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Hilbert's Foundation of Physics: from a Theory of Everything to A MAX-PLANCK-INSTITUT FÜR WISSENSCHAFTSGESCHICHTE Max Planck Institute for the History of Science PREPRINT 118 (1999) Jürgen Renn and John Stachel Hilbert’s Foundation of Physics: From a Theory of Everything to a Constituent of General Relativity ISSN 0948-9444 HILBERT’S FOUNDATION OF PHYSICS: FROM A THEORY OF EVERYTHING TO A CONSTITUENT OF GENERAL RELATIVITY JÜRGEN RENN AND JOHN STACHEL EDITED BY STEFAN HAJDUK 1. ON THE COMING INTO BEING AND FADING AWAY OF AN ALTERNATIVE POINT OF VIEW The legend of a royal road to general relativity Hilbert is commonly seen as having publicly presented the derivation of the field equa- tions of general relativity five days before Einstein on 20 November 1915 – after only half a year’s work on the subject in contrast to Einstein’s eight years of hardship from 1907 to 1915.1 We thus read in Kip Thorne’s fascinating account of recent developments in gen- eral relativity:2 Remarkably, Einstein was not the first to discover the correct form of the law of warpage [of space-time, i.e. the gravitational field equations], the form that obeys his relativity principle. Recognition for the first discovery must go to Hilbert. In autumn 1915, even as Einstein was struggling toward the right law, making mathematical mistake after mistake, Hilbert was mulling over the things he had learned from Einstein’s summer visit to Göt- tingen. While he was on an autumn vacation on the island of Rugen in the Baltic the key idea came to him, and within a few weeks he had the right law–derived not by the arduous trial-and-error path of Einstein, but by an elegant, succinct mathematical route. Hilbert presented his derivation and the resulting law at a meeting of the Royal Academy of Sci- ences in Göttingen on 20 November 1915, just five days before Einstein’s presentation of the same law at the Prussian Academy meeting in Berlin. Hilbert himself emphasized that he actually had two separate starting points for his approach, Mie’s electromagnetic theory of matter as well as Einstein’s attempt to base a theory of gravitation on the metric tensor. Hilbert’s superior mastery of mathematics apparently allowed him to arrive quickly and independently at combined field equations for the electromagnetic and gravitational fields. Although his use of Mie’s ideas initially led Hilbert to a theory that was, from the point of view of later general relativity, restricted to a particular source for the gravitational field, the electromagnetic field, he is nevertheless regarded by many historians of science and physicists as the first to have 1 For discussions of Einstein’s path to general relativity see Norton 1984 and Renn and Sauer 1998. For historical reviews of Hilbert’s contribution, see Guth 1970; Mehra 1974; Earman and Glymour 1978; Pais 1982, pp. 257-261; Corry 1996; Corry, Renn, and Stachel 1997; Corry 1997; Corry 1999a; Sauer 1999, Corry (Forthcoming); and Corry 1999b. 2 Thorne 1994, p. 117; for a similar account see Fölsing 1997, pp. 375-376. 1 2 FROM A THEORY OF EVERYTHING TO GENERAL RELATIVITY established a mathematical framework for general relativity that provides both essential results of the theory, such as the field equations, and a clarification of previously obscure conceptual issues, such as the nature of causality in generally covariant field theories.3 His contributions to general relativity, although initially inspired by Mie and Einstein, hence appear as a unique and independent achievement. In addition, Hilbert is seen by some as initiating the subsequent search for unified field theories of gravitation and elec- tromagnetism.4 In view of all these results, established within a very short time, it appears that Hilbert indeed had found an independent “royal road” to general relativity and beyond. In a recent paper we have shown that Hilbert actually did not anticipate Einstein in pre- senting the field equations.5 Our argument is based on the analysis of a set of proofs of Hilbert’s first paper; in the following they are referred to as the “Proofs.”6 These Proofs not only do not include the explicit form of the field equations of general relativity, but they also show Hilbert’s original theory to be in many ways closer to the earlier, non- covariant versions of Einstein’s theory of gravitation than to general relativity. It was only after the publication on 2 December 1915 of Einstein’s definitive paper that Hilbert mod- ified his theory in such a way that his results were in accord with those of Einstein.7 The final version of his first paper, which was not published until March 1916, now includes the explicit field equations and has no restriction on general covariance.8 Hilbert’s second paper, a sequel to his first communication, in which he first discussed causality, appar- ently also underwent a major revision before eventually being published in 1917.9 The transformation of the meaning of Hilbert’s work Hilbert presented his contribution as emerging from a research program that was entirely his own – the search for an axiomatization of physics as a whole creating a synthesis of electromagnetism and gravitation. This view of his achievement was shared by Felix 3 See Howard and Norton 1993. 4 See, for example, Vizgin 1989, who refers to “Hilbert’s 1915 unified field theory, in which the attempt was first made to unite gravitation and electromagnetism on the basis of the general theory of relativity“ (see p. 301). 5 Corry, Renn, and Stachel 1997. 6 A copy of the proofs of Hilbert’s first paper is preserved at Göttingen, in SUB Cod. Ms. 634. They comprise 13 pages and are virtually complete, apart from the fact that roughly the upper quarter of two pages (7 and 8) is cut. The Proofs are dated “submitted on 20 November 1915.” The Göttingen copy bears a printer’s stamp dated 6 December 1915 and is marked in Hilbert’s own hand “First proofs of my first note.” In addition, the Proofs carry several marginal notes in Hilbert’s hand which are discussed below. 7 The conclusive paper is Einstein 1915e. 8 Hilbert 1915. In the following referred to as Paper 1. 9 Hilbert 1917. In the following referred to as Paper 2. HILBERT’S FOUNDATION OF PHYSICS 3 Klein who took the distinctiveness of Hilbert’s approach also as an argument against see- ing it from the perspective of a priority competition with Einstein:10 Von einer Prioritätsfrage kann dabei keine Rede sein, weil beide Autoren ganz verschie- dene Gedankengänge verfolgen (und zwar so, daß die Verträglichkeit der Resultate zunächst nicht einmal sicher schien). Einstein geht induktiv vor und denkt gleich an belie- bige materielle Systeme. Hilbert deduziert, indem er übrigens die [...] Beschränkung auf Elektrodynamik eintreten läßt, aus voraufgestellten obersten Variationsprinzipien. Hilbert hat dabei insbesondere auch an Mie angeknüpft. It is indeed clear that both Hilbert’s original programmatic aims as well as the interpreta- tion he gave of his own results do not fit into the framework of general relativity as we understand it today, even if one disregards the non-covariant version of his theory as pre- sented in the proofs version of his first paper. As we shall discuss in detail below, in the context of Hilbert’s attempt at a synthesis of electromagnetism and gravitation theory, he interpreted, for instance, the contracted Bianchi identities as substituting for the funda- mental equations of electromagnetism, an interpretation that was soon recognized to be problematic by Hilbert himself. With hindsight, however, there can be little doubt that a number of important contribu- tions to the development of general relativity do have roots in Hilbert’s work, not so much the mere fact of a variational formulation of gravitational field equations, which had already been introduced by Einstein, but, for instance, the relation of the gravitational Lagrangian to the Ricci scalar, and first hints of Noether’s theorem. The intrinsic plausibility of both of these two perspectives, seeing Hilbert’s work as aim- ing at a theory different from general relativity or as a contribution to general relativity, respectively, represents a puzzle. In fact, how can Hilbert’s contributions be interpreted as making sense only within an independent research program, different in essence from that of Einstein, if they ultimately came to be seen, at least by most physicists, as constituents of general relativity? This puzzle raises a profound historical question concerning the nature of scientific development: how were Hilbert’s results, produced within a research program originally aiming at an electrodynamic foundation of all of physics, eventually transformed into constituents of general relativity, a theory of gravitation? The pursuit of this question promises insights into the processes by which scientific results acquire and change their meaning and, in particular, into the process by which a viewpoint emerges and eventually fades away that is different from the one eventually accepted as main- stream. Hilbert’s work on the foundations of physics turns out to be especially suited for such an analysis, not only because the proofs version of his first paper provides us with a previ- ously–unknown point of departure for following his development, but also because he 10 Klein 1921, p. 566. (The text was originally published in 1917; see Klein 1917. The quote is from a footnote to remarks added to the 1921 republication.) For a recent reconstruction of Hil- bert’s perspective, see Sauer 1999. 4 FROM A THEORY OF EVERYTHING TO GENERAL RELATIVITY came back time and again to his original papers, rewriting them in terms of the insights he had meanwhile acquired and in the light of the developments of Einstein’s “mainstream” program.
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