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Einstein, Nordström and the Early Demise of Scalar, Lorentz Covariant Theories of Gravitation

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Einstein, Nordström and the Early Demise of Scalar, Lorentz Covariant Theories of Gravitation JOHN D. NORTON EINSTEIN, NORDSTRÖM AND THE EARLY DEMISE OF SCALAR, LORENTZ COVARIANT THEORIES OF GRAVITATION 1. INTRODUCTION The advent of the special theory of relativity in 1905 brought many problems for the physics community. One, it seemed, would not be a great source of trouble. It was the problem of reconciling Newtonian gravitation theory with the new theory of space and time. Indeed it seemed that Newtonian theory could be rendered compatible with special relativity by any number of small modifications, each of which would be unlikely to lead to any significant deviations from the empirically testable conse- quences of Newtonian theory.1 Einstein’s response to this problem is now legend. He decided almost immediately to abandon the search for a Lorentz covariant gravitation theory, for he had failed to construct such a theory that was compatible with the equality of inertial and gravitational mass. Positing what he later called the principle of equivalence, he decided that gravitation theory held the key to repairing what he perceived as the defect of the special theory of relativity—its relativity principle failed to apply to accelerated motion. He advanced a novel gravitation theory in which the gravitational potential was the now variable speed of light and in which special relativity held only as a limiting case. It is almost impossible for modern readers to view this story with their vision unclouded by the knowledge that Einstein’s fantastic 1907 speculations would lead to his greatest scientific success, the general theory of relativity. Yet, as we shall see, in 1 In the historical period under consideration, there was no single label for a gravitation theory compat- ible with special relativity. The Einstein of 1907 would have talked of the compatibility of gravitation and the principle of relativity, since he then tended to use the term “principle of relativity” where we would now use “theory of relativity”. See (CPAE 2, 254). Minkowski (1908, 90) however, talked of reform “in accordance with the world postulate.” Nordström (1912, 1126), like Einstein, spoke of “adapting ... the theory of gravitation to the principle of relativity” or (Nordström 1913, 872) of “treat- ing gravitational phenomena from the standpoint of the theory of relativity,” emphasizing in both cases that he planned to do so retaining the constancy of the speed of light in order to distinguish his work from Einstein’s and Abraham’s. For clarity I shall describe gravitation theories compatible with special relativity by the old-fashioned but still anachronistic label “Lorentz covariant.” It describes exactly the goal of research, a gravitation theory whose equations are covariant under Lorentz trans- formation. For a simplified presentation of the material in this chapter, see also (Norton 1993). Jürgen Renn (ed.), The Genesis of General Relativity Vol. 3: Theories of Gravitation in the Twilight of Classical Physics. Part I. © 2005 Kluwer Academic Publishers: Printed in the Netherlands. 2JOHN D. NORTON 1907 Einstein had only the slenderest of grounds for judging all Lorentz covariant gravitation theories unacceptable. His 1907 judgement was clearly overly hasty. It was found quite soon that one could construct Lorentz covariant gravitation theories satisfying the equality of inertial and gravitational mass without great difficulty. Nonetheless we now do believe that Einstein was right in so far as a thorough pursuit of Lorentz covariant gravitation theories does lead us inexorably to abandon special relativity. In the picturesque wording of Misner et al. (1973, Ch.7) “gravity bursts out of special relativity.” These facts raise some interesting questions. As Einstein sped towards his general theory of relativity in the period 1907–1915 did he reassess his original, hasty 1907 judgement of the inadequacy of Lorentz covariant gravitation theories? In particular, what of the most naturally suggested Lorentz covariant gravitation theory, one in which the gravitational field was represented by a scalar field and the differential operators of the Newtonian theory were replaced by their Lorentz covariant counter- parts? Where does this theory lead? Did the Einstein of the early 1910s have good reason to expect that developing this theory would lead outside special relativity? This paper provides the answers to these questions. They arise in circumstances surrounding a gravitation theory, developed in 1912–1914, by the Finnish physicist Gunnar Nordström. It was one of a number of more conservative gravitation theories advanced during this period. Nordström advanced this most conservative scalar, Lorentz covariant gravitation theory and developed it so that it incorporated the equality of inertial and gravitation mass. It turned out that even in this most conserva- tive approach, odd things happened to space and time. In particular, the lengths of rods and the rates of clocks turn out to be affected by the gravitational field, so that the spaces and times of the theory’s background Minkowski spacetime ceased to be directly measurable. The dénouement of the story came in early 1914. It was shown that this conservative path led to the same sort of gravitation theory as did Einstein’s more extravagant speculations on generalizing the principle of relativity. It lead to a theory, akin to general relativity, in which gravitation was incorporated into a dynam- ical spacetime background. If one abandoned the inaccessible background of Minkowski spacetime and simply assumed that the spacetime of the theory was the one revealed by idealized rod and clock measurements, then it turned out that the gravitation theory was actually the theory of a spacetime that was only conformally flat—gravitation had burst out of special relativity. Most strikingly the theory’s gravi- tational field equation was an equation strongly reminiscent to modern readers of the field equations of general relativity: RkT= where RT is the Riemann curvature scalar and the trace of the stress-energy tensor. This equation was revealed before Einstein had advanced the generally covariant field equations of general relativity, at a time in which he believed that no such field equa- tions could be physically acceptable. What makes the story especially interesting are the two leading players other than Nordström. The first was Einstein himself. He was in continued contact with Nord- EINSTEIN, NORDSTRÖM AND THE EARLY DEMISE ... 3 ström during the period in which the Nordström theory was developed. We shall see that the theory actually evolved through a continued exchange between them, with Einstein often supplying ideas decisive to the development of the theory. Thus the theory might more accurately be called the “Einstein-Nordström theory.” Again it was Einstein in collaboration with Adriaan Fokker who revealed in early 1914 the connection between the theory and conformally flat spacetimes. The second leading player other than Nordström was not a person but a branch of special relativity, the relativistic mechanics of stressed bodies. This study was under intensive development at this time and had proven to be a locus of remarkably non- classical results. For example it turned out that a moving body would acquire addi- tional energy, inertia and momentum simply by being subjected to stresses, even if the stresses did not elastically deform the body. The latest results of these studies— most notably those of Laue—provided Einstein and Nordström with the means of incorporating the equality of inertial and gravitational mass into their theory. It was also the analysis of stressed bodies within the theory that led directly to the conclu- sion that even idealized rods and clocks could not measure the background Minkowski spacetime directly but must be affected by the gravitational field. For Ein- stein and Nordström concluded that a body would also acquire a gravitational mass if subjected to non-deforming stresses and that one had to assume that such a body would alter its size in moving through the gravitational field on pain of violating the law of conservation of energy. Finally we shall see that the requirement of equality of inertial and gravitational mass is a persistent theme of Einstein’s and Nordström’s work. However the require- ment proves somewhat elastic with both Einstein and Nordström drifting between conflicting versions of it. It will be convenient to prepare the reader by collecting and stating the relevant versions here. On the observational level, the equality could be taken as requiring: • Uniqueness of free fall:2 The trajectories of free fall of all bodies are independent of their internal constitution. Einstein preferred a more restrictive version: • Independence of vertical acceleration: The vertical acceleration of bodies in free fall is independent of their constitutions and horizontal velocities. In attempting to devise theories compatible with these observational requirements, Einstein and Nordström considered requiring equality of gravitational mass with • inertial rest mass • the inertial mass of closed systems • the inertial mass of complete static systems • the inertial mass of a a complete stationary systems3 2 This name is drawn from (Misner et al. 1973, 1050). 3 The notions of complete static and complete stationary systems arise in the context of the mechanics of stressed bodies and are discussed in Sections 9 and 12 below. 4JOHN D. NORTON More often than not these theoretical requirements failed to bring about the desired observational consequences. Unfortunately it is often unclear precisely which requirement is intended when the equality of inertial and gravitational mass was invoked. 2. THE PROBLEM OF GRAVITATION IMMEDIATELY AFTER 1905 In the years immediately following 1905 it was hard to see that there would be any special problem in modifying Newtonian gravitation theory in order to bring it into accord with the special theory of relativity. The problem was not whether it could be done, but how to choose the best of the many possibilities perceived, given the expec- tation that relativistic corrections to Newtonian theory might not have measurable consequences even in the very sensitive domain of planetary astronomy.
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