General Relativity (j µ and T µν for point particles) van Nieuwenhuizen, Spring 2018

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1 Consistency of conservation laws in SR and GR General Relativity j µ and for point particles van Nieuwenhuizen, Spring Introduction The Einstein equations for matter coupled to gravity read Einstein 1915 G µν = kt matter µν 1 where G µν = R µν 1 2 g µνr and k = 8πG/c 4. Further, R = g µν R µν, and R is positive for a sphere. The Einstein tensor G µν satisfies an identity: for any g µν, also g µν which do not satisfy the field equations, one has the twice contracted Bianchi identity Bianchi 1902 D µ G µν = 0. 2 One obtains this identity by contracting the Bianchi identity D [µ R νρ]σ τ = 0 with δ τ µ and g ρσ. The consistency of 1 and 2 requires that the matter stress tensor should satisfy on-shell matter = 0. 3 One should find that this covariant conservation law holds if one uses the matter equations of motion and the matter field equations. We shall check this by considering a particular matter system: point particles coupled to electromagnetism. We wrote covariant conservation law in quotation marks because a real conservation law should be of the form µ V µ = 0 with an ordinary derivative µ. From a physical point of view one would expect that energy and momentum can be exchanged between the matter system and the gravitational field, but in such a way that the total energy and momentum is conserved. Thus one expects that there exists another conservation law, again valid only on-shell, of the following form where stands for point particle. µ T µν + + grav = 0 We shall begin with conservation laws and consistency in flat space, but with special relativity SR. We derive the expansion for the relativistic current density j ν x, t and stress tensor density µν x, t for a point particle, and the expression for T. Then we shall check that in flat space µ + = 0 if one uses the Maxwell equation µf µν = 4πj ν x, t/c and the equation of motion d m dx µ = 1 F µν J c ν t. We shall define the current J ν t and check that j ν x, t and J ν t are both vectors in SR. Then we turn to GR and repeat these constructions in curved space.

2 2 Conservation laws and consistency in SR Electromagnetism coupled to point particles in flat space In this section we consider flat space, and construct the electromagnetic 4-vector current density j µ x, t and the 4-vector current J µ t for a point particle coupled to electromagnetism, and the stress tensor density µν x, t for point particles and T x, t for electromagnetic fields. Consistency of the Maxwell equations requires the conservation law µ j µ x, t = 0, and we shall check that it holds for point particles. We shall also check that the total stress tensor density is conserved: µ + = 0, but this is only true if the equation of motion for the point particle and the Maxwell equations for the electromagnetic field hold. In one defines the electric density of a point particle by and the current density ρ x, t = eδ 3 r rt, j i x, t = e dxi t δ 3 r rt. These densities appear in the Maxwell equations 1 µ F µν = 4πj ν /c. It follows that j ν should be a 4-vector in SR, and this suggests to write it in 4-vector notation as follows j µ x, t = n dx µ e nt n δ 4 x xt 4 Here τ is the relativistic Lorentz invariant proper time = c 1 v 2 /c 2 = dx 0 2 d x 2 = η µν dx µ dx ν 5 and x 0 = ct. The 4-dimensional delta function is a scalar under Lorentz transformations since δ 4 x µ x µ τ = δ 4 L µ νx ν x ν τ = 1 det L δ4 x ν x ν τ = δ 4 x ν x ν τ 6 where we used det L = 1. So j µ x, t in 4 is indeed a Lorentz 4-vector. Using that x 0 τ = ctτ we can perform the integral over τ in 4 using δx 0 x 0 τ = 1. 7 dx 0 Inserting this result into 4 recovers the familiar expression with which we started j 0 x, t = cρ x, t = ecδ 3 x xt, j i = e dxi δ3 x xt. 8 1 In Gaussian units j 0 = ecδ 3 r rt and E i = F 0i = 1 c ta i i φ with A 0 = A 0 = φ. 2

3 The Maxwell equations read µ F µν = 4π c jν x, t. The Bianchi identity for in flat space reads ν µ F µν = 0, and for consistency j ν should satisfy the conservation law ν j ν = 0 on-shell. This is easily proven. Beginning with the space divergence one finds for a set of point particles with electric charges e n x i ji x, t = n = n = n e n dx i n e n dx i n This proves that j µ is conserved. x i δ3 r r n t δ 3 r r x i n t n e n t δ3 r rt = x 0 j0 x, t use chain rule We shall also need the stress tensor density for point particles and for the field. For point particles with masses m n it reads x, t = n = n dx ν n δ3 r r n t m n c 2 dxµ n dx ν n δ4 x x n t Note the d and d in the first line and the d and d in the second line. Using 5, the second line goes over into the first line. It is clear that is a Lorentz tensor. In the nonrelativistic limit T 00 m nc 2 δ 3 r rt since x 0 = ct and τ ct, and thus is normalized such that T 00 x, t is the energy density. The stress tensor for the field with the same normalization is given by F µλ F λ ν 14 η µνf ρσ F ρσ. For example, T µν = 1 4π T00 = 1 E π 2 E 2 + B 2 = 1 E 2 + B 8π 2, which is the well-known expression for the energy in Gaussian units. The divergence of times 4π reads 4π µ = µf µλ F νλ + F µλ µ F νλ 1 2 νf ρσ F ρσ = µ F µλ F νλ 1 2 F ρσ ν F ρσ + σ F νρ + ρ F νσ 3

4 We replaced µ F νλ by 1 2 µf νλ 1 2 λf µν and relabelled indices. The last three terms vanish due to the Bianchi identity [ν F ρσ] = 0 of. We are left with part of the field equation, namely µ F µλ times F νλ. Since the system plus point particles is a closed system, its total energy and momentum should be conserved. Let us check this. We begin again with the space divergence conservation T µi x i x i T µi x, t = n So we have found = n = t x, t and perform the same steps as when we checked current [ n dx i n = x 0 T µ0 + n t x i δ3 r r n t δ 3 r r n t we replaced δ3 r r n t d m n c d m n c x µ = n Let us now add the contribution of the field ] + n d m n c dx µ n δ 3 r r n t. dx µ n with x i x i n δ 3 r r n t we used the chain rule dx µ n δ 3 r r n t. 9 µ = 1 4π µf µλ F ν λ. 10 The relativistic field equation and the equation of motion for a point particle coupled to reads µ + = n µ F µν = 4π c jν see before mc d dx µ = 1 c F µν J ν ; J ν t = e dxν 11 If one adds 9 and 10 and uses 11, one finds that the total stress tensor is conserved d dx ν n m n c δ 3 r r n t + 1 4π µf µλ F ν λ = n 1 c F ν λe dxλ n δ 3 r r n t + 1 4π 4π c edxλ n δ3 r rt F ν λ = 0. Comment: We did not need to use the equation of motion of the point particle to prove electromagnetic current density conservation, but we needed it to prove the conservation of the local stress tensor. The reason is that if one deforms the classical trajectory of a point particle in some arbitrary way, the point particle continues to carry electric chage e, but its energy and momentum charge are no longer conserved. 4

5 3 Conservation laws and consistency in general relativity Maxwell fields and point particles coupled to gravity In this section we extend our discussion of the interplay between Bianchi identities and current density conservation to include Einstein gravity. We consider the Einstein-Maxwellpoint particle system, including the coupling of Maxwell fields to point particles. To begin with we must define the 4-dimensional delta function δ 4 x. We define it for any function fx as follows δ 4 x yfx = fy. 12 This is the usual definition in mathematics, but some relativists insert a factor g or 1/ g into the definition. We do not add such a factor and use 12. However, whereas δ 4 x xτ was a Lorentz scalar in SR, it is an Einstein density in GR because it transforms the same way as g δ 4 x xτ = δ 4 xx xxτ = 1 det x x x ḡ x = det gx. x Thus the following is a scalar in GR an Einstein scalar δ 4 x xτ gx. δ 4 x xτ. The parameter τ is now the proper time in GR an Einstein scalar τ = g µν dx µ dx ν and generalizes the flat space result τ = ct 1 v2 of SR. c 2 The Maxwell equations in GR are obtained by replacing ordinary derivatives by covariant derivatives D µ F µν = 4π c jν x, t 13 where F µν is obtained from F µν by raising indices with the metric, and F µν is still µ A ν ν A µ because in D µ A ν D ν A µ the Christoffel connection term cancels due to symmetry. In order that 13 is covariant, j ν x, t must be a vector density, and this can be achieved by including a factor 1/ g j ν x, t = e dxν δ 4 x xτ. g The left hand side satisfies an identity which we could call the Bianchi identity for in curved space D ν D µ F µν 0. 5

6 It follows from [D µ, D ν ]F µν = R µνλ µ F λν + R µνλ ν F µλ = R νλ F λν R µλ F µλ = 0 which vanishes because R µν is symmetric and F µν antisymmetric. Hence for consistency the following conservation equation should hold D ν j ν x, t = 1 g ν gj ν x, t = 0. We shall check ν gj ν x, t = 0 for a point particle because that is easier than D µ j µ x, t = 0. As in flat space we begin with x i gj i x, t and see what happens. Straightaway we see a nice simplification: the g cancels the 1/ g associated with the delta function. Following familiar steps replace /x i by /x i t and use the chain rule we find x i gj i x, t = dx i x e i δ4 x xτ = This proves that µ gj µ x, t = 0. x i edxi δ3 x xt = e t δ3 x xt = t 1 gj 0 x, t. c We proved the consistency of the Maxwell equations in curved space, and now turn to the consistency of the Einstein equations. As discussed in the introduction, the consistency of the Einstein field equations G µν = k matter with the Bianchi identity D µ G µν requires that the total local matter stress tensor satisfies matter = 0. The expression for the tensor x, t is as follows dx µ x, t = dx µ mc2 δ 4 x xτ = mc 2 dxµ dx ν δ 3 x xt. g c g It contains the factor 1/ g which we can partially eliminate, as suggested by our earlier treatment of the electromagnetic case, by rewriting as follows = 1 µ gt µν + ν Γµλ T µλ g We then use this expression and begin again with a space divergence 1 g i gt iν and converting the /x i into a /t 1 gt iν g x i = 1 mc 2 dxν g = 1 g x 0 1 c t δ 3 x xt mc 2 dxν δ3 x xt g mc d dx ν δ 3 x xt

7 The term with /x 0 completes the space divergence to a 4-divergence, and we arrive at = 1 d dx ν mc δ 3 x xt + Γ ν µλ mc dxµ dx λ δ 3 x xt g g = 1 D dx ν mc δ 3 x xt. g Dt It is now clear how to continue: we use the geodesic equation for a point particle in curved space coupled to electromagnetism, and separately evaluate, in the hope that the sum of these two expressions cancels. The geodesic equation reads mc 2 D Dτ dx µ = 1 c F µ νj ν, J ν = e dxν. The factors of c are correct: for µ = i the nonrelativistic limit yields m d2 x i = 1 2 c Ei ec The covariant divergence of the stress tensor reads 4π = D µ F µρ F ν ρ 14 gµν F ρσ F ρσ = D µ F µρ F ν ρ + F µρ g νσ D µ F σρ + D σ F ρµ + D ρ F µσ. The last three terms constitute the Bianchi identity in curved space all Christoffel symbols cancel, so we get upon using the Maxwell equation total = 1 D dx ν mc δ 3 x xt + 1 ] 4πc δ [ 3 x xt edxρ F ν ρ. g Dt 4π g Finally we use the equation of motion mc D dx ν Dt = 1 c F µ νe dxν and see that all terms indeed cancel. Thus the total matter stress tensor of matter is on-shell convariantly conserved matter, total = 0. There remains a last issue, which is, in fact, the most interesting: how to incorporate grav into these considerations, and related to that, the definition of grav. We shall discuss these issues when we reach gravitational radiation. 7