Analytic Black Hole Perturbation Approach to Gravitational Radiation
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Abstract
We review the analytic methods used to perform the postNewtonian expansion of gravitational waves induced by a particle orbiting a massive, compact body, based on black hole perturbation theory. There exist two different methods of performing the postNewtonian expansion. Both are based on the Teukolsky equation. In one method, the Teukolsky equation is transformed into a ReggeWheeler type equation that reduces to the standard Klein Gordon equation in the flatspace limit, while in the other method (which was introduced by Mano, Suzuki, and Takasugi relatively recently, the Teukolsky equation is used directly in its original form. The former’s advantage is that it is intuitively easy to understand how various curved space effects come into play. However, it becomes increasingly complicated when one goes to higher and higher postNewtonian orders. In contrast, the latter’s advantage is that a systematic calculation to higher postNewtonian orders can be implemented relatively easily, but otherwise, it is so mathematical that it is hard to understand the interplay of higher order terms. In this paper, we review both methods so that their pros and cons may be seen clearly. We also review some results of calculations of gravitational radiation emitted by a particle orbiting a black hole.
Keywords
Black Hole Gravitational Wave Circular Orbit Kerr Black Hole Coulomb Wave Function1 Introduction
1.1 General
In the past several years, there has been substantial progress in the projects of groundbased laser interferometric gravitational wave detectors, which include LIGO [65], VIRGO [108], GE0600 [50], and TAMA300 [102, 3, 103]. TAMA300 was in operation from 1999 until 2004. LIGO and GE0600 began operating in 2002. LIGO has performed it’s fifth science run from November 2005 to October 2007, and over one year of sciencequality data was taken with all of it’s three LIGO interferometers in simultaneous operation. The Virgo detector performed its first science run in 2007 and 4.5 months of joint datataking with LIGO was done. There are several future projects as well. Most importantly, the Laser Interferometer Space Antenna (LISA) project is in progress [67, 66]. The DECIGO [2] and BBO [36] are more ambitious space interferometer proposals which aim to cover the frequency gap between the ground based interferometers and LISA. For a review of ground and space laser interferometers, see, e.g., the respective Living Reviews article [88].
The detection of gravitational waves will be done by extracting gravitational wave signals from a noisy data stream. In developing the data analysis strategy, detailed knowledge of the gravitational waveforms will help us greatly to detect a signal, and to extract the physical information about its source. Thus, it has become a very important problem for theorists to predict with sufficiently good accuracy the waveforms from possible gravitational wave sources.
Gravitational waves are generated by dynamical astrophysical events, and they are expected to be strong enough to be detected when compact stars such as neutron stars (NS) or black holes (BH) are involved in such events. In particular, coalescing compact binaries are considered to be the most promising sources of gravitational radiation that can be detected by the groundbased laser interferometers. The last inspiral phase of a coalescing compact binary, in which the binary stars orbit each other for ∼ 10^{4} cycles, will be in the bandwidth of the interferometers, and this phase may not only be detectable: it could provide us with important astrophysical information about the system, if the theoretical templates are sufficiently accurate.
Unfortunately, it seems difficult to attain the sensitivity to detect NSNS binary inspirals with the first generation of interferometric detectors. However, the coalescence of BHNS/BHBH binaries with a black hole mass of ∼ 10–20 M_{⊙} may be detected out to the distance of the VIRGO cluster if we are lucky enough. In any case, it will be necessary to wait for the next generation of interferometric detectors to see these coalescing events more frequently [73, 82].
To predict the waveforms, a conventional approach is to formulate the Einstein equations with respect to the flat Minkowski background and apply the postNewtonian expansion to the resulting equations (see the Section 1.2).
In this paper, however, we review a different approach, namely the black hole perturbation approach. In this approach, binaries are assumed to consist of a massive black hole and a small compact star which is taken to be a point particle. Hence, its applicability is constrained to the case of binaries with large mass ratio. Nevertheless, there are several advantages here that cannot be overlooked.
Most importantly, the black hole perturbation equations take full account of general relativistic effects of the background spacetime and they are applicable to arbitrary orbits of a small mass star. In particular, if a numerical approach is taken, gravitational waves from highly relativistic orbits can be calculated. Then, if we can develop a method to calculate gravitational waves to a sufficiently high PN order analytically, it can give insight not only into how and when general relativistic effects become important, by comparing with numerical results, but it will also give us a knowledge, complementary to the conventional postNewtonian approach, about as yet unknown higherorder PN terms or general relativistic spin effects.
Moreover, one of the main targets of LISA is to observe phenomena associated with the formation and evolution of supermassive black holes in galactic centers. In particular, a gravitational wave event of a compact star spiraling into such a supermassive black hole is indeed a case of application for the black hole perturbation theory.
1.2 PostNewtonian expansion of gravitational waves
The postNewtonian expansion of general relativity assumes that the internal gravity of a source is small so that the deviation from the Minkowski metric is small, and that velocities associated with the source are small compared to the speed of light, c. When we consider the orbital motion of a compact binary system, these two conditions become essentially equivalent to each other. Although both conditions may be violated inside each of the compact objects, this is not regarded as a serious problem of the postNewtonian expansion, as long as we are concerned with gravitational waves generated from the orbital motion, and, indeed, the two bodies are usually assumed to be pointlike objects in the calculation.
In fact, Itoh, Futamase, and Asada [57, 58] developed a new postNewtonian method that can deal with a binary system in which the constituent bodies may have strong internal gravity, based on earlier work by Futamase and Schutz [44, 45, 42]. They derived the equations of motion to 2.5PN order and obtained a complete agreement with the DamourDeruelle equations of motion [26, 25], which assumes the validity of the pointparticle approximation. In the FutamaseSchutz method, each star in a binary is first expressed as an extended object and then the limit is taken to set the radius to zero in a specific manner first proposed by Futamase [42]. At the same time, the surface integral approach (a la EinsteinInfeldHoffmann [32]) is taken to derive the equations of motion. More recently Itoh and Futamase [56, 54] derived the 3PN equations of motion based on the FutamaseSchutz method, and they are again in agreement with those derived by Damour, Jaranowski and Schafer [27] and by Blanchet et al. [10] in which the pointparticle approximation is used.
There are two existing approaches of the postNewtonian expansion to calculate gravitational waves: one developed by Blanchet, Damour, and Iyer (BDI) [12, 7] and another by Will and Wiseman (WW) [111] based on previous work by Epstein, Wagoner, and Will [33, 109]. In both approaches, the gravitational waveforms and luminosity are expanded in time derivatives of radiative multipoles, which are then related to some source multipoles (the relation between them contains the “tails”). The source multipoles are expressed as integrals over the matter source and the gravitational field. The source multipoles are combined with the equations of motion to obtain explicit expressions in terms of the source masses, positions, and velocities.
The lowest order of the gravitational waves is given by the Newtonian quadrupole formula. It is standard to refer to the postNewtonian formulae (for the waveforms and luminosity) that contain terms up to \(\mathcal{O}\left( {{{(v/c)}^n}} \right)\) beyond the Newtonian quadrupole formula as the (n/2)PN formulae. Evaluation of gravitational waves emitted to infinity from a compact binary system has been successfully carried out to the 3.5 postNewtonian (PN) order beyond the lowest Newtonian quadrupole formula in the BDI approach [12, 7, 18, 19, 15, 16, 11]. Up to now, the WW approach gives the same answer for the gravitational waveforms and luminosity to 2PN order.
The computation of the 3.5PN flux requires the 3PN equations of motion. As mentioned in the above, the 3PN equations of motion have been derived by three different methods. The first is the direct postNewtonian iteration in the harmonic coordinates [14, 28, 10]. The second employs the ArnowittDeserMisner (ADM) coordinates within the Hamiltonian formalism of general relativity [59, 60, 27]. The third is based on the FutamaseSchutz method [56, 54].
Since the first two methods use the point particle approximation while the third one is not, let us first focus on the first two. In both methods, since the stars are represented by the Dirac delta functions, the divergent selffields must be regularized. In earlier papers, they used the Hadamard regularization method [59, 60, 14, 28]. However, it turned out that there remains an unknown coefficient which cannot be determined within the regularization method. This problem was solved by Damour, Jaranowski and Schäfer [27] who successfully derived the 3PN equations of motion without undetermined numerical coefficients by using the dimensional regularization within an ADM Hamiltonian approach. Then the 3PN equations of motion in the harmonic coordinates were also derived without undetermined coefficients by using a combination of the Hadamard regularization and the dimensional regularization in [10]. The 3.5PN radiation reaction terms in the equations of motion are also derived in both approaches [76, 62]. See reviews by Blanchet [8, 9] for details and summaries on postNewtonian approaches.
In the case of FutamaseSchutz method, as mentioned in the beginning of this subsection, the 3PN equations of motion is derived by Itoh and Futamase [56, 54], and the 3.5PN terms are derived by Itoh [55]. See a review article by Futamase and Itoh [43] for details on this method.
There are other methods in which stars are treated as fluid balls [51, 63, 80, 81]. Pati and Will [80, 81] use an method which is an extension of the WW approach in which the retarded integral is evaluated directly. With these method, the 2PN equations of motion as well as 2.5PN and 3.5PN radiation reaction effects are derived.
1.3 Linear perturbation theory of black holes
In the black hole perturbation approach, we deal with gravitational waves from a particle of mass μ orbiting a black hole of mass M, assuming μ ≪ M. The perturbation of a black hole spacetime is evaluated to linear order in μ/M. The equations are essentially in the form of Equation (2) with η_{ μν }, replaced by the background black hole metric g _{ μν } ^{BH} and the higher order terms Λ(h)_{ μν }, neglected. Thus, apart from the assumption μ ≪ M, the black hole perturbation approach is not restricted to slowmotion sources, nor to small deviations from the Minkowski spacetime, and the Green function used to integrate the Einstein equations contains the whole curved spacetime effect of the background geometry.
The black hole perturbation theory was originally developed as a metric perturbation theory. For nonrotating (Schwarzschild) black holes, a single master equation for the metric perturbation was derived by Regge and Wheeler [87] for the socalled odd parity part, and later by Zerilli [112] for the even parity part. These equations have the nice property that they reduce to the standard KleinGordon wave equation in the flatspace limit. However, no such equation has been found in the case of a Kerr black hole so far.
Then, based on the NewmanPenrose nulltetrad formalism, in which the tetrad components of the curvature tensor are the fundamental variables, a master equation for the curvature perturbation was first developed by Bardeen and Press [6] for a Schwarzschild black hole without source (T^{ μν } = 0), and by Teukolsky [106] for a Kerr black hole with source (T^{ μν } ≠ 0). The master equation is called the Teukolsky equation, and it is a wave equation for a nulltetrad component of the Weyl tensor ψ_{0} or ψ_{4}. In the sourcefree case, Chrzanowski [23] and Wald [110] developed a method to construct the metric perturbation from the curvature perturbation.
The Teukolsky equation has, however, a rather complicated structure as a wave equation. Even in the flatspace limit, it does not reduce to the standard KleinGordon form. Later, Chandrasekhar showed that the Teukolsky equation can be transformed to the form of the ReggeWheeler or Zerilli equation for the sourcefree Schwarzschild case [21]. A generalization of this to the Kerr case with source was done by Sasaki and Nakamura [92, 93]. They gave a transformation that brings the Teukolsky equation to a ReggeWheeler type equation that reduces to the ReggeWheeler equation in the Schwarzschild limit. It may be noted that the SasakiNakamura equation contains an imaginary part, suggesting that either it is unrelated to a (yettobefound) master equation for the metric perturbation for the Kerr geometry or implying the nonexistence of such a master equation.
As mentioned above, an important difference between the blackhole perturbation approach and the conventional postNewtonian approach appears in the structure of the Green function used to integrate the wave equations. In the blackhole perturbation approach, the Green function takes account of the curved spacetime effect on the wave propagation, which implies complexity of its structure in contrast to the flatspace Green function. Thus, since the system is linear in the blackhole perturbation approach, the most nontrivial task is the construction of the Green function.
There are many papers that deal with a numerical evaluation of the Green function and calculations of gravitational waves induced by a particle. See Breuer [20], Chandrasekhar [22], and Nakamura, Oohara, and Kojima [72] for reviews and for references on earlier papers.
Here, we are interested in an analytical evaluation of the Green function. One way is to adopt the postMinkowski expansion assuming GM/c^{2} ≪ r. Note that, for bound orbits, the condition GM/c^{2} ≪ r is equivalent to the condition for the postNewtonian expansion, v^{2}/c^{2} ≪ 1. If we can calculate the Green function to a sufficiently high order in this expansion, we may be able to obtain a rather accurate approximation of it that can be applicable to a relativistic orbit fairly close to the horizon, possibly to a radius as small as the innermost stable circular orbit (ISCO), which is given by r_{ISCO} = 6GM/c^{2} in the case of a Schwarzschild black hole.
It turns out that this is indeed possible. Though there arise some complications as one goes to higher PN orders, they are relatively easy to handle as compared to situations one encounters in the conventional postNewtonian approaches. Thus, very interesting relativistic effects such as tails of gravitational waves can be investigated easily. Further, we can also easily investigate convergence properties of the postNewtonian expansion by comparing a numerically calculated exact result with the corresponding analytic but approximate result. In this sense, the analytic blackhole perturbation approach can provide an important test of the postNewtonian expansion.
1.4 Brief historical notes
Let us briefly review some of the past work on postNewtonian calculations in blackhole perturbation theory. Although the literature on numerical calculations of gravitational waves emitted by a particle orbiting a black hole is abundant, there are not so many papers that deal with the postNewtonian expansion of gravitational waves, mainly because such an analysis was not necessary until recently, when the construction of accurate theoretical templates for the interferometric gravitational wave detectors became an urgent issue.
In the case of orbits in the Schwarzschild background, one of the earliest papers was by Gal’tsov, Matiukhin and Petukhov [47], who considered the case when a particle is in a slightly eccentric orbit around a Schwarzschild black hole, and calculated the gravitational waves up to 1PN order. Poisson [83] considered a circular orbit around a Schwarzschild black hole and calculated the waveforms and luminosity to 1.5PN order at which the tail effect appears. Cutler, Finn, Poisson, and Sussman [24] worked on the same problem numerically by applying the leastsquare fitting technique to the numerically evaluated data for the luminosity, and obtained a postNewtonian formula for the luminosity to 2.5PN order. Subsequently, a highly accurate numerical calculation was carried out by Tagoshi and Nakamura [99]. They obtained the formulae for the luminosity to 4PN order numerically by using the leastsquare fitting method. They found the log v terms in the luminosity formula at 3PN and 4PN orders. They concluded that, although the convergence of the postNewtonian expansion is slow, the luminosity formula accurate to 3.5PN order will be good enough to represent the orbital phase evolution of coalescing compact binaries in theoretical templates for groundbased interferometers. After that, Sasaki [91] found an analytic method and obtained formulae that were needed to calculate the gravitational waves to 4PN order. Then, Tagoshi and Sasaki [100] obtained the gravitational waveforms and luminosity to 4PN order analytically, and confirmed the results of Tagoshi and Nakamura. These calculations were extended to 5.5PN order by Tanaka, Tagoshi, and Sasaki [105]. Fujita and Iyer [39] extended this work and derived 5.5PN waveforms.
In the case of orbits around a Kerr black hole, Poisson calculated the 1.5PN order corrections to the waveforms and luminosity due to the rotation of the black hole, and showed that the result agrees with the standard postNewtonian effect due to spinorbit coupling [84]. Then, Shibata, Sasaki, Tagoshi, and Tanaka [94] calculated the luminosity to 2.5PN order. They calculated the luminosity from a particle in circular orbit with small inclination from the equatorial plane. They used the SasakiNakamura equation as well as the Teukolsky equation. This analysis was extended to 4PN order by Tagoshi, Shibata, Tanaka, and Sasaki [101], in which the orbits of the test particles were restricted to circular ones on the equatorial plane. The analysis in the case of slightly eccentric orbit on the equatorial plane was also done by Tagoshi [95, 96] to 2.5PN order.
Tanaka, Mino, Sasaki, and Shibata [104] considered the case when a spinning particle is in a circular orbit near the equatorial plane of a Kerr black hole, based on the Papapetrou equations of motion for a spinning particle [79] and the energy momentum tensor of a spinning particle by Dixon [29]. They derived the luminosity formula to 2.5PN order which includes the linear order effect of the particle’s spin.
The absorption of gravitational waves into the black hole horizon, appearing at 4PN order in the Schwarzschild case, was calculated by Poisson and Sasaki for a particle in a circular orbit [85]. The black hole absorption in the case of a rotating black hole appears at 2.5PN order [46]. Using a new analytic method to solve the homogeneous Teukolsky equation found by Mano, Suzuki, and Takasugi [68], the black hole absorption in the Kerr case was calculated by Tagoshi, Mano, and Takasugi [98] to 6.5PN order beyond the quadrupole formula.
If gravity is not described by the Einstein theory but by the BransDicke theory, there will appear scalartype gravitational waves as well as transversetraceless gravitational waves. Such scalartype gravitational waves were calculated to 2.5PN order by Ohashi, Tagoshi, and Sasaki [77] in the case when a compact star is in a circular orbit on the equatorial plane around a Kerr black hole.
In the above works the energy and angular momentum flux at infinity or the absorption rate at the horizon were evaluated. In the Kerr case, in order to specify the evolution of particle’s trajectory under the influence of radiation reaction, we need to determine the rate of change of the Carter constant which is not directly related to the asymptotic gravitational waves. Mino [70] proved that the average rate of change of the Carter constant can be evaluated by using the radiative field (i.e., retarded minus advanced field) in the adiabatic approximation. An explicit calculation of the rate of change of the Carter constant was done in the case of a scalar charged particle in [30]. Sago et al. [90] extended Mino’s work and found a simpler formula for the average rate of change of the Carter constant. They derived analytically the rate of change of the Carter constant as well as the energy and the angular momentum of a particle for orbits with small eccentricities and inclinations up to O(v^{5}) [89]. In Ref. [48], the method was extended to the case of the orbits with small eccentricity but arbitrary inclination angle, and the rate of change of the energy, angular momentum and the Carter constant up to O(v^{5}) were derived.
In the rest of the paper, we use the units c = G = 1.
2 Basic Formulae for the Black Hole Perturbation
2.1 Teukolsky formalism
If we set s = 2 in Equation (6), with appropriate change of the source term, it becomes the perturbation equation for ψ_{0}. Moreover, it describes the perturbation for a scalar field (s = 0), a neutrino field (s = 1/2), and an electromagnetic field (s = 1) as well.
We note that the homogeneous Teukolsky equation is invariant under the complex conjugation followed by the transformation m → m and ω → ω. Thus, we can set \(\bar R_{\ell m\omega }^{\text{in},\;\text{up}}\), where the bar denotes the complex conjugation.
2.2 ChandrasekharSasakiNakamura transformation
As seen from the asymptotic behaviors of the radial functions given in Equations (24) and (25), the Teukolsky equation is not in the form of a canonical wave equation near the horizon and infinity. Therefore, it is desirable to find a transformation that brings the radial Teukolsky equation into the form of a standard wave equation.
In the Schwarzschild case, Chandrasekhar found that the Teukolsky equation can be transformed to the ReggeWheeler equation, which has the standard form of a wave equation with solutions having regular asymptotic behaviors at horizon and infinity [21]. The ReggeWheeler equation was originally derived as an equation governing the odd parity metric perturbation [87]. The existence of this transformation implies that the ReggeWheeler equation can describe the even parity metric perturbation simultaneously, though the explicit relation of the ReggeWheeler function obtained by the Chandrasekhar transformation with the actual metric perturbation variables has not been given in the literature yet.
Later, Sasaki and Nakamura succeeded in generalizing the Chandrasekhar transformation to the Kerr case [92, 93]. The ChandrasekharSasakiNakamura transformation was originally introduced to make the potential in the radial equation shortranged, and to make the source term wellbehaved at the horizon and at infinity. Since we are interested only in bound orbits, it is not necessary to perform this transformation. Nevertheless, because its flatspace limit reduces to the standard radial wave equation in the Minkowski spacetime, it is convenient to apply the transformation when dealing with the postMinkowski or postNewtonian expansion, at least at low orders of expansion.
3 PostNewtonian Expansion of the ReggeWheeler Equation
In this section, we review a postNewtonian expansion method for the Schwarzschild background, based on the ReggeWheeler equation. We focus on the gravitational waves emitted to infinity, but not on those absorbed by the black hole. The black hole absorption is deferred to Section 4, in which we review the ManoSuzukiTakasugi method for solving the Teukolsky equation.
Since we are interested in the waves emitted to infinity, as seen from Equation (25), what we need is a method to evaluate the ingoing wave Teukolsky function \(R_{\ell m\omega }^{\text{in}}\) or its counterpart in the ReggeWheeler equation, \(X_{\ell m\omega }^{\text{in}}\), which are related by Equation (59). In addition, we assume ω > 0 whenever it is necessary throughout this section. Formulae and equations for ω < 0 are obtained from the symmetry \(\bar X_{\ell m\omega }^{{\text{in}}} = X_{\ell  m  \omega }^{\text{in}}\).
3.1 Basic assumptions
We consider the case of a test particle with mass μ in a nearly circular orbit around a black hole with mass M ≫ μ. For a nearly circular orbit, say at r ∼ r_{0}, what we need to know is the behavior of \(R_{\ell m\omega }^{\text{in}}\;\text{at}\;r \sim {r_0}\). In addition, the contribution of ω to \(R_{\ell m\omega }^{\text{in}}\) comes mainly from ω ∼ mΩ_{ ϕ }, where Ω_{ ϕ } ∼ (M/r _{0} ^{3} )^{1/2} is the orbital angular frequency.
Thus, if we express the ReggeWheeler equation (63) in terms of a nondimensional variable z ≡ ωr, with a nondimensional parameter ε ≡ 2Mω, we are interested in the behavior of \(X_{\ell m\omega }^{\text{in}} (z)\) at z ∼ ωr_{0} ∼ m(M/r_{0})^{1/2} ∼ v with ε ∼ 2m(M/r_{0})^{3/2} ∼ v^{3}, where v ≡ (M/r_{0})^{1/2} is the characteristic orbital velocity. The postNewtonian expansion assumes that v is much smaller than the velocity of light: v ≪ 1. Consequently, we have ε ≪ v ≪ 1 in the postNewtonian expansion.
It should be noted that if we reinstate the gravitational constant G, we have ε = 2GMω. Thus, the expansion in terms of e corresponds to the postMinkowski expansion, and expanding the ReggeWheeler equation with the assumption ε ≪ 1 gives a set of iterative wave equations on the flat spacetime background. One of the most significant differences between the black hole perturbation theory and any theory based on the flat spacetime background is the presence of the black hole horizon in the former case. Thus, if we naively expand the ReggeWheeler equation with respect to e, the horizon boundary condition becomes unclear, since there is no horizon on the flat spacetime. To establish the boundary condition at the horizon, we need to treat the ReggeWheeler equation near the horizon separately. We thus have to find a solution near the horizon, and the solution obtained by the postMinkowski expansion must be matched with it in the region where both solutions are valid.
It may be of interest to note the difference between the matching used in the BDI approach for the postNewtonian expansion [7, 12] and the matching used here. In the BDI approach, the matching is done between the postMinkowskian metric and the nearzone postNewtonian metric. In our case, the matching is done between the postMinkowskian gravitational field and the gravitational field near the black hole horizon.
3.2 Horizon solution; z ≪ 1
As we will see below, the above solution is accurate enough to determine the boundary condition of the outer solution up to the 6PN order of expansion.
3.3 Outer solution; ε ≪ 1
We now solve Equation (70) in the limit ε ≪ 1, i.e., by applying the postMinkowski expansion to it. In this section, we consider the solution to \(\mathcal{O}(\epsilon)\). Then we match the solution to the horizon solution given by Equation (80) at ε ≪ z ≪ 1.
3.4 More on the inner boundary condition of the outer solution
In this section, we discuss the inner boundary condition of the outer solution in more detail. As we have seen in Section 3.3, the boundary condition on ξ_{ℓ} is that it is regular at z → 0, at least to \(\mathcal{O}(\epsilon)\), while in the full nonlinear level, the horizon boundary is at z = ε. We therefore investigate to what order in ε the condition of regularity at z = 0 can be applied.
It is also useful to keep in mind the above fact when we solve for ξ_{ℓ} under the postMinkowski expansion. It implies that we may choose a phase such that A _{ℓ} ^{inc} and A _{ℓ} ^{ref} are complex conjugate to each other, to \(\mathcal{O}\left( {{\epsilon^{2\ell + 1}}} \right)\). With this choice, the imaginary part of X_{ℓ}, which reflects the boundary condition at the horizon, does not appear until \(\mathcal{O}(\epsilon^{2\ell+2})\) because the ReggeWheeler equation is real. Then, recalling the relation of ξ_{ℓ} to X_{ℓ}, Equation (71), Im (ξ _{ℓ} ^{(n)} ) for a given n ≤ 2ℓ+ 1 is completely determined in terms of Re (ξ _{ℓ} ^{(r)} ) for r ≤ n  1. That is, we may focus on solving only the real part of Equation (84).
3.5 Structure of the ingoing wave function to \(\mathcal{O}(\epsilon^2)\)
With the boundary condition discussed in Section 2, we can integrate the ingoing wave ReggeWheeler function iteratively to higher orders of ε in the postMinkowskian expansion, ε ≪ 1. This was carried out in [91] to \(\mathcal{O}(\epsilon^2)\) and in [105] to \(\mathcal{O}(\epsilon^3)\) (See [71] for details). Here, we do not recapitulate the details of the calculation since it is already quite involved at \(\mathcal{O}(\epsilon^2)\), with much less space for physical intuition. Instead, we describe the general properties of the ingoing wave function to \(\mathcal{O}(\epsilon^2)\).
Given a postNewtonian order to which we want to calculate, by setting \(z=\mathcal{O}(v)\) and \(\epsilon=\mathcal{O}(v^3)\), the above asymptotic behaviors tell us the highest order of X _{ℓ} ^{(n)} we need. We also see the presence of In z terms in X _{ℓ} ^{(2)} . The logarithmic terms appear as a consequence of the mathematical structure of the ReggeWheeler equation at z ≪ 1. The simple power series expansion of X _{ℓ} ^{(n)} in terms of z breaks down at \(\mathcal{O}(\epsilon^2)\), and we have to add logarithmic terms to obtain the solution. These logarithmic terms will give rise to In v terms in the waveform and luminosity formulae at infinity, beginning at \(\mathcal{O}(v^6)\) [99, 100]. It is not easy to explain physically how these In v terms appear. But the above analysis suggests that the In v terms in the luminosity originate from some spatially local curvature effects in the nearzone.
Note that the above form of A _{ℓ} ^{inc} implies that the socalled tail of radiation, which is due to the curvature scattering of waves, will contain In v terms as phase shifts in the waveform, but will not give rise to such terms in the luminosity formula. This supports our previous argument on the origin of the ln v terms in the luminosity. That is, it is not due to the wave propagation effect but due to some nearzone curvature effect.
4 Analytic Solutions of the Homogeneous Teukolsky Equation by Means of the Series Expansion of Special Functions
In this section, we review a method developed by Mano, Suzuki, and Takasugi [68], who found analytic expressions of the solutions of the homogeneous Teukolsky equation. In this method, the exact solutions of the radial Teukolsky equation (14) are expressed in two kinds of series expansions. One is given by a series of hypergeometric functions and the other by a series of the Coulomb wave functions. The former is convergent at horizon and the latter at infinity. The matching of these two solutions is done exactly in the overlapping region of convergence. They also found that the series expansions are naturally related to the low frequency expansion. Properties of the analytic solutions were studied in detail in [69]. Thus, the formalism is quite powerful when dealing with the postNewtonian expansion, especially at higher orders.
In many cases, when we study the perturbation of a Kerr black hole, it is more convenient to use the SasakiNakamura equation, since it has the form of a standard wave equation, similar to the ReggeWheeler equation. However, it is not quite suited for investigating analytic properties of the solution near the horizon. In contrast, the ManoSuzukiTakasugi (MST) formalism allows us to investigate analytic properties of the solution near the horizon systematically. Hence, it can be used to compute the higher order postNewtonian terms of the gravitational waves absorbed into a rotating black hole.
We also note that this method is the only existing method that can be used to calculate the gravitational waves emitted to infinity to an arbitrarily high postNewtonian order in principle.
4.1 Angular eigenvalue
In the postNewtonian expansion, the parameter aω is assumed to be small. Then, it is straightforward to obtain a spheroidal harmonic _{ s }S_{ℓm} of spinweight s and its eigenvalue λ perturbatively by the standard method [86, 101, 94].
It is also possible to obtain the spheroidal harmonics by expansion in terms of the Jacobi functions [35]. In this method, if we calculate numerically, we can obtain them and their eigenvalues for an arbitrary value of aω.
4.2 Horizon solution in series of hypergeometric functions
The Teukolsky equation has two regular singularities at r = r_{±}, and one irregular singularity at r = ∞. This implies that it cannot be represented in the form of a single hypergeometric equation. However, if we focus on the solution near the horizon, it may be approximated by a hypergeometric equation. This motivates us to consider the solution expressed in terms of a series of hypergeometric functions.
Thus, {a _{ n } ^{(1)} } is minimal as n → ∞ and {b _{ n } ^{(1)} } is minimal as n →  ∞.
Finally, we note that if ν is a solution of Equation (133) or (136), ν + k with an arbitrary integer k is also a solution, since ν appears only in the combination of n+ν. Thus, Equation (133) or (136) contains an infinite number of roots. However, not all of these can be used to express a solution we want. As noted in the earlier part of this section, in order to reproduce the solution in the limit ε → 0, Equation (118), we must have ν → ℓ (or ν → ℓ  1 by symmetry). Thus, we impose a constraint on ν such that it must continuously approach ℓ as ε → 0.
4.3 Outer solution as a series of Coulomb wave functions
The solution as a series of hypergeometric functions discussed in Section 4.2 is convergent at any finite value of r. However, it does not converge at infinity, and hence the asymptotic amplitudes, B^{inc} and B^{ref}, cannot be determined from it. To determine the asymptotic amplitudes, it is necessary to construct a solution that is valid at infinity and to match the two solutions in a region where both solutions converge. The solution convergent at infinity was obtained by Leaver as a series of Coulomb wave functions [64]. In this section, we review Leaver’s solution based on [69].
In this section again, by noting the symmetry \({\bar R_{\ell m\omega }} = {R_{\ell  m  \omega }}\), we assume ω > 0 without loss of generality.
The fact that we can use the same ν as in the case of hypergeometric functions to obtain the convergence of the series of the Coulomb wave functions is crucial to match the horizon and outer solutions.
4.4 Matching of horizon and outer solutions
4.5 Low frequency expansion of the hypergeometric expansion
To solve Equation (174), we first note the following. Unless the value of ν is such that the denominator in the expression of α _{ n } ^{ ν } or γ _{ n } ^{ ν } happens to vanish, or β _{ n } ^{ ν } happens to vanish in the limit ε → 0, we have \(\alpha _n^v = \mathcal{O}(\epsilon),\gamma _n^v = \mathcal{O}(\epsilon)\), and \(\beta _n^v = \mathcal{O}(1)\). Also, from the asymptotic behavior of the minimal solution f _{ n } ^{ ν } as n → ±t8 given by Equation (134), we have \({R_n}(v) = \mathcal{O}(\epsilon)\) and \({L_{  n}}(v) = \mathcal{O}(\epsilon)\) for sufficiently large n. Thus, except for exceptional cases mentioned above, the order of a _{ n } ^{ ν } in ε increases as n increases. That is, the series solution naturally gives the postMinkowski expansion.
First, let us consider the case of R_{ n }(ν) for n > 0. It is easily seen that \(\alpha _n^v = \mathcal{O}(\epsilon),\gamma _n^v = \mathcal{O}(\epsilon)\), and \(\beta _n^v = \mathcal{O}(1)\) for all n > 0. Therefore, we have \({R_n}(v) = \mathcal{O}(\epsilon)\) for all n > 0.
On the other hand, for n < 0, the order of L_{n}(ν) behaves irregularly for certain values of n. For the moment, let us assume that \({L_{  1}}(v) = \mathcal{O}(\epsilon)\). We see from Equations (124) that \(\alpha _0^v = \mathcal{O}(\epsilon)\), \(\gamma _0^v = \mathcal{O}(\epsilon)\), since \(v = \ell + \mathcal{O}(\epsilon)\). Then, Equation (174) implies \(\beta _0^v = \mathcal{O}({\epsilon^2})\). Using the expansion of λ given by Equation (110), we then find \(v = \ell + \mathcal{O}({\epsilon^2})\) (i.e., there is no term of \(\mathcal{O}(\epsilon)\) in ν). With this estimate of ν, we see from Equation (128) that \({L_{  1}}(v) = \mathcal{O}(\epsilon)\) is justified if L_{2} (ν) is of order unity or smaller.

For n = 2ℓ  1, we have \({\alpha _n} = \mathcal{O}(\epsilon),\beta = \mathcal{O}({\epsilon^2})\), and \({\gamma _n} = \mathcal{O}(\epsilon)\).

For n = ℓ  1, we have \(\alpha _n^v = \mathcal{O}(1/\epsilon),\beta _n^v = \mathcal{O}(1/\epsilon)\), and \({\gamma _n} = \mathcal{O}(\epsilon)\).

For n = ℓ, we have \(\mathcal{O}(\epsilon),{\beta _n} = \mathcal{O}(1/\epsilon)\), and \({\gamma _n} = \mathcal{O}(1/\epsilon)\).
The postMinkowski expansion of homogeneous Teukolsky functions can be obtained with arbitrary accuracy by solving Equation (123) to a desired order, and by summing up the terms to a sufficiently large n. The first few terms of the coefficients f _{ n } ^{ ν } are explicitly given in [68]. A calculation up to a much higher order in \(\mathcal{O}(\epsilon)\) was performed in [98], in which the black hole absorption of gravitational waves was calculated to \(\mathcal{O}({v^8})\) beyond the lowest order.
4.6 Property of ν
The value of ν for various value of Mω in the case s = −2, l = m = 2 and q = 0.
Mω  Re(ν)  IM(ν) 

0.1  1.9793154547208  0.0000000000000 
0.2  1.9129832302687  0.0000000000000 
0.3  1.7792805424199  0.0000000000000 
0.4  1.5000000000000  0.1862468531447 
0.5  1.5000000000000  0.3618806153941 
0.6  1.7878302655744  0.0000000000000 
0.7  2.0000000000000  0.8003377636925 
0.8  2.0000000000000  1.1099466644118 
0.9  2.0000000000000  1.3699138540831 
1.0  2.0000000000000  1.6085538776570 
2.0  2.0000000000000  3.6867890278893 
3.0  2.0000000000000  5.5939000509184 
It becomes possible to determine v in the wide range of ω by allowing Im(v) ≠ 0. The MST formalism is now very useful in the fully numerical evaluation of homogeneous solutions of the Teukolsky equation. As a first step, Fujita, Hikida and Tagoshi [38] considered generic bound geodesic orbits around a Kerr black hole and evaluate the energy and angular momentum flux to infinity as well as the rate of change of the Carter constant in a wide range of orbital parameters.
The critical value of ω when v becomes complex is not very small. The complex v does not appear in the analytic evaluation of v in the low frequency expansion in powers of ε  2Mω. Thus, at the first glance, it seems impossible to express the complex v in the power series expansion of ε. However, Hikida et al. [52] pointed out that it is possible to evaluate sin^{2} (πν) very accurately in terms of the power series expansion of ε, even if ω is larger than a critical value and v is complex. Such an analytical expression of v is very useful in the numerical root finding of Equation (174) as well as in the analytical calculation of the homogeneous solutions.
5 Gravitational Waves from a Particle Orbiting a Black Hole
Based on the ingoing wave functions discussed in Section 3 and 4, we can derive the gravitational wave energy and angular momentum flux emitted to infinity. The formula for the energy and the angular momentum luminosity to infinity are given by Equations (48) and (49). Since most of the calculations are very long, we show only the final results. In [71], some details of the calculations are summarized. We define the postNewtonian expansion parameter by x  (MΩ_{ ϕ })^{1/3}, where M is the mass of the black hole and Ω_{ ϕ }. is the orbital angular frequency of the particle. Since the parameter x is directly related to the observable frequency, this result can be compared with the results by another method easily.
5.1 Circular orbit around a Schwarzschild black hole
5.2 Circular orbit on the equatorial plane around a Kerr black hole
5.3 Waveforms in the case of circular orbit
In the previous two subsections, we only considered the luminosity formulas for the energy and the angular momentum. Here, focusing on circular orbits, we review the previous calculation of the gravitational waveforms.
On the other hand, the gravitational waveforms have also been calculated. In the case of circular orbit around a Schwarzschild black hole, Poisson [83] derived the 1.5PN waveform and Tagoshi and Sasaki [100] derived the 4PN waveform. These were done by using the postNewtonian expansion of the ReggeWheeler equation discussed in Section 3. Recently, Fujita and Iyer [39] derived the 5.5PN waveform by using the MST formalism. They also discussed factorized resummed waveforms which is useful to obtain better agreement with accurate numerical data.
In the case of circular orbit around a Kerr black hole, Poisson [83] derived the 1.5PN waveform under the assumption of slow rotation of the black hole. In [94] and [101], although the luminosity was derived up to 2.5PN and 4PN order respectively, the waveform was not derived up to the same order. Recently, Tagoshi and Fujita [97] computed the all multipolar modes \({\tilde Z_{lm\omega }}\) necessary to derive the waveform up to 4PN order, and the results were used to derive the factorized, resummed, multipolar waveform in [78].
5.4 Slightly eccentric orbit around a Schwarzschild black hole
5.5 Slightly eccentric orbit around a Kerr black hole
5.6 Circular orbit with a small inclination from the equatorial plane around a Kerr black hole
5.7 Absorption of gravitational waves by a black hole
We note that the leading terms in (dE/dt)_{H} are negative for q > 0, i.e., the black hole loses energy if the particle is corotating. This is because of the superradiance for modes with k < 0.
5.8 Adiabatic evolution of Carter constant for orbit with small eccentricity and small inclination angle around a Kerr black hole
In the Schwarzschild case, the particle’s trajectory is characterized by the energy E and the zcomponent angular momentum L_{ z }. In the adiabatic approximation, the rates of change of E and L_{ z } are equated with those radiated to infinity as gravitational waves or with those absorbed into the black hole horizon, in accordance with the conservation of E and L_{ z }. On the other hand, in the Kerr case, the Carter constant, Q, is also necessary to specify the particle’s trajectory. In this case, the rate of change of Q is not directly related to the asymptotic gravitational waves. Mino [70] proposed a new method for evaluating the average rate of change of the Carter constant by using the radiative field in the adiabatic approximation. He showed that the average rate of change of the Carter constant as well as the energy and angular momentum can be obtained by the radiative field of the metric perturbation. The radiative field is defined as half the retarded field minus half the advanced field. Mino’s work gave a proof of an earlier work by Gal’tsov [46] in which the radiative field is used to evaluate the average rate of change of the energy and the angular momentum without proof. Inspired by this new development, it was demonstrated in [53] and [31] that the timeaveraged rates of change of the energy and the angular momentum can be computed numerically for generic orbits. A first step toward explicit calculation of the rate of change of the Carter constant was done in the case of a scalar charged particle in [30]. After that a simpler formula for the average rate of change of the Carter constant was found in [90, 89]. This new formula relates the rate of change of the Carter constant to the flux evaluated at infinity and on the horizon. Based on the new formula, in [89], the rate of change of the Carter constant for orbits with small eccentricities and inclinations is derived analytically up to O(v^{5}) by using the MK method discussed in Section 4. In Ref. [48], the method was extended to the case of the orbits with small eccentricity but arbitrary inclination angle.
5.9 Adiabatic evolution of constants of motion for orbits with generic inclination angle and with small eccentricity around a Kerr black hole
If we assume that the inclination angle is small and Y = 1  y′/2 + O(y′^{2}), we find that Equations (229) – (231) reduce respectively to (220) – (222) in Section 5.8. As discussed in [48], in the case of largely inclined orbits, the fundamental frequency of gravitational waves is expressed not only with Ω_{ ϕ } but also the frequency of θocillation, Ω_{ θ }.
6 Conclusion
In this article, we described analytical approaches to calculate gravitational radiation from a particle of mass μ orbiting a black hole of mass M with M ≫ μ, based upon the perturbation formalism developed by Teukolsky. A review of this formalism was given in Section 2. The Teukolsky equation, which governs the gravitational perturbation of a black hole, is too complicated to be solved analytically. Therefore, one has to adopt a certain approximation scheme. The scheme we employed is the postMinkowski expansion, in which all the quantities are expanded in terms of a parameter ε = 2Mω where ω is the Fourier frequency of the gravitational waves. For the source term given by a particle in bound orbit, this naturally gives the postNewtonian expansion.
In Section 3, we considered the case of a Schwarzschild background. For a Schwarzschild black hole, one can transform the Teukolsky equation into the ReggeWheeler equation. The advantage of the ReggeWheeler equation is that it reduces to the standard KleinGordon equation in the flatspace limit, and hence it is easier to understand the postMinkowskian or postNewtonian effects. Therefore, we adopted this method in the case of a Schwarzschild background. However, the postMinkowski expansion of the ReggeWheeler equation is not quite systematic, and as one goes to higher orders, the equations to be solved become increasingly complicated. Furthermore, for a Kerr background, although one can perform a transformation similar to the Chandrasekhar transformation, it can be done only at the expense of losing the reality of the equation. Thus, the resulting equation is not quite suited for analytical treatments.
In Section 4, we described a different method, developed by Mano, Suzuki, and Takasugi [69, 68] that directly deals with the Teukolsky equation, and we considered the case of a Kerr background with this method. Although the method is mathematically rather complicated and it is hard to obtain physical insights into relativistic effects, it has great advantage in that it allows a systematic postMinkowski expansion of the Teukolsky equation, even on the Kerr background. We gave a thorough review on how this method works and how it gives a systematic postMinkowski expansion.
Finally, in Section 5, we recapitulated the results of calculations of the gravitational waves for various orbits that had been obtained by various authors using the methods described in Sections 3 and 4. These results are useful not only by themselves for the actual case of a compact star orbiting a supermassive black hole, but also because they give us useful insights into higher order postNewtonian effects even for a system of equalmass binaries.
Footnotes
 1.
^{1} In the first version of this article, these asymptotic amplitudes contained errors in the phase. We thank W. Throwe and S.A. Hughes for pointing out these errors.
Notes
Acknowledgements
It is our great pleasure to thank Shuhei Mano, Yasushi Mino, Takashi Nakamura, Eric Poisson, Masaru Shibata, Eiichi Takasugi and Takahiro Tanaka for collaborations and fruitful discussions. We are grateful to Luc Blanchet and Bala Iyer for useful suggestions and comments. We are also grateful to Ryuichi Fujita, Norichika Sago and Hiroyuki Nakano for pointing out several typos and errors in the first version of this article. We also thank N. Sago for confirming the formulas in Section 5.8 and 5.9. This work was supported in part by GrantinAid for Scientific Research, Nos. 14047214, 12640269, 16540251, 20540271 and 21244033 of the Ministry of Education, Culture, Sports, Science, and Technology of Japan.
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