmann (1913), p. 225. “‘See also Lanczos (1972), p. 149.

Definition of Rigidity in the Special Theory of Relativity

investigate the role played by Born in this important research.

293

phase of Einstein’s

5. On the Influence of Born’s Work on the Genesis of the General Theory of Relativity

We now wish to discuss the possible influence exerted by Born’s work on Einstein’s path from SR to GR. In principle, we think that there are grounds to conjecture that the problem of the rigid body in the context of the principle of relativity played a major role in stimulating those reflections that were to put Einstein on the right road towards GR. Such influence is not simply limited to the formalism employed by Born and Einstein, but also concerns the deepest features of their theories and will eventually lead us to discuss the connection between physics and geometry. Let us now summarize the most important points, before proposing a number of analogies between Born’s and Einstein’s theories. According to Born, when considering a flow of matter, the line element in Eq. (11) ds2 = c, dx;,

(where xq = ict), whose initial coordinates through such coordinates by Eq. (12):

(11) are r,, . ... & can be expressed

(12) Furthermore, since initial coordinates are arbitrary, Eq. (12) is the gene& way to express the distance between two infinitesimally close world-lines. Thus, the relativistic analogous of the rigid behaviour in old mechanics is shown by the locus of the points such that in Eq. (12) the terms containing the proper time can be eliminated; this gives Eq. (14):

d2 = i P&Z&k, i.k =

(14)

I

from which, as we have already seen, it is immediately possible to draw the condition of rigidity [Eq. (15)]. In other words, the need for a new definition of the condition of rigidity when passing from Newtonian to relativistic mechanics arises from the fact that it is not always possible to eliminate the term containing dxi from Eq. (11) through suitable coordinate transformations. This

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happens since, as we have already pointed out, in relativity a subspace t = const does not become a subspace t’ = const with Lorentz transformations. In Newtonian mechanics the problem does not arise since the relationship t = const holds everywhere and the transformation equations do not depend on time, so that the line element can always be written as a sum of the squares of spatial coordinates. In Einstein’s generalization procedure, Eq. (11) is just a particular case of the expression d? = c g,,dx,,dx,

(23)

where the g,, are to be determined according to the properties of the gravitational field to be built. According to Einstein, Eq. (11) holds when, applying suitable coordinate transformations, it is possible to pass into a frame of reference where the space has Euclidean properties. However, this is possible only locally. Both Einstein and Born have to face the transition from properties of Euclidean space, which hold locally, to properties valid in general. Formally, this means that the relationship between ds2 and the coordinate differentials, formerly expressible through a diagonal matrix, can now be written in general via a non-diagonal matrix. The concept of local validity is a good starting point to grasp an important analogy between the problems solved by Born and Einstein, namely the definition of rigid body and the transition to the tensorial theory, respectively. Here we must note the conceptual analogy between the transition from the classical to the relativistic definition of rigid body and the transition from the metric properties of Euclidean space to the metric properties of the space-time continuum in a gravitational field. As a matter of fact, the classical definition of rigidity, which holds in a finite region of space, has to be replaced with a differential definition and, on account of the principle of relativity, the concept itself of rigid body has sense only locally, since the size of the region of space where Born’s definition of rigidity holds depends on the value of the acceleration of the rigid body itself. Einstein knew Born’s theory and considered it very important: he may well have discerned an analogy between the two cases in the general -+ local transition. Born’s theory may therefore have contributed to making Einstein meditate about the local validity of Euclidean geometry and on the resulting loss of the immediate physical sense of coordinates. In fact, as we have seen before, Born himself had put his theory in the context of a new (relativistic) mechanics to be considered as a generalization of Newtonian mechanics which acquired the status of a limiting case of the new mechanics.91 It was therefore clear from the outset that the definition of rigidity in SR was “See note 38

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Definition of Rigidity in the Special Theory of Relativity

a generalization of a corresponding definition acting as a limiting case in Newtonian mechanics. It is likely that this helped Einstein see the way to construct the new gravitational theory, i.e. that it was necessary to look for a theoretical framework where the old theory was to be covered by the new one as a particular case. Born’s definition of rigidity may have therefore influenced Einstein, who was trying to devise a scheme where Euclidean geometry held locally and coordinates were to lose their immediate geometrical meaning; such an effect might therefore be framed within that course of generalizations the crucial steps of which were transition from the particular to the general case and the treatment of the ‘old’ as a specific case of the newly devised ‘new’ general case. In other words, Born operates in a methodological context grounded on the generalization of a classical definition: he expounds a theoretical scheme where the classical definition is a special case. In the case in point, Born generalizes the old definition of rigidity and obtains a new definition that holds locally and includes the old one as a limiting case which holds when the velocity of propagation of signals is infinite. If we replace the phrase ‘old definition of rigidity’ with the expression ‘theory of relativity of 1905’ and we assume that the special case is the one without any gravitational field, we actually obtain Einstein’s approach in the years 1907-1915.92 Furthermore, we see another aspect of importance for GR emerging from Born’s definition of rigidity, namely the central position attributed to the concept of invariance. Born was actually led to seek a new (differential) definition of rigidity since the classical one was not Lorentz-invariant. To achieve this goal, he looks for that quantity associated with two points that is invariant with respect to Lorentz transformations and may be considered as the distance between the points. Born acts in the spirit of the new theory, where the guideline for formulating physical laws is no longer represented by Galileau but by Lorentz transformations: by doing so he clearly indicates the significant role played by the concept of invariance in this way of operating. The question of the physical meaning of coordinates represents another analogy in the problems dealt with by Born and Einstein. Stachel pointed out that both of Einstein’s most famous presentations of GR show the same sequence of conceptual steps leading to the origin of the theory: ‘First the discussion of the disk, leading to the conclusion that a non-Euclidean geometry holds on the disk, and therefore space (and time) coordinates cannot be given a direct physical meaning as in the special theory. The two-dimensional Gaussian theory of curved surfaces is recalled, based upon the possibility of introducing entirely arbitrary coordinate systems, and then using the metric tensor to describe the metrical properties of the surface. The analogy with the arbitrary use of space-time coordinates and the metric tensor to mathematically 92We might therefore make out traces work, too (see note 7).

of the generalized

principle

of correspondence

in Born’s

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characterize the gravitational field is then drawn’.93 Coordinates no longer have an immediate physical meaning and this implies that: ‘In the general theory of relativity, space and time cannot be defined in such a way that differences of spatial co-ordinates can be directly measured by the unit measuring-rod, or differences in the time co-ordinate by a standard clock’.94 Now, this impossibility has an immediate consequence on the way we can represent the line element. Even if we can locally ‘transform away’ the gravitational field through a suitable choice of coordinates and can express the line element through Eq. (1 I), this is not possible in the general case, in which Eq. (23) holds; since the circumstance can, as we have seen, be ascribed to the gravitational field, we can attribute to the terms g,, the significance of field potentials. In other words, while Eq. (11) ‘has, by the special theory of relativity, a value which is independent of the orientation of the local system of co-ordinates, and is ascertainable by measurements of space and time’,95 within GR Eq. (11) may hold only locally, i.e. in an infinitesimal region of space-time. This means that it does not hold for an arbitrary frame of reference xi, .... xi related in an infinitesimal region of space-time to the ‘local’ frame by the following linear homogeneous relation between the coordinate differentials: dxi = x a,dxi

(24)

which, when inserted in Eq. (1 l), in fact gives Eq. (23) ‘where the gPV[...I can no longer be dependent on the orientation and the state of motion of the “local” system of co-ordinates, for ds2 is a quantity ascertainable by rod-clock measurements of point-events infinitely proximate in space-time, and defined independently of any particular choice of co-ordinates’.96 The line of reasoning is therefore: (1) start from the ‘old’ expression of the line element, which in the new theory holds only locally; 93Stachel (1980), p. 6. The presentations we refer to are in Einstein (1916a), pp. 773-779, English translation, pp. 115-120, and Einstein (1922), pp. 58863. See also Einstein (1916b), English translation, pp. 79-86. 94‘In der allgemeinen Relativitltstheorie kijnnen Raum- und ZeitgroDen nicht so defmiert werden, da8 raumliche Koordinatendifferenzen unmittelbar mit dem EinheitsmaBstab, zeitliche mit einer Normaluhr gemessen werden kbnnten’, Einstein (1916a), p. 775, English translation, p. 117. 9s‘1...]hat dann nach der soeziellen Relativitltstheorie einen von der Orientieruna des lokalen Koordinatensystems unabh&igen, durch Raum-Zeitmessung ermittelbaren W&t’, Einstein (1916a), p. 778, English translation, p. 119. 96‘[...] wobei die g,,” [...I nicht mehr von der Orientierung und dem Bewegungzustand des “lokalen” Koordinatensystems abhiingen konnen; denn d? ist eine durch MaOstab-Uhremnessung ermittelbare, zu den betrachteten, zeitrlumlich unendlich benachbarten Bunktereignissen gehorige, unabhlngig von jeder besonderen Koordinatenwahl definierte GrSe’, Einstein (1916a), p. 778, English translation, p. 119.

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(2) compare the ‘local’ coordinate system to an arbitrary system, related to the local one through a linear homogeneous relation between the coordinate differentials; (3) derive a general expression for the line element by substituting the relation derived at point (2) in the expression of the line element mentioned at point (1). This sequence is interestingly analogous to Born’s path toward a definition of rigidity in the framework of the principle of relativity. In this case the ‘old’ expression of the line element is: ds2 = dx; + dx; + dx;,

(25)

and holds only locally, while, in general, it has to be replaced with Eq. (11) which is connected to the ‘initial’ system of coordinates rr, . ... t4 through a linear homogeneous relation involving the coordinate differentials:

If we insert Eq. (26) into Eq. (25) we obtain an equation perfectly analogous to Eq. (23), i.e. Eq. (12), which we have already examined. Just as in the transition from SR to GR the expression for ds2 is generalized and the coordinates lose their direct physical meaning, according to Born’s new definition of rigidity the direct physical meaning of the line element is lost since in its expression there is, in general, a term containing the differential of the proper time. The new definition of rigidity can be obtained only by recovering the physical meaning of coordinates, i.e. by imposing that only the differentials of spatial coordinates are to be contained in the expression for ds2, written using the initial coordinates of the flow of matter. Thus, in both cases, due to generalization of the expression for the line element, the meaning of some parts of the old conceptual framework is lost, such as the distance of two points infinitesimally close in a flow of matter or even the physical meaning of coordinates themselves. The problem of the immediate physical meaning of coordinates is linked to another important feature of Einstein’s conception of the physical theory: the possibility of measuring the physical quantities, since the measurability of a physical quantity guarantees that it is physically meaningful. In his lecture delivered to the Nordic Assembly of Naturalists at Gothenburg, on 11 July 1923, Einstein declared that, in the theory of relativity, ‘concepts and distinctions are only admissible to the extent that observable facts can be assigned to them without ambiguity (stipulation that concepts and distinctions should

298

Studies in History and Philosophy of Modern Physics

have meaning). This postulate, pertaining to epistemology, fundamental importance’.97 He then goes on to add:

proves to be of

of the rigid body (and that of the clock) has a key bearing on the foregoing consideration of the fundamentals of mechanics, a bearing which there is some justification for challenging. The rigid body is only approximately achieved in Nature, not even with desired approximation; this concept does not therefore strictly satisfy the ‘stipulation of meaning’. It is also logically unjustifiable to base all physical consideration on the rigid or solid body and then finally reconstruct that body atomically by means of elementary physical laws which in turn have been determined by means of the rigid measuring body. I am mentioning these deficiencies of method because in the same sense they are also a feature of the relativity theory. [...I Certainly it would be logically more correct to begin with the whole of the laws and to apply the ‘stipulation of meaning’ to this whole first, i.e. to put the unambiguous relation to the world of experience last instead of already fulfilling it in an imperfect form for an artificially isolated part, namely the space-time metric. We are not, however, sufficiently advanced in our knowledge of Nature’s elementary laws to adopt this more perfect method without going out of The concept

our depth.98 As we have seen, Einstein had already tackled such a problem in 1907; however, in that case (the paper published in the Jahrbuch) he had rather avoided such an obstacle, assuming that rods and clocks were not influenced by accelerations. We have also seen that Born deemed such a solution inadequate, and that his analysis eventually led to a concept of local validity of the definition of rigidity. In his 1912 theory of the static gravitational field Einstein again avoided directly tackling the problem since, on the one hand, he admitted that rigid bodies cannot really exist, and, on the other hand, he employed as rigid body a set of small non-rigid bodies so aligned alongside each other that they do not exert any pressure forces on each other. However, after the transition to tensors in August 1912, the problem appeared so urgent that it was no longer possible to postpone its solution: in fact, when the expression of d.s becomes, in general, Eq. (23), coordinates no longer possess a direct metric meaning (‘differences of co-ordinates = measurable lengths, viz., times’),99 it is no longer possible to interpret d.s as the infinitesimal distance between two points and, finally, the concept itself of rigid body disappears. The issue was thus tackled during his collaboration with Grossmann; in the section entitled ‘Meaning of the fundamental tensor gPUfor measurements of space and time’ of their joint paper, Einstein remarks: We can assume that between space-time coordinates [...I and clocks and rods we cannot establish such a simple connection as holds in the old theory of relativity (SR). [...I We must therefore face the problem of the physical meaning (measurability in principle) of coordinates. [...I We observe that d.s must be considered as an invariant 97Einstein (1923), English translation, p. 482. 9*Einstein (1923), English translation, pp. 483484. “Schilpp (1949), p. 69.

Definition of Rigidity in the Special Theory of Relativity

299

measure of the distance between two space-time points infinitely close to each other; ds must therefore have a physical meaning independent of the frame of reference chosen. Let us therefore consider ds as the distance ‘naturally measured’ between space-time points.lw

Einstein then introduces a set of coordinate transformations that locally eliminates the gravitational field: ‘In such an elementary reference system the usual theory of relativity holds and the physical meaning of lengths and times is that of the usual theory of relativity, i.e. ds2 is the square of the fourdimensional distance between infinitely close space-time points, measured by means of rigid bodies not accelerated with respect to the reference system’.“” The sense of Einstein’s line of reasoning is the following. In SR we know how to perform measurements, using clocks and rods at rest in the frame of reference where the measurement is being made. In GR we have to take into account the influences of the accelerations on clocks and rods. As we have seen, however, we can always ‘transform away’ locally the gravitational field through a suitable choice of coordinates: in infinitely small regions of space it is therefore possible to refer to the ds of SR (which makes sense as a rigid body) to make the measurements. Einstein postulatedlo that this had always to be possible, i.e. in any gravitational field: this is tantamount to postulating that two infinitesimal rods of equal length, in absence of a gravitational field, are still equal even if one of them is subjected to an arbitrary gravitational field: this is the core of the definition of rigidity in GR.10’

6. Concluding Remarks: The Role of Geometry in the Creation of a Physical Theory Einstein passed to the tensorial theory of gravitation some time between the spring and the summer of 1912. He had probably been aware of the need for such a transition since his last weeks in Prague, before coming back to Zurich ““‘[...]kann man schon entnehmen, dal3 zwischen den Raum-Zeit-Koordinaten [...I und den mittelst Magstlben und Uhren zu erhaltenden MeBergebnissen keine so einfachen Beziehungen bestehen kiinnen, wie in der alten Relativitatstheorie. [...I Es erhebt sich deshalb die Frage nach der physikalischen Bedeutung (prinzipiellen Megbarkeit) der Koordinaten. [...I Hierzu bemerken wir, dal3 ds als invariantes Mat3 fiir den Abstand zweier unendlich benachbarter Raumzeitpunkte aufzufassen ist. Es mu13daher ds such eine vom gewahlten Bezugsystem unabhlngige physikalische Bedeutung zukommen. Wir nehmen an, ds sei der “nattirlich gemessene” Abstand beider Raumzeitpunkte’, Einstein and Grossmann (1913), p. 230. t”“In diesem elementaren System gilt dann die gewohnliche Relativitltstheorie, und es sei in diesem System die physikalische Bedeutung von Llngen und Zeiten dieselbe wie in der gewohnlithen Relativittitstheorie, d.h. d.? ist das Quadrat des vierdimensionalen Abstandes beider unendlich benachbarten Raumzeitpunkte, gemessen mittelst eines im System nicht beschleunigten starren Kiirpers’, ibid. “‘See Einstein (1916a), pp. 777-779, English translation, pp. 118-120. “‘%ee D’Agostino and Orlando (1994). See also Einstein (1916a), pp. 7733779, English translation, pp. 115-120, Einstein (1922) pp. 59-62, and Einstein and Grossmann (1913) pp. 230-231.

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in August 1912. Apart from the other evidence,‘04 Einstein himself bears out this reconstruction in a reminiscence of 1923 where, however, he seems to ascribe the genesis of the decisive idea to the period subsequent to his return to Zurich: I had the decisive idea of the analogy between the mathematical problem connected with the theory [of GR] and the Gaussian theory of surfaces only after my return to Zurich in 1912, without then knowing Riemann’s and R&i’s or Levi-Civita’s investigations. I was made aware of these [investigations] only by my friend Grossmann in Zurich, when I posed to him the problem of finding generally covariant tensors whose components depend only on derivatives of the coefficients [g,J of the quadratic fundamental invariant lgflYdx/IdxV].105 When Einstein first spoke to Grossmann about the problem of gravitation, he already knew what he had to ask, and, in his mind, the velocity of the light c had already been replaced by the terms gpy as the quantities needed to describe the gravitational field. This transition was most likely motivated by a line of thought grounded on the analogies between the problem to be solved and other already known problems showing some common features (local validity of Euclidean geometry, physical meaning of coordinates). We have so far sought to analyze the role played by Born’s work on rigid bodies in SR in putting Einstein on the right track toward the theory of the gravitational field. We grounded our analysis on analogies concerning both the formalism and the substance of the two theoretical schemes. If one opts for the hypothesis of a mere ‘formal coincidence’, it may be possible to restrict oneself to the assumption that the matrix-tensor notation was suggested to Einstein by the need to find a general expression for a property of Euclidean space. In other words, it would have been a necessary solution, given the kind of problem under examination. Alternatively (and this is Pais’s thesis), Einstein might have been inspired by Born’s formalism: in this case, the course on matrix algebra Born attended as a student’06 would have been decisive not only for the birth of quantum mechanics, but also for the creation of GR. If, on the contrary-as we have endeavoured to do-analysis is to be grounded on the substantial analogies between Born’s and Einstein’s ways of reasoning, it is necessary to focus-again, as we have done-on the following aspects: lWPais (1982), pp. 210-211; Stachel (1980), p. 11. i’s‘Den entscheidenden Gedanken von der Analogie des mit der Theorie verbundenen mathematischen Problems mit der Gauss’chen Flbhentheorie hatte ich allerdings erst 1912 nach meiner Rtickkehr nach Zurich, ohne zunbhst Riemanns und Riccis sowie Levi-Civitas Forschungen zu kennen. Auf diese wurde ich erst durch meinen Freund Grossmann in Zurich aufmerksam, als ich ihm das Problem stellte, allgemein kovariante Tensoren aufzusuchen, deren Komponenten nur von Ableitungen der Koellizienten der quadratischen Fundamental-Invariante abhangen’, ‘Vorwort des Autors zur Tschechischen Ausgabe’, in BiEak (1979), p. 42; written in 1922 for a Czech edition of Einstein (1916b). Translation as in Stachel (1986), p. 65. ‘“The above-mentioned course is the one given by Professor Rosanes, and attended by Born at the University of Breslau in 1901 (Born, 1978, pp. 52-53).

Definition of Rigidity in the Special Theory of Relativity

301

(1) Local validity of the condition of rigidity, to be considered as an example of a property generally valid in the old mechanics, holding however only locally in the new one. (2) Loss of meaning of the Newtonian definition of rigidity, due to the presence of terms containing the differential of the proper time in the expression of the line element. This implies that the coordinates lose their immediate physical meaning. It seems quite reasonable to us to assume that Einstein’s attention was drawn to the causes of these analogies, i.e. to deeper and more substantial analogies. In short, the feature ultimately common to the two theories, which we may say embodies their core, is that, in both of them, a physical entity in the Newtonian theory cannot be fully understood in the new theory without taking into account the relationship between geometry and physics. This can be clearly seen both in the loss of direct physical meaning of the coordinates, which Einstein underlined over and over again and considered a crucial transition in the path from the old to the new theory,lo7 and in the new definition of rigid body sought by Born: ‘The rigid body, which in Newtonian physics plays such a decisive role, is actually equivalent to the hypothesis that the distance between any two points of the body remains unchanged during its motion+learly a combination of purely geometrical concepts’.ros All this suggests that we should take into account-and we do so as a concluding remark-the connection between physics and geometry, an issue that Einstein always considered crucial in the development of a physical theory. We wish to begin our analysis by quoting a passage in a paper Einstein published in 1914, entitled ‘Die formale Grundlage der allgemeinen Relativitatstheorie’, where he first discussed the foundations of GR after the transition to the tensorial formalism. We wish to focus on the arguments that Einstein put forward to justify, in principle, the transition to the tensorial theory; these arguments are closely connected with the analysis developed so far and clearly indicate that the rigid body concept played a crucial role in Einstein’s analysis from the very beginning of his work devoted to constructing GR. In the section entitled ‘Critical remarks on the foundations of the theory’, we can read the following remarks about his ideas at the time on the foundations of physics: Before Maxwell, the laws of nature with respect to space were, as a rule, integral laws; we usually express in this way the fact that finite distances between points appeared in the elementary laws. This description of nature is grounded upon Euclidean geometry. First of all, this is nothing other than the substance of the consequences of the geometrical axioms and consequently it does not possess any geometrical content. But geometry becomes a physical science by adding the definition [according to which] “‘See, for example, Schilpp (1949), p. 67 ‘08Lanczos

(1972), p. 7.

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two points of a ‘rigid’ body must be separated by a certain distance, which is independent of the position of the bodies; the propositions of geometry completed with this statement are (in a physical sense) either right or wrong. Geometry is, in this enlarged sense, that thing which is the foundation of physics. From this viewpoint, the geometrical propositions are to be considered integral laws of physics, since they deal with finite distances between points. Through Maxwell’s work, physics has experienced a radical revolution by gradually introducing the requirement that the finite distances between points could no longer appear in the elementary laws, i.e. theories of ‘action at a distance’ are replaced by theories relying upon ‘action by contiguity’. In this process sight was lost of the fact that also Euclidean geometry-as employed in physics-consists of physical propositions, which from a physical viewpoint are to be set precisely at the same level as the integral laws of the Newtonian mechanics of the material point. In my opinion, this represents an inconsistency, which we must get rid of.log Einstein’s

remarks

are particularly

meaningful,

since

they

show

us the

ultimate source, even if not the details of his chain of reasoning concerning the theory of gravitation. First of all, the connection between physics and geometry must

be taken

definition geometry Einstein’s acquires

into

of rigidity

account:

it is accomplished

and it is precisely

through

by means

the ‘bridge’

becomes a physical theory. A fundamental and ever-present thought, the clues to which are not lacking, was that physical

identity

when

coordinates

of the

of such a definition

are considered

that

aspect of geometry

as the result

of

measures carried out with rigid rods. 1lo In his lecture ‘Geometry and Experience’, given in 1921 to the Prussian Academy of Sciences, Einstein stated explicitly that such a completion is necessary for the development

of geometry (through the definition of rigidity) of physics and of GR in particular:

It is clear that the system of concepts of axiomatic geometry alone cannot make any assertion as to the behaviour of real objects [...I which we call practically-rigid bodies. ““Vor Maxwell waren die Naturgesetze in raumlicher Beziehung im Prinzip Integralgesetze: damit sol1 ausgedrtickt werden, dal3 in den Elementargesetzen die Abstlnde zwischen endlich voneinander entfernten Punkten auftraten. Dieser Naturbeschreibung liegt die euklidische Geometrie zugrunde. Letztere bedeutet zunachst nichts als den Inbegriff der Folgerungen aus den geometrischen Axiomen; sie hat insofern keinen physikalischen Inhalt. Die Geometrie wird aber dadurch zu einer physikalischen Wissenschaft, daB man die Bestimmung hinzufiigt, zwei Punkte eines “starren” Kiirpers sollen einen bestimmten von der Lage des Korpers unabhlngigen Abstand realisieren; die Satze der durch diese Festsetzung erglnzten Geometrie sind (im physikalischen Sinne) entweder zutreffend oder unzutreffend. Die Geometrie in diesem erweiterten Sinne ist es, welche der Physik zugrunde liegt. Die Sltze der Geometrie sind von diesem Gesichtpunkte aus als physikalische Integralgesetze anzusehen, indem sie von den Abstanden endlich entfemter Punkte handeln. Durch und seit Maxwell hat die Physik eine durchgreifende Umwalzung erfahren, indem sich allmlhlich die Forderung durchsetzte, daD in den Elementargesetzen Abstlnde endlich entfemter Punkte nicht mehr auftreten diirften; d.h. die ‘Femwirkungs-Theorien’ werden durch ‘Nahewirkungs-Theorien’ ersetzt. Bei diesem ProzeB vergal3 man, dal3 such die euklidische Geomettie-wie sie in der Physik verwendet wird-aus physikalischen Satzen besteht, die den Integralgesetzen der Newtonschen Punktmechanik vom physikalischen Gesichtspunkte aus durchaus an die Seite zu stellen sind. Dies bedeutet nach meiner Ansicht eine Inkonsequenz, von der wir uns befreien miissen’, Einstein (1914), pp. 1079-1080. “‘See for example Schilpp (1949), p. 5.5, Einstein (1956a), essay ‘Geometry and Experience’, and Einstein (1956b), essay ‘Physics and Reality’.

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Definition of Rigidity in the Special Theory of Relativity

To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the coordination of real objects of experience with the empty conceptual schemata of axiomatic geometry. To accomplish this, we need only add the proposition: solid bodies are related, with respect to their possible dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the behavior of practically-rigid bodies. Geometry thus completed is evidently a natural science; we may regard it in fact as the most ancient branch of physics. [...I I attach special importance to the view of geometry which I have just set forth, because without it I should have been unable to formulate the theory of relativity. Without it the following reflection would have been impossible: in a system of reference rotating relatively to an inertial system, the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction; thus if we admit non-inertial systems on an equal footing, we must abandon Euclidean geometry.ili

In a later essay, Einstein formulated the ‘logical and axiomatic, but also [...I the empirical content’ of Euclidean geometry as follows: The whole content of Euclidean geometry can axiomatically be founded upon the following statements: (1) Two specified points of a rigid body determine a distance. (2) We may coordinate triplets of numbers x,,x2,x3 to points of space in such a manner that for every distance P’P” under consideration, the coordinates of whose end points are x;,X;,x;; x’;,x’&x’& the expression

s2= (x;’ is independent bodies.ii*

-

q2 + (xi’

-

q2 + (xi’

-

xg2

of the position of the body and of the positions of any and all other

The fact is that abandonment is fatal for physical

or oblivion

of the empirical

content

of geometry

theories:

The fatal error that the necessity of thinking, preceding all experience, was at the basis of Euclidean geometry and the concept of space belonging to it, this fatal error arose from the fact the empirical basis, on which the axiomatic construction of Euclidean geometry rests, had fallen into oblivion. In so far as one can speak of the existence of rigid bodies in nature, Euclidean geometry is a physical science, the usefulness of which must be shown by application to sense experiencesir Returning to the concepts expressed in the 1914 paper ‘Die formale Grundlage . ..’ it seems to us that if, as Einstein asserts, geometry is a branch of physics, then it will have to suffer from the changes in the latter. A substantial change was introduced into physics by Maxwell’s theory, through the transition from integral to differential laws. If, then, physics shifts from integral to “‘Einstein (1956a), “*Einstein (1956b), the Franklin Institute ‘13Einstein (1956b),

essay ‘Geometry and Experience’, pp. 23k235. essay ‘Physics and Reality’, p. 67. This essay was published in the Journal of 221, No. 3, March 1936. essay ‘Physics and Reality’, p. 68.

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differential laws, this must also affect geometry, connected to physics through the definition of rigidity. In the framework of a physical science resting upon differential laws, geometry can no longer be connected to it through a definition of rigidity based upon finite distances between points: ‘an inconsistency, which we have to get rid of’. All this inevitably leads us to the problem of defining rigidity in SR-which had clearly emerged much before Born’s work-and to Born’s solution to this problem, involving transition from an integral to a differential definition of rigidity. If the definition of rigidity needed by physical theory must change, then this must have repercussions on geometry, connected to physics by means of this definition. The existence of inconsistencies in the foundations of current physical theory always spurred Einstein to devise theoretical schemes where such discrepancies could be removed: it may suffice to consider the role played in the genesis of SR and GR by the inconsistencies bound up with the privileged position of the ether rest frame and of inertial frames of reference, respectively. As far as the definition of rigidity itself is concerned, acting as it did as a link between physics and geometry it stimulated the change in the geometry that was to be associated with the space-time continuum after the physics to be described in such a continuum had changed. It is in this last and more general sense of removing inconsistencies from the foundations of the theory that, in our opinion, lies the general influence of Born’s definition of rigidity on the genesis of general relativity. Acknowledgements-We thank Silvio Bergia and Michelangelo De Maria for reading an early version of the manuscript and for providing us with helpful suggestions. We are also grateful to Giorgio Dragoni for some bibliographical help.

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