General Relativity Fall 2018 Lecture 17: Energy-momentum of gravitational radiation Yacine Ali-Ha¨ımoud Just like electromagnetic radiation, gravitational radiation carries energy and momentum. The electromagnetic stress-energy tensor is quadratic in the vector potential Aµ, and similarly, we expect that the stress-energy of GWs contains terms quadratic in the metric perturbations hµν . There are important differences, however. First, GW energy cannot be localized. Indeed, at any spacetime event, we may choose coordinates in which hµν = @λhµν = 0. As a consequence, terms of order ∼ h@2h; (@h)2 can always be made to vanish locally. The stress-energy of GWs is therefore only meaningful after averaging over a finite region of spacetime, typically several GW wavelengths. Secondly, while the electromagnetic field is not charged, the gravitational field self-interacts: it is the stress- energy that sources spacetime curvature, so if the gravitational field carries stress-energy, it will source itself. This implies that the full gravitational \stress-energy tensor" has terms at all orders of hµν , not just quadratic. Gravitational field stress-energy \tensor" Consider an asymptotically flat spacetime, i.e. such that sufficiently far from sources, the metric is nearly Minkowski. Adopt coordinates that reflect this, i.e. in which gµν = ηµν + hµν , where hµν 1 far enough away from sources, but (1) need not be small in the vicinity of sources. The Einstein tensor Gµν is a non-linear functional of hµν . Define Gµν to be the linearized Einstein tensor (which we studied in the last few lectures), and define the symmetric quantity 1 t ≡ − G − G(1) : (1) µν 8π µν µν (1) Note that tµν is not a generally covariant tensor, since Gµν is not (its definition depends on the chosen coordinate system). The pseudo-tensor tµν consists of terms of arbitrary order in hµν , but with at most two derivatives. To 2 2 second order in hµν , it consists of terms of the form h @ h,(@h) . We may then rewrite Einstein's field equations as (1) Gµν = 8π(Tµν + tµν ) ≡ 8πτµν : (2) µ (1) The linearized Einstein tensor satisfied @ Gµν = 0, so the total effective stress energy tensor τµν is conserved in the ordinary sense, i.e. µν @µτ = 0: (3) µ This ordinary conservation equation is exact and equivalent to the covariant conservation law r Tµν = 0. One can think of the latter as describing the exchange of energy-momentum between matter and radiation. We may formally solve Eq. (2) exactly like we did for a fully Newtonian source1, and obtain the metric far from the source, 4 ds2 = −(1 − 2M=r)dt2 + (^x × J~) · d~xdt + (1 + 2M=r)δ + hTT dxidxj; (4) r2 ij ij where the mass and angular momentum are now defined as volume integrals of τµν rather than Tµν : Z Z 3 00 3 j 0k M = d y τ ;Ji ≡ d y ijky τ ; (5) and the TT part of the metric is sourced by the inertia tensor, defined as Z 3 00 Iij ≡ d y yiyj τ : (6) 1 µν µν We still have to assume that the sources are quasi-stationary for our previous derivation to hold, i.e. @tτ @xτ 2 As we explained earlier, one should not make too much case of these volume integrals: the integrands are clearly gauge-dependent, and the split in background plus perturbations is meaningless inside relativistic sources. The integrals themselves are gauge-invariant, however (although it is not straightforward to prove), and do have meaning provided spacetime is asymptotically flat2. They are physically measurable quantities through their effect of geodesics in the asymptotically flat region, for instance, Kepler's law, or the Lense-Thirring precession of gyroscopes. The rates of change of the mass and angular momentum can be re-expressed in terms of surface integrals, using Stokes' theorem and the ordinary conservation of τµν : Z Z Z 3 00 3 0i 0i M_ = d y @0τ = − d y @iτ = − dSi τ ; (7) Z Z Z Z 3 j 0k 3 j lk 3 j lk j lk J_i = d y ijky @0τ = − d y ijky @lτ = − d y ijk@l(y τ ) = − dSl ijk y τ ; (8) where S is a surface far away from the source, and dSl ≡ dAnl, where dA is the element of area, and nl is the normal to the surface. Since these integrals are computed outside the source, where Tµν = 0, we may replace τµν ! tµν . Moreover, provided the surface is in the asymptotially flat region where hµν 1, the pseudo-tensor tµν is dominated 2 2 by its piece quadratic in hµν . Since this contains pieces of order h@ h and (@h) , we see that all the non-GW pieces of the metric lead to contributions that die off as 1=r4 at least. Only the GW part has contributions of order 1=r2, which dominate at large distance, and moreover give a result independent of the surface S, for large enough r. Effective stress-energy tensor of GWs (2) It is rather cumbersome to Taylor-expand the Einstein tensor to second order in hµν to compute Gµν . Upon doing so, and averaging over several wavelengths (which allows us to replace hY@µ@ν Xi ! −h@µY@ν Xi), one obtains, if only considering the GW contribution, 1 tGW = h@ hTT@ hTTi: (9) µν 32π µ ij ν ij The formal averaging procedure and proof of gauge invariance are detailed in Isaacson 1968. Application: power radiated by a time-varying mass quadrupole We saw that the GWs are sourced by the moment of inertia tensor. At large distances: 2 hTT = PTT I¨ (t − r); (10) ij r ijkl kl where, for general, relativistic (but still quasi-stationary) sources, Iij is given by Eq. (6). Let us compute the energy flux of GWs in the a direction: 1 T 0a = − hh_ TT@ hTTi: (11) GW 32π ij a ij The dominant term is when the spatial derivative is applied to the retarded time (other terms are suppressed by TT a _ TT another factor of 1=r). This means @ahij ≈ −n^ hij . So we get n^a n^a ... ... T 0a = hh_ TTh_ TTi = PTT PTT h I I i: (12) GW 32π ij ij 8πr2 ijkl ijmn kl mn Let us compute the product of TT projection tensors: 1 1 1 PTT PTT = P P − P P P P − P P = P P − P P = PTT : (13) ijkl ijmn ik jl 2 ij kl im jn 2 ij mn km ln 2 kl mn klmn 2 It turns out that the integrands of these volume integrals is an ordinary divergence, so M, J~ can be rewitten as surface integrals, which can be computed far away from the source, in the asymptotically flat spacetime. See Misner, Thorne & Wheeler Chapter 20 3 This is to be expected, as PTT is itself a projection operator 0a The power radiated by GWs is then obtained by integrating T n^a over a sphere at large distance from the source. 2 We see that the r factors cancel... out, and we are left with the angle-average of the projection operator times the time-average of the square of I : 1 Z 1 ... ... P GW = d2n^ P P − P P h I I i: (14) 8π km ln 2 kl mn kl mn ... From this expression already, it is easy to see... that the trace of I kl does not contribute, so we can replace it by the triple derivative of the quadrupole moment Qkl. Now, recall that Pij = δij − n^in^j. We need to average over angles combinations of the form Z d2n^ P P = αδ δ + β (δ δ + δ δ ) ; (15) 4π km ln km ln kl mn kn lm where the right-hand-side follows from the requirement that, after averaging, the tensor is isotropic, so it must be expressed in terms of Kronecker deltas. We have also enforced the sysmmetries of this tensor. To determine the coefficients α and β, just take some traces: Z d2n^ Z d2n^ 4 P P = 2 P = δ = (3α + 2β)δ ; (16) 4π kk ln 4π ln 3 ln ln Z d2n^ Z d2n^ 2 P P = P = δ = (α + 4β)δ : (17) 4π km kn 4π mn 3 mn mn This gives us α = 2=5, β = 1=15, so that Z d2n^ ... ... 2 1 ... ... 7 ... ... P P hQ Q i = δ δ + (δ δ + δ δ ) h I I i = hQ Q i; (18) 4π km ln kl mn 5 km ln 15 kl mn kn lm kl mn 15 kl kl Z d2n^ ... ... 2 ... ... P P hQ Q i = hQ Q i; (19) 4π kl mn kl mn 15 kl kl hence 1 ... ... P GW = hQ Q i : (20) 5 kl kl This is the power radiated by a time-varying quadrupole moment. It is the gravitational analog of the power radiated ~ EM 2 ¨ ¨ by an electric dipole moment d, P = 3 hdidii. The first derivative of the mass dipole moment is just the linear momentum, and its second derivative vanishes, which is why there is no gravitational mass-dipole radiation. Application to a circular binary For a circular binary with semi-major axis a, we have already computed ¨I = Q¨ . The third derivative is then ... 4M M Q = − 1 2 Ω3a2 [sin(2Ωt)(^v ⊗ v^ − u^ ⊗ u^) + cos(2Ωt)(^u ⊗ v^ +v ^ ⊗ u^)] ; (21) M so that the power radiated is 32 M 2M 2 32 M 2M 2M 32 M M 2 P GW = 1 2 Ω6a4 = 1 2 = 1 2 Ω10=3; (22) 5 M 2 5 a5 5 M 1=3 where we used Ω2a3 = M.