A Brief History of the Energy-Momentum Tensor; 1900-1912
aa r X i v : . [ phy s i c s . h i s t - ph ] N ov June 27, 2018 20:59 WSPC - Proceedings Trim Size: 9.75in x 6.5in JPP A BRIEF HISTORY OF THE ENERGY-MOMENTUM TENSOR;1900-1912
J.-P. PROVOST
Emeritus Professor, University of Nice-Sophia Antipolis, France ∗ E-mail: [email protected] appear in the 13th Marcel Grossmann Proceedings, Stockholm st − th July T µν ) in a crucial way.When turning from a scalar to a tensor theory, Einstein acknowledges that “ the gen-eral validity of the conservation laws [(C.L.) ∂ µ T µν = 0 or f ν ] and the law of inertia [ T i = T i ] is the most important new advance in the theory of relativity . . . Theproblem to be solved always consists of finding how T µν is to be found from thevariables characterizing the processes under consideration ”. As discussed in Ref. 1this will explain in a large part his bumpy road to G.R between 1913 and 1915.1900 is the time all ingredients of the future e.m T µν are known. CertainlyMaxwell has already explained in his 1873 Treatise “ the forces [ f = ρE + j ∧ B ] which act on an element of a body placed in an e.m field “by” the hypothesis of amedium in a state of stress” [ f i = − ∂ i ( eδ ij − E i E j − B i B j ) with e = ( E + B ) / E ∧ B . In Lorentz 1900 Jubilee
Poincar´e gives a final touch to the mechanical properties of the e.m field,providing it with a momentum density E ∧ B (in order to satisfy the action reactionprinciple with the charges), and submitting it to a new relativity (Galilean change x ′ = x − V t completed by Lorentz proper time t ′ = t − V x interpreted as a conventionpreserving the velocity of light at first order in V ). In the same Jubilee
Wien proposesan “ e.m foundation of mechanics ” first developed by Abraham where the energy E , momentum p and Lagrangian L of an electrostatic spherical electron in globalmotion are deduced from the spatial integration of e , E ∧ B and ( B − E ) / corresponding states ” for e.m at any order in V ), it is contractedalong the direction of motion; but for both, Hamilton eq. p = ∂L/∂v is violated (inLorentz model p = (4 / γE v , L = − E √ − v , E = E γ (1 + v /
3) where E isthe electrostatic energy at rest). This problem which is related to the non trivialquestion of the assimilation of extended systems to punctual ones will be clarified byvon Laue in 1911 (true end of R.D). It is not by chance that the citation histogramof Einstein’s 1905 relativity papers presents a bump between 1907 and 1913. une 27, 2018 20:59 WSPC - Proceedings Trim Size: 9.75in x 6.5in JPP In 1905, Poincar´e writes his
Palermo memoir (59 pages, published in January1906), which exhibits the mathematical essence (“ les rapports vrais ”) of Lorentzwork. In particular: (i) he verifies explicitly that Lorentz eq. of dynamics F = d ( mγv ) /dt is covariant with respect to Lorentz group which he introduces andstudies (this allows him to propose many relativistic gravitational forces and toattack the Mercury perihelion problem in 1906); (ii) he shows that L = − E √ − v arises from the invariance of the action and is linked to contraction (in short, forus today: S = R φd rdt = R φ d r dt −→ φ = φ −→ L ); (iii) he attributesthe failure of Hamilton eq. to the instability of Lorentz electron and cures it byadding δL = − Ar √ − v with A = E / E ( r ) + Ar with E ( r ) ∝ r − ), but he ignores the contribution of A to the mass.This contrasts with Einstein who hesitates in June concerning the eq. of dynamicsbut argues in September that ∆ m = ∆ E if a body emits opposite plane waves. InMarch 1906, using Einstein’s results on acceleration, Planck shows that Newton lawat rest ma = qE more generally reads γma = q ( E + v ∧ B − v ( vE )) or “ to put it insimpler form d ( γmv ) /dt = q ( E + v ∧ B ) = F ”; he deduces the Lagrangian and theenergy up to constants, i.e. ignoring the explicit covariance of the new dynamics.If in 1906 Planck criticizes the consideration of extended systems necessitat-ing to estimate “ the work of deformation ”, he soon comes back to them. Awarethat one can no longer separate kinetic from internal energy because radiation isomnipresent in matter, he asks Mosengeil (dead at 22 years in September 1906)to reconsider Hasenh¨orl theory of a moving black body under the light of Ein-stein relativity. Remarkable advances of his 1907 paper are the invariance of en-tropy, of the “ number of action elements [ h ] in nature ” and the mass defect∆( E + p V ) in chemistry. Technically, he uses Mosengeil results to deduce theentropy S = γ V T and Helmholtz energy H = vp − K + T S which entersthe Least Action Principle R Hdt . A lot of tedious calculations (among whichthe transformation law of the force F = d ( ∂H/∂v ) /dt ) leads him to the in-variance of γ ( H − Cste ); Mechanics is recovered for
Cste = 0 , S = 0. In hisDecember 1907 review paper, Einstein considers the possibility that energy andmomentum are provided to an extended moving system of charges by an ap-plied field, under the condition that the system acquires no momentum at rest: R dE = R dt R ρvE a d r ; R dp = R dt R ρ ( E a + v ∧ B a ) d r ; R dp = R dt R f d r = 0 . The transformation of e.m quantities leads him by integration (and up to constantsinside E ) to R dE = γ R dE ; R dp = γ [ R dE + R f d r dt ] i.e if f = 0 to E = γ ( m + E ) , p = vE, (1)“ a result of an extraordinary theoretical importance ” (equivalent role of mass andinternal energy). Being an expert in simultaneity, he is the first to understand thatif R f d r = 0 at t fixed, with f = 0 (for instance the pressure force on the wallsof the black body cavity), it is no longer true at t fixed. He recovers in that wayPlanck 1907 relation E + pV = γ ( E + p V ) with p = p .1908 is the year Minkowski introduces the 4d formalism, showing in particular une 27, 2018 20:59 WSPC - Proceedings Trim Size: 9.75in x 6.5in JPP that Newton’s law dP µ /dτ = F µ reads ∂ µ T µν = f ν with T µν = ρ u µ u ν (freematter), and Planck brings in physics a major conceptual unification. In Remarkson the action and reaction principle in general dynamics he notes that energy is bothvarious (kinetic, gravitational, calorific, chemical, e.m . . . ) and unique (through itsC.L) whereas momentum is known only for mechanics and e.m. Through severalexamples, he shows that the energy current is nothing but the momentum density(law of inertia T i = T i generalizing Eq. (1)). He also claims that the stress tensor,e.g. Maxwell’s one, is a momentum current which must be examined for gravitation.The physical consequences of this unification and the various expressions of T µν for e.m. media (Minkowski, Abraham . . . ), hydrodynamics and elasticity (Born,Herglotz . . . ) will be developed between 1908 and 1911 (see Ref. 2,3).The synthesis of point particles and continuous media mechanics is made byvon Laue in 1911. He shows that the above odd results are simple consequences(provided T i rest = 0) of the law of transformation of T µν : T = γ ( T + v T ) rest ; T = vγ ( T + T ) rest ; T − vT = T . (2)He also gives a sufficient guarantee for R T µ d r to be a quadrivector P µ , namelythat the system be static at rest and that R T ij rest d r = 0. This implies ∂ µ T µν = 0which is clearly not satisfied for a static electron or gas; in 1918 Klein will provethat the reciprocal is true (H. Ohanian private communication). In addition, manyprevious “paradoxes” such as Ehrenfest 1909 paradox (a body with T = 0 gets p y = 0 if it is boosted along x ), or concerning open systems such as Trouton-Noble1903 condensator or Lewis-Tolman 1909 lever, which are submitted at rest to azero torque but not when boosted, making them rotate if one forgot that internalstresses lead to an energy flow, i.e. to a momentum density in parts of the system.In conclusion and although it is not the standard way to look at it, the history ofR.D has been implicitly or explicitly concerned with issues relative to T µν : integra-tions of e.m. densities and their insufficiency, Planck 1908 formulation of Einstein1905-1907 inertiae, importance of its transformation law and its C.L. This is thereason why in his 1911 book on relativity, the first one on this subject, von Lauepresents its chapter 7 Dynamics as “ a new exposure studying the influence of elasticstresses on energy momentum and their transformation laws ” (whereas today R.Dcan be introduced and developed in a few lines for students). Is the story of T µν finished in 1912? Of course not; from 1912 to 1921, G.R has been concerned withmany issues dealing with this tensor, in particular with the status of its gravita-tional part in relation to the new extended covariance of the theory. Still today weignore the status of the dark energy tensor. References
1. J. P. PROVOST and C. BRACCO, in Gravitation Quantique (Paris, Herman, 2013)and references therein.2. M. v. LAUE,
Principle of relativity (Leipzig, 1911).3. W. PAULI,