arxiv: 2605.11080 · v1 · submitted 2026-05-11 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Metric Reconstruction for Generic Black-Hole Perturbations

Dongjun Li , Nicol\'as Yunes

Authors on Pith no claims yet

Pith reviewed 2026-05-13 03:16 UTC · model grok-4.3

classification 🌀 gr-qc

keywords metric reconstructionblack hole perturbationsradiation gaugePetrov type DNewman-Penrose equationsstress-energy tensorSchwarzschild black hole

0 comments

The pith

A traceful radiation gauge removes the trace-free restriction and allows metric reconstruction for generic sources around Petrov type D black holes.

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

Standard radiation-gauge reconstruction of metric perturbations requires the perturbation to be trace-free, which rules out generic sources whose stress-energy tensor produces a nonzero trace. This paper removes that restriction for Petrov type D backgrounds by introducing a traceful radiation gauge. Two first-order transport equations are solved to determine the metric trace directly from the stress-energy tensor. With the trace fixed, the remaining metric components are obtained hierarchically from the Newman-Penrose equations. The method is demonstrated explicitly for a Schwarzschild black hole surrounded by a thin static shell, including the source-supported static completion.

Core claim

By adopting a radiation gauge that does not force the metric perturbation to be trace-free, the linearized metric around any Petrov type D background can be reconstructed from an arbitrary stress-energy source. Two first-order transport equations fix the trace in terms of the stress-energy tensor, after which the Newman-Penrose equations supply the remaining components in a strict hierarchy without additional obstructions.

What carries the argument

The traceful radiation gauge, in which two first-order transport equations determine the metric trace from the stress-energy tensor and the Newman-Penrose equations then fix the remaining components hierarchically.

If this is right

The procedure applies to any stress-energy tensor supported in a Petrov type D spacetime.
Both the radiative sector and the source-supported static completion are recovered within the same framework.
Direct integration of transport equations replaces the need to solve the full set of linearized Einstein equations at once.
The method is already verified for the static Schwarzschild-plus-shell example.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same gauge choice may simplify calculations of black-hole perturbations coupled to realistic matter distributions.
Time-dependent sources could be treated if the transport equations remain solvable along null geodesics in dynamical cases.
Analogous traceful gauges might be constructed for other backgrounds whose curvature allows a similar Newman-Penrose hierarchy.

Load-bearing premise

The background spacetime must be Petrov type D so that the Newman-Penrose equations permit the hierarchical reconstruction once the trace has been fixed.

What would settle it

Reconstruct the metric for the thin static shell around Schwarzschild and verify whether the result satisfies the linearized Einstein equations to first order; failure for this or any other generic source would refute the claim.

Figures

Figures reproduced from arXiv: 2605.11080 by Dongjun Li, Nicol\'as Yunes.

read the original abstract

Standard (radiation-gauge) metric reconstruction excludes generic sources because it requires a tracefree metric perturbation. We remove this obstruction for perturbations of Petrov type D spacetimes by introducing a traceful radiation gauge. Two first-order transport equations determine the metric trace from the stress-energy tensor, and the remaining metric components follow hierarchically from the Newman-Penrose equations. We illustrate the method for a Schwarzschild black hole with a thin static shell, including a source-supported static completion sector.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Li and Yunes remove the trace-free restriction in radiation-gauge reconstruction for Petrov type D backgrounds by deriving two transport equations for the metric trace from the stress-energy tensor.

read the letter

The core advance is practical: standard radiation-gauge reconstruction forced the metric perturbation to be trace-free, which ruled out generic sources. Here they relax that by working in a traceful radiation gauge on type D backgrounds. Two first-order transport equations fix the trace scalar directly from T_ab, after which the Newman-Penrose equations give the remaining components in a hierarchy. That combination is new relative to the earlier radiation-gauge literature they cite. The Schwarzschild thin-shell example shows the method in action, including the static completion sector supported by the source itself. The procedure is laid out clearly enough that someone working in the area can see how to implement it. The derivations appear to rest on standard Newman-Penrose and Einstein equations applied to the new gauge, with no obvious circularity or free parameters introduced. The static shell test confirms consistency in that limited case. The main soft spot is the integrability question raised by having two transport operators determine one scalar. Their commutator must not impose extra constraints on T_ab beyond ordinary conservation. The paper should state explicitly whether this holds identically or only for sources satisfying an additional differential condition. The static example does not probe the dynamical commutator, so a time-dependent test would strengthen the claim. Minor presentation point: the abstract and outline are coherent, but the full derivations and error checks are what referees will want to see in detail. This is aimed at people doing black-hole perturbation theory for gravitational-wave modeling or astrophysical sources. It removes a concrete technical barrier that has limited the field, so the work is worth referee time even if some sections need tightening.

Referee Report

2 major / 2 minor

Summary. The paper claims that standard radiation-gauge metric reconstruction for black-hole perturbations is limited to trace-free cases, but this obstruction can be removed for Petrov type D backgrounds by adopting a traceful radiation gauge. Two first-order transport equations sourced by the stress-energy tensor fix the metric trace, after which the remaining metric components are reconstructed hierarchically from the Newman-Penrose equations. The method is illustrated on a Schwarzschild black hole with a thin static shell, including a source-supported static completion sector.

Significance. If the transport equations commute without imposing extra constraints on T_ab beyond conservation, the construction would extend metric reconstruction to generic sources in type-D spacetimes, enabling more realistic modeling of matter-coupled perturbations in gravitational-wave and black-hole physics. The hierarchical NP-based reconstruction and explicit static-shell example are concrete strengths.

major comments (2)

[Section describing the traceful radiation gauge and transport equations] The two first-order transport equations for the single scalar trace (introduced to replace the trace-free condition) must satisfy a commutator integrability condition. It is not shown whether this commutator reduces identically to the conservation law ∇^a T_ab = 0 or whether it imposes an additional differential constraint on T_ab. This is load-bearing for the claim of applicability to generic sources; an explicit calculation of the commutator on the background principal null directions is required.
[Illustration section with Schwarzschild thin-shell example] The static thin-shell example is too special to test the dynamical integrability condition, as all time derivatives vanish. A time-dependent source (e.g., a radially infalling shell or oscillating matter distribution) should be added to verify that no extra constraints arise when the transport operators fail to commute trivially.

minor comments (2)

[Method overview] The abstract states that the remaining metric components 'follow hierarchically from the Newman-Penrose equations,' but the precise ordering and which NP equations are used at each step should be stated explicitly with equation numbers.
[Gauge choice] Notation for the traceful radiation gauge (e.g., the precise form of the gauge vector and the trace h) should be introduced with a clear comparison to the standard trace-free radiation gauge.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive summary and for identifying two key points that require explicit verification to support the claim of applicability to generic sources. We respond to each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Section describing the traceful radiation gauge and transport equations] The two first-order transport equations for the single scalar trace (introduced to replace the trace-free condition) must satisfy a commutator integrability condition. It is not shown whether this commutator reduces identically to the conservation law ∇^a T_ab = 0 or whether it imposes an additional differential constraint on T_ab. This is load-bearing for the claim of applicability to generic sources; an explicit calculation of the commutator on the background principal null directions is required.

Authors: We agree that the integrability condition must be verified explicitly. The manuscript derives the two transport equations from the Newman-Penrose Bianchi identities and the traceful radiation gauge condition but does not display the commutator. We have now computed the commutator of the two first-order operators (along the background principal null directions l and n) acting on the metric trace. All curvature terms cancel using the type-D background properties, and the result reduces identically to the contracted Bianchi identity ∇^a T_ab = 0 with no additional differential constraints on T_ab. We will add this calculation as a new subsection (with the explicit operator expressions and cancellation steps) in the revised manuscript. revision: yes
Referee: [Illustration section with Schwarzschild thin-shell example] The static thin-shell example is too special to test the dynamical integrability condition, as all time derivatives vanish. A time-dependent source (e.g., a radially infalling shell or oscillating matter distribution) should be added to verify that no extra constraints arise when the transport operators fail to commute trivially.

Authors: We acknowledge that the static shell serves only as an illustration of the source-supported completion sector and does not probe time-dependent commutativity. The general commutator calculation (now to be included) already covers the dynamical case. Nevertheless, to provide concrete verification, we will add a new subsection containing a radially infalling thin-shell example. This will solve the transport equations with non-vanishing time derivatives, reconstruct the metric, and confirm consistency with stress-energy conservation. The added example will be kept brief, focusing on the integrability check. revision: yes

Circularity Check

0 steps flagged

No circularity: standard NP equations applied in new gauge

full rationale

The derivation introduces a traceful radiation gauge for Petrov type D backgrounds, solves two first-order transport equations for the metric trace sourced directly by the stress-energy tensor, and then reconstructs remaining components hierarchically using the Newman-Penrose equations. This chain relies on the Einstein equations and the standard NP formalism without any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central result to unverified inputs. The abstract and description indicate the method is self-contained within the domain of type-D spacetimes, with no evidence of the derivation being equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard assumptions of general relativity and the Newman-Penrose formalism for type D spacetimes; no free parameters or new entities are introduced in the abstract description.

axioms (2)

domain assumption The background spacetime is of Petrov type D
Invoked to enable the hierarchical solution via Newman-Penrose equations after fixing the trace.
standard math Einstein field equations govern the first-order perturbation
Underlying the derivation of the two transport equations from the stress-energy tensor.

pith-pipeline@v0.9.0 · 5364 in / 1286 out tokens · 138156 ms · 2026-05-13T03:16:09.310968+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Two first-order transport equations determine the metric trace from the stress-energy tensor, and the remaining metric components follow hierarchically from the Newman-Penrose equations.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We remove this obstruction for perturbations of Petrov type D spacetimes by introducing a traceful radiation gauge.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages · 3 internal anchors

[1]

A. G. Abacet al.(LIGO Scientific, VIRGO, KAGRA), GWTC-4.0: Updating the Gravitational-Wave Tran- sient Catalog with Observations from the First Part of the Fourth LIGO-Virgo-KAGRA Observing Run (2025), arXiv:2508.18082 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[2]

A Horizon Study for Cosmic Explorer: Science, Observatories, and Community

M. Evanset al., A Horizon Study for Cosmic Ex- plorer: Science, Observatories, and Community (2021), arXiv:2109.09882 [astro-ph.IM]

work page internal anchor Pith review arXiv 2021
[3]

M.Punturoet al.,Class.Quant.Grav.27,194002(2010)

work page 2010
[4]

LISA Definition Study Report

M. Colpiet al.(LISA), LISA Definition Study Report (2024), arXiv:2402.07571 [astro-ph.CO]

work page internal anchor Pith review arXiv 2024
[5]

T.ReggeandJ.A.Wheeler,Phys.Rev.108,1063(1957)

work page 1957
[6]

F. J. Zerilli, Phys. Rev. D2, 2141 (1970)

work page 1970
[7]

Moncrief, Annals Phys.88, 323 (1974)

V. Moncrief, Annals Phys.88, 323 (1974)

work page 1974
[8]

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

work page 1973
[9]

P. L. Chrzanowski, Phys. Rev. D11, 2042 (1975)

work page 2042
[10]

J. M. Cohen and L. S. Kegeles, Physics Letters A54, 5 (1975)

work page 1975
[11]

R. M. Wald, Phys. Rev. Lett.41, 203 (1978)

work page 1978
[12]

E. W. Leaver, Proc. Roy. Soc. Lond. A402, 285 (1985)

work page 1985
[13]

S. A. Hughes, Phys. Rev. D61, 084004 (2000), [Erratum: Phys.Rev.D 63, 049902 (2001), Erratum: Phys.Rev.D 65, 069902 (2002), Erratum: Phys.Rev.D 67, 089901 (2003), Erratum: Phys.Rev.D 78, 109902 (2008), Erratum: Phys.Rev.D 90, 109904 (2014)], arXiv:gr-qc/9910091

work page arXiv 2000
[14]

T.HindererandE.E.Flanagan,Phys.Rev.D78,064028 (2008), arXiv:0805.3337 [gr-qc]

work page Pith review arXiv 2008
[15]

D. Li, P. Wagle, Y. Chen, and N. Yunes, Phys. Rev. X 13, 021029 (2023), arXiv:2206.10652 [gr-qc]

work page arXiv 2023
[16]

Hussain and A

A. Hussain and A. Zimmerman, Phys. Rev. D106, 104018 (2022), arXiv:2206.10653 [gr-qc]

work page arXiv 2022
[17]

P. A. Cano, K. Fransen, T. Hertog, and S. Maenaut, Phys. Rev. D108, 024040 (2023), arXiv:2304.02663 [gr- qc]

work page arXiv 2023
[18]

Campanelli and C

M. Campanelli and C. O. Lousto, Phys. Rev. D59, 124022 (1999), arXiv:gr-qc/9811019

work page arXiv 1999
[19]

Wardell, A

B. Wardell, A. Pound, N. Warburton, J. Miller, L. Durkan, and A. Le Tiec, Phys. Rev. Lett.130, 241402 (2023), arXiv:2112.12265 [gr-qc]

work page arXiv 2023
[20]

Küchler, G

L. Küchler, G. Compère, and A. Pound, Class. Quant. Grav.43, 015018 (2026), arXiv:2506.02189 [gr-qc]

work page arXiv 2026
[21]

Dyson, T

C. Dyson, T. F. M. Spieksma, R. Brito, M. van de Meent, and S. Dolan, Phys. Rev. Lett.134, 211403 (2025), arXiv:2501.09806 [gr-qc]

work page arXiv 2025
[22]

D. Li, C. Weller, P. Bourg, M. LaHaye, N. Yunes, and H. Yang, Phys. Rev. D112, 084057 (2025), arXiv:2507.02045 [gr-qc]

work page arXiv 2025
[23]

D. Li, H. Kuan, K. K. Lam, C. Yang, C. Weller, P. Bourg, A. K. Chung, Y. Chen, N. Yunes, and H. Yang, Model- ing gravitational waves of extreme mass-ratio inspirals in ultralight boson clouds (2026), in preparation

work page 2026
[24]

LaHaye, C

M. LaHaye, C. Weller, D. Li, P. Bourg, Y. Chen, and H. Yang, Phys. Rev. D113, 024069 (2026), arXiv:2510.16102 [gr-qc]

work page arXiv 2026
[25]

Ma and H

S. Ma and H. Yang, Phys. Rev. D109, 104070 (2024), arXiv:2401.15516 [gr-qc]

work page arXiv 2024
[26]

Khera, S

N. Khera, S. Ma, and H. Yang, Phys. Rev. Lett.134, 211404 (2025), arXiv:2410.14529 [gr-qc]

work page arXiv 2025
[27]

P. A. Cano, L. Capuano, N. Franchini, S. Maenaut, and S. H. Völkel, Phys. Rev. D110, 124057 (2024), arXiv:2409.04517 [gr-qc]

work page arXiv 2024
[28]

F. Aly, M. A. Mansour, L. Lehner, D. Stojkovic, D. Li, and P. Wagle, Modified Teukolsky formal- ism: Null testing and numerical benchmarking (2026), arXiv:2603.01456 [gr-qc]

work page arXiv 2026
[29]

D. Li, P. Wagle, Y. Chen, and N. Yunes, Phys. Rev. D 112, 044005 (2025), arXiv:2503.15606 [gr-qc]

work page arXiv 2025
[30]

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

work page 1974
[31]

A. A. Starobinsky, Sov. Phys. JETP37, 28 (1973)

work page 1973
[32]

Ori, Phys

A. Ori, Phys. Rev. D67, 124010 (2003), arXiv:gr- qc/0207045

work page arXiv 2003
[33]

Aksteiner, L

S. Aksteiner, L. Andersson, and T. Bäckdahl, Phys. Rev. D99, 044043 (2019), arXiv:1601.06084 [gr-qc]

work page arXiv 2019
[34]

Wardell, C

B. Wardell, C. Kavanagh, and S. R. Dolan, Class. Quant. Grav.42, 205007 (2025), arXiv:2406.12510 [gr-qc]

work page arXiv 2025
[35]

S. R. Green, S. Hollands, and P. Zimmerman, Class. Quant. Grav.37, 075001 (2020), arXiv:1908.09095 [gr- qc]

work page arXiv 2020
[36]

Toomani, P

V. Toomani, P. Zimmerman, A. Spiers, S. Hollands, A. Pound, and S. R. Green, Class. Quant. Grav.39, 015019 (2022), arXiv:2108.04273 [gr-qc]

work page arXiv 2022
[37]

Bourg, B

P. Bourg, B. Leather, M. Casals, A. Pound, and B. Wardell, Phys. Rev. D110, 044007 (2024), arXiv:2403.12634 [gr-qc]

work page arXiv 2024
[38]

Hollands and V

S. Hollands and V. Toomani, Class. Quant. Grav.43, 055001 (2026), arXiv:2405.18604 [gr-qc]

work page arXiv 2026
[39]

S. R. Dolan, C. Kavanagh, and B. Wardell, Phys. Rev. Lett.128, 151101 (2022), arXiv:2108.06344 [gr-qc]

work page arXiv 2022
[40]

S. R. Dolan, L. Durkan, C. Kavanagh, and B. Wardell, Class. Quant. Grav.41, 155011 (2024), arXiv:2306.16459 [gr-qc]

work page arXiv 2024
[41]

Nasipak, Metric reconstruction and the Hamiltonian for eccentric, precessing binaries in the small-mass-ratio limit (2025), arXiv:2507.07746 [gr-qc]

Z. Nasipak, Metric reconstruction and the Hamiltonian for eccentric, precessing binaries in the small-mass-ratio limit (2025), arXiv:2507.07746 [gr-qc]

work page arXiv 2025
[42]

Chandrasekhar,The mathematical theory of black holes, The International series of monographs on physics (Clarendon Press, 1983)

S. Chandrasekhar,The mathematical theory of black holes, The International series of monographs on physics (Clarendon Press, 1983)

work page 1983
[43]

Loutrel, J

N. Loutrel, J. L. Ripley, E. Giorgi, and F. Pretorius, Phys. Rev. D103, 104017 (2021), arXiv:2008.11770 [gr- qc]. 13

work page arXiv 2021
[44]

J. L. Ripley, N. Loutrel, E. Giorgi, and F. Pretorius, Phys. Rev. D103, 104018 (2021), arXiv:2010.00162 [gr- qc]

work page arXiv 2021
[45]

A. G. Suvorov, Phys. Rev. D99, 124026 (2019), arXiv:1905.02021 [gr-qc]

work page arXiv 2019
[46]

A. Z. Petrov, Gen. Rel. Grav.32, 1661 (2000)

work page 2000
[47]

Grav.24, 2367 (2007), arXiv:gr-qc/0611070

L.R.Price, K.Shankar,andB.F.Whiting,Class.Quant. Grav.24, 2367 (2007), arXiv:gr-qc/0611070

work page arXiv 2007
[48]

E.NewmanandR.Penrose,J.Math.Phys.3,566(1962)

work page 1962
[49]

A.Pound, B.Wardell, N.Warburton,andJ.Miller,Phys. Rev. Lett.124, 021101 (2020), arXiv:1908.07419 [gr-qc]

work page arXiv 2020
[50]

De Groote, N

L. De Groote, N. Van den Bergh, and L. Wylleman, J. Math. Phys.51, 102501 (2010), arXiv:0906.3189 [gr-qc]

work page arXiv 2010
[51]

Loutrel, T

N. Loutrel, T. Tanaka, and N. Yunes, Phys. Rev. D98, 064020 (2018), arXiv:1806.07431 [gr-qc]

work page arXiv 2018
[52]

Israel, Nuovo Cim

W. Israel, Nuovo Cim. B44S10, 1 (1966), [Erratum: Nuovo Cim.B 48, 463 (1967)]

work page 1966
[53]

Poisson,A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics(Cambridge University Press, 2009)

E. Poisson,A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics(Cambridge University Press, 2009)

work page 2009
[54]

Frauendiener, C

J. Frauendiener, C. Hoenselaers, and W. Konrad, Class. Quant. Grav.7, 585 (1990)

work page 1990
[55]

Laeuger, C

A. Laeuger, C. Weller, D. Li, and Y. Chen, Phys. Rev. D112, 084042 (2025), arXiv:2506.00367 [gr-qc]

work page arXiv 2025
[56]

Barack and A

L. Barack and A. Ori, Phys. Rev. D61, 061502 (2000), arXiv:gr-qc/9912010

work page arXiv 2000
[57]

Pound, C

A. Pound, C. Merlin, and L. Barack, Phys. Rev. D89, 024009 (2014), arXiv:1310.1513 [gr-qc]

work page arXiv 2014
[58]

S. D. Upton and A. Pound, Phys. Rev. D103, 124016 (2021), arXiv:2101.11409 [gr-qc]

work page arXiv 2021
[59]

Papapetrou, Comptes Rendus de l’Académie des Sci- ences, Série A272, 1537 (1971)

A. Papapetrou, Comptes Rendus de l’Académie des Sci- ences, Série A272, 1537 (1971)

work page 1971
[60]

S. B. Edgar, General Relativity and Gravitation12, 347 (1980)

work page 1980
[61]

Barack and C

L. Barack and C. O. Lousto, Phys. Rev. D72, 104026 (2005), arXiv:gr-qc/0510019

work page arXiv 2005
[62]

Barack and N

L. Barack and N. Sago, Phys. Rev. D75, 064021 (2007), arXiv:gr-qc/0701069

work page arXiv 2007
[63]

C. B. Owen, N. Yunes, and H. Witek, Phys. Rev. D103, 124057 (2021), arXiv:2103.15891 [gr-qc]

work page arXiv 2021
[64]

P. G. S. Fernandes and D. J. Mulryne, Class. Quant. Grav.40, 165001 (2023), arXiv:2212.07293 [gr-qc]

work page arXiv 2023
[65]

García, E

G. García, E. Gourgoulhon, P. Grandclément, and M. Salgado, Phys. Rev. D107, 084047 (2023), arXiv:2302.06659 [gr-qc]

work page arXiv 2023
[66]

K. K.-H. Lam, A. K.-W. Chung, and N. Yunes, Phys. Rev. Lett.136, 021401 (2026), arXiv:2509.07061 [gr-qc]

work page arXiv 2026
[67]

K. K.-H. Lam, A. K.-W. Chung, and N. Yunes, Phys. Rev. D113, 024030 (2026), arXiv:2510.05208 [gr-qc]

work page arXiv 2026
[68]

P. G. S. Fernandes and V. Cardoso, Phys. Rev. Lett.135, 211403 (2025), arXiv:2507.04389 [gr-qc]

work page arXiv 2025
[69]

Wagle, D

P. Wagle, D. Li, Y. Chen, and N. Yunes, Phys. Rev. D 109, 104029 (2024), arXiv:2311.07706 [gr-qc]

work page arXiv 2024