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The Fourteenth Marcel Grossmann Meeting on Recent September 4, 2017 9:33 ws-procs961x669 MG-14 – Proceedings (Part D) D429 page 3348 3348 A brief history of the energy-momentum tensor 1912-1915: Einstein physicist’s logic in compromising gravitation with relativity Jean-Pierre Provost∗ Universit´e de Nice Sophia Antipolis, Nice, 06108/France ∗E-mail: [email protected] In MG13 (Stockholm 2012) we discussed how the history (1900-1911) of relativistic dy- namics was intimately linked to issues concerning energy momentum tensors, in particu- lar the electromagnetic one. Here we examine how, for Einstein, the expected properties of the matter and gravitational field tensors have been sources of inspiration strongly conditioning his quest for a relativistic description of gravitation. It is in this physical framework already present in 1912 that in competition with Hilbert at the fall of 1915, he thinks to have justified general covariance. Keywords: Einstein; General Relativity; Energy-momentum tensor. 1. Introduction: The point of view of energy momentum tensors (E.M.T) The history of general relativity (G.R) has been extensively studied by historians and philosophers and has become a ‘well written subject’ as T. Sauer said in his MG13’s talk 1 referring to Ref. 2. But in our opinion Einstein’s path to G.R has too often been written from the mathematical perspective of the final result, with Einstein ‘stumbling’ (J.D. Norton) or following a ‘bumpy road’ (M. Janssen) before touching the destination of general covariance. As we shall emphasize, the physical point of view of the role played by the energy momentum tensor (E.M.T) in Ein- stein’s quest for conciliating relativity with gravitation is unavoidable because it is at the basis of his logic from 1912 to 1915. Let us recall that Einstein is primarily a physicist who has always apprehended mathematics in his own way, as much as possible in physical terms (see for example his correspondence with E. Cartan 3). In addition, as presented in Ref. 4, the point of view of the E.M.T allows an interesting unified treatment of the period 1900-1915 which has seen the study of electromag- The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.com netic and gravitational processes leading respectively to S.R and G.R with in both cases the central issue of inertial space time and relativity, the pre-existence of well adapted mathematics (group theory, general differential calculus) and the col- laboration of great physicists and mathematicians (Lorentz, Planck, Einstein . , Poincar´e,Levi-Civita,Hilbert,...). by GERMAN ELECTRON SYNCHROTRON on 04/25/21. Re-use and distribution is strictly not permitted, except for Open Access articles. 2. 1912: Einstein realizes the importance of EMTs As we recalled in MG13 5, 1912 is a remarkable date for relativity and gravitation because it marks the end of the Relativistic Dynamics and the beginning of G.R, both involving the E.M.T in a crucial way. When turning from a scalar to a tensor theory, Einstein acknowledges that: September 4, 2017 9:33 ws-procs961x669 MG-14 – Proceedings (Part D) D429 page 3349 3349 μν ν ‘the general validity of the conservation laws [(C.L) ∂μT = f generalizing New- μ μ μν μ ν ton’s law dP /dτ = F for T = ρ0u u (incoherent masses for Einstein)] and the law of inertia [identity T i0 = T 0i of energy current and momentum density extending p = vE] suggest that they are to be ascribed a general significance even though they are obtained in a very special case [electromagnetism (e.m)]. We owe this generalization which is the most important new advance in the theory of rel- ativity to the investigations of Minkowski, Planck and Laue [(cf. Ref. 6, chap. 7) μν 0μ μ which discusses the necessity of a condition ∂μT =0for T = P to be a quadrivector, i.e. for connecting continuous and point mechanics 5]. The problem to be solved always consists of finding how ...[Tμν ]...istobefoundfromvariables characterizing the processes under consideration’ 7. In Einstein’s mind, these processes are surely gravitational ones to which Planck already referred in 1908. In the following, tμν will denote the gravitational tensor. 3. 1915: Official birth of G.R. 1915, 100 years ago, corresponds to the first writing, on Nov. 25th of Einstein’s equations 8 λ α λ 1 Rμν =Γ +Γ Γ = − Tμν − gμν T , μν,λ μλ να 2 although this writing is doubly non covariant since the trace term concerns Tμν and not Rμν (Ricci tensor) and since Rμν is written for coordinates such that g = −1 (g determinant of the metric). With respect to this non covariance, Einstein’s con- clusion of his paper is much surprising: αβ ‘Our additional term ensures that these equations [∂μ[∂α∂βg − (T + t)] = 0 , new writing of two years old vivid constraints on the theory presented below] carry no additional conditions [other than the field eq.]. We thus need not make other hypotheses about the energy momentum tensor of matter [such as T =0onNov. 4th, erroneously justified on Nov. 11th] other than it complies with the energy mo- ν ν mentum theorem [∂ν (Tμ + tμ) = 0]. With this we have finally completed the general The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.com theory of relativity as a logical structure [underlined by us]’. At that time Einstein is in competition with Hilbert and it is probably to distin- guish his physical approach from what he has heard from Hilbert (Weltparameter, Weltfunction, Axiom von der allgemeinen Invarianz, axiomatischen method) that he adds that ‘[The theory] makes space time coordinates into physically meaningless parameters by GERMAN ELECTRON SYNCHROTRON on 04/25/21. Re-use and distribution is strictly not permitted, except for Open Access articles. ...leadswithcompelling necessity to a specific theory of gravitation that explains ...theperihelionofMercury[success of the 18th Nov. paper]...[It] cannot reveal to us anything new ...thatthespecialtheorytoldusalready’. If Einstein feels his completion of G.R to be logical, it is necessary in priority to examine this logic. September 4, 2017 9:33 ws-procs961x669 MG-14 – Proceedings (Part D) D429 page 3350 3350 4. Einstein’s logic in the period 1913-1915 From 1913 to 1915, this logic lies on three physical ideas which, as we discuss below, are already present in 1912: first Matter, whatever it is, must gravitate in the way μ α β particles do, i.e. according to the geodesic eq. dP /dτ =(∂μgαβ)u u /2whichin tensor language reads √ ν 1√ αβ ν ∂ν ( −gT )= −g(∂μgαβ)T (∇ν T = 0 for Grossmann). (1) μ 2 μ Second, the gravitational field must contribute to the C.L of energy momentum √ √ ν ν 0 0 ∂ν ( −g(Tμ + tμ)) = 0 (hence −g(Tμ + tμ)=Pμ = quadrivector). (2) Finally Poisson eq. Δϕ = ρ must be a limit case of an equation involving the tensors Γμν = Tμν (3) or better Δμν = Tμν + tμν . (4) This logic is accompanied by two procedures for deriving (3). In 1913 (the Entwurf) 7 Einstein guesses (4), i.e. both Δμν and tμν on the basis of Grossmann’s identity (to which we shall come back) and he affects to discover (2); then he realizes the √ ν constraint ∂ν ( −gΔμ) = 0 due to (2) and (4) and he tries to think of it in terms of relativity: ‘adapted’ coordinates changes extending affine relativity on one side; restriction of general covariance through the hole argument on the other side. In th 1915 he postulates (3), setting Γμν = −Rμν on Nov. 4 and introducing the trace term on Nov. 25th and he also postulates g = −1, which is not much in the spirit of general covariance, but will reveal to be crucial; then he calculates αβ αβ ν (∂μgαβ)Γ and using (1) which involves (∂μgαβ)T , he determines tμ by imposing (2). Everything works on Nov. 25th (no constraint), which implies that something was wrong before; was it an unfortunate physical hypothesis or more seriously, a mathematical inconsistency? Clearly, it was the latter since if Tμν satisfies (1), whatever its physical or geometrical writing, one must absolutely verify that Γμν , equal to Tμν , satisfies (1). Maybe Grossmann has been aware of the problem and this The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.com would be a reason why he makes a separate contribution in the Entwurf. In 1913, ν calculating ∇ν Tμ with Tμν =Δμν − tμν and complicate expressions for Δμν and tμν was impossible, but in 1915 this became tractable thanks to the simplification g = −1. In our opinion, Einstein may have realized the above inconsistency as a contradiction in his calculations between Nov. 11th and Nov. 25th and this is logically also the origin for his introduction of the trace term. Of course, he and by GERMAN ELECTRON SYNCHROTRON on 04/25/21. Re-use and distribution is strictly not permitted, except for Open Access articles. historians after him have never claimed it, but the way Einstein clouds the issue on Nov. 25th testifies his trouble: ‘Thereasonsthatmotivatedmetointroducethesecondterm...willonlybecome transparent in what follows, but they are completely analogous to those just quoted [on Nov. 4th, which is contradictory]’. September 4, 2017 9:33 ws-procs961x669 MG-14 – Proceedings (Part D) D429 page 3351 3351 In particular the first reason is playing with words: ‘...ouradditionaltermissuchthatthetensorofthegravitationalfieldoccursin αβ (9) [eq. ∂α∂βg = T + t which is the contraction of the field eq.] on footing equal νλ α with the one of matter, which was not the case in eq. 21.1.c [eq. (∂αg )Γνμ − 1 δλgρσ α β − T λ tλ th 2 μ Γρβ Γσα = ( μ + μ)whichistheNov.4 form of (4) where the left hand λ side could be taken as the field part Δμ (although the second term in it is also part λ of tμ )]’.
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