Classical Gravitational Bremsstrahlung from a Worldline Quantum Field Theory
Gustav Uhre Jakobsen, Gustav Mogull, Jan Plefka, Jan Steinhoff
HHU-EP-21/03-RTG
Classical Gravitational Bremsstrahlung from a Worldline Quantum Field Theory
Gustav Uhre Jakobsen,
1, 2, ∗ Gustav Mogull,
1, 2, † Jan Plefka, ‡ and Jan Steinhoff § Institut f¨ur Physik und IRIS Adlershof, Humboldt-Universit¨at zu Berlin,Zum Großen Windkanal 2, 12489 Berlin, Germany Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨uhlenberg 1, 14476 Potsdam, Germany
Using the recently established formalism of a worldline quantum field theory (WQFT) descriptionof the classical scattering of two spinless black holes, we compute the far-field time-domain waveformof the gravitational waves produced in the encounter at leading order in the post-Minkowskian (weakfield, but generic velocity) expansion. We reproduce previous results of Kovacs and Thorne in ahighly economic way. Then using the waveform we extract the leading-order total radiated angularmomentum, reproducing a recent result of Damour. Our work may enable crucial improvements ofgravitational-wave predictions in the regime of large relative velocities.
When two compact objects (black holes, neutron starsor stars) fly past each other their gravitational in-teractions not only deflect their trajectories but theyalso produce gravitational radiation, or gravitationalBremsstrahlung in analogy to the electromagnetic case.The resulting waveform in the far field at leading or-der in Newton’s constant G has been constructed (in thespinless case) in a series of papers by Kovacs, Thorne,and Crowley in the 1970s [1–4] — see Refs. [5] for re-cent work on slow-motion sources. Today’s gravitationalwave (GW) observatories routinely detect quasi-circular inspirals and mergers of binary black holes and neutronstars [6]. Yet Bremsstrahlung events currently appear tobe out of reach as the signal is not periodic and typicallyless intensive [7]. Still, they represent interesting targetsfor GW searches, calling for accurate waveform models.Indeed, the experimental success of GW astronomybrings up the need for high-precision theoretical predic-tions for the classical relativistic two-body problem [8].A number of complementary classical pertubative ap-proaches have been established over the years [9]. Yetquantum-field-theory based techniques founded in a per-turbative Feynman-diagrammatic expansion of the pathintegral in the classical limit have proven to be highlyefficient. These come in two alternative approaches.The first approach, the effective field theory (EFT)formalism [10], models the compact objects as point-likemassive particles coupled to the gravitational field. It hasmostly been applied to a nonrelativistic post-Newtonian(PN) scenario for bound orbits, in which an expansionin powers of Newton’s constant G implies an expansionin velocities ( Gmc r ∼ v c ). Recently it has also been ex-tended to the post-Minkowskian (PM) expansion for un-bound orbits [11, 12] relevant for this work, an expansionin G for arbitrary velocities. In these EFT settings thegraviton field h µν ( x ) is integrated out successively (fromsmall to large length scales) in the path integral, while ∗ [email protected] † [email protected] ‡ [email protected] § jan.steinhoff@aei.mpg.de the worldline trajectories of the black holes x µi ( τ i ) arekept as classical background sources — see Refs. [13] forreviews.The second now blossoming approach starts out fromscattering amplitudes of massive scalars — avatars ofspinless black holes — minimally coupled to general rel-ativity [14], thereby putting the younger innovations inon-shell techniques for scattering amplitudes (e.g. gen-eralized unitarity [15] or the double copy [16]) to work.In order to obtain the conservative gravitational poten-tial one performs a subtle classical limit of the scatteringamplitudes [17] in order to match to a non-relativisticEFT for scalar particles with the desired potential [18](see also Refs. [14, 18]), which is known to 3PM or-der [14] (complemented by certain radiation-reaction ef-fects [17, 19, 20]). Very recently the 4PM conservativepotential was also reported [21]. The so-obtained effec-tive potential is then used to compute observables suchas the scattering angle or the (PM-resummed) periastronadvance in the bound system [11, 21]. Further recent PMresults exist for non-spinning particles [22], for spin ef-fects [23], tidal effects [24], and radiation effects [25].In a recent work of three of the present authors thesynthesis of these two quantum-field-theory based ap-proaches to classical relativity was provided in the formof a worldline quantum field theory (WQFT) [26]: quan-tizing both the graviton field h µν and the fluctuationsabout the bodies’ worldline trajectories z µi were shownto yield an efficient approach yielding only the relevantclassical contributions. In essence the WQFT formalismprovides an efficient diagrammatic framework for solvingthe equations of motion of gravity-matter systems per-turbatively.In this Letter we employ this novel formalism tocompute the time-domain gravitational waveform of aBremsstrahlung event at leading order in G , demonstrat-ing its effectiveness. To our knowledge the seminal resultof Kovacs and Thorne [4] has not been verified in its en-tirety to date. As we shall see, our approach is far moreefficient than the one employed back then, paving theway for calculations of higher orders. We stress that weare able to determine the far-field waveforms which areof direct relevance for GW observatories. As a check on a r X i v : . [ g r- q c ] F e b µ,ν −→ ω kq ↑ (a) µ,ν ω −→ kq ↓ (b) µ,ν kq ↓ q ↑ (c) FIG. 1. The three diagrams contributing to theBremsstrahlung at 2PM order, where ω i = k · v i by energy con-servation at the worldline vertices. All three diagrams havethe integral measure in Eq. (16); in the rest frame of blackhole 1 diagram (a) does not contribute as soon as the outgo-ing graviton is contracted with a purely spatial polarizationtensor. these waveforms we furthermore reproduce Damour’s re-cent result for the total radiated angular momentum [19]at 2PM order. Our results also complement the recentresult of the total radiated momentum at leading order in G (3PM) established with amplitude techniques [27]. Wecomment on how to achieve this result from our methods. Worldline Quantum Field Theory. —
The classicalgravitational scattering of two massive objects m i mov-ing on trajectories x µi ( τ i ) = b µi + v µi τ i + z µi ( τ i ) is describedby the worldline quantum field theory (WQFT) with par-tition function [26] Z WQFT := const × (cid:90) D [ h µν ] (cid:90) (cid:89) i =1 D [ z i ] e i ( S EH + S gf ) (1)exp (cid:104) − i (cid:88) i =1 (cid:90) ∞−∞ d τ i m i η µν + κh µν ( x )] ˙ x µi ˙ x νi (cid:105) , where S EH + S gf is the gauge-fixed Einstein-Hilbert action S EH + S gf = (cid:90) d x (cid:0) − κ √− gR + ( ∂ ν h µν − ∂ µ h νν ) (cid:1) , (2)with κ = 32 πG the gravitational coupling; we have sup-pressed the ghost contributions in Eq. (1) as they areirrelevant in the classical setting. We work in mostlyminus signature, η µν = diag(1 , − , − , − (cid:104)O ( h, { x i } ) (cid:105) WQFT result from an insertion of the operator O in the pathintegral and dividing by Z WQFT . Moving to momentumspace for the graviton h µν ( k ) and energy space for thefluctuations z µ ( ω ) we have the retarded propagators kµν ρσ = i P µν ; ρσ ( k + i(cid:15) ) − k , (3a) ωµ ν = − i η µν m ( ω + i(cid:15) ) , (3b)with P µν ; ρσ := η µ ( ρ η σ ) ν − η µν η ρσ . The relevant verticesfor the emission of a graviton off the worldline read h µν ( k ) = − i mκ e ik · b δ − ( k · v ) v µ v ν , (4)with k outgoing, δ − ( ω ) := (2 π ) δ ( ω ) and h µν ( k ) z ρ ( ω ) = mκ e ik · b δ − ( k · v + ω ) (5) × (cid:16) ωv ( µ δ ν ) ρ + v µ v ν k ρ (cid:17) . The energy ω is also taken as outgoing. One also has thestandard bulk graviton vertices, of which we shall needonly the three-graviton vertex — see e.g. Ref. [28].To determine the Bremsstrahlung of two travers-ing black holes we compute the expectation value k (cid:104) h µν ( k ) (cid:105) WQFT . At leading (2PM) order there are threediagrams contributing, cp. Fig. 1. We integrate over themomenta or energies of internal gravitons or fluctuationsrespectively; lack of three-momentum conservation at theworldline vertices leaves unresolved integrals for the tree-level diagrams.Diagram (a) of Fig. 1 then takes the form k (cid:104) h µν ( k ) (cid:105) WQFT (cid:12)(cid:12)(cid:12) ( a ) = − m m κ (cid:90) q ,q µ , ( k ) (2 ω v ( µ δ ν ) ρ − v µ v ν k ρ )(2 ω v ( σ η λ ) ρ − v σ v λ q ρ )( ω + i(cid:15) ) P σλ ; αβ [( q + i(cid:15) ) − q ] v α v β , (6) In principle we should also contract with P µν ; ρσ for an outgo-ing graviton line; however as the polarization tensors e µν + , × are traceless we find it unnecessary. where ω = k · v , (cid:82) q i := (cid:82) d q i (2 π ) and the integral measure is µ , ( k ) = e i ( q · b + q · b ) δ − ( q · v ) δ − ( q · v ) δ − ( k − q − q ) , (7)with δ − ( k ) := (2 π ) δ (4) ( k ). The diagram (b) is naturally obtained by swapping 1 ↔
2. Diagram (c) includes thethree-graviton vertex V ( µν )( ρσ )( λτ )3 ( k, − q , − q ): k (cid:104) h µν ( k ) (cid:105) WQFT (cid:12)(cid:12)(cid:12) ( c ) = − m m κ (cid:90) q ,q µ , ( k ) V ( µν )( ρσ )( λτ )3 P ρσ ; αβ [( q + i(cid:15) ) − q ] P λτ ; γδ [( q + i(cid:15) ) − q ] v α v β v γ v δ . (8)These integrands were already given in Ref. [26]. Thesum of the three integrands also agrees with a previousamplitudes-based result [14] and is gauge-invariant.The waveform in spacetime in the wave zone is ob-tained from (cid:104) h µν ( k ) (cid:105) WQFT as follows: we may identify k (cid:104) h µν ( k ) (cid:105) WQFT = κ S µν ( k ) , (9)where S µν = τ µν − η µν τ λλ and τ µν is the combinedenergy-momentum pseudo-tensor of matter and the grav-itational field. Consider S µν ( k ) for a fixed GW frequency k = Ω. In the wave zone ( r (cid:29) {| b i | , Ω − , Ω | b i | } ) themetric perturbation h µν ( x , t ) takes the form of a planewave (see e.g. Chapter 10.4 of Weinberg [29]): κh µν ( x , t ) = 4 Gr S µν (Ω , k = Ω ˆx ) e − ik µ x µ + c.c , (10)with the wave vector k µ = Ω(1 , ˆx ); ˆx = x /r is the unitvector pointing in the direction of the observation point(hence k = 0).The total gauge-invariant frequency-domain waveformcan be read off as 4 G S TT ij (Ω , k = Ω ˆx ), where TT de-notes the transverse-traceless projection. The corre-sponding time-domain waveform f ij ( u, θ, φ ) is essentiallyits Fourier transform in Ω: κh TT ij = f ij r = 4 Gr (cid:90) Ω e − ik · x S TT ij ( k ) (cid:12)(cid:12)(cid:12) k µ =Ω (1 , ˆ x ) , (11)where (cid:82) Ω := (cid:82) ∞−∞ dΩ2 π . Note that k · x = Ω( t − r ) yieldsthe retarded time u = t − r . Our task now is to performthe integrals; in a PM expansion f ij = (cid:80) n G n f ( n ) ij , andwe seek the 2PM component f (2) ij . By focusing on thetime-domain instead of the frequency-domain waveformwe considerably simplify the integration step — as weshall see, the integration over frequency Ω of the outgoingradiation coincides neatly with energy conservation alongeach worldline. Kinematics. —
We describe the waveform in a Carte-sian coordinate system ( t, x, y, z ) where black hole 1 isinitially at rest v µ = (1 , , ,
0) and located at the spa-tial origin, i.e. we set b µ = 0. The orbit of blackhole 2 we put in the x – y plane with initial velocity v µ = ( γ, γ v, ,
0) in the x -direction; the impact param-eter b µ = (0 , , b,
0) =: b µ points in the y -direction. In-troducing the polar angles θ and φ we may write the unit (spatial) vector ˆ x µ pointing from black hole 1 to ourobservation point asˆ x µ = ˆ e µ cos θ + sin θ (cid:0) ˆ e µ cos φ + ˆ e µ sin φ (cid:1) , (12)where ˆ e µi = (0 , ˆ e i ) are spatial unit vectors. Also, we put ρ µ = v µ + ˆ x µ .The two additional unit spatial vectors orthogonal toˆ x µ areˆ θ µ = ∂ θ ˆ x µ = (0 , ˆ θ ) , ˆ φ µ = θ ∂ φ ˆ x µ = (0 , ˆ φ ) . (13)Together with ˆ x µ they form a right-handed spatial co-ordinate system. GWs travel in the direction of ˆ x µ andwe use ˆ θ µ and ˆ φ µ to define our polarization tensors in alinear basis: e µν + = ˆ θ µ ˆ θ ν − ˆ φ µ ˆ φ ν , e µν × = ˆ θ µ ˆ φ ν + ˆ φ µ ˆ θ ν . (14)The waveform f ij ( u, θ, φ ) is thus decomposed as f ij = f + ( e + ) ij + f × ( e × ) ij (15)with f + , × = ( e + , × ) ij f ij .The polarization tensors have zero time components,which conveniently implies the vanishing of diagram (a)in Fig. 1 once contracted with them. This observationfollows directly from the expression for vertex (5): in thecase of diagram (a) the instance of this vertex that con-tracts with the outgoing graviton line carries an overallfactor of v µ = (1 , , , Integration. —
The two non-zero diagrams in Fig. 1share the integration measure µ , ( k ) (7). Including alsothe integration with respect to Ω in Eq. (11) the fullmeasure becomes (cid:90) Ω ,q ,q µ , ( k ) e − ik · x = 1 ρ · v (cid:90) q e i q · (cid:101) b , (16)where we recall that k µ = Ω ρ µ ; using the delta functionconstraints in µ , ( k ) we can now identify q = k − q , q = (0 , q ) , Ω = − vγρ · v q · ˆ e . (17)We are left with a three-dimensional Euclidean integralinvolving the shifted τ -dependent impact parameter: (cid:101) b ( τ ) = b + τ ˆ e , τ = vγρ · v ( u + b · ˆx ) , (18)noting that ρ · v = γ (1 − v cos θ ). The polarizations ofthe waveform from Eq. (11) now take the schematic form(also using Eq. (15)) f (2)+ , × m m (19)= 4 π (cid:90) q e i q · (cid:101) b (cid:32) N i + , × q i q ( q · ˆ e − i(cid:15) ) + M ij + , × q i q j q ( q + q · L · q ) (cid:33) , with the two terms corresponding to the non-zero dia-grams (b) and (c) in Fig. 1 respectively. The rank-2matrix L introduced here is L ij = 2 vγρ · v ˆ e ( i ˆ x j ) . (20)Finally the vector and matrix insertions are explicitlygiven as the real and imaginary parts of N i = 2 γ sin θρ · v (cid:16) γ (1 − v ) ρ · v + (1 + v ) (cid:17) ˆ e i (21a)+ 2 γ (1 + v ) sin θρ · v (cid:16) ( ρ · v ) − v ( ρ · v ) cos φ + 2 iγ sin φ (cid:17) ˆ e i , M ij = 8 γ v sin θ ( ρ · v ) ˆ e i ˆ e j + 16 γ v sin θ ( ρ · v ) ˆ e ( i ( ˆ θ + i ˆ φ ) j ) + 4 γ (1 + v ) ρ · v ( ˆ θ + i ˆ φ ) i ( ˆ θ + i ˆ φ ) j , (21b)where N i = N i + + i N i × and M ij = M ij + + i M ij × . Theinsertions N i and M ij correspond to a helicity basis inwhich they have a particular simple expression. We in-tegrate the two diagrams separately.Integration of the first diagram is achieved using thesimple result (true regardless of the vector (cid:101) b ) (cid:90) q e i q · (cid:101) b q i q ( q · ˆ e − i(cid:15) ) (22)= 14 π (cid:32) ˆ e i | (cid:101) b | − (cid:101) b i − ( (cid:101) b · ˆ e )ˆ e i (cid:101) b − ( (cid:101) b · ˆ e ) (cid:32) (cid:101) b · ˆ e | (cid:101) b | (cid:33)(cid:33) , which we prove in the Appendix. The other integral re-quired corresponding to diagram (c) is somewhat moreinvolved. The denominator of this integral is composedof an isotropic propagator together with an anisotropicone. The physical interpretation is a convolution betweenthe potentials of the two black holes, where the potentialof black hole 2 is boosted and leads to the anisotropic To compactify these results we have used the generalized gaugeinvariance N + , × → N + , × + X ˆ e , M + , × → M + , × − X ( + L )for an arbitrary function X of external kinematics. We havealso dropped a term from N ± in the ˆ e direction which does notcontribute to the final integrated result (27). propagator. One compact representation is (cid:90) q e i q · (cid:101) b q i q j q ( q + q · L · q ) (23)= 12 π ∆( G ) (cid:34) ( G + αG ) A ij − ( G + αG ) B ij (cid:112) G ( α ) (cid:35) α =1 α =0 , where we have introduced the quadratic polynomial G ( α ) = G + 2 αG + α G , (24) G = (cid:101) b , G = (cid:101) b i (cid:101) b j δ ij L kk − L ij , G = ( (cid:101) b · ˆ φ ) Det L , and ∆( G ) = 4( G − G G ) is the polynomial discrimi-nant. We have also introduced the two matrices A ij = Det ( L ) (cid:16) − (cid:101) b · ˆ φ )( L − · (cid:101) b ) ( i ˆ φ j ) (25a)+ ( (cid:101) b · ˆ φ ) ( L − ) ij + ( (cid:101) b · L − · (cid:101) b ) ˆ φ i ˆ φ j (cid:17) ,B ij = (cid:101) b δ ij − (cid:101) b i (cid:101) b j . (25b)Det ( L ) and L − are computed in the 2-dimensional sub-space spanned by L , while ˆ φ is the unit vector orthog-onal to this subspace. This integral is also discussed inthe Appendix; again, both of the integrals (22) and (23)are solved for arbitrary (cid:101) b i and L ij (with the assumptionthat L ij is rank 2). In the present case where L ij is givenby Eq. (20) we find thatDet ( L ) = − (cid:16) γv sin θρ · v (cid:17) , (26a)( L − ) ij = ρ · v γv (cid:88) ± ± ( ˆ x ± ˆ e ) i ( ˆ x ± ˆ e ) j (1 ± cos θ ) , (26b)summing over the two signs in the latter case. Leading-Order Waveform. —
By combining Eq. (19)with the insertions N i + , × and M ij + , × and the integralsabove we get the full 2PM waveform: f (2)+ , × m m = ˆ e i N i + , × √ b + τ − b i N i + , × b (cid:18) τ √ b + τ (cid:19) + 2 M ij + , × ∆( G ) (cid:20) ( G + αG ) A ij − ( G + αG ) B ij (cid:112) G ( α ) (cid:21) α =1 α =0 . (27)This is a rather compact representation of the gravi-tational Bremsstrahlung waveform, which we have con-firmed agrees with the (rather lengthy) result of Kovacsand Thorne [4]. The two values of α in the second linecorrespond to contributions from the two black holes.Note that there is also a leading (and non-radiating) 1PMcontribution to the waveform which is independent of theretarded time u = t − r : f (1)+ = 2 m γv sin θ − v cos θ , f (1) × = 0 . (28) FIG. 2. Plots of the wave memories ∆ f + , × for v = 0 .
2. For avisualisation of the complete waveforms as they evolve withretarded time u see f + ( u, θ, φ ) | v =0 . and f × ( u, θ, φ ) | v =0 . . Diagrammatically this consists only of the vertex (4) withemission from worldline 2; the contribution from world-line 1 again vanishes in our frame due to ( v · e ± · v ) = 0.To illustrate this result in Fig. 2 we present the grav-itational wave memories ∆ f + , × := [ f + , × ] u =+ ∞ u = −∞ . Thebeauty of our result (27) lies in the fact that the memo-ries only receive contributions from the second term, andread ∆ f + , × = − G m m b i N i + , × b + O ( G ) . (29)Diagrammatically they exclusively emerge from diagram(b) of Fig. 1. So they are manifestly insensitive to grav-itational self-interactions — this was also pointed out inRef. [19]. Radiated Angular Momentum. —
One may now useour result for the waveform (27) to compute the total ra-diated momentum and angular momentum. Expressionsfor these quantities in terms of the asymptotic waveformare given in Refs. [19, 30]: P µ rad = 132 πG (cid:90) d u d σ [ ˙ f ij ] ρ µ , (30) J rad ij = 18 πG (cid:90) d u d σ (cid:18) f k [ i ˙ f j ] k − x [ i ∂ j ] f kl ˙ f kl (cid:19) , (31)where ˙ f ij := ∂ u f ij and d σ = sin θ d θ d φ is the unit spheremeasure. Here we concentrate on J rad ij as it contributesat leading order O ( G ) due to ˙ f (1) ij = 0 and was recentlyobtained in the center-of-mass frame [19].At leading order the static nature of f (1) ij (28) allowsone to trivially perform the u -integration and express theradiated angular momentum in terms of the wave mem-ories ∆ f + , × . Inserting the basis of polarization tensors(15) (and using f (1) × = 0) gives J rad xy = 18 π (cid:90) d σ (cid:104) sin φ sin θ f (1)+ ∆ f × −
12 cos φ ∂ θ f (1)+ ∆ f + (cid:105) + O ( G ) . (32) The spherical integral is elementary and yields J rad xy J init xy = 4 G m m b (2 γ − (cid:112) γ − I ( v ) + O ( G ) , (33a) I ( v ) = −
83 + 1 v + (3 v − v arctanh( v ) , (33b)where we have normalized our result with respect to theinitial angular momentum in the rest frame of black hole1: J init xy = m | v || b | = m γvb . Compared with Damour’scenter-of-mass frame result [19] we find perfect agree-ment. Similarly evaluating Eq. (30) should reproducethe recent result of Ref. [27] for the radiated momentum.
Conclusions and Outlook. —
Searching for GWs fromscattering events over the full range of impact veloci-ties requires precision predictions in the PM approxima-tion. Furthermore, PM results may even improve GWpredictions for bound systems observed by current GWdetectors. Indeed, while the potential and radiation ofbound systems was calculated to high PN order [31] (seeRefs. [32] for spinning bodies), a resummation of PN re-sults in the strong-field and fast-motion regimes is essen-tial for building accurate waveform models and avertingthe imminent domination of systematic errors in observa-tions [8]. The PM resummation is one promising recentattempt [18, 33, 34].We believe our results provide a stepping stone forhigher-order calculations, where a repertoire of advancedintegration techniques can be put to use [14, 21, 27, 35].This work provides an explicit link, via the WQFT [26],between quantum scattering amplitudes and classicalwaveforms ultimately used by GW observatories. An ex-tension to spin and finite-size effects within the WQFTappears quite possible, and would involve integrals sim-ilar to the non-spinning case. Deriving an analytic ex-pression for frequency-domain PM waveforms would alsobe very useful; in fact the WQFT readily leads to one-dimensional integrals involving Bessel functions. Yet, theclass of special functions describing these waveforms re-mains to be identified.Finally, our eventual aim is an extension to bound or-bits. Recent work has shown that mappings betweenbound and unbound orbits exist for both conservativeand dissipative observables [21, 36]; finding a similarmapping for the waveform would be of great utility.
Acknowledgments. —
We would like to thank A. Buo-nanno and J. Vines for helpful discussions. We arealso grateful for use of G. K¨alin’s C ++ graph library.GUJ’s and GM’s research is funded by the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) Projektnummer 417533893/GRK2575 “Rethink-ing Quantum Field Theory”. As the two frames are related by a boost in the x direction thisimplies that J rad0 y = 0 in both frames. INTEGRALS
We begin with the simpler integral in Eq. (22), corre-sponding to diagram (b) in Fig. 1. Working in Cartesiancomponents with b = ( b , b , b ) (in the main text we re-place b → (cid:101) b ) and q = ( q , q , q ) it is sufficient to show (cid:90) q e i q · b q q ( q − i(cid:15) ) = − b π ( b − b ) (cid:18) b | b | (cid:19) . (34)The corresponding result with numerator q is related bysymmetry, and the one with q is trivially given by (cid:90) q e i q · b q = 14 π | b | . (35)We make convenient use of the fact that (cid:90) ω e iωτ f ( ω ) ω − i(cid:15) = i (cid:90) τ −∞ d τ (cid:48) (cid:90) ω e iωτ (cid:48) f ( ω ) , (36)the i(cid:15) prescription implying that our integrals are bound-ary fixed at τ → −∞ . This is to be expected given ouruse of retarded propagators. The original integral cantherefore be re-written as (cid:90) q e i q · b q q ( q − i(cid:15) ) = ∂∂b (cid:90) b −∞ d b (cid:48) (cid:90) q e i q · b (cid:48) q , (37)where b (cid:48) = ( b (cid:48) , b , b ). From here using Eq. (35) it is asimple exercise to reproduce Eq. (34). Next, let us consider the anisotropic integral (23) corre-sponding to the gravitational self-interaction of the twoblack hole potentials. The numerator can be obtainedby differentiation with respect to the impact parameter;also introducing a Feynman parameter α the integral isre-expressed as I ij = (cid:90) q e i q · b q i q j q ( q + q · L · q )= − ∂∂ b i ∂∂ b j (cid:90) d α (cid:90) q e i q · b ( q + α q · L · q ) . (38)The q -integration and b derivatives are straightforward;we are left with the Feynman parameter integral I ij = (cid:90) d α det( M ) b k b l ( M − ) k [ l ( M − ) i ] j πG ( α ) / , (39)where the quadratic function G ( α ) = det( M ) b · M − · b was given in Eq. (24) and the α -dependent matrix M ij is defined as simply M ij = δ ij + αL ij . (40)The Feynman parameter integral is solved by recognizingthat the numerator in Eq. 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