General Relativity Fall 2019 Lecture 10: charge conservation; electromagnetism; stress-energy tensor Yacine Ali-Ha¨ımoud October 1st 2019 CHARGE CONSERVATION Last lecture we derived the 4-current J µ of an ensemble of particles, in a LICS. Its 0-th component is the charge density (where \charge" can be electric charge, baryon number, etc...), 0 X (3) J (t; ~x) = qnδD (~x − ~xn(t)) n and its spatial component is the charge current, X d~xn (3) J~(t; ~x) = q δ (~x − ~x (t)): n dt D n n We showed that the two expressions can be grouped in the 4-vector Z µ Z X dx (4) X (4) J µ(xν ) = q dt n δ (xν − xν ) = q dτ uµ δ (xν − xν ); n n dt D n n n n D n n n n µ where un is the 4-velocity of particle n. This is clearly a Lorentz 4-vector. Let us compute the following: X d~xn (3) X d~xn (3) @ J µ = @ J 0 + r~ · J~ = − q · (r~ δ )(~x − ~x (t)) + q · (r~ δ )(~x − ~x (t)) = 0; µ t n dt D n n dt D n n n regardless of the trajectoris of the particles. Let us now rewrite these expressions in a way that is generally covariant, i.e. define generally tensorial expressions. First, we can rewrite Z X 1 (4) J µ = q dτ uµ p δ (xν − xν ); (1) n n n −g D n n (4) p which is now a bona fide 4-vector, since we saw that δD = −g is invariant under arbitrary coordinate transforma- 0 ~ tions.p You can work backwards and find that the expressions for J and J are the same as before, just with a factor 1= −g. µ Second, the conservation of charge @µJ = 0 becomes µ rµJ = 0 : Both expressions match the ones we derived in a LICS, and are moreover generally covariant. Let us write the latter equation explicitly: µ µ µ ν 0 = rµJ = @µJ + Γµν J : Now the relevant contraction of Christoffel symbols is 1 1 Γµ = gµλ (g + g − g ) = gµλg + gµλg : µν 2 µλ,ν νλ,µ µν,λ 2 µλ,ν ν[λ,µ] The last term vanishes: it is the contraction of a symmetric tensor with an antisymmetric tensor! Now, for a matrix M, Jacobi's identity tells us 1 @ @ @ det(M) = tr M −1 M = (M −1)µν M : det(M) @x @x @x µν 2 Applying this to Mµν = gµν , we thus find p @ −g Γµ = pν : µν −g Therefore the conservation of charge can be written as p @ −g 1 p 0 = @ J µ + pν J ν = p @ ( −gJ µ); µ −g −g µ which we could directly have derived from Eq. (1). MAXWELL'S EQUATIONS Maxwell's equations for the electric field E~ and magnetic field B~ can be written in a simple, compact form, in terms µν 0i i ij ijk of the rank-2 antisymmetric electromagnetic tensor F , such that F = E and F = Bk. Note that I no longer say rank-(0,2) or rank-(2,0) or rank-(1, 1), as such tensors can be transformed into one another with the metric and inverse metric. The special-relativistic Maxwell's equations (thus, in a LICS) are νµ ν @µF = J ;@[µFνλ] = 0; where J ν is the electric 4-current. Note that the second equation is equivalent to @µFνλ + @ν Fλµ + @λFµν = 0; because @(µFνλ) = 0, and the above combination is proprotional to @(µFνλ) + @[µFνλ]. Given an electromagnetic field, the motion of a particle of mass m and charge q, in a LCIS, is given by duµ m = qF µν u : dτ ν The generally covariant equations are αβ α β α αβ rβF = J ; r[αFβγ] = 0; mu rβu = qF uβ : STRESS-ENERGY TENSOR α α Consider a set of massive particles (labeled by n) with 4-momenta pn = mnun. Remember that the 4-momentum is such that p0 is the energy and ~p is the 3-momentum. Just like we defined the density and flux of charge, we may define the density and flux of 4-momentum. In a LICS, they are given by µ0 X µ (3) T (t; ~x) = mnunδD (~x − ~xn(t)); n and the flux of 4-momentum is i X dx (3) T µi(t; ~x) = m uµ n δ (~x − ~x (t)): n n dt D n n Just like before, these can be grouped into ν Z X dx (3) X (4) T µν (xσ) = m uµ n δ (~x − ~x (t)) = m uµuν dτ δ (xσ − xσ): n n dt D n n n n n D n n n µν This expression is manifestly a Lorentz tensor, andp moreover shows that T is symmetric. This can be rewritten in a generally covariant way by multiplying by 1= −g. 3 Let us now compute, in a LICS: µ X du (3) @ T µν = @ T µ0 + @ T µi = m n δ (~x − ~x (t)); ν 0 i n dt D n n ~ (3) where the contributions proportional to rδD cancel out as before. Again, we may rewrite this in a manisfestly Lorentz-covariant form: Z µ Z X du (4) X (4) @ T µν = dτ m n δ (xσ − xσ) = dτ f µδ (xσ − xσ); ν n n dτ D n n n D n n n n µ where fn is the 4-force acting on particle n. The covariant version of this equation is Z X 1 (4) r T µν = dτ f µ p δ (xσ − xσ) ≡ F µ; ν n n −g D n n where the right-hand-side is a 4-force density. More generally, we can define the stress-energy tensor of any substance, T µν , as the symmetric tensor such that: T 00 = energy density T i0 = density of i-th component of momentum 0j T = energy flux along @(j) ij T = flux of i−th component of momentum along @(j) The total stress energy tensor of all matter fields is conserved, i.e. there is no net creation or destruction of overal 4-momentum µν rµT(total) = 0 : However, as we saw in the case of a swarm of particles, the stress-energy tensor of any particular species s is not necessarily conserved: µν X µ rν T(s) = Fs0!s: s06=s The conservation of the total stress-energy tensor is just a re-expression of the law of action-reaction: µν X X µ rν T(total) = Fs0!s = 0: s s06=s STRESS-ENERGY TENSOR OF THE ELECTROMAGNETIC FIELD µ µν Let us apply the above results to charged particles in an electromagnetic field: the force is then fn = qnF uν , thus X 1 (4) r T µν = q F µν u p δ (xσ − xσ) = F µν J ; ν (particles) n ν −g D n ν n where Jν is the electric current density defined earlier. Let us define the following tensor: 1 T αβ ≡ F α F βδ − gαβ F ρδF : (em) δ 4 ρδ We then find 1 r T αβ = (r F α )F βδ + F α r F βδ − gαβ(r F )F ρδ; β (em) β δ δ β 2 β ρδ 4 where we used metric compatibility of the covariant derivative, and the symmetry of the last term. Let's now use Maxwell's equations to simplify: 1 r T αβ = −F α J δ + gαγ F βδr F + F δρr F ; β (em) δ β γδ 2 γ ρδ where I remnamed dummy indices and shuffled up-and-down indices with the metric. Doing some more index renam- ing, and using the antisymmetry of Fµν , we get 1 r T αβ = −F α J δ + gαγ F δβ r F + r F : β (em) δ β δγ 2 γ βδ βδ Now because F is antisymmetric, we may replace rβFδγ by r[βFδ]γ , i.e. its antisymmetric part in β; δ. Using the antisymmetry of Fµν , the term in parenthesis then becomes 1=2(rβFδγ + rδFγβ + rγ Fβδ) = 0, from the second of Maxwell's equations. Thus we find αβ αδ αβ rβT(em) = −F Jδ = −∇βT(particles) : αβ We thus see that it makes sense to define T(em) as the stress-energy tensor of the electromagnetic field, since its satisfies αβ αβ rβ T(em) + T(particles) = 0 : STRESS-ENERGY TENSOR OF AN IDEAL FLUID An ideal fluid is isotropic in a preferred LICS, called the fluid’s rest frame. This means that, in this frame, 0i ij ij there is no preferred direction, thus T = 0 and T / δ . We define ρrf to be the energy density in the fluid’s 00 ij ij rest-frame, i.e. such that T = ρrf , and denote by Prf the pressure in the fluid’s rest-frame, i.e. T = Prf δ in that frame. Let us now denote by uµ the 4-velocity of the fluid with respect to some arbitrary frame. In particular, in the fluid’s µ µν µ ν µν µ ν rest-frame, u = (1; 0; 0; 0) by definition, thus we can rewrite, in that frame, T = ρrf u u + Prf (η + u u ) since in that frame, ηµν + uµuν = diag(0; 1; 1; 1). This is a Lorentz-covariant expression. The generally covariant version is αβ α β αβ α β T = ρrf u u + Prf (g + u u ) : It is important to remember that ρrf and Prf are the fluid’s energy density and pressure in its restframe.