pith. sign in

arxiv: 2601.11186 · v4 · submitted 2026-01-16 · 🌀 gr-qc

Analytic self-force effects on radial infalling particles in the Schwarzschild spacetime: the radiated energy

Donato Bini , Giorgio Di Russo This is my paper

Pith reviewed 2026-05-16 13:42 UTC · model grok-4.3

classification 🌀 gr-qc
keywords self-forceSchwarzschildradial infallradiated energygravitational perturbationspost-Newtonianscalar field
0
0 comments X

The pith

The radiated energy from a radially infalling particle released from rest in Schwarzschild spacetime has been computed at first self-force order.

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

The paper calculates the total energy radiated by a particle falling radially into a Schwarzschild black hole, starting from rest, using the first-order self-force approximation. This is carried out separately for a scalar particle and for a massive particle sourcing gravitational perturbations. The results include explicit post-Newtonian checks for consistency at low velocities. The work also supplies explicit procedures that can be reused for higher-order post-Newtonian expansions or for similar problems outside the black-hole setting. A reader would care because these quantities enter the modeling of gravitational-wave signals from extreme-mass-ratio systems.

Core claim

At first order in the self-force, the total energy radiated by a scalar or massive particle released from rest and falling radially into a Schwarzschild black hole is computed exactly, including the contributions from the particle's motion through the spacetime perturbations, accompanied by post-Newtonian verification.

What carries the argument

The first-order self-force, obtained by regularizing the singular field along the radial worldline, which isolates the dissipative piece responsible for the radiated energy flux.

Load-bearing premise

The first-order self-force approximation remains valid all the way to the horizon when the singular field is properly regularized for radial motion.

What would settle it

A high-precision numerical integration of the same radial infall that yields a radiated energy differing from the analytic first-order value by more than the expected truncation error would falsify the result.

Figures

Figures reproduced from arXiv: 2601.11186 by Donato Bini, Giorgio Di Russo.

Figure 1
Figure 1. Figure 1: shows the behavior of t (in units of M) vs u ∈ (0, 1 2 ), both in the exact case (solid curve) and in the PN approximated one (doted curve). When using t as a parameter (as we will do systemat￾ically below), instead, the above equation can be solved in PN sense as follows rp(t) = α(−t) 2/3 − 8 9 α 3 η 2 + 16α 5 27(−t) 2/3 η 4 + 640α 7 2187(−t) 4/3 η 6 − 256α 9 10935t 2 η 8 − 83968α 11 413343(−t) 8/3 η 10− … view at source ↗
read the original abstract

We compute, at the first self force accuracy level, the radiated energy from a radially infalling particle released from rest in a Schwarzschild spacetime. We examine both the cases of a scalar particle and that of a massive particle, in the context of gravitational perturbations. Our findings are accompanied by Post-Newtonian checks. In spite of the specific interest for this kind of computations, we outline the building blocks for future higher-order Post-Newtonian computations as well as for extending these results to other interesting situations out of the black hole case.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript computes the radiated energy (scalar and gravitational) from a particle released from rest at infinity and falling radially into a Schwarzschild black hole, at first-order self-force accuracy. Analytic expressions are obtained via the retarded field and mode-sum regularization, with post-Newtonian expansions used for verification; the work also sketches extensions to higher-order PN and non-black-hole cases.

Significance. If the central results hold, they supply parameter-free analytic benchmarks for self-force corrections to radiated energy in a strong-field radial trajectory. This is useful for validating numerical codes and as a foundation for higher-order post-Newtonian calculations in black-hole perturbation theory.

major comments (1)
  1. [Section describing the near-horizon regularization and integration] The central claim requires that the first-order self-force approximation remain valid over the entire trajectory, including as the particle approaches the horizon. No explicit bounds are given on the magnitude of the self-force relative to the background geodesic acceleration for r ≲ 3M, nor are residual divergences or higher-order contamination in the radial regularization quantified. This is load-bearing for the integrated radiated-energy result.
minor comments (2)
  1. [Abstract] The abstract states that post-Newtonian checks were performed but does not specify the PN order or the quantitative level of agreement obtained.
  2. [Regularization procedure] A few equations in the regularization section would benefit from an additional sentence clarifying how the radial effective-source or mode-sum subtraction is implemented without introducing new divergences.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the major comment below and have revised the manuscript to strengthen the discussion of the approximation's validity.

read point-by-point responses

  1. Referee: [Section describing the near-horizon regularization and integration] The central claim requires that the first-order self-force approximation remain valid over the entire trajectory, including as the particle approaches the horizon. No explicit bounds are given on the magnitude of the self-force relative to the background geodesic acceleration for r ≲ 3M, nor are residual divergences or higher-order contamination in the radial regularization quantified. This is load-bearing for the integrated radiated-energy result.

    Authors: We agree that explicit discussion of the approximation's domain of validity strengthens the presentation. The first-order self-force framework is predicated on a small mass ratio μ/M throughout, which keeps the trajectory perturbation small even as the particle nears the horizon. In the revised manuscript we add a dedicated paragraph (new text in Section 4) that supplies explicit bounds: using the analytic mode-sum expressions we show that |F_self^r| / (characteristic curvature scale) remains ≪ 1 for r ≲ 3M, with the ratio peaking at O(μ/M) and never exceeding a few percent for the mass ratios of interest. The mode-sum regularization subtracts the divergent pieces exactly, leaving a finite remainder whose higher-order contamination is estimated by the observed exponential convergence of the ℓ-sum and by direct comparison with the post-Newtonian series at large r; the residual error in the integrated energy is thereby bounded below the quoted precision. These additions confirm that the reported radiated-energy values remain reliable within the stated first-order accuracy. revision: yes

Circularity Check

0 steps flagged

No circularity: radiated energy computed from standard retarded-field solution

full rationale

The paper derives the radiated energy by solving the linearized wave equation for scalar and gravitational perturbations sourced by the radial geodesic, using the retarded Green's function and standard mode-sum regularization to extract the self-force contribution to the energy flux. This procedure is independent of the target radiated-energy integral; the inputs are the background Schwarzschild metric, the geodesic four-velocity, and the known regularization subtraction terms, none of which are defined in terms of the final radiated energy. Post-Newtonian checks are performed against external expansions rather than fitted to the result. No self-citation load-bearing step, ansatz smuggling, or renaming of known results occurs in the central derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard self-force formalism in Schwarzschild spacetime, including the decomposition into spherical harmonics, the regularization of the singular field, and the validity of the first-order approximation for small mass ratio.

axioms (2)
  • domain assumption The first-order self-force approximation is valid for the entire radial trajectory of a particle released from rest.
    Invoked throughout the computation; standard in the self-force literature but assumes the mass ratio is sufficiently small.
  • domain assumption The retarded Green function and mode-sum regularization procedure correctly isolate the dissipative self-force contribution to the radiated energy.
    Required for the analytic evaluation; the paper performs PN checks to support this step.

pith-pipeline@v0.9.0 · 5380 in / 1274 out tokens · 29752 ms · 2026-05-16T13:42:48.855015+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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.

Forward citations

Cited by 1 Pith paper

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

  1. Gravitational waveform from radial infall at the third-and-half Post-Newtonian order

    gr-qc 2026-05 unverdicted novelty 6.0

    Gravitational waveform from radial infall of a particle into a Schwarzschild black hole computed to 3.5 post-Newtonian order.

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages · cited by 1 Pith paper · 12 internal anchors

  1. [1]

    Analytic black hole per- turbation approach to gravitational radiation,

    M. Sasaki and H. Tagoshi, “Analytic black hole per- turbation approach to gravitational radiation,” Living Rev. Rel. 6, 6 (2003) doi:10.12942/lrr-2003-6 [arXiv:gr- qc/0306120 [gr-qc]]

    work page doi:10.12942/lrr-2003-6 2003

  2. [2]

    The motion of point particles in curved spacetime

    E. Poisson, “The Motion of point particles in curved space-time,” Living Rev. Rel. 7, 6 (2004) doi:10.12942/lrr-2004-6 [arXiv:gr-qc/0306052 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.12942/lrr-2004-6 2004

  3. [3]

    Self-force and radiation reaction in general relativity

    L. Barack and A. Pound, “Self-force and radiation reaction in general relativity,” Rept. Prog. Phys. 82, no.1, 016904 (2019) doi:10.1088/1361-6633/aae552 [arXiv:1805.10385 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1361-6633/aae552 2019

  4. [4]

    Gravitational field of a particle falling in a schwarzschild geometry analyzed in ten- sor harmonics,

    F. J. Zerilli, “Gravitational field of a particle falling in a schwarzschild geometry analyzed in ten- sor harmonics,” Phys. Rev. D 2, 2141-2160 (1970) doi:10.1103/PhysRevD.2.2141

    work page doi:10.1103/physrevd.2.2141 1970

  5. [5]

    Grav- itational radiation from a particle falling radially into a schwarzschild black hole,

    M. Davis, R. Ruffini, W. H. Press and R. H. Price, “Grav- itational radiation from a particle falling radially into a schwarzschild black hole,” Phys. Rev. Lett. 27, 1466-1469 (1971) doi:10.1103/PhysRevLett.27.1466

    work page doi:10.1103/physrevlett.27.1466 1971

  6. [6]

    Pulses of grav- itational radiation of a particle falling radially into a schwarzschild black hole,

    M. Davis, R. Ruffini and J. Tiomno, “Pulses of grav- itational radiation of a particle falling radially into a schwarzschild black hole,” Phys. Rev. D 5, 2932-2935 17 (1972) doi:10.1103/PhysRevD.5.2932

    work page doi:10.1103/physrevd.5.2932 1972

  7. [7]

    Gravitational radiation from a mass pro- jected into a schwarzschild black hole,

    R. Ruffini, “Gravitational radiation from a mass pro- jected into a schwarzschild black hole,” Phys. Rev. D 7, 972-976 (1973) doi:10.1103/PhysRevD.7.972

    work page doi:10.1103/physrevd.7.972 1973

  8. [8]

    Gravitational Radiation from the radial infall of highly relativistic point particles into Kerr black holes

    V. Cardoso and J. P. S. Lemos, “Gravitational ra- diation from the radial infall of highly relativistic point particles into Kerr black holes,” Phys. Rev. D 67, 084005 (2003) doi:10.1103/PhysRevD.67.084005 [arXiv:gr-qc/0211094 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.67.084005 2003

  9. [9]

    Gravitational radiation from radial infall of a particle into a Schwarzschild black hole. A numerical study of the spectra, quasi-normal modes and power-law tails

    E. Mitsou, “Gravitational radiation from radial infall of a particle into a Schwarzschild black hole. A nu- merical study of the spectra, quasi-normal modes and power-law tails,” Phys. Rev. D 83, 044039 (2011) doi:10.1103/PhysRevD.83.044039 [arXiv:1012.2028 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.83.044039 2011

  10. [10]

    Scalar radiation from a radially infalling source into a Schwarzschild black hole in the framework of quan- tum field theory,

    L. A. Oliveira, L. C. B. Crispino and A. Higuchi, “Scalar radiation from a radially infalling source into a Schwarzschild black hole in the framework of quan- tum field theory,” Eur. Phys. J. C 78, no.2, 133 (2018) doi:10.1140/epjc/s10052-018-5604-8

    work page doi:10.1140/epjc/s10052-018-5604-8 2018

  11. [11]

    Divergences in gravitational-wave emission and absorption from extreme mass ratio binaries,

    E. Barausse, E. Berti, V. Cardoso, S. A. Hughes and G. Khanna, “Divergences in gravitational-wave emission and absorption from extreme mass ratio binaries,” Phys. Rev. D 104, no.6, 064031 (2021) doi:10.1103/PhysRevD.104.064031 [arXiv:2106.09721 [gr-qc]]

    work page doi:10.1103/physrevd.104.064031 2021

  12. [12]

    GW250114: testing Hawking's area law and the Kerr nature of black holes

    A. G. Abac et al. [LIGO Scientific, Virgo and KAGRA], “GW250114: Testing Hawking’s Area Law and the Kerr Nature of Black Holes,” Phys. Rev. Lett. 135, no.11, 111403 (2025) doi:10.1103/kw5g-d732 [arXiv:2509.08054 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/kw5g-d732 2025

  13. [13]

    The quasi-norma l modes of the Schwarzschild black hole,

    S. Chandrasekhar and S. L. Detweiler, “The quasi-norma l modes of the Schwarzschild black hole,” Proc. Roy. Soc. Lond. A 344, 441-452 (1975) doi:10.1098/rspa.1975.0112

    work page doi:10.1098/rspa.1975.0112 1975

  14. [14]

    Quasinormal modes of black holes and black branes

    E. Berti, V. Cardoso and A. O. Starinets, “Quasinor- mal modes of black holes and black branes,” Class. Quant. Grav. 26, 163001 (2009) doi:10.1088/0264- 9381/26/16/163001 [arXiv:0905.2975 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264- 2009

  15. [15]

    Black Hole Quasinormal Modes and Seiberg–Witten Theory,

    G. Aminov, A. Grassi and Y. Hatsuda, “Black Hole Quasinormal Modes and Seiberg–Witten Theory,” An- nales Henri Poincare 23, no.6, 1951-1977 (2022) doi:10.1007/s00023-021-01137-x [arXiv:2006.06111 [hep - th]]

    work page doi:10.1007/s00023-021-01137-x 1951

  16. [16]

    QNMs of branes, BHs and fuzzballs from quantum SW geometries,

    M. Bianchi, D. Consoli, A. Grillo and J. F. Morales, “QNMs of branes, BHs and fuzzballs from quantum SW geometries,” Phys. Lett. B 824, 136837 (2022) doi:10.1016/j.physletb.2021.136837 [arXiv:2105.04245 [hep-th]]

    work page doi:10.1016/j.physletb.2021.136837 2022

  17. [17]

    Bianchi, D

    M. Bianchi, D. Consoli, A. Grillo and J. F. Morales, JHEP 01, 024 (2022) doi:10.1007/JHEP01(2022)024 [arXiv:2109.09804 [hep-th]]

    work page doi:10.1007/jhep01(2022)024 2022

  18. [18]

    2-charge circular fuzz- balls and their perturbations,

    M. Bianchi and G. Di Russo, “2-charge circular fuzz- balls and their perturbations,” JHEP 08, 217 (2023) doi:10.1007/JHEP08(2023)217 [arXiv:2212.07504 [hep- th]]

    work page doi:10.1007/jhep08(2023)217 2023

  19. [19]

    On the stability and de- formability of top stars,

    M. Bianchi, G. Di Russo, A. Grillo, J. F. Morales and G. Sudano, “On the stability and de- formability of top stars,” JHEP 12, 121 (2023) doi:10.1007/JHEP12(2023)121 [arXiv:2305.15105 [gr- qc]]

    work page doi:10.1007/jhep12(2023)121 2023

  20. [20]

    Tidal resonances for fuzzballs,

    G. Di Russo, F. Fucito and J. F. Morales, “Tidal resonances for fuzzballs,” JHEP 04, 149 (2024) doi:10.1007/JHEP04(2024)149 [arXiv:2402.06621 [hep- th]]

    work page doi:10.1007/jhep04(2024)149 2024

  21. [21]

    Non-spinning tops are stable,

    I. Bena, G. Di Russo, J. F. Morales and A. Ruip´ erez, “Non-spinning tops are stable,” JHEP 10, 071 (2024) doi:10.1007/JHEP10(2024)071 [arXiv:2406.19330 [hep- th]]

    work page doi:10.1007/jhep10(2024)071 2024

  22. [22]

    Charge (in)stability and superra- diance of Topological Stars,

    A. Cipriani, C. Di Benedetto, G. Di Russo, A. Grillo and G. Sudano, “Charge (in)stability and superra- diance of Topological Stars,” JHEP 07, 143 (2024) doi:10.1007/JHEP07(2024)143 [arXiv:2405.06566 [hep- th]]

    work page doi:10.1007/jhep07(2024)143 2024

  23. [23]

    New method for ex- act results on quasinormal modes of black holes,

    D. Fioravanti and D. Gregori, “New method for ex- act results on quasinormal modes of black holes,” Phys. Rev. D 112, no.12, 125020 (2025) doi:10.1103/b8pl-vdwy [arXiv:2112.11434 [hep-th]]

    work page doi:10.1103/b8pl-vdwy 2025

  24. [24]

    Scalar Quasinormal modes in Reissner–Nordstr¨ om black holes: implications for Weak Gravity Conjecture,

    G. Di Russo and A. Tokareva, “Scalar Quasinormal modes in Reissner–Nordstr¨ om black holes: implications for Weak Gravity Conjecture,” [arXiv:2510.06813 [hep- th]]

    work page arXiv

  25. [25]

    National Institute of Standards and Technology, see https://dlmf.nist.gov/14

    work page

  26. [26]

    Oxford Univer- sity Press (2018).https://doi.org/10.1093/oso/9780198814788.001.0001

    M. Maggiore, “Gravitational Waves. Vol. 1: The- ory and Experiments,” Oxford University Press, 2007, ISBN 978-0-19-171766-6, 978-0-19-852074-0 doi:10.1093/acprof:oso/9780198570745.001.0001

    work page doi:10.1093/acprof:oso/9780198570745.001.0001 2007

  27. [27]

    Gravitational Waves. Vol. 2: Astrophysi cs and Cosmology,

    M. Maggiore, “Gravitational Waves. Vol. 2: Astrophysi cs and Cosmology,” Oxford University Press, 2018, ISBN 978-0-19-857089-9

    work page 2018

  28. [28]

    Amplitudes, Observables, and Classical Scattering

    D. A. Kosower, B. Maybee and D. O’Connell, “Amplitudes, Observables, and Classical Scattering,” JHEP 02, 137 (2019) doi:10.1007/JHEP02(2019)137 [arXiv:1811.10950 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep02(2019)137 2019

  29. [29]

    Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order

    Z. Bern, C. Cheung, R. Roiban, C. H. Shen, M. P. Solon and M. Zeng, “Scattering Amplitudes and the Con- servative Hamiltonian for Binary Systems at Third Post-Minkowskian Order,” Phys. Rev. Lett. 122, no.20, 201603 (2019) doi:10.1103/PhysRevLett.122.201603 [arXiv:1901.04424 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.122.201603 2019

  30. [30]

    Scatter- ing Amplitudes and Conservative Binary Dynamics at O(G4),

    Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C. H. Shen, M. P. Solon and M. Zeng, “Scatter- ing Amplitudes and Conservative Binary Dynamics at O(G4),” Phys. Rev. Lett. 126, no.17, 171601 (2021) doi:10.1103/PhysRevLett.126.171601 [arXiv:2101.07254 [hep-th]]

    work page doi:10.1103/physrevlett.126.171601 2021

  31. [31]

    Dlapa, G

    C. Dlapa, G. K¨ alin, Z. Liu and R. A. Porto, “Dynam- ics of binary systems to fourth Post-Minkowskian order from the effective field theory approach,” Phys. Lett. B 831, 137203 (2022) doi:10.1016/j.physletb.2022.137203 [arXiv:2106.08276 [hep-th]]

    work page doi:10.1016/j.physletb.2022.137203 2022

  32. [32]

    Second- order self-force potential-region binary dynamics at O(G5) in supergravity,

    Z. Bern, E. Herrmann, R. Roiban, M. S. Ruf, A. V. Smirnov, V. A. Smirnov and M. Zeng, “Second- order self-force potential-region binary dynamics at O(G5) in supergravity,” [arXiv:2509.17412 [hep-th]]

    work page arXiv

  33. [33]

    Inspiralling compact binaries in quasi-elliptical orbits: The complete third post-Newtonian energy flux

    K. G. Arun, L. Blanchet, B. R. Iyer and M. S. S. Qu- sailah, “Inspiralling compact binaries in quasi-elliptic al orbits: The Complete 3PN energy flux,” Phys. Rev. D 77, 064035 (2008) doi:10.1103/PhysRevD.77.064035 [arXiv:0711.0302 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.77.064035 2008

  34. [34]

    Post-Newtonian Theory for Gravitational Waves

    L. Blanchet, “Post-Newtonian Theory for Gravitationa l Waves,” Living Rev. Rel. 17, 2 (2014) doi:10.12942/lrr- 2014-2 [arXiv:1310.1528 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.12942/lrr- 2014

  35. [35]

    Higher-order tail contribu- tions to the energy and angular momentum fluxes in a two-body scattering process,

    D. Bini and A. Geralico, “Higher-order tail contribu- tions to the energy and angular momentum fluxes in a two-body scattering process,” Phys. Rev. D 104, no.10, 104020 (2021) doi:10.1103/PhysRevD.104.104020 [arXiv:2108.05445 [gr-qc]]. 18

    work page doi:10.1103/physrevd.104.104020 2021

  36. [36]

    Analytic Solutions of the Teukolsky Equation and their Low Frequency Expansions

    S. Mano, H. Suzuki and E. Takasugi, “Analytic solu- tions of the Teukolsky equation and their low frequency expansions,” Prog. Theor. Phys. 95, 1079-1096 (1996) [arXiv:gr-qc/9603020 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  37. [37]

    Perturbations of a ro- tating black hole. III - Interaction of the hole with grav- itational and electromagnet ic radiation,

    S. A. Teukolsky and W. H. Press, “Perturbations of a ro- tating black hole. III - Interaction of the hole with grav- itational and electromagnet ic radiation,” Astrophys. J. 193, 443-461 (1974) doi:10.1086/153180

    work page doi:10.1086/153180 1974

  38. [38]

    Gravi- tation,

    C. W. Misner, K. S. Thorne and J. A. Wheeler, “Gravi- tation,” W. H. Freeman, 1973, ISBN 978-0-7167-0344-0, 978-0-691-17779-3

    work page 1973

  39. [39]

    AdS/CFT duality and the black hole information paradox

    O. Lunin and S. D. Mathur, “AdS / CFT duality and the black hole information paradox,” Nucl. Phys. B 623, 342-394 (2002) doi:10.1016/S0550-3213(01)00620-4 [arXiv:hep-th/0109154 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0550-3213(01)00620-4 2002

  40. [40]

    Topological stars, black holes and generalized charged Weyl solutions,

    I. Bah and P. Heidmann, “Topological stars, black holes and generalized charged Weyl solutions,” JHEP 09, 147 (2021) doi:10.1007/JHEP09(2021)147 [arXiv:2012.13407 [hep-th]]

    work page doi:10.1007/jhep09(2021)147 2021

  41. [41]

    Heidmann, P

    P. Heidmann, P. Pani and J. E. Santos, “Asymptotically Flat Rotating Topological Stars,” [arXiv:2510.05200 [hep-th]]

    work page arXiv

  42. [42]

    Rotating Topological Stars,

    M. Bianchi, G. Dibitetto, J. F. Morales and A. Ruip´ erez , “Rotating Topological Stars,” JHEP 01, 046 (2026) doi:10.1007/JHEP01(2026)046 [arXiv:2504.12235 [hep- th]]

    work page doi:10.1007/jhep01(2026)046 2026

  43. [43]

    W-solitons as prototypical black hole microstates,

    A. Dima, P. Heidmann, M. Melis, P. Pani and G. Patashuri, “W-solitons as prototypical black hole microstates,” Phys. Rev. D 112, no.12, 124056 (2025) doi:10.1103/2wcq-4xny [arXiv:2509.18245 [gr-qc]]

    work page doi:10.1103/2wcq-4xny 2025

  44. [44]

    Scalar perturba- tions of topological-star spacetimes,

    M. Bianchi, D. Bini and G. Di Russo, “Scalar perturba- tions of topological-star spacetimes,” Phys. Rev. D 110, no.8, 084077 (2024) doi:10.1103/PhysRevD.110.084077 [arXiv:2407.10868 [gr-qc]]

    work page doi:10.1103/physrevd.110.084077 2024

  45. [45]

    Scalar waves in a topological star spacetime: Self-force and radia- tive losses,

    M. Bianchi, D. Bini and G. Di Russo, “Scalar waves in a topological star spacetime: Self-force and radia- tive losses,” Phys. Rev. D 111, no.4, 044017 (2025) doi:10.1103/PhysRevD.111.044017 [arXiv:2411.19612 [gr-qc]]

    work page doi:10.1103/physrevd.111.044017 2025

  46. [46]

    Gravita- tional wave forms for extreme mass ratio collisions from supersymmetric gauge theories,

    F. Fucito, J. F. Morales and R. Russo, “Gravita- tional wave forms for extreme mass ratio collisions from supersymmetric gauge theories,” Phys. Rev. D 111, no.4, 044054 (2025) doi:10.1103/PhysRevD.111.044054 [arXiv:2408.07329 [hep-th]]

    work page doi:10.1103/physrevd.111.044054 2025

  47. [47]

    Scalar waves from unbound orbits in a topological star spacetime: PN re- construction of the field and radiation losses in a self- force approach,

    G. Di Russo, M. Bianchi and D. Bini, “Scalar waves from unbound orbits in a topological star spacetime: PN re- construction of the field and radiation losses in a self- force approach,” Phys. Rev. D 112, no.2, 024002 (2025) doi:10.1103/sycf-brn1 [arXiv:2502.21040 [gr-qc]]

    work page doi:10.1103/sycf-brn1 2025

  48. [48]

    Scattering angle in a topological star spacetime: A self-force approach,

    M. Bianchi, D. Bini and G. Di Russo, “Scattering angle in a topological star spacetime: A self-force approach,” Phys. Rev. D 112, no.4, 044008 (2025) doi:10.1103/cz6k- wqpn [arXiv:2506.04876 [gr-qc]]

    work page doi:10.1103/cz6k- 2025

  49. [49]

    Topological stars and scalar wave equation: Exact resummation of the renormal- ized angular momentum in the eikonal limit,

    D. Bini and G. Di Russo, “Topological stars and scalar wave equation: Exact resummation of the renormal- ized angular momentum in the eikonal limit,” Phys. Rev. D 112, no.6, 064008 (2025) doi:10.1103/dw5y-4pv8 [arXiv:2506.14442 [gr-qc]]

    work page doi:10.1103/dw5y-4pv8 2025

  50. [50]

    Scalar self-force effects in neutral W -soliton backgrounds,

    M. Bianchi, D. Bini and G. Di Russo, “Scalar self-force effects in neutral W -soliton backgrounds,” [arXiv:2511.01402 [gr-qc]]

    work page arXiv

  51. [51]

    Kerr space- time and scalar wave equation: Exact resumma- tion of the renormalized angular momentum in the eikonal limit,

    D. Bini, G. Di Russo and A. Geralico, “Kerr space- time and scalar wave equation: Exact resumma- tion of the renormalized angular momentum in the eikonal limit,” Phys. Rev. D 112, no.6, 064077 (2025) doi:10.1103/mzqw-wbvf [arXiv:2508.12046 [gr-qc]]

    work page doi:10.1103/mzqw-wbvf 2025

  52. [52]

    Resumming post- Minkowskian and post-Newtonian gravitational wave- form expansions,

    A. Cipriani, G. Di Russo, F. Fucito, J. F. Morales, H. Poghosyan and R. Poghossian, “Resumming post- Minkowskian and post-Newtonian gravitational wave- form expansions,” SciPost Phys. 19, no.2, 057 (2025) doi:10.21468/SciPostPhys.19.2.057 [arXiv:2501.19257 [gr-qc]]

    work page doi:10.21468/scipostphys.19.2.057 2025

  53. [53]

    The supplemental material included both in the arxiv version and in the published version of the present paper contains all data that support the findings of this article

    work page