Pith. sign in

REVIEW 2 major objections 6 minor 2 cited by

Post-Newtonian expansion of gravitational energy and angular momentum fluxes: inclined spherical orbits about a Kerr black hole

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper derives post-Newtonian expansions through 12PN for the energy and angular-momentum fluxes radiated by a point mass on an inclined spherical orbit around a Kerr black hole, with coefficients exact in spin and inclination.

desk verdict Solid, high-value calculation paper giving the first 12PN flux formulas for inclined spherical Kerr orbits; the main soft spot is an unproven k-mode truncation rule, but the numerical validation largely compensates. read the letter →

arxiv 2411.09700 v2 pith:2F6KBBNH submitted 2024-11-14 gr-qc

Jezreel C. Castillo , Charles R. Evans , Chris Kavanagh , Jakob Neef , Adrian Ottewill , Barry Wardell This is my paper · ORCID
classification gr-qc MSC 83C2583C3583C57 PACS 04.25.Nx04.30.-w04.70.Bw
keywords post-NewtonianexpansiongravitationalwavefluxKerrblackholeinclinedsphericalorbitextreme-mass-ratioinspiralTeukolskyequationself-forceangularmomentum
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper derives closed-form post-Newtonian (PN) expressions for the energy and angular-momentum fluxes radiated by a point mass on an inclined spherical orbit around a Kerr black hole. The expansions run through 12PN order, and every coefficient is an exact function of the spin $\tilde a$ and the inclination $x$, with no further expansion in either parameter. If the calculation is right, these are the first high-order PN flux formulas for inclined spherical Kerr orbits, a configuration that matters directly for extreme-mass-ratio inspiral models because such orbits are generically inclined. The authors check the series by reproducing known Schwarzschild and equatorial limits and by comparing with direct numerical Teukolsky calculations, finding the expected residual fall-off as more PN terms are included.

What carries the argument

The argument is carried by the MST method, an analytic representation of the homogeneous Teukolsky radial solutions as convergent sums of Coulomb wave functions, expanded order by order in $1/p$; by the expansion of the spin-weighted spheroidal harmonics in powers of $a\omega$ times spin-weighted spherical harmonics; and by the truncation rule $K=\{k\in\mathbb{Z} : -\ell-m-2\lfloor N/4\rfloor \le k \le \ell-m+2\lfloor N/4\rfloor\}$ for the polar harmonics that contribute at PN order $N$. After parameterizing the polar motion by $\cos\theta = \sqrt{1-x^2}\cos\chi$, the mode integral collapses to a single Fourier coefficient $2\pi c^{(0)}_{\ell mk}$, so the formally infinite double sum over $\ell$ and $k$ becomes finite at each PN order. That finiteness is what makes a fully algebraic 12PN calculation possible, with the dependence on inclination and spin left exact.

What would settle it

Take one parameter point from the paper's own comparison, say $a=0.9M$ and $x=1/4$, and evaluate the 12PN expansion at a large separation such as $p=100$. Independently compute the same flux with a Teukolsky code that sums all polar modes $k$ up to a large cutoff (well beyond the set in Eq. (3.22)) and all relevant $\ell$. If the difference does not fall off as the next missing PN order, or if modes outside the set in Eq. (3.22) contribute at the 12PN level, the truncation rule is wrong.

Watch

Extended reading notes

Core claim

The central claim is that the asymptotic fluxes of energy and angular momentum for an inclined spherical orbit can be written as PN series, e.g. $$\left\langle\frac{dE}{dt}\right\rangle_\infty = \frac{32}{5}\left(\frac{\mu}{M}\right)^2 $p^{{-5}}$\left[1 - \frac{1247}{336}$p^{{-1}}$ + \left(4\pi - \frac{73}{12}\tilde a x\right)$p^{{-3/2}}$ + \cdots\right],$$ in which the coefficients are polynomials in $x$ times powers of $\tilde a$ at low orders, with combinations of polygamma functions of $i\sigma\tilde a$ appearing at higher orders. The same construction gives the angular-momentum flux with an overall $x p^{-7/2}$ prefactor, and horizon fluxes with their own PN structure. In the Schwarzschild and equatorial limits the series reduce to previously established results, and direct numerical comparison shows the residual after subtracting the 12PN expansion falls off at the expected rate. The paper also identifies structural regularities: odd powers of spin appear at half-integer PN orders, even powers at integer orders, and the leading-spin terms are simple polynomials in $x$ up to the orders shown.

Load-bearing premise

The whole calculation depends on an unproven rule that, at a given post-Newtonian order, only a specific finite list of polar oscillation modes contributes to the flux; if any mode outside that list has a nonzero contribution, the 12PN formulas would be incomplete.

Editorial extensions

If this is right

  • For extreme-mass-ratio inspiral models, these expressions supply the dissipative fluxes that drive the adiabatic evolution of inclined spherical inspirals, as exact functions of spin and inclination rather than as double expansions in both.
  • The known Schwarzschild and equatorial circular-orbit flux results are recovered as the limits $x\to1$ and $\tilde a\to0$, so the new formulas interpolate between previously separate regimes.
  • The horizon flux series quantify the energy and angular momentum absorbed by the hole, giving the back-reaction on the primary's mass and spin during an inclined inspiral.
  • Because the formulas are valid for prograde, polar, and retrograde orbits ($-1\le x\le1$), they cover the full range of inclinations, including the polar case where the angular-momentum flux at infinity arises entirely from frame dragging.

Reading between the lines

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

  • If the finite-mode truncation rule continues to hold at higher order, the same algebraic procedure should push the expansion beyond 12PN (for example to 14PN) without a change of method, since nothing in the construction is tied to order 12.
  • The angular-function expansion used here is likely transferable to the conservative sector of the self-force problem, where worldline-regularized quantities are needed; that would be a natural next step that the paper leaves to future work.
  • The appearance of polygamma functions at high PN order hints that a resummation in spin might expose all-order structure, much as leading-log sequences were extracted for eccentric orbits; the paper does not attempt this resummation.
  • Beyond inspirals, the same flux expressions could serve as a benchmark for calibrating faster phenomenological or surrogate EMRI waveform models before full post-adiabatic waveforms with inclination become available; this data-analysis use is not discussed in the paper.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper presents a calculation of the gravitational energy and angular momentum fluxes radiated by a point mass on an inclined, spherical geodesic orbit about a Kerr black hole. Using the Mano-Suzuki-Takasugi analytical solutions of the Teukolsky equation, the authors derive post-Newtonian expansions of the fluxes at infinity and at the horizon, exact in the black hole spin parameter a and the inclination parameter x. The text and tables show coefficients through roughly 5PN, while the full expansions through 12PN at infinity (and 9.5PN at the horizon relative to the horizon leading order) are made available in public repositories. The paper also characterizes the general PN structure, identifies leading-spin terms, discusses the polar-orbit limit, and validates the expansions against numerical Teukolsky data for several spins, inclinations, and orbital radii.

Significance. If correct, these are the first high-order PN flux formulas for inclined spherical Kerr orbits that are exact in both spin and inclination, extending earlier 5PN inclined-orbit results to much higher order in the spherical case. The results should be useful for adiabatic EMRI waveform modeling and for benchmarking future PN expansions of conservative self-force quantities. The paper has several notable strengths: the derivation has no fitted parameters; the expansions are cross-checked against the a=0 Schwarzschild limit to 12PN, the x=1 equatorial limit, retrograde-orbit comparisons, and independent numerical Teukolsky data; two independently written codes were used; and the full expressions are publicly released. The main weakness is that the completeness of the k-mode truncation, which is essential to the 12PN claim, is asserted rather than rigorously established.

major comments (2)
  1. [Section III.D, Eq. (3.22)] The finite k-mode truncation rule is load-bearing for the claim that the flux expansions are complete through 12PN, but it is supported only by a sketch. The integrand leading to Eq. (3.21) contains the non-periodic factor (1+2iaχ/p^{3/2}+...) from Eq. (3.16), and the argument that multiplication by χ merely widens the Fourier range by 2 only every 4PN is not self-evident. The authors should either provide a rigorous bound on the Fourier support of the PN-expanded angular integrand at each order, or supply an independent check that all excluded k-modes have identically vanishing contributions at the claimed PN orders. Without this, the completeness of the 12PN results is not established by the text alone.
  2. [Section IV.E, Figs. 1-4] The numerical comparisons validate the total flux on a relatively small set of (a,x,p) samples, so good agreement there does not by itself rule out a missing k-mode whose contribution is small and similar across those samples. I ask for at least one targeted test, such as comparison of individual k-mode flux contributions at representative (a,x,p) points, or numerical convergence checks at extreme parameter values (high spin, retrograde, near-polar orbits), so that the completeness of the k-sum is tested in a way that does not rely on the unproven rule in Eq. (3.22).
minor comments (6)
  1. [Section III.C.2, Eq. (3.16)] The '±' notation is not defined precisely for the reader; since the sign is determined by the sign of m, it would help to write sign(m) explicitly in Eqs. (3.16) and (3.20).
  2. [Equations (4.3)-(4.4)] The sum notation 'X_{k=0}' is ambiguous; it is presumably intended as a sum over the spin-power k defining ASk and CSk, but the range of k should be stated explicitly.
  3. [Appendix B, Fig. 3] The caption explains that p=7 is excluded for retrograde orbits because it lies near the last stable orbit, but the main text should also mention this restriction when discussing retrograde compatibility.
  4. [Table I] The entry 'AS4 4' appears both in Table I and immediately after it in Eq. (4.12); one of the duplicate presentations should be removed or cross-referenced.
  5. [Equation (3.22)] The symbol N denoting the PN order should be defined explicitly, since the expansions contain both integer and half-integer PN orders and the floor(N/4) rule is otherwise ambiguous.
  6. [Abstract and Appendix A] The abstract's '12PN' refers to the infinity-side flux, while the horizon expansions are complete only to 9.5PN relative to the horizon leading order; the equivalence stated in Appendix A should be reflected in the abstract or introduction to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 12PN flux formulas are derived analytically from the Teukolsky equation and validated against independent numerics.

full rationale

The paper's central derivation is self-contained and non-circular. The flux expressions are obtained by constructing PN expansions of MST solutions of the Teukolsky radial equation, PN-expanding the orbital parameters and Mino-time frequencies, PN-expanding the spin-weighted spheroidal harmonics, and then summing the asymptotic amplitudes through Eqs. (3.10)-(3.13). No parameter is fitted to the flux output: the inputs are the geodesic constants and the Teukolsky equation itself, and the resulting flux coefficients are new closed-form functions of p, a and x. Validation is external: the paper compares against independent numerical Teukolsky results (Figs. 1-4) and against known Schwarzschild and equatorial circular limits, e.g. 'We confirm that the terms that survive in the a = 0 limit match known results [49] all the way to 12 PN order.' Self-citations such as [18,41] for the radial-function PN expansion procedure and [46] for spheroidal-harmonic coefficients are to independently established technical machinery, not to the flux results themselves, and the MST method itself is an external result [15,16]. The one flagged gap, the k-mode truncation rule Eq. (3.22), is asserted with a brief sketch rather than a rigorous proof; however, this is a completeness/correctness concern about possible omitted k-modes, not a circularity. The rule is a claim about Fourier support of the integrand, not a restatement of the flux formulas or a fitted input. Therefore the derivation chain does not reduce to its own inputs, and the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; the expansion parameter p and the exact inputs a and x are physical orbital and spacetime parameters. The derivation relies on established mathematical machinery (MST solutions, spheroidal harmonics) and standard flux formulas, all cited. The only paper-specific assumption is the finite k-mode truncation rule in Eq. (3.22), which is sketched but not fully proven. No new physical entities are introduced.

assumptions (5)
  • domain assumption The MST method provides uniformly convergent analytical solutions to the Teukolsky radial equation, and the PN expansion of these solutions is valid to the required order.
    Invoked in Sec. III.C.1; the paper relies on Refs. [15,16,18,41] for the correctness of the MST solutions and their PN expansion without re-deriving them.
  • domain assumption The spin-weighted spheroidal harmonics can be expanded in spin-weighted spherical harmonics with coefficients computed by the BHPT package, with each power of aω adding two harmonics.
    Sec. III.C.2, Eq. (3.18); the paper uses the SpinWeightedSpheroidalHarmonics package [46] without deriving the expansion coefficients.
  • domain assumption The point-particle stress-energy tensor in Eq. (3.9) with delta functions is the correct source model for the secondary mass.
    Standard in self-force theory; used in Sec. III.B.
  • ad hoc to paper The finite k-mode truncation rule in Eq. (3.22) is correct at each PN order, so the flux sums are complete.
    This paper-specific rule is presented with a sketch in Sec. III.D and underpins the completeness of the 12PN flux expressions; it is not rigorously proven.
  • standard math The standard Teukolsky-Press flux formulas (Eqs. 3.10-3.13) give the radiated energy and angular momentum from the asymptotic amplitudes.
    Sec. III.B; well-established result in black hole perturbation theory.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Post-Newtonian expansion of gravitational energy and angular momentum fluxes: inclined spherical orbits about a Kerr black hole." pith.science (2026). https://pith.science/paper/2F6KBBNH

@misc{pith2026241109700,
  author       = {Pith},
  title        = {Pith review of: Post-Newtonian expansion of gravitational energy and angular momentum fluxes: inclined spherical orbits about a Kerr black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2F6KBBNH}},
  note         = {Machine review of arXiv:2411.09700}
}
abstract

We present analytical expressions for the fluxes of energy and angular momentum from a point mass on an inclined spherical orbit about a Kerr black hole. The expressions are obtained using the method of Mano, Suzuki and Takasugi to construct analytical solutions of the Teukolsky equation, and are given as post-Newtonian expansions valid through 12PN, with arbitrary values for the inclination parameter $x$ and black hole spin $a$. We characterize the structure of the PN expansions in terms of their dependence on $x$ and $a$, and we validate our results against numerical calculations.

Figures

Figures reproduced from arXiv: 2411.09700 by the authors.

Figure 1
Figure 1. FIG. 1. Relative error in the infinity-side energy flux when comparing the numerical evaluation of PN expansions versus [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Relative error in the horizon energy flux when comparing the numerical evaluation of PN expansions versus numerical [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison of relative errors in the infinity-side energy flux for prograde versus retrograde orbits when comparing [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of relative errors in the horizon energy flux for prograde versus retrograde orbits when comparing the [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Secular evolution of orbital parameters for general bound orbits in Kerr spacetime

    gr-qc 2026-03 conditional novelty 6.0 of 10

    Analytic formulas for the orbit-averaged gravitational-wave fluxes of energy, angular momentum, and Carter constant for generic bound Kerr orbits are extended to 6PN order and O(e^16) in eccentricity, with numerical T...

  2. Post-Newtonian templates for phase evolution of spherical extreme mass ratio inspirals

    gr-qc 2024-11 conditional novelty 6.0 of 10

    A 12PN analytic phase model for quasi-spherical inclined EMRIs in Kerr spacetime is presented, and TaylorT1 is found to converge fastest among the time-domain approximants.

Reference graph

Works this paper leans on

64 extracted references · 26 canonical work pages · cited by 2 Pith papers

  1. [1]

    A self-force expansion, which recasts the problem in terms of solutions of the Teukolsky equation for perturbations of Kerr spacetime

    work page

  2. [2]

    As shown by Mano, Suzuki, and Takasugi (MST) [15, 16], the Teukolsky equation admits analytical solutions in terms of an infinite, uniformly convergent sum of special functions

    A post-Newtonian expansion of the self-force equa- tions of motion. As shown by Mano, Suzuki, and Takasugi (MST) [15, 16], the Teukolsky equation admits analytical solutions in terms of an infinite, uniformly convergent sum of special functions. At any given order in a post-Newtonian (PN) expansion the sum is in fact finite, and the method yields closed-f...

    arXiv 2025

  3. [3]

    (3.5), can be found using the methods pioneered by Mano, Suzuki, and Takasugi (MST) [15, 16]

    Teukolsky radial function Solutions to the radial Teukolsky equation, Eq. (3.5), can be found using the methods pioneered by Mano, Suzuki, and Takasugi (MST) [15, 16]. The MST method represents the homogeneous solutions as a convergent sum of special functions. There is some freedom in the particular choice of special functions. For the purpose of produci...

    work page

  4. [4]

    tidal response function

    Angular function We next address the calculation of the PN expansion of the spin-weighted spheroidal harmonic. First, we note that the azimuthal dependence factors out: sSℓm(θ, φ; aω) = sSℓm(θ, 0; aω)eimφ. We thus start by considering the PN expansion of this dependence on the azimuthal coordinate function φ(χ). For the pur- poses of producing expressions...

    work page 2016

  5. [5]

    Amaro-Seoane, J

    P. Amaro-Seoane, J. R. Gair, M. Freitag, M. C. Miller, I. Mandel, C. J. Cutler, and S. Babak, Classical and Quantum Gravity 24, R113 (2007), astro-ph/0703495

    arXiv 2007

  6. [6]

    Abbott, B. P. et al. (The LIGO Scientific Collabora- tion and VIRGO Collaboration), Physical Review Letters 116, 061102 (2016), arXiv:1602.03837 [gr-qc]

    arXiv 2016

  7. [7]

    Abbott, B. P. et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 119, 161101 (2017)

    work page 2017

  8. [8]

    Abbott et al

    R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. X 11, 021053 (2021), arXiv:2010.14527 [gr-qc]

    arXiv 2021

Show all 64 references
  1. [9]

    Abbott et al

    R. Abbott et al. (KAGRA, VIRGO, LIGO Scientific), Phys. Rev. X 13, 041039 (2023), arXiv:2111.03606

    2023 arXiv

  2. [10]

    J. R. Gair, S. Babak, A. Sesana, P. Amaro-Seoane, E. Ba- rausse, C. P. L. Berry, E. Berti, and C. Sopuerta, Journal of Physics: Conference Series 840, 012021 (2017)

    2017

  3. [11]

    Barack, Class

    L. Barack, Class. Quant. Grav. 26, 213001 (2009), arXiv:0908.1664 [gr-qc]. 15

    2009 arXiv

  4. [12]

    Poisson, A

    E. Poisson, A. Pound, and I. Vega, Living Rev. Rel. 14, 7 (2011), arXiv:gr-qc/1102.0529

    2011 arXiv

  5. [13]

    Wardell, A

    B. Wardell, A. Pound, N. Warburton, J. Miller, L. Durkan, and A. Le Tiec, Phys. Rev. Lett. 130, 241402 (2023)

    2023

  6. [14]

    Burke, G

    O. Burke, G. A. Piovano, N. Warburton, P. Lynch, L. Speri, C. Kavanagh, B. Wardell, A. Pound, L. Durkan, and J. Miller, Phys. Rev. D 109, 124048 (2024), arXiv:2310.08927 [gr-qc]

    2024 arXiv

  7. [15]

    P. C. Peters, Physical Review 136, B1224 (1964)

    1964

  8. [16]

    Glampedakis and D

    K. Glampedakis and D. J. Kennefick, Phys. Rev. D 66, 044002 (2002)

    2002

  9. [17]

    S. A. Hughes, Phys. Rev. D 61, 084004 (2000), gr- qc/9910091

    2000

  10. [18]

    Della Rocca, S

    M. Della Rocca, S. Barsanti, L. Gualtieri, and A. Maselli, (2024), arXiv:2401.09542 [gr-qc]

    2024 arXiv

  11. [19]

    S. Mano, H. Suzuki, and E. Takasugi, Progress of The- oretical Physics 96, 549 (1996), gr-qc/9605057

    1996 arXiv

  12. [20]

    S. Mano, H. Suzuki, and E. Takasugi, Progress of The- oretical Physics 95, 1079 (1996), gr-qc/9603020

    1996 arXiv

  13. [21]

    Isoyama, R

    S. Isoyama, R. Fujita, A. J. K. Chua, H. Nakano, A. Pound, and N. Sago, Phys. Rev. Lett. 128, 231101 (2022)

    2022

  14. [22]

    Kavanagh, A

    C. Kavanagh, A. C. Ottewill, and B. Wardell, Phys. Rev. D 93, 124038 (2016), arXiv:1601.03394 [gr-qc]

    2016 arXiv

  15. [23]

    Munna, Phys

    C. Munna, Phys. Rev. D 108, 084012 (2023)

    2023

  16. [24]

    Postnewtonianselfforce,

    N. Warburton, B. Wardell, C. Munna, and C. Kavanagh, “Postnewtonianselfforce,” (2023), https://doi.org/10.5281/zenodo.8112975

    2023 doi

  17. [25]

    Black Hole Perturbation Toolkit,

    “Black Hole Perturbation Toolkit,” bhptoolkit.org

  18. [26]

    UNC Gravitational Physics Group,

    “UNC Gravitational Physics Group,” https://github. com/UNC-Gravitational-Physics

  19. [27]

    Carter, Phys

    B. Carter, Phys. Rev. 174, 1559 (1968)

    1968

  20. [28]

    Schmidt, Class

    W. Schmidt, Class. Quant. Grav. 19, 2743 (2002), arXiv:gr-qc/0202090

    2002 arXiv

  21. [29]

    Drasco and S

    S. Drasco and S. A. Hughes, Phys. Rev. D 73, 024027 (2006), gr-qc/0509101

    2006 arXiv

  22. [30]

    Mino, Phys

    Y. Mino, Phys. Rev. D 67, 084027 (2003), arXiv:gr- qc/0302075

    2003

  23. [31]

    Fujita and W

    R. Fujita and W. Hikida, Classical and Quantum Gravity 26, 135002 (2009), arXiv:0906.1420 [gr-qc]

    2009 arXiv

  24. [32]

    S. A. Hughes, Phys. Rev. D 64, 064004 (2001)

    2001

  25. [33]

    Darwin, Proc

    C. Darwin, Proc. R. Soc. Lond. A 249, 180 (1959)

    1959

  26. [34]

    Darwin, Proceedings of the Royal Society of London Series A 263, 39 (1961)

    C. Darwin, Proceedings of the Royal Society of London Series A 263, 39 (1961)

    1961

  27. [35]

    Drasco and S

    S. Drasco and S. A. Hughes, Phys. Rev. D 69, 044015 (2004)

    2004

  28. [36]

    Warburton, Phys

    N. Warburton, Phys. Rev. D 91, 024045 (2015), arXiv:1408.2885 [gr-qc]

    2015 arXiv

  29. [37]

    S. A. Teukolsky, Phys. Rev. Lett. 29, 1114 (1972)

    1972

  30. [38]

    Teukolsky, Astrophys

    S. Teukolsky, Astrophys. J. 185, 635 (1973)

    1973

  31. [39]

    Wardell, C

    B. Wardell, C. Kavanagh, and S. R. Dolan, (2024), arXiv:2406.12510 [gr-qc]

    2024

  32. [40]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun, Handbook of Mathe- matical Functions, New York: Dover, 1972 (1972)

    1972

  33. [41]

    Sasaki and H

    M. Sasaki and H. Tagoshi, Living Reviews in Relativity 6, 6 (2003), gr-qc/0306120

    2003 arXiv

  34. [42]

    S. A. Teukolsky and W. H. Press, Astrophys. J. 193, 443 (1974)

    1974

  35. [43]

    Y. F. Bautista, G. Bonelli, C. Iossa, A. Tanzini, and Z. Zhou, Phys. Rev. D 109, 084071 (2024), arXiv:2312.05965 [hep-th]

    2024 arXiv

  36. [44]

    High precision calculation of generic extreme mass ratio inspirals,

    W. Throwe, “High precision calculation of generic extreme mass ratio inspirals,” (2010), http://hdl.handle.net/1721.1/61270

    2010

  37. [45]

    Kavanagh, A

    C. Kavanagh, A. C. Ottewill, and B. Wardell, Phys. Rev. D 92, 084025 (2015), arXiv:1503.02334 [gr-qc]

    2015 arXiv

  38. [46]

    J. Neef, A. Ottewill, and C. Kavanagh, https://gitlab.com/jakobneef/sfpn.git

  39. [47]

    K. Ganz, W. Hikida, H. Nakano, N. Sago, and T. Tanaka, Progress of Theoretical Physics 117, 1041 (2007), gr-qc/0702054

    2007 arXiv

  40. [48]

    Sago and R

    N. Sago and R. Fujita, Progress of Theoretical and Exper- imental Physics 2015, 073E03 (2015), arXiv:1505.01600 [gr-qc]

    2015 arXiv

  41. [49]

    Fujita and M

    R. Fujita and M. Shibata, Phys. Rev. D 102, 064005 (2020)

    2020

  42. [50]

    SpinWeightedSpheroidalHarmonics,

    B. Wardell, N. Warburton, K. Fransen, S. D. Up- ton, K. Cunningham, A. Ottewill, and M. Casals, “SpinWeightedSpheroidalHarmonics,” (2024), https://doi.org/10.5281/zenodo.11199019

    2024 doi

  43. [51]

    Munna, Phys

    C. Munna, Phys. Rev. D 102, 124001 (2020), arXiv:2008.10622 [gr-qc]

    2020 arXiv

  44. [52]

    Blanchet, Living Reviews in Relativity 17, 2 (2014), arXiv:1310.1528 [gr-qc]

    L. Blanchet, Living Reviews in Relativity 17, 2 (2014), arXiv:1310.1528 [gr-qc]

    2014 arXiv

  45. [53]

    Fujita, Progress of Theoretical Physics 128, 971 (2012), arXiv:1211.5535 [gr-qc]

    R. Fujita, Progress of Theoretical Physics 128, 971 (2012), arXiv:1211.5535 [gr-qc]

    2012 arXiv

  46. [54]

    Fujita, Prog

    R. Fujita, Prog. Theor. Exp. Phys. 2015 (2015), 10.1093/ptep/ptv012, arXiv:1412.5689 [gr-qc]

    2015 arXiv

  47. [55]

    NIST Digital Library of Mathematical Func- tions,

    DLMF, “NIST Digital Library of Mathematical Func- tions,” http://dlmf.nist.gov/, Release 1.0.10 of 2015-08- 07, online companion to [60]

    2015

  48. [56]

    Taracchini, A

    A. Taracchini, A. Buonanno, S. Hughes, and G. Khanna, Phys. Rev. D 88, 044001 (2013), arXiv:1305.2184 [gr-qc]

    2013 arXiv

  49. [57]

    A. G. Shah, Phys. Rev. D 90, 044025 (2014), arXiv:1403.2697 [gr-qc]

    2014 arXiv

  50. [58]

    N. K. Johnson-McDaniel, A. G. Shah, and B. F. Whit- ing, Phys. Rev. D 92, 044007 (2015), arXiv:1503.02638 [gr-qc]

    2015 arXiv

  51. [59]

    Munna and C

    C. Munna and C. R. Evans, Phys. Rev. D 100, 104060 (2019), arXiv:1909.05877 [gr-qc]

    2019 arXiv

  52. [60]

    Munna and C

    C. Munna and C. R. Evans, Phys. Rev. D 102, 104006 (2020), arXiv:2009.01254 [gr-qc]

    2020 arXiv

  53. [61]

    Teukolsky,

    B. Wardell, N. Warburton, K. Cunningham, L. Durkan, B. Leather, Z. Nasipak, C. Kavanagh, T. Torres, A. Ottewill, and M. Casals, “Teukolsky,” (2023), https://doi.org/10.5281/zenodo.10040501

    2023 doi

  54. [62]

    N. Sago, R. Fujita, and H. Nakano, arXiv e-prints , arXiv:2411.09147 (2024), arXiv:2411.09147 [gr-qc]

    2024 arXiv

  55. [63]

    L. C. Stein and N. Warburton, Phys. Rev. D 101, 064007 (2020)

    2020

  56. [64]

    F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions (Cambridge University Press, New York, NY, 2010) print companion to [51]

    2010

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.