Modulation of the gravitational waveform by the effect of radiation reaction
aa r X i v : . [ g r- q c ] J a n Modulation of the gravitational waveform by the effect of radiation reaction
Yasushi Mino ∗ mail code 130-33, California Institute of Technology, Pasadena, CA 91125 When we calculate gravitational waveforms from extreme-mass-ratio inspirals (EMRIs) by metricperturbation, it is a common strategy to use the adiabatic approximation. Under that approxima-tion, we first calculate the linear metric perturbation induced by geodesics orbiting a black hole,then we calculate the adiabatic evolution of the parameters of geodesics due to the radiation reactioneffect through the calculation of the self-force. This procedure is considered to be reasonable, how-ever, there is no direct proof that it can actually produce the correct waveform we would observe. Inthis paper, we study the formal expression of the second order metric perturbation and show that itbe expressed as the linear metric perturbation modulated by the adiabatic evolution of the geodesic.This evidence supports the assumption that the adiabatic approximation can produce the correctwaveform, and that the adiabatic expansion we propose in Ref.[1] is an appropriate perturbationexpansion for studying the radiation reaction effect on the gravitational waveform.
I. INTRODUCTION
We study the gravitational waveforms from particles moving around Kerr black holes by using a metric perturbationmethod. There is an established method for calculating these waveforms from the linear metric perturbation ofKerr black holes. By the consistency of the Einstein equation, the source stress-energy tensor of the linear metricperturbation must satisfy the conservation law with respect to the background. As a result, the source of the linearmetric perturbation moves along a geodesic of the background Kerr metric. Due to the integrability of the geodesicequation for the Kerr metric, its bound solutions are stable and have periodic features. To understand some featureof gravitational waves from such a stable system, we develop a technique of formal calculation[2]. The advantage ofthis technique is that, one can easily grasp some key features of the waveform without a complicated calculation, andit helps us to construct the strategy for an explicit calculation. Using this technique, we find that the waveform fromthe linear metric perturbation has a periodic feature[2] as we review in Sec.II. Based on our understanding of thisfeature, a present numerical code is trying to identify which of the gravitational wave modes are strong enough to beobserved by gravitational wave detectors[3].Because gravitational waves carry away energy and angular momentum, the system must have a dissipative evo-lution. This should change the periodic features of the gravitational waveform that we know from the linear metricperturbation. New features must be seen in the second order metric perturbation because the source term mustinclude this dissipative effect. The goal of this paper is to find these new features. That is, the goal is to find how thewave amplitude and phase are modulated by gravitational radiation reaction. However, a serious calculation of thesecond order metric perturbation still has a lot of technical difficulties. We therefore extend the technique of formalcalculation we developed to study the linear metric perturbation. We expect that the result of this formal calculationwill be useful in future explicit calculations of the second order metric perturbation.This problem has attracted the attention of the gravity community[4] because its relevance to gravitational wavedetectors, especially LISA. Among the primary targets for LISA are extreme-mass ratio inspirals (EMRIs), theinspiralling binary systems of supermassive black holes (with mass ∼ − M ⊙ ) and stellar mass compact objects(with mass ∼ − M ⊙ ). Because of the extreme mass ratios, the metric perturbation of the black hole is effectivefor studying the dynamics of the system; we use the Kerr geometry of the supermassive black hole as a background,and approximate the compact object as a point particle. The linear metric perturbation may predict the waveformat an instant. However, LISA will detect gravitational waves for several years. This time scale is comparable to theradiation reaction time scale of EMRIs and is essential to consider the dissipation effect of the gravitational radiationreaction because the effect will accumulate during the observation time.Most of the relevant investigations made so far discuss the calculation of the self-force [5] which needs regularization[6]. The underlying idea for such calculations is that: (1) At each instant, the orbit is approximated by a geodesicsince, due to the extreme mass ratio, gravitational radiation reaction is a small effect. (2) As the effect of radiationreaction accumulates over time, the orbit changes from one geodesic to another. (3) These changes can be deducedfrom the self-force. This idea is usually referred as the ‘adiabatic’ approximation. This calculation strategy has some ∗ Electronic address: [email protected] theoretical problems. Since the self-force is gauge dependent, we may not be able to make a unique prediction aboutthe orbital evolution. In Ref. [1], we showed that, for times over which the standard metric perturbation expansionis valid, the self-force can be arbitrarily adjusted via gauge changes. As an extreme example, we showed that there isa gauge transformation which completely eliminates the self-force.Do we still need to calculate the self-force? In Ref. [1], we argue that the answer may be yes. Under a certain gaugecondition, the self-force may include the correct radiation reaction effect With this gauge condition, the calculationby the ‘adiabatic’ approximation will give us the correct prediction of the gravitational waveform. The present paperdiscusses the self-force under that ‘physically reasonable’ class of gauge conditions proposed in Ref. [1, 2] in thecontext of the second order metric perturbation. We will focus on gauge invariant quantities. The basic idea is that,because the gravitational waveform is observable, the features we can see from the waveform have invariant meanings.We will show that the self-force correctly describes the radiation reaction effect through the second order metricperturbation.The field equations for the linear and second order metric perturbations are derived as follows. We expand themetric in a small parameter ǫ g µν = g (0) µν + ǫg (1) µν + ǫ g (2) µν + O ( ǫ ) , (1.1)where g (0) µν is the background metric. We insert the ǫ -expansion of the metric into the Einstein tensor and expand itin powers of ǫ . We formally obtain G µν = ǫG [1] µν [ g (1) ] + ǫ n G [1] µν [ g (2) ] + G [2] µν [ g (1) , g (1) ] o + O ( ǫ ) , (1.2)where G [1] µν [ h ] is linear in h µν and G [2] µν [ h (1) , h (2) ] is bi-linear in h [1] µν and h (2) µν . By similarity expanding the sourceterm T µν = ǫT [1] µν + ǫ T [2] µν + O ( ǫ ) , (1.3)we can formally write the perturbed Einstein equations to second order ǫG [1] µν [ g (1) ] = 8 πT [1] µν , (1.4) G [1] µν [ g (2) ] = − G [2] µν [ g (1) , g (1) ] + 8 πT [2] µν . (1.5)In Sec.II, we review the derivation of (1.4) [2]. Due to the consistency of the Einstein equation, the source of thelinear metric perturbation, T [1] µν , must be a geodesic. Because we are interested in a bound geodesic, bound geodesicsare triperiodic, and we will see that the linear metric perturbation induced by it is as well. Before discussing thesecond order equation (1.5), we make a formal argument about the self-force and the orbital evolution due to thiseffect in Sec. III because those effects should be included in the source of the second order equation, T [2] µν . In Sec.IV, we discuss the formal analysis of the second order Einstein equation (1.5). The second order metric perturbation isinduced by the quadratic term of the linear metric perturbation, and by the second order source term. We emphasizethese two effects leads to substantially different features of the second order metric perturbation. Sec. V concludesour result.Our background geometry will be a Kerr black hole with mass M and spin parameter a . We will a point particlewith mass µ as the source of the metric perturbation. Throughout this paper, we use Boyer-Lindquist coordinates, { t, r, θ, φ } = { x α } . We adopt the geometrized units which are defined such that G = c = 1. II. LINEAR METRIC PERTURBATION
In this section, we review the periodic feature of the linear metric perturbation. It is well known that, when thebackground is a vacuum solution, the source term of the linear metric perturbation must be conserved with respectto the background. Since we use a point mass source, this means that its world line is a geodesic of the backgroundgeometry. Because we are interested in gravitational waves from binaries in the inspiralling phase, we only considerbound geodesics.We denote the orbital coordinates of the geodesic by { ¯ x α } and the 4-velocity by ¯ v α := d ¯ x α /dτ , where τ is the propertime. The geodesic equation for a Kerr black hole has three nontrivial constants of motion: the energy E = µη Eα ¯ v α ,the z -component of angular momentum L = µη Lα ¯ v α and the Carter constant C = ( µ/ η αβ ¯ v α ¯ v β . Here η Eα and η Lα aretemporal and rotational Killing vectors, respectively, and η αβ is the Killing tensor. Hereafter we collectively denotethese constants by E a . The geodesic equations can then be formally written as (cid:18) d ¯ rdλ (cid:19) = R (¯ r ; E a ) , (cid:18) d ¯ θdλ (cid:19) = Θ(¯ θ ; E a ) , (2.1) d ¯ tdλ = T r (¯ r ; E a ) + T θ (¯ θ ; E a ) , d ¯ φdλ = Φ r (¯ r ; E a ) + Φ θ (¯ θ ; E a ) , (2.2)where λ is the affine parameter of the orbit related with the proper time by λ = R dτ / (¯ r + a cos ¯ θ ). For boundgeodesics, the r -motion and the θ -motion are periodic and one can write them as a discrete Fourier series. We formallywrite ¯ r = X k r k e ikχ r , χ r = Υ r ( λ − λ r ) , (2.3)¯ θ = X l θ l e ilχ θ , χ θ = Υ θ ( λ − λ θ ) , (2.4)where λ r and λ θ are constants for integration. The expansion coefficients, r k and θ l , and the effective frequencies, Υ r and Υ θ , are functions of the constants of motion E a . Using (2.3) and (2.4), we can formally integrate (2.2) as¯ t = χ t + X k t rk e ikχ r + X l t θl e ilχ θ , χ t = Υ t ( λ − λ t ) , (2.5)¯ φ = χ φ + X k φ rk e ikχ r + X l φ θl e ilχ θ , χ φ = Υ φ ( λ − λ φ ) , (2.6)where λ t and λ φ are constants of integration. The expansion coefficients, t rk , t θk , φ rk and φ θl , and the effective frequencies,Υ t and Υ φ , are functions of the constants of motion E a . In summary, one can specify a geodesic by three constantsof motion E a and four constants of integration λ α . Because one is free to choose the zero of the affine parameter λ ,one of four integral constants λ α is not physically significant, and the gravitational waveform induced by the geodesicdepends only on three differences of the four integral constants [See (2.14).].Under a certain gauge condition, one can define a tensor Green’s function for the linear Einstein equation (1.4).Because the background is stationary and axisymmetric, this tensor Green’s function has the form G αβµν ( x, x ′ ) = G αβµν ( t − t ′ , φ − φ ′ ; r, r ′ , θ, θ ′ ) . (2.7)For the linear metric perturbation, the source term is described by the stress-energy tensor of a point particle [5]written as T [1] µν = µ Z dτ ¯ v µ ¯ v ν δ ( x − ¯ x ( τ )) p −| g [0] | , (2.8)where | g [0] | is the determinant of the background metric. The corresponding linear metric perturbation is given by g [1] αβ ( x ) = 8 πµ Z dλ (cid:18) dτdλ (cid:19) G αβµν ( x, ¯ x ( λ )) ¯ v µ ¯ v ν . (2.9)Using the formal expression for the geodesic, (2.3)-(2.6), we obtain the following formal expression for this linearmetric perturbation g [1] αβ ( x ) = X k,l,m e − iω ( k,l,m ) t + imφ g [1] αβ ( k,l,m ) ( r, θ ) e i ( kω r t r + lω θ t θ + mω φ t φ ) , (2.10) ω ( k,l,m ) = kω r + lω θ + mω φ , ω r = Υ r Υ t , ω θ = Υ θ Υ t , ω φ = Υ φ Υ t , (2.11) t r = Υ t ( λ r − λ t ) , t θ = Υ t ( λ θ − λ t ) , t φ = Υ t ( λ φ − λ t ) , (2.12)where the expansion coefficients, g [1] αβ ( k,l,m ) ( r, θ ), depend only on the constants of motion, E a , and are independent of λ r , λ θ , λ t and λ φ .The Green’s function (2.7) can be obtained under the Lorenz gauge or the Harmonic gauge. However, it can bedefined under a larger class of gauge conditions which we call by the ‘physically reasonable class’. Its definition canbe understood by considering the residual gauge transformation. By the gauge transformation x α → x α + ξ α withthe gauge field of the form ξ α ( x ) = X k,l,m e − iω ( k,l,m ) t + imφ ξ α ( k,l,m ) ( r, θ ) e i ( kω r t r + lω θ t θ + mω φ t φ ) , (2.13)the expansion coefficients h αβ ( k,l,m ) ( r, θ ) are transformed, but, the formal expression of the linear metric perturbation(2.10) is invariant.The main features of the gravitational waveforms can be read off from the formal expression (2.10). For any distantobserver at r → ∞ and a specific angular position, the waveform can be written as h ( t ) = X k,l,m h ( k,l,m ) e − ikω r ( t − t r ) − ilω θ ( t − t θ ) − imω φ ( t − t φ ) . (2.14)Because the characteristic frequencies of waves ω r , ω θ , and ω φ are all associated with geodesics, we may concludethat the waveform (2.14) does not reflect dissipative effects occurring on the radiation reaction time scale. This is tobe expected since the source of the linear metric perturbation moves on a stable geodesic. III. SELF-FORCE AND THE ORBITAL EVOLUTION
Because the gravitational wave part of the linear metric perturbation carries away energy, the orbit of the particledeviates away from a geodesic. The effect of the orbital deviation is described by the so-called self-force, that isboth induced by and acting on the particle itself. Because the metric perturbation induced by the particle divergesalong the orbit, a regularization prescription is necessary to obtain the finite self-force [5]. Up to leading order in theparticle’s mass, we developed the regularization prescription by the technique of the matched asymptotic expansion.The final result is now called the MiSaTaQuWa self-force [5], and is formally written as
Ddτ v α := µf α ( τ ) = lim x → ¯ x ( τ ) µf α [ g [1] − g [1] sing. ]( x ) , (3.1)where the bare term g [1] represents the full linear metric perturbation induced by the point particle, and the counterterm g [1] sing. is the singular part of the linear metric perturbation to be subtracted for regularization. f α [] is aderivative operator for the self-force. Because both terms g [1] and g [1] sing. are divergent along the geodesic, it isnecessary to evaluate them at a field point x α , not at the geodesic. After the subtraction, g [1] − g [1] sing. becomesregular along the geodesic, and we may take the limit x α → ¯ x α to obtain the finite self-force.One can use the formal expression (2.10) to evaluate the bare term g [1] . The formal expression (2.10) is alsoapplicable for the counter term g [1] sing. because one can derive the Green’s function for the counter term in the sameform (2.7) as we discussed in Ref. [2]. After some formal calculations, the self-force can be written as µf α = X k,l µf αk,l e ikχ r + ilχ θ . (3.2)It is crucial to observe that the orbital equations (2.1)and (2.2) are still true for time dependent E a , and we willuse these equations to calculate the orbital evolution due to the self-force. Quantities that evolve due to the self-forceare denoted with tilde. The evolution equations for ˜ E a are then ddλ ˜ E = (cid:18) dτdλ (cid:19) µ η Eα f α , ddλ ˜ L = (cid:18) dτdλ (cid:19) µ η Lα f α , ddλ ˜ C = (cid:18) dτdλ (cid:19) µ η αβ ¯ v α f β . (3.3)If each of the ˜ E a is expanded in a manner similar to (3.2). these equations can be formally integrated to give˜ E = E + D ˙ E E λ + X k,l E k,l e ikχ r + ilχ θ , ˜ L = L + D ˙ L E λ + X k,l L k,l e ikχ r + ilχ θ , (3.4)˜ C = C + D ˙ C E λ + X k,l C k,l e ikχ r + ilχ θ , (3.5)where E a denote the initial values at λ = 0 . D ˙ E a E and E ak,l are of order ( µ/M ) and due to the self-force.The orbital evolution can be derived. from (2.1), (2.2), (3.4) and (3.5). Following our notation convention introducedabove, the inspiralling world line is denoted by ˜ x α . We define the r - and θ -motions by˜ r = X k ˜ r k e ik ˜ χ r , ˜ θ = X l ˜ θ l e il ˜ χ θ , (3.6)where the expansion coefficients ˜ r k and ˜ θ l are the same as those in (2.3) and (2.4), but they are functions of ˜ E a insteadof E a . The evolution equations for ˜ χ r and ˜ χ θ are given by ddλ ˜ χ r = ˜Υ r − P k ( d λ ˜ r k ) e ik ˜ χ r P k ′ ik ′ ˜ r k ′ e ik ′ ˜ χ r , ddλ ˜ χ θ = ˜Υ θ − P l ( d λ ˜ θ l ) e il ˜ χ θ P l ′ il ′ ˜ θ l ′ e il ′ ˜ χ θ , (3.7)where d λ = ( d/dλ ) acts on ˜ E a of ˜ r k and ˜ θ l . The effective frequencies ˜Υ r and ˜Υ θ are the same as Υ r and Υ θ but arefunctions of ˜ E a instead of E a . Similarly, we define the t - and φ -motions by˜ t = ˜ χ t + X k ˜ t rk e ik ˜ χ r + X l ˜ t θl e ilχ θ , ˜ φ = ˜ χ φ + X k ˜ φ rk e ikχ r + X l ˜ φ θl e ilχ θ . (3.8)Then, the evolution equations for ˜ χ t and ˜ χ φ are ddλ ˜ χ t = ˜Υ t + X k n ik (cid:16) d λ ˜ χ r − ˜Υ r (cid:17) ˜ t rk − d λ ˜ t rk o e ik ˜ χ r + X l n il (cid:16) d λ ˜ χ θ − ˜Υ θ (cid:17) ˜ t θl − d λ ˜ t θl o e il ˜ χ θ , (3.9) ddλ ˜ χ φ = ˜Υ φ + X k n ik (cid:16) d λ ˜ χ r − ˜Υ r (cid:17) ˜ φ rk − d λ ˜ φ rk o e ik ˜ χ r + X l n il (cid:16) d λ ˜ χ θ − ˜Υ θ (cid:17) ˜ φ θl − d λ ˜ φ θl o e il ˜ χ θ , (3.10)where the effective frequencies ˜Υ t and ˜Υ φ are the same as Υ t and Υ φ but are functions of ˜ E a instead of E a .We now derive the formal expression of ˜ χ α by perturbation. Recall that ˜ χ α − χ α is of order µ/M by definition.The formal expressions (3.7), (3.9) and (3.10) can therefore be written in the form ddλ ˜ χ α = Υ α (0) + D ˙Υ α E λ + X k,l Υ α ( k,l ) e ikχ r + ilχ θ , (3.11)where Υ α (0) = Υ α | E = E is for the background orbital evolution. The formal expression of ˜ χ α becomes˜ χ α = Υ α (0) ( λ − λ α ) + 12 D ˙Υ α E λ + Υ α (0 , λ + X k,l ˜ χ α ( k,l ) e ikχ r + ilχ θ , (3.12)where we set such that ˜ χ α = χ α when λ = 0 . This expression has two key features. One is the linear growth from D ˙Υ α E λ . This effect comes from the linear growth of E a since < ˙Υ α > = ( ∂ Υ α /∂ E a ) < ˙ E a > . This is expected togive the dominant phase evolution of the gravitational waveform because of the quadratic growth of the phase. Itis the reason that the orbit’s phase deviates from that of a geodesic on the dephasing time scale ( ∝ / √ µ ), whileits frequencies deviate on the radiation reaction time scale ( ∝ /µ ). The second feature is a small shift of the timeaveraged frequencies by Υ α (0 , . The phase evolution due to this effect will accumulate in time [7], however it remainssmall over the radiation reaction time and is therefore not likely to be observable [1]. We choose E a , , such that P k,l E ak,l e ikχ r + ilχ θ = 0 when λ = 0. This is achieved by choosing ˜ χ α (0 , such that P k,l ˜ χ α ( k,l ) e ikχ r + ilχ θ = 0 at λ = 0. IV. THE SECOND ORDER METRIC PERTURBATION
In this section we derive formal solutions of the second order Einstein equation (1.5). By using the same gaugecondition as we solve (1.4), we can formally integrate (1.5) with the tensor Green’s function (2.7). We separate thesecond order metric perturbation into pieces g [2] αβ = g [2] NLαβ + g [2] SFαβ , (4.1)where g [2] NLαβ is due to the nonlinearity of the Einstein equation and g [2] SFαβ is from the perturbation of the source.(1.5) then becomes G [1] µν [ g (2) NL ] = − G [2] µν [ g (1) , g (1) ] , G [1] µν [ g (2) SF ] = 8 πT [2] µν . (4.2)We first discuss the formal calculation of g [2] NL . By the Green’s method, we have g [2] NLαβ ( x ) = − Z −∞ In this paper, we discuss the modulation of the gravitational waveform due to gravitational radiation reaction. Sincethe linear metric perturbation is induced only by the background geodesic, it is necessary to calculate the second ordermetric perturbation in order to see the radiation reaction effect. A more rigorous calculation of the second order metricperturbation is not yet available. For this reason, we have considered only a qualitative study, extending the techniqueof formal calculation used to obtain the linear metric perturbation in Ref. [2]. The advantage of this technique isthat one can grasp some key features of the gravitational waveform without a complicated calculation.There are two kinds of source terms for the second order metric perturbation. One is the non-linear term of thelinear metric perturbation G [2] µν [ g (1) , g (1) ]. Because the formal calculation of this term is triperiodic, the part ofthe second order metric perturbation (4.5) is also triperiodic. The waveform induced by this part has exactly thesame spectral form as that of the linear metric perturbation. This part simply changes the amplitude of the waves,therefore, describing the correction to the wave propagation due to the non-linearity of the Einstein equation. Theother kind of source term is the point source due to the orbital deviation from the background geodesic, T [2] µν . Thepart of the the second order metric perturbation induced by this source has two new features; the linear growth ofthe wave amplitude over the radiation reaction time scale [See (4.11).] and the quadratic growth of the wave phaseover the dephasing time scale [See (4.14).]. From the derivations of (4.11) and (4.14), it should be clear that thesefeatures reflect the secular effect of the orbital evolution due to the self-force discussed in Sec. III. The actual procedure of integration is Z dωe − iωt Z dt ′ t ′ e i ( ω − ω ) t ′ = Z dωe − iωt ( − d dω Z dt ′ e i ( ω − ω ) t ′ = (2 π ) Z dωe − iωt ( − d dω δ ( ω − ω )= (2 π ) Z dωδ ( ω − ω )( − d dω e − iωt = (2 π ) t e − iω t . (4.13) These results suggest that the effect of radiation reaction on the waveform are included in templates of the form(4.17), which is the linear metric perturbation with the geodesic constants replaced by those which evolve adiabaticallydue to the self-force. This result agrees with the procedure of the adiabatic approximation. However, strictly speakingthe perturbation scheme is valid over a dephasing time [1], and over that time scale, only the feature of the quadraticgrowth of the wave phase can be identified as a secular effect. Beyond this time scale, it is necessary to account forthe secular effect of the self-force due to the second order metric perturbation. One can only discuss those linearlygrowing features in a gauge invariant way after calculating the third order metric perturbation.Because the perturbation scheme we use is limited to the dephasing time scale, it is not an appropriate methodfor calculating gravitational waveforms. In order to solve this problem, we propose an adiabatic expansion that isa systematic perturbation of a field theory coupled to a particle [1]. Because this expansion recovers the adiabaticapproximation at leading order, our result here supports the validity of this new expansion method. The applicationof this technique to the adiabatic expansion shall be discussed elsewhere. Acknowledgement We thank Prof. Richard Price for fruitful discussion. We thank Dr. Steve Drasco for carefully reading the material.This work is supported by NSF grant PHY-0601459, NASA grant NNX07AH06G, NNG04GK98G and the BrinsonFoundation. APPENDIX A: FORMAL EXPRESSION OF THE LINEAR METRIC PERTURBATION In this appendix, we review the formal calculation of the linear metric perturbation given as (2.9). With the ansatz(2.7), we decompose the Green’s function as G αβ µν ( x, x ′ ) = X ω,m g ω,mαβ µν ( r, r ′ ; θ, θ ′ ) e − iω ( t − t ′ )+ im ( φ − φ ′ ) . (A1)We plug this into (2.9) and we have g [1] αβ ( x ) = 8 πµ Z dλ (cid:18) dτdλ (cid:19) X ω,m g ω,mαβ µν ( r, ¯ r ; θ, ¯ θ ) v µ v ν e − iω ( t − ¯ t )+ im ( φ − ¯ φ ) . . (A2)At this point, we recall the periodic feature of the orbit (2.3)-(2.6). It is easy to see that dτ /dλ and g ω,mαβ µν ( r, ¯ r ; θ, ¯ θ )can be expanded as a discrete Fourier series in e − ikχ r − ilχ θ . From (2.1) and (2.2), v µ v ν can be similarly expanded.As for e iω ¯ t − im ¯ φ , one may expand by the same discrete Fourier series except the factor e iωχ t − imχ φ . In summary, onecan formally rewrite (A2) as g [1] αβ ( x ) = 8 πµ Z dλ X ω,k,l,m e − iωt + imφ h ω,k,l,mαβ ( r, θ ) e iωχ t − imχ φ − ikχ r − ilχ θ . (A3)By integrating over λ , we obtain g [1] αβ ( x ) = 2 π Υ t πµ X k,l,m e − iω ( k,l,m ) t + imφ h ω ( k,l,m ) ,k,l,mαβ ( r, θ ) e im Υ φ ( λ φ − λ t )+ ik Υ r ( λ r − λ t )+ il Υ θ ( λ θ − λ t ) , (A4)which is equal to (2.10). APPENDIX B: REGULARIZATION OF THE ULTRAVIOLET DIVERGENCE We use the regularization prescription for the ultraviolet divergence proposed in Ref. [8]. By the consistency ofthe matched asymptotic expansion, we know that the divergence behavior of the second order metric perturbationbecomes g [2] NL ∝ µ /R , with respect to the local inertial coordinates in the neighborhood of the geodesic. ( R is thespatial distance from the particle in the local inertia frame.) The idea in Ref. [8] is to subtract this divergence by0using the quadratic combination of the scalar field induced by a point particle and calculate the remaining regularpart by the Green’s method. We use the scalar field that satisfies (cid:3) Φ = Z dτ δ ( x − ¯ x ( τ )) p −| g [0] | , (B1)under the retarded boundary condition, where ¯ x µ ( τ ) is the same geodesic used in (2.8) for the linear metric pertur-bation. The resulting scalar field has the divergence behavior Φ ∝ /R near the geodesic. With appropriate tensors k [2] αβ and k [1] αβ , which are regular around the orbit, one may construct g ( S )[2] NLαβ = k [2] αβ Φ + k [1] αβ Φ , (B2)such that g [2] NL and g ( S )[2] NL have the same singular behavior near the geodesic. We define the remaining part by g ( R )[2] NLαβ = g [2] NLαβ − g ( S )[2] NLαβ , (B3)which satisfies G [1] µν [ g ( R )(2) NL ] = − G [1] µν [ g ( S )(2) NL ] − G [2] µν [ g (1) , g (1) ] . (B4)Because g ( R )[2] NLαβ is regular, the RHS of (B4) must be regular near the geodesic, and g ( R )(2) NL can be written as g ( R )[2] NLαβ ( x ) = − Z −∞ 2. In summary, the RHS of (B4) has the same formal structure as (4.4). APPENDIX C: REGULARIZATION OF THE INFRARED DIVERGENCE Though we can remove the ultraviolet divergence by calculating (B5) instead of calculating (4.3), it does notguarantee that we can also remove the infrared divergence, i.e. the integration over the spatial volume of (B5)becomes divergent at the large spatial radius and at the black hole horizon.A formal method of avoiding this divergence is to introduce the radial regulator Γ to (B5) as g ( R )[2] NLαβ ( x ; Γ) = − Z − Γ