General Relativity Fall 2018 Lecture 17: Energy-Momentum of Gravitational Radiation

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 ∇ 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 + (ˆx × 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 ≈ −nˆ hij . So we get nâ nâ ...... 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 nˆa 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 = d2nˆ 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 − nˆinˆj. We need to average over angles combinations of the form

Z d2nˆ 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 d2nˆ Z d2nˆ 4 P P = 2 P = δ = (3α + 2β)δ , (16) 4π kk ln 4π ln 3 ln ln Z d2nˆ Z d2nˆ 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 d2nˆ ...... 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 d2nˆ ...... 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)(ˆv ⊗ vˆ − uˆ ⊗ uˆ) + cos(2Ωt)(ˆu ⊗ vˆ +v ˆ ⊗ uˆ)] , (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. The energy of the binary (homework) is 1 M M 1 M M E = − 1 2 = − 1 2 Ω2/3. (23) 2 a 2 M 1/3 Therefore, we ﬁnd that the orbital frequency increases as 96 M M 96 Ω˙ = 1 2 Ω11/3 ≡ M5/3 Ω11/3, (24) 5 M 1/3 5 chirp PHYSICAL REVIEW LETTERS week ending PRL 119, 161101 (2017) 20 OCTOBER 2017

∼100 s (calculated starting from 24 Hz) in the detectors’ sensitive band, the inspiral signal ended at 12 41:04.4 UTC. In addition, a γ-ray burst was observed 1.7∶ s after the coalescence time [39–45]. The combination of data from the LIGO and Virgo detectors allowed a precise sky position localization to an area of 28 deg2. This measure- PHYSICAL REVIEW LETTERS week ending 4 mentPRL 119, enabled161101 an electromagnetic (2017) follow-up campaign that 20 OCTOBER 2017 identified a counterpart near the galaxy NGC 4993, con- sistent∼100 s with (calculated the localization starting from and 24 distance Hz) in inferred the detectors from’ sensitive band, the inspiral signal ended at 12 41:04.4 UTC. gravitational-wave data [46–50]. ∶ InFrom addition, the gravitational-wave a γ-ray burst was signal, observed the 1.7 best s measured after the combinationcoalescence time of[39 the–45] masses. The is combination the chirp of mass data from[51] the LIGO and0.004 Virgo detectors allowed a precise sky M 1.188−þ0.002 M . From the union of2 90% credible intervalsposition¼ localization obtained using⊙ to an different area of 28 waveformdeg . This models measure- (see Sec.mentIV enabledfor details), an electromagnetic the total mass follow-up of the system campaign is between that 2.73identified and 3 a.29 counterpartM . The individual near the galaxy masses NGC are in 4993, the broad con- rangesistent of with 0.86 the to 2⊙. localization26 M , due andto correlations distance inferred between from their uncertainties.gravitational-wave This data suggests⊙[46–50] a BNS. as the source of the gravitational-waveFrom the gravitational-wave signal, as the signal, total the masses best of measured known combination of the masses is the chirp mass [51] BNS systems0 are.004 between 2.57 and 2.88 M with compo- ⊙ nentsM between1.188−þ0.002 1.17M and. From1.6 theM union[52]. of Neutron 90% credible stars in ¼ ⊙ ∼ generalintervals have obtained precisely using measured different masses⊙ waveform as large models as 2.01 (see 0Sec..04 IVM for[53] details),, whereas the total stellar-mass mass of the black system holes is found betweenÆ in 2.73 and⊙ 3.29 M . The individual masses are in the broad binaries in our galaxy⊙ have masses substantially greater thanrange the of 0.86 components to 2.26 M of GW170817, due to correlations[54 56] between. their ⊙ – uncertainties.Gravitational-wave This suggests observations a BNS alone as the are source able to of mea- the suregravitational-wave the masses of the signal, two objectsas the totaland set masses a lower of limit known on theirBNS systemscompactness, are between but the 2.57 results and presented2.88 M with here compo- do not ⊙ excludenents between objects 1.17 more and compact∼1.6 M than[52] neutron. Neutron stars suchstarsas in ⊙ general have precisely measured massesFIG. as 1. large “Chirp” as 2 of.01 the GWFIG. frequency 1. Time-frequency emitted by the representations neutron star binary[65] of system data containing observed by LIGO on August 17, 2017. quark stars, black holes, or more exotic objects [57–61]Æ. The0.04 detectionM [53], of whereas GRB 170817A stellar-mass and black subsequent holes found electro- in the gravitational-wave event GW170817, observed by the LIGO- ⊙ magneticbinaries in emission our galaxy demonstrates have masses the substantially presence of greatermatter. Hanford (top), LIGO-Livingston (middle), and Virgo (bottom) than the components of GW170817where[54 we56] have. defined the chirpdetectors. mass Timesas the are following shown relative combination to August of 17, masses 2017 12 (it41:04 has dimensions of a mass): Moreover, although a neutron star–black– hole system is not ∶ Gravitational-wave observations alone are able to mea- UTC. The amplitude scale in each detector is normalized to that ruled out, the consistency of the mass estimates with the 3/5 detector’s noise amplitude spectralM1M density.2 In the LIGO data, dynamicallysure the masses measured of the two masses objects of knownand set neutrona lower limit stars on in Mchirp ≡ . (25) their compactness, but the results presented here do not independently observable noise sourcesM 1/3 and a glitch that occurred binaries, and their inconsistency with the masses of known in the LIGO-Livingston detector have been subtracted, as exclude objects more compact than neutron stars such as black holes in galactic binary systems,Solving suggests this gives the source us described in the text. This noise mitigation is the same as that FIG. 1. Time-frequency representations [65] of data containing wasquark composed stars, black of two holes, neutron or more stars. exotic objects [57–61]. used for the results presented in Sec. IV. the gravitational-wave event GW170817, observed− by3/8 the LIGO- The detection of GRB 170817A and subsequent electro- 256 Hanford (top),Ω(t) = LIGO-LivingstonΩ−8/3 − M (middle),5/3 (t − andt ) Virgo, (bottom) (26) magnetic emission demonstratesII. DATA the presence of matter. 0 3 chirp 0 Moreover, although a neutron star black hole system is not detectors. Times are shown relative to August 17, 2017 12 41:04 – UTC. The amplitude scale in each detector is normalized∶ to that ruledAtthe out, time the consistencyof GW170817, of the the mass Advancedthatis, estimates the LIGO orbital with detec- frequency the in diverges the Virgo (and data so does due tothe the gravitational lower BNS frequency), horizon and or “chirps”, the after a finite time tors and the Advanced Virgo detector were in observing directiondetector’s of noise the sourceamplitude with spectral respect density. to the detector In the LIGO’s antenna data, dynamically measured masses of known neutron stars in independently observable noise sources and a glitch that occurred mode. The maximum distances at which the LIGO- pattern. 3 −5/3 −8/3 binaries, and their inconsistency with the masses of known t − t0 = tchirp ≡ M Ω . (27) inFigure the LIGO-Livingston1 illustrates the detector data256 as have theychirp been were0 subtracted, analyzed toas Livingstonblack holes andin galactic LIGO-Hanford binary systems, detectors suggests could the detect source a described in the text. This noise mitigation is the same as that BNSwas composed system (SNR of two8 neutron), known stars. as the detector horizon 2 determine2/3 astrophysical source properties. After data col- ¼ The GW strain if ∝ (Ωa) ∝usedΩ foralso the formally results presented diverges in at Sec.tchirpIV. . [32,62,63], were 218 Mpc and 107 Mpc,Of course, while we for cannot Virgo use thelection, Newtonian several approximationindependently measured to relate E terrestrialand Ω all contribu- the way to Ω → ∞: this calculation the horizon was 58 Mpc.II. The DATA GEO600only holds detector as long[64] aswasa Mtions, and tov ∼ thep detectorM/a noise1. were subtracted from the LIGO also operating at the time, but its sensitivityThe decay was insufficient of the orbital perioddata using due Wiener to GW filtering radiation[66] was, asfirst described measured in [67 in–70] the. This Hulse-Taylor binary pulsar, and, toAt contribute the time to of the GW170817, analysis of the the Advancedafter inspiral. 30 years The LIGO of configu- data, detec- is stillsubtractionin in the perfect Virgo agreement removed data due calibration with to the GR’s lower lines prediction. BNS and 60horizon Hz ac and power the rationtors and of the the Advanced detectors Virgo at the detector timeMore wereof recently, GW170817 in observing LIGO is detectedmainsdirection a harmonics handful of the source of from binary with both black respect LIGO hole datato mergers, the streams. detector and The’sthe antenna sensi- merger of a binary neutron star. summarizedmode. The in maximum[29]. distancesThe at which “chirp” the in frequency LIGO- istivitypattern. well visible of the in the LIGO-Hanford time-frequency detector figure shown was below.particularly This allows to measure the chirp +0.004 LivingstonA time-frequency and LIGO-Hanford representation detectorsmass[65] toof exquisitecould the data detect precision: from a improved forFigure this system, by1 illustrates the it subtraction was measured the data of laser as to theybe pointingM werechirp noise;= analyzed 1.188 several−0.002 to M . BNS system (SNR 8), known as the detector horizon determine astrophysical source properties. After data col- all three detectors around¼ the time of the signal is shown in broad peaks in the 150–800 Hz region were effectively Fig[32,62,63]1. The, weresignal 218 is clearly Mpc and visible 107 inMpc, the while LIGO-Hanford for Virgo removed,lection, several increasing independently the BNS measured horizon terrestrial of that contribu- detector andthe horizon LIGO-Livingston was 58 Mpc. data. The The GEO600 signal detector is not[64] visiblewas bytions 26%. to the detector noise were subtracted from the LIGO also operating at the time, but its sensitivity was insufficient data using Wiener filtering [66], as described in [67–70]. This to contribute to the analysis of the inspiral. The configu- subtraction removed calibration lines and 60 Hz ac power ration of the detectors at the time of GW170817161101-2 is mains harmonics from both LIGO data streams. The sensi- summarized in [29]. tivity of the LIGO-Hanford detector was particularly A time-frequency representation [65] of the data from improved by the subtraction of laser pointing noise; several all three detectors around the time of the signal is shown in broad peaks in the 150–800 Hz region were effectively Fig 1. The signal is clearly visible in the LIGO-Hanford removed, increasing the BNS horizon of that detector and LIGO-Livingston data. The signal is not visible by 26%.

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