Post-Newtonian Expansion of Gravitational Waves from a Particle in Circular Orbits around a Rotating Black Hole :Effects of Black Hole Absorption

(2)Osaka Univ.); Eiichi Takasugi(2) ((1)Nation. Astron. Obs. Japan; Hideyuki Tagoshi(1); Shuhei Mano(2)

REVIEW

Post-Newtonian Expansion of Gravitational Waves from a Particle in Circular Orbits around a Rotating Black Hole :Effects of Black Hole Absorption

Hideyuki Tagoshi(1), Shuhei Mano(2), Eiichi Takasugi(2) ((1)Nation. Astron. Obs. Japan, (2)Osaka Univ.)

General Relativity and Quantum Cosmology · Submitted 25 Nov 1997 · 17 references

Pith's verdict: The v^5-versus-v^8 spin result is structurally sound and worth citing; the soft spots are in the higher-order coefficients and one cross-check that doesn't obviously close.

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arXiv IDgr-qc/9711072 (version 1)

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Citation(2)Osaka Univ.), Eiichi Takasugi(2) ((1)Nation. Astron. Obs. Japan, Hideyuki Tagoshi(1), Shuhei Mano(2). "Post-Newtonian Expansion of Gravitational Waves from a Particle in Circular Orbits around a Rotating Black Hole :Effects of Black Hole Absorption." arXiv:gr-qc/9711072 (1997). https://pith.science/paper/TCOYJPE5

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Editor's note

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The thing to take away: the headline claim is right, and it's right for a reason you can see in one line. Horizon absorption scales with k = ω − mΩ_H through the Teukolsky-Press flux factor k(k²+4ε̃²)(k²+16ε̃²). At q=0 that's O(v³/M); with spin it's O(1/M). Three powers of v, exactly as they say. You don't need any of the long algebra in Eq. (4.6) to believe the order counting, and the leading coefficient −q/4 − 3q³/4 comes from ℓ=m=2 at a_0=1, so the MST truncation worry doesn't touch it. The reader's concern about summing to 'appropriately large |n|' is real but sits on the v⁶–v⁸ coefficients, not the claim in the abstract (abstract:s5). The stress-test note is correct on this and I'd say so to the reader too: the truncation issue is a coefficient-accuracy issue, not a load-bearing one. The q=0 reduction to the Poisson-Sasaki v⁸ result (Eq. 4.8) is a genuine independent check, and the odd-in-q leading term is what makes superradiance show up, which is a good sign the bookkeeping is sane. What I'd actually push on: the claimed agreement with Gal'tsov (Eq. 6.1). As printed, Gal'tsov's bracket gives −q²/2 − 3q⁴/2 at leading order, even in q, while Eq. (4.6) gives −q/4 − 3q³/4, odd in q. Those can't both be right, and an even-in-q formula can't show superradiance at all. Most likely the transcription of Gal'tsov is off, but the paper asserts agreement without showing it, and a referee should ask. Second, §5 and Table I are thin — sub-1.5 cycle shifts, and the q<0 explanation leans on 'the total cycle is much smaller' without showing the numbers. That's a rough application section, fine as a sketch, not fine as a result. Third, no numerical cross-check of the q-dependent intermediate coefficients anywhere; the whole thing is hand algebra with no code. For a paper whose value is a long polynomial, that's the thing I'd want. Who should read it: anyone building PN templates for spinning holes, and anyone who needs the Kerr absorption reference. Send it out. The order-counting argument is solid and the coefficients are worth having in the literature even if a referee makes them tighten the Gal'tsov comparison.

Referee report

Major objections (2)

[§6, Eq. (6.1)] The printed Gal'tsov formula, expanded at leading order in v, gives -(q^2/2 + 3q^4/2)v^5, whereas Eq. (4.6) gives -(q/4 + 3q^3/4)v^5. The claim that 'This formula agrees with our formula Eq. (4.6) to lowest order' is therefore not verifiable from the equation as printed. Either the transcription of Eq. (6.1) is in error (plausibly a q versus q^2 and a factor of two) or the claim of agreement needs to be qualified. Please correct the equation and state precisely the sense in which the two results agree.
[§3, truncation of the MST series] The paper states that the post-Newtonian order of the expansion is not monotonic for n<0 and that one obtains the desired accuracy 'by summing up the terms up to the appropriately large |n|', but it does not state how many terms were retained or give any convergence or truncation-error estimate. Since the v^6-v^8 coefficients in Eq. (4.6) are part of the stated result, the paper should state the truncation criterion and give evidence (for example, stability of the quoted coefficients under increasing |n|) that they are converged. The leading v^5 coefficient is unaffected because it requires only a_0=1, so this does not threaten the headline claim, but it does bear on the higher-order deliverable.

Minor objections (6)

[Abstract and §6] The abstract says the rate is obtained 'to O((v/c)^8) compared to the lowest order', while §6 says 'to O(v^13) beyond the Newtonian quadrapole luminosity'; these use different reference points (the leading absorption term versus the quadrupole luminosity) and should be reconciled or explained so the reader is not left with an apparent inconsistency.
[Abstract and §6] The word 'quadrapole' should be 'quadrupole' throughout.
[Table I] The q=0 entries mix an inequality ('< 10^{-2}') with equalities ('0.02', '0.1'), and the caption does not state the units (presumably cycles) or the precision justified by the approximations in §5. Please make the entries uniform and state the units and the indicative nature of the numbers.
[§5] The paper acknowledges that finite-mass effects are ignored and that r_f=6M is a choice; given those approximations, the two-significant-figure values in Table I should be explicitly labelled as indicative rather than as quantitative predictions.
[§4, Eq. (4.6)] The sign convention for (dE/dt)_H should be stated explicitly near Eq. (4.6): positive means energy flowing into the hole, so a negative value indicates superradiant extraction. The sentence 'Eq. (4.6) can be negative when q>0' would then read more clearly as a statement about the superradiant regime Ω<Ω_H, which in the PN regime (large r) is automatically satisfied for q>0.
[§6, reference 17] The conditions and approximation under which Gal'tsov's formula applies (for example, whether it is the absorption piece of the radiation reaction and in which velocity variable) should be stated, so that the comparison with Eq. (4.6) can be checked by the reader.

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The paper

The paper claims

For a particle orbiting a spinning black hole, the share of gravitational-wave energy swallowed by the horizon first appears at order $v^5$ beyond the Newtonian quadrupole formula, three powers of the orbital velocity earlier than for a non

The reading

A small mass spiraling around a black hole radiates gravitational waves, and a fraction of that radiation falls back through the horizon instead of escaping to distant observers. This paper computes that absorbed fraction for a test particle on a circular equatorial orbit around a Kerr black hole, in the limit where the orbital velocity $v$ is small compared with light. Using an analytic scheme for the Teukolsky equation, the authors obtain the horizon energy flux as a power series in $v$ carried through $O(v^8)$ relative to the standard Newtonian quadrupole luminosity. The result changes the bookkeeping: with spin, absorption first appears at order $v^5$, three powers of $v$ earlier than the $v^8$ entry point found for a non-rotating hole, so the effect is much less negligible for spinning systems than the earlier estimate suggested. The authors then feed the rate into an inspiral model and estimate how many wave cycles are lost or gained in the observing band of a ground-based interferometer.

What carries the argument

The engine is the Mano–Suzuki–Takasugi analytic representation of homogeneous Teukolsky solutions: one expansion in hypergeometric functions that converges near the horizon and one in Coulomb wave functions that converges at large radius, joined by a constant $K_ν$ that relates them. The post-Newtonian expansion becomes tractable because the natural small parameters are ordered, $ε=2Mω ∼ v^3$ and $z=ωr ∼ v$, so truncating the series in $ε$ and $z$ is the same as truncating in $v$. The radial solution is then assembled by the Green function method with a source term built from the geodesic circular orbit, and the absorbed flux is read off from the Teukolsky–Press horizon flux formula, whose spin-dependent factor $|C|^2$ supplies the leading $q$-dependence.

Extended reading notes

Core claim

The paper establishes that black hole absorption of gravitational waves from a circular equatorial orbit around a Kerr black hole begins at $O(v^5)$ relative to the Newtonian quadrupole luminosity $(dE/dt)_N = (32/5)(M/r_0)^5 (M/r_0)^{5}$-normalized rate, with leading coefficient $-3q^3/4 - q/4$ where $q=a/M$ is the dimensionless spin. The full total rate is given through $O(v^8)$ in Eq. (4.6), built from per-mode contributions listed in Appendix C and rewritten in the orbital-frequency variable $x=(M/Omega)^{1/3}$ in Appendix D. The authors check that setting $q=0$ collapses the expression to the previously known Schwarzschild result $(dE/dt)_H = (dE/dt)_N v^8$, and that the lowest-order term agrees with Gal'tsov's radiation-reaction formula. They note that the rate can be negative for $q>0$, which they identify as superradiance, and they give the finite $|q| o 1$ limit explicitly. Finally, they estimate the accumulated cycle difference for neutron-star–black-hole binaries, finding values from below $10^{-2}$ cycles up to about 1.5 cycles depending on spin and black hole mass.

Load-bearing premise

The coefficients quoted to $O(v^8)$ are trustworthy only if summing the hypergeometric and Coulomb series to a finite but unspecified number of terms really delivers that accuracy; the authors themselves note in §3 that the post-Newtonian order of the terms is not monotonic for negative index $n$ and that the expansion there must be handled with care, and no truncation-error bound is given.

Editorial extensions

If this is right

For a spinning hole, the absorption term enters at the same 2.5-post-Newtonian order as standard radiation-reaction effects, so templates for spinning binaries that omit it are formally incomplete at that order, whereas for a non-spinning hole the $O(v^8)$ entry makes it negligible in the interferometer band.
Because the rate can be negative for co-rotating orbits, the horizon can return energy to the orbit rather than draining it, a superradiant contribution that grows in relative importance as the spin approaches extremality.
For neutron-star–black-hole binaries the accumulated cycle shift rises with black hole mass, reaching roughly 1.5 cycles for a $70 M_\odot$ hole with $q=0.9$, which is large enough to matter for matched filtering.
The machinery is order-by-order in $v$, so extending the series past $O(v^8)$ requires only more terms in the same expansions rather than a new method.
The finite $|q| \to 1$ limit given in Eq. (4.11) provides a closed-form check for any future extremal-spin calculation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the leading absorption term is odd in spin and starts at $v^5$, it sits at the same post-Newtonian order as the standard 2.5PN radiation-reaction term; a template that includes one and not the other is internally inconsistent at that order, which suggests the absorption term should be quoted alongside the 2.5PN flux in any spinning-template comparison.
The cycle-shift table implies the effect is only marginal for stellar-mass holes but reaches order unity for holes of tens of solar masses; the natural observational target is therefore an intermediate-mass or massive black hole inspiral, where the same $v^5$ term would be far larger.
Since the series truncation carries no stated error bound, the cheapest independent check would be a numerical frequency-domain or time-domain Teukolsky solver evaluating the horizon flux at moderate $v$ and high spin, which would either confirm the quoted coefficients or expose a truncation artifact.
The restriction to equatorial circular orbits leaves inclined orbits open; there the spin-orbit coupling redistributes power among modes, so the $v^5$ coefficient could pick up an inclination-dependent piece that the present calculation cannot see.

Original abstract

When a particle moves around a Kerr black hole, it radiates gravitational waves.Some of these waves are absorbed by the black hole. We calculate such absorption of gravitational waves induced by a particle of mass mu in a circular orbit on an equatorial plane around a Kerr black hole of mass M. We assume that the velocity of the particle v is much smaller than the speed of light c and calculate the energy absorption rate analytically. We adopt an analytic technique for the Teukolsky equation developed by Mano, Suzuki and Takasugi. We obtain the energy absorption rate to O((v/c)^8) compared to the lowest order. We find that the black hole absorption occurs at O((v/c)^5) beyond the Newtonian-quadrapole luminosity at infinity in the case when the black hole is rotating, which is O((v/c)^3) lower than the non-rotating case. Using the energy absorption rate, we investigate its effects on the orbital evolution of coalescing compact binaries.

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When a particle moves around a Kerr black hole, it radiates gravitational waves.Some of these waves are absorbed by the black hole. We calculate such absorption of gravitational waves induced by a particle of mass mu in a circular orbit on an equatorial plane around a Kerr black hole of mass M. We assume that the velocity of the particle v is much smaller than the speed of light c and calculate the energy absorption rate analytically. We adopt an analytic technique for the Teukolsky equation developed by Mano, Suzuki and Takasugi. We obtain the energy absorption rate to O((v/c)^8) compared to the lowest order. We find that the black hole absorption occurs at O((v/c)^5) beyond the Newtonian-quadrapole luminosity at infinity in the case when the black hole is rotating, which is O((v/c)^3) lower than the non-rotating case. Using the energy absorption rate, we investigate its effects on the orbital evolution of coalescing compact binaries.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	When a particle moves around a Kerr black hole, it radiates gravitational waves.Some of these waves are absorbed by the black hole.	background 1.00	0.06	0.03	asserted 1.00
s1	We calculate such absorption of gravitational waves induced by a particle of mass mu in a circular orbit on an equatorial plane around a Kerr black hole of mass M.	contribution 1.00	0.95	0.04	asserted 0.96
s2	We assume that the velocity of the particle v is much smaller than the speed of light c and calculate the energy absorption rate analytically.	method 0.93	0.90	0.03	asserted 0.97
s3	We adopt an analytic technique for the Teukolsky equation developed by Mano, Suzuki and Takasugi.	method 1.00	0.40	0.02	asserted 0.96
s4	We obtain the energy absorption rate to O((v/c)^8) compared to the lowest order.	result 0.72	0.96	0.09	asserted 1.00
s5	We find that the black hole absorption occurs at O((v/c)^5) beyond the Newtonian-quadrapole luminosity at infinity in the case when the black hole is rotating, which is O((v/c)^3) lower than the non-rotating case.	result 1.00	0.96	0.78	asserted 1.00
s6	Using the energy absorption rate, we investigate its effects on the orbital evolution of coalescing compact binaries.	contribution 0.92	0.93	0.08	asserted 0.66

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read

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References

Works cited (17)

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T. Tanaka, H. Tagoshi, and M. Sasaki, Prog. Theor. Phys. 96 (1996), 1087. See also E. Poisson, Phys. Rev. D47 (1993), 1497: C. Cutler, L.S. Finn, E. Poisson and G.J. Sussman, Phys. Rev. D 47 (1993), 1511. H. Tagoshi and T. Nakamura, Phys. Rev. D49 (1994), 4016: M. Sasaki, Prog. Theor. Phys. 92 (1994), 17: H. Tagoshi and M. Sasaki, Prog. Theor. Phys. 92 (1...

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Binary-inspiral gravitational-wave emission can be reorganized as a tower of effective field theories, one per length scale, with a definite velocity power-counting rule at every stage.
The Effective Field Theorist's Approach to Gravitational Dynamics
hep-th 2016-01 accept novelty 1.0 of 10

A comprehensive review of non-relativistic general relativity (NRGR), the EFT framework for computing binary inspiral dynamics and gravitational waveforms to high post-Newtonian order.

How later papers used it

49 later papers cite it

Evidence: substantial. 49 of the citing papers share no author with it, and 10 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 98 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 compared its own results with it.

Compares with it 1 of 7 papers with this label

Demonstrated the consistency of its own expression for the tidal dissipative flux with this paper's tidal heating formula in the EMRI limit.

A New Spin on Dissipative Tides: First-Post-Newtonian Effects in Compact Binary Inspirals (2026) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

45 arXiv papers cite this in 89 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 61 sentences without a confident label

Contests it 1 paper

When taking the test-particle limit of our results we do not recover the results of [ref] .

Improved next-to-leading order tidal heating and torquing of a Kerr black hole · Conclusions · contest 0.74

Builds on it 1 paper

ddt E_h = ∑^2_α=1 v⃗_α·F⃗_α= -325 G_N^-1 (G_N M|x⃗|)^8 (μM)^2 (1 - 4μM + 2 μ^2M^2) [v⃗^2 + 2 (v⃗·x⃗)^2|x⃗|^2], which agrees with and generalizes the result of [ref] valid in the limit $μ≪M$ where black hole perturbation theory holds.

Horizon radiation reaction forces · Post-Newtonian equations of motion for binary dynamics · 0.91

Uses its method or tool 2 papers

For analytical expressions of the scattering phase shifts in GR, we use the black hole perturbation theory (BHPT) methods of Mano, Suzuki, and Takasugi (MST) for the analytic solution of the Teukolsky master equation [ref] [Note that similar in spirit matching calculations were recently performed by [ref] in worldline theory and [ref] in an on-shell-amplitudes approach.].

Dynamical Tidal Response of Kerr Black Holes from Scattering Amplitudes · Introduction · 0.99
This calculation is performed with the analytical module of the Black Hole Perturbation Club (BHPC) code [ref] developed in Refs. [ref] .

Adiabatic waveforms from extreme-mass-ratio inspirals: an analytical approach · Introduction · 0.97

Compares with it 6 papers

The difference between our Eq. [ref] and the test-mass result of [ref] arises at the 1.5PN order, and is given by

Improved next-to-leading order tidal heating and torquing of a Kerr black hole · Small mass ratios · 0.98
We conclude this section by comparing these results with those obtained earlier in Refs. [ref] .

Modeling horizon absorption in spinning binary black holes using effective worldline theory · General expressions for evolution equations of mass and spin · 1.00
There is a relation between the coefficients $Z^H_ℓm,-2$ and $Z^H_ℓm,+2$ given by [ref] [To facilitate comparison with [ref] , we recall that their $Z_hole$ is our $Z_ℓm,-2^H(ω)/(2M)$ and their $Y_hole$ is our $Z_ℓm,+2^H(ω)/(2M)$.

"Waveforms" at the Horizon · Behavior at the horizon · 0.85

Cites it as background 9 papers

Complementary work employing the effective-one-body approach has enabled the modeling of the spin evolution in dense clusters in a wide range of parameter space [ref] The broader phenomenon of spin changes due to the re-absorption of angular momentum that was emitted in gravitational radiation is called “tidal-torquing” and has been explored using (semi-) analytic models [ref] .

Spin-up and mass-gain in hyperbolic encounters of spinning black holes · Introduction · 0.87
The tidal heating of black holes was recently investigated by Alvi [ref] , Poisson [ref] , Taylor and Poisson [ref] , and Comeau and Poisson [ref] , building on earlier work by Poisson and Sasaki [ref] and Tagoshi, Mano, and Takasugi [ref] .

Geometry and dynamics of a tidally deformed black hole · Tidal heating · 0.90
Previous theoretical studies of horizon absorption effects include [ref] .

Absorptive Effects and Classical Black Hole Scattering · Introduction · 1.00

Discussion

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