The Fourteenth Marcel Grossmann Meeting on Recent

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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 dynamics was intimately linked to issues concerning energy momentum tensors, in particular 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

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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 relativity 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 conclusion 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

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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

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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 )]’. The second argument (absence of constraints here above mentioned) is of course more serious.

5. 1907-1913: The priority to physics As Einstein said after 1915, he and Grossmann were close to the solution in their numerous attempts to adapt tensor theory to Poisson eq. quoted in the Zürich notebook 2. But it is precisely because of their failure that he came back to physics. As we now see, Grossmann’s capital identity in the Entwurf is the result of Ein- stein’s previous reflexions on gravitation and acceleration, the so called ‘principle of equivalence’. For physicists, this principle goes back to Galileo and Newton and Einstein’s originality has been to apply it to clocks (1907), to e.m (1907, 1912) and even to quanta (1911), thanks to his expertise in (non) simultaneity (a necessity for apprehending physics in an accelerated frame Σ considered as a succession of inertial ones). From the relation between proper times and gravitational potential τ(r)=τ(0)(1 + ϕ(r)) and the convention t = τ(0) for time in Σ, he writes Maxwell eq. in Σ as rot(cE)=−∂tB, rot(cB)=∂tE with c(r)=1+ϕ(r) and notes with satisfaction that the e.m field, like matter, experiences gravitational energy ρemϕ and force −ρem∇ϕ. Physics is also present when Einstein explains the Doppler paradox ν(0) = ν(h)(1 + gh) = ν(h) (for light falling in a gravitational field) by the homogeneous time t in Σ not being given by identical ‘pocket clocks’. Even in 1912 7, when he looks for the changes of coordinates bringing ds2 = c2(r)dt2 −dr2 to 2 dt2 − dr , his major aim is to deduce the Mechanics in Σ from the inertial motions in S. In particular, the relation E = cmγ m(1 − v2/2+ϕ) in the Newtonian

The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.com limit will lead him to rewrite Poisson eq. in the form Δc = ρ+ρϕ,orΔc = ρc.This eq. will soon appear in 1912 to be a serious error, and its solution (the introduction ν of the E.M.T for gravitation tμ in the Entwurf) to be for Einstein physically more pertinent than chasing changes of dubious coordinates. Thanks to Laue, Einstein realizes that the total force −ρ∇ϕ diﬀers from zero if Δc = cρ, which violates the action-reaction principle raising ‘doubts as to the admissibility of the entire theory’.

by GERMAN ELECTRON SYNCHROTRON on 04/25/21. Re-use and distribution is strictly not permitted, except for Open Access articles. After envisaging some escapes, he suggests the new eq.

1 (∇c)2 Δc = cρ + = e (scalar form of (4)) 2 c September 4, 2017 9:33 ws-procs961x669 MG-14 – Proceedings (Part D) D429 page 3352

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and justiﬁes it by the energy conservation ρ∇ϕδr = δe (=0 if δr is a constant) and by the momentum conservation (which also implies ρ∇c =0):

(∂ic)(∂j c) (∂ic)(∂ic) Δc (∂ic)(∂ic) ∂i − ∂j =(∂j c) − (= ρ∇c). (5) c 2c c 2c2

This identity, which is not quite written explicitly in this form by Einstein in his paper, but is implicitly present, is the only way one can understand his indications in the Entwurf for obtaining the ﬁeld eq., namely

1√ αβ μν ‘sum of diﬀerential quotients’ = −g(∂σgμν )[∂α(g ∂βg ) + ‘corrections’] 2 αβ μν (in as much as he shows that ∂α(g ∂βg )goestoΔϕ in the Newtonian limit). It is also the only way one can understand his miraculous reading of Grossmann’s identity (for very complicated tμν , Δμν see Ref. 4)

√ α 1√ μν μν √ ν ν ∂α( −gt )=− −g(∂σgμν )[t − Δ ] leading to ∂ν ( −g(T + t )) = 0. σ 2 μ μ Certainly (5) does not occur in the Entwurf . In the latter, Einstein prefers to assert that in electrostatics the force −ρ∇ϕ and the identity ∂i[(∂iϕ)(∂j ϕ)] − ∂ 1 ∂ ϕ ∂ ϕ ∂ ϕ ϕ ϕ −ρ j 2 ( i )( i ) =( j )Δ uniquely imply Δ = (without realizing that an implicit hypothesis on geometry, here the Euclidean one, is central for this identity, which makes it not unique).

6. Conclusion: Einstein’s completion of G.R as an interesting ‘negative result’ In conclusion, and although it is not the standard way to look at it, Einstein’s ‘completion’ of G.R as a ‘logical structure ’ has dealt with issues concerning energy momentum tensors: the gravitational one introduced in 1912 in order to satisfy conservation laws, a scenario used again in the Entwurf and verified soon after in Nord- ström theory 4; the matter one whose fundamental property (1) has become crucial The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.com in Nov. 1915 for Einstein realizing a serious inconstancy in his previous work. If at the fall of 1912, tensor calculus has appeared to him as an urgent necessity (‘Help me...’toGrossmann),thishasnotbeentheideaofgeneralcovariance(whichcor- responds to what Poincaré in 1912 called ‘psychological relativity ’ and opposed to ‘Lorentz physical relativity’) but for technical reasons. After the failure quoted in the Zürich notebook, paradoxically due not to an insufficient mathematical knowl- by GERMAN ELECTRON SYNCHROTRON on 04/25/21. Re-use and distribution is strictly not permitted, except for Open Access articles. edge but to a lack of issue in 1911 and a wrong hypothesis in 1912 concerning the spatial metric, Einstein has made relativity to depend on his physical approach. Relations such as R = −T for Nordström theory in 1914 or his call for Rμν and g = −1 in 1915 are far from an adhesion to general covariance. When Einstein claims it on Nov. 25th it is not as a research program like for Hilbert but rather as September 4, 2017 9:33 ws-procs961x669 MG-14 – Proceedings (Part D) D429 page 3353

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an interesting ‘negative result’ (according to T. Paul’s terminology) making ‘space and time coordinates into physically meaningless parameters’ or ‘which takes away from space and time the last remnant of physical relativity’ (1916). Modern physicists like to see in this claim the necessity for defining local inertial spacetime by a tetrad field, which replaces the metric one (standard gauge theory of gravitation) or which acts as a connection on more fundamental objects such as spinors (gauge theories with a cosmological constant). Considering general covariance in physics as a revolution (in Bachelard’s sense) would be surely an epistemological error. The issue of inertia first raised by Kepler and Galileo is far from having come to an end.

References 1. T. Sauer, Marcel Grossmann and his contribution to the General Theory of Relativity, in MG 13 proceedings,eds.R.T.Jantzen,K.RosquistandR.Ruffini(WordScientific, Singapore, 2015), pp. 456-503. 2. J. Renn (ed.), The genesis of general relativity (Springer, 2007). 3. Albert Einstein, Elie Cartan, Letters on Absolute parallelism, 1929-1932 (Princeton University Press, Princeton, 1979). 4. J.-P. Provost and C. Bracco, Une brève histoire du tenseur énergie-impulsion: 1900- 1915, in Théorie quantique des champs: Méthodes et applications, Actes de l’école de physique théorique de Jijel 2009 (Hermann, Paris, 2013), pp. 417-449. 5. J.-P. Provost, A brief history of the energy-momentum tensor; 1900-1912, in MG 13 proceedings, eds. R. T. Jantzen, K. Rosquist and R. Ruffini (Word Scientific, Singapore, 2015), pp. 2054-2056. 6. Max von Laue, Die Relativitätstheorie, Bd 1: Die spezielle Relativitätstheorie (Vieweg & Sohn, Braunschweig, 1911). 7. M. J. Klein, A. J. Kox, J. Renn and R. Schulman (eds.), The collected papers of Albert Einstein (CPAE), vol.4, The Swiss years: Writings 1912-1914 (Princeton University Press, Princeton, 1995). 8. M. J. Klein, A. J. Kox and R. Schulman (eds.), CPAE, vol.6, The Berlin years: Writings 1914-1917 (Princeton University Press, Princeton, 1996). The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.com by GERMAN ELECTRON SYNCHROTRON on 04/25/21. Re-use and distribution is strictly not permitted, except for Open Access articles.