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arxiv: 2603.07714 · v2 · submitted 2026-03-08 · 🌀 gr-qc · astro-ph.HE· hep-th

Recognition: no theorem link

Scattering from compact objects: Debye series and Regge-Debye poles

Mohamed Ould El Hadj
Authors on Pith no claims yet

Pith reviewed 2026-05-15 14:32 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-th
keywords Debye seriesRegge polesscattering amplitudecompact starsSchwarzschild spacetimescalar wavespole spectruminterior transmission
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The pith

An exact Debye-series decomposition of the scattering matrix separates direct surface reflection from interior transmission contributions for waves scattering off compact stars.

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

The paper develops a Debye-series expansion for the scattering of massless scalar waves by a uniform-density star in Schwarzschild spacetime. This expansion breaks the scattering amplitude into terms corresponding to direct reflection at the surface and terms that involve transmission into the star's interior followed by multiple internal reflections or propagations. By identifying the poles in the complex angular momentum plane associated with these contributions, the work shows how different families of Regge-Debye poles dominate the scattering in neutron-star-like and ultracompact regimes. Reconstructing the amplitude from these Debye terms matches direct calculations, revealing that rainbow enhancements at high frequency arise from interior resonances in one regime but are pole-dominated in the other. This approach provides a trajectory-based interpretation useful for understanding wave scattering in strong gravity.

Core claim

The central claim is that an exact Debye-series decomposition of the scattering matrix for a uniform-density star separates surface reflection from interior transmission terms, and the associated Regge-Debye poles explain the scattering amplitude with two pole families for R>3M and split branches for R<3M, leading to pole-dominated amplitudes in the ultracompact case.

What carries the argument

The Debye-series decomposition of the scattering matrix, which isolates direct reflection and multiple interior transmission contributions, together with the Regge-Debye poles in the complex angular-momentum plane that encode surface waves and interior resonances.

If this is right

  • The scattering amplitude can be reconstructed order by order from Debye contributions with high accuracy.
  • In the neutron-star regime, rainbow enhancements at high frequency arise from the first interior-transmission term dominated by interior-resonance poles.
  • In the ultracompact regime, the amplitudes are overwhelmingly pole dominated.
  • Different pole branches appear: surface-wave and interior-resonance for larger radii, with splitting for smaller radii.

Where Pith is reading between the lines

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

  • This decomposition could extend to other fields or potentials beyond the scalar case, potentially unifying descriptions of scattering in black hole and star spacetimes.
  • High-frequency scattering features like rainbows might be observable in gravitational wave echoes from compact objects if similar decompositions apply to tensor perturbations.
  • The trajectory interpretation suggests a semiclassical picture where rays bounce inside the star, which could link to quasinormal mode calculations.
  • Testing the pole spectrum numerically for specific compactness values would confirm the branch splitting at R=3M.

Load-bearing premise

The star has a uniform density interior matched continuously to a Schwarzschild exterior, with regularity at the center.

What would settle it

A direct computation of the scattering amplitude for a specific frequency and impact parameter that deviates significantly from the sum of the first few Debye terms plus their Regge poles.

Figures

Figures reproduced from arXiv: 2603.07714 by Mohamed Ould El Hadj.

Figure 1
Figure 1. Figure 1: FIG. 1. Regge–Debye pole spectrum for the massless scalar [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Regge–Debye pole spectrum for the massless scalar [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Differential scattering cross section for a neutron-star-like compact body with [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Differential scattering cross section for an ultracompact object with [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Sommerfeld–Watson contour [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. CAM reconstruction of Debye contributions for a neutron-star-like compact body ( [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. CAM reconstruction of Debye contributions for a neutron-star-like compact body ( [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. CAM reconstruction of the Debye contributions for an ultracompact object ( [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. CAM reconstruction of the Debye contributions for an ultracompact object ( [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Role of the broad-resonance Regge–Debye poles in [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
read the original abstract

We investigate elastic scattering by a compact, horizonless body in curved spacetime, considering a massless scalar wave incident on a static, spherically symmetric, uniform-density star of radius $R$ and mass $M$ with a Schwarzschild exterior. We introduce an exact Debye-series decomposition of the scattering matrix, in the spirit of Debye expansions in Mie scattering. This decomposition separates direct surface reflection from contributions involving transmission into the interior and subsequent propagation, and admits a natural trajectory interpretation. We then determine the associated Regge-Debye pole spectrum in the complex angular-momentum plane. For neutron-star-like tenuities ($R>3M$), the spectrum exhibits two pole families: a surface-wave branch associated with the surface matching condition and a broad-resonance branch associated with the interior regularity condition. For ultracompact objects ($R<3M$), the surface-wave branch persists, while the interior-resonance sector splits into broad- and narrow-resonance branches. We next reconstruct the scattering amplitude from the Debye partial-wave contributions and find excellent agreement with direct partial-wave calculations. Finally, we develop complex angular-momentum representations order by order in the Debye series, making explicit how the pole families and non-pole sectors contribute to each Debye term. In the neutron-star-like regime, we find a genuine competition between Regge-Debye pole sums and branch-cut contributions, and show that, at high frequency, the rainbow-like enhancement already arises from the first interior-transmission contribution and is dominated by the interior-resonance Regge-Debye poles. By contrast, in the ultracompact regime, the Debye amplitudes are overwhelmingly pole dominated.

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

0 major / 2 minor

Summary. The paper claims to introduce an exact Debye-series decomposition of the partial-wave scattering matrix for a massless scalar wave incident on a static, spherically symmetric uniform-density star of radius R and mass M matched to a Schwarzschild exterior. This decomposition separates direct surface reflection from interior transmission and propagation contributions, admits a trajectory interpretation, and is used to identify Regge-Debye pole families (surface-wave and resonance branches) whose structure differs for R>3M versus R<3M. The scattering amplitude is reconstructed from the Debye terms and reported to agree excellently with direct partial-wave sums; complex angular-momentum representations are then developed order-by-order in the Debye series, showing explicit pole and branch-cut contributions, including rainbow-like enhancements from interior resonances at high frequency.

Significance. If the central claims hold, the work supplies a valuable analytic framework that bridges partial-wave sums with geometric and resonance interpretations for scattering from horizonless compact objects. The exact decomposition, trajectory picture, and explicit CAM representations order-by-order constitute genuine strengths, while the reported numerical agreement provides direct support for the decomposition's validity. These tools could prove useful for high-frequency scattering, rainbow phenomena, and modeling of potential gravitational-wave echoes from ultracompact objects.

minor comments (2)
  1. [Reconstruction of the scattering amplitude] The section presenting the numerical reconstruction should include quantitative error metrics (e.g., relative L2 discrepancy or maximum pointwise error versus frequency) rather than relying solely on visual agreement to substantiate the 'excellent' claim.
  2. [Setup and Debye decomposition] Clarify the precise matching and regularity conditions used to define the interior solution at r=0 and at the surface r=R; a brief explicit statement of the radial wave equation inside the star would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. The report correctly identifies the central contributions: the exact Debye-series decomposition of the scattering matrix, the separation of surface reflection from interior transmission, the trajectory interpretation, the identification of distinct Regge-Debye pole families for R>3M and R<3M, the numerical validation against partial-wave sums, and the order-by-order complex-angular-momentum representations. We appreciate the recommendation for minor revision and will incorporate improvements to presentation and clarity.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The central derivation constructs an exact Debye-series decomposition of the partial-wave scattering matrix S_l for the uniform-density star matched to Schwarzschild exterior, separating surface reflection from interior transmission terms via the standard matching conditions at r=R and regularity at the center. This decomposition is then summed to reconstruct the full scattering amplitude, with direct numerical comparison to independent partial-wave summation serving as verification rather than a tautology. No load-bearing step reduces to a fitted parameter renamed as prediction, no self-citation chain is invoked for uniqueness, and the pole spectra are extracted from the analytic continuation of the same S_l without circular redefinition. The assumptions (static spherical symmetry, uniform density) are external to the decomposition itself and do not force the result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard general-relativistic assumptions for the metric and wave equation together with uniform-density and boundary-matching conditions; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption The spacetime consists of a static, spherically symmetric uniform-density interior matched to a Schwarzschild exterior.
    Explicitly stated as the model for the compact body.
  • standard math The incident field is a massless scalar wave obeying the wave equation with regularity at the center and continuity at the surface.
    Standard setup for scalar scattering problems in curved spacetime.

pith-pipeline@v0.9.0 · 5604 in / 1405 out tokens · 46432 ms · 2026-05-15T14:32:34.386863+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages · 16 internal anchors

  1. [1]

    (56) and use Cauchy’s theorem to extract the contributions from the Regge–Debye poles associated with the zeros of αin λ−1/2(ω)in the first quadrant of the CAM plane

    CAM representation ofp= 0 We now deform the contourCin Eq. (56) and use Cauchy’s theorem to extract the contributions from the Regge–Debye poles associated with the zeros of αin λ−1/2(ω)in the first quadrant of the CAM plane. The derivation follows the standard contour-deformation strategy of CAM theory, here applied to the modified Sommerfeld–Watson repr...

    work page

  2. [2]

    (57) and use Cauchy’s theorem to extract the Regge–Debye pole terms associated with the analytically continued Debye element S(p) λ−1/2(ω)forp≥1[see Eq

    CAM representation ofp≥1 We now deform the contourCin Eq. (57) and use Cauchy’s theorem to extract the Regge–Debye pole terms associated with the analytically continued Debye element S(p) λ−1/2(ω)forp≥1[see Eq. (59b)]. The derivation fol- lows the same CAM contour-deformation strategy as for p= 0, adapted to the modified Sommerfeld–Watson rep- resentation...

    work page

  3. [3]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett.116, 061102 (2016), arXiv:1602.03837 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  4. [4]

    B. P. Abbottet al.(LIGO Scientific, Virgo), GW170817: Observation of Gravitational Waves from a Binary Neu- tron Star Inspiral, Phys. Rev. Lett.119, 161101 (2017), arXiv:1710.05832 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  5. [5]

    GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run

    R. Abbottet al.(KAGRA, VIRGO, LIGO Scien- tific), GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run, Phys. Rev. X13, 041039 (2023), arXiv:2111.03606 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  6. [6]

    First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole

    K. Akiyamaet al.(Event Horizon Telescope), First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole, Astrophys. J. Lett.875, L1 (2019), arXiv:1906.11238 [astro-ph.GA]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  7. [7]

    Akiyamaet al.(Event Horizon Telescope), First Sagit- tarius A* Event Horizon Telescope Results

    K. Akiyamaet al.(Event Horizon Telescope), First Sagit- tarius A* Event Horizon Telescope Results. VI. Test- ing the Black Hole Metric, Astrophys. J. Lett.930, L17 (2022), arXiv:2311.09484 [astro-ph.HE]

    work page arXiv 2022

  8. [8]

    Testing the nature of dark compact objects: a status report

    V. Cardoso and P. Pani, Testing the nature of dark com- pact objects: a status report, Living Rev. Rel.22, 4 (2019), arXiv:1904.05363 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  9. [9]

    J. A. H. Futterman, F. A. Handler, and R. A. Matzner, Scattering from black holes(Cambridge University Press, 2012)

    work page 2012

  10. [10]

    W. W. Hildreth,The Interaction of Scalar Gravita- tional Waves with the Schwarzschild Metric., Ph.D. the- sis, Princeton University (1964)

    work page 1964

  11. [11]

    singularity

    R. A. Matzner, Scattering of massless scalar waves by a schwarzschild“singularity”, Journal of Mathematical Physics9, 163 (1968)

    work page 1968

  12. [12]

    Vishveshwara, Scattering of gravitational radiation by a Schwarzschild black-hole., Nature227, 936 (1970)

    C. Vishveshwara, Scattering of gravitational radiation by a Schwarzschild black-hole., Nature227, 936 (1970)

    work page 1970

  13. [13]

    Mashhoon, Scattering of Electromagnetic Radiation from a Black Hole, Phys

    B. Mashhoon, Scattering of Electromagnetic Radiation from a Black Hole, Phys. Rev.D7, 2807 (1973)

    work page 1973

  14. [14]

    Fabbri, Scattering and absorption of electromagnetic waves by a schwarzschild black hole, Physical Review D 12, 933 (1975)

    R. Fabbri, Scattering and absorption of electromagnetic waves by a schwarzschild black hole, Physical Review D 12, 933 (1975)

    work page 1975

  15. [15]

    N. G. Sanchez, Elastic Scattering of Waves by a Black Hole, Phys. Rev.D18, 1798 (1978)

    work page 1978

  16. [16]

    R. A. Matzner and M. P. J. Ryan, Scattering of gravi- tational radiation from vacuum black holes., The Astro- physical Journal Supplement Series36, 451 (1978)

    work page 1978

  17. [17]

    F. A. Handler and R. A. Matzner, Gravitational wave scattering, Phys. Rev.D22, 2331 (1980)

    work page 1980

  18. [18]

    R. A. Matzner, C. DeWitt-Morette, B. Nelson, and T.-R. Zhang, Glory scattering by black holes, Phys. Rev.D31, 1869 (1985)

    work page 1985

  19. [19]

    Andersson, Scattering of massless scalar waves by a Schwarzschild black hole: A Phase integral study, Phys

    N. Andersson, Scattering of massless scalar waves by a Schwarzschild black hole: A Phase integral study, Phys. Rev.D52, 1808 (1995)

    work page 1995

  20. [20]

    Scattering of scalar waves by rotating black holes

    K. Glampedakis and N. Andersson, Scattering of scalar waves by rotating black holes, Class. Quant. Grav.18, 1939 (2001), arXiv:gr-qc/0102100 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 1939

  21. [21]

    S. R. Dolan, Scattering and Absorption of Gravitational Plane Waves by Rotating Black Holes, Class. Quant. Grav.25, 235002 (2008), arXiv:0801.3805 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  22. [22]

    L. C. B. Crispino, S. R. Dolan, and E. S. Oliveira, Electromagnetic wave scattering by Schwarzschild black holes, Phys. Rev. Lett.102, 231103 (2009), arXiv:0905.3339 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  23. [23]

    Rainbow scattering of gravitational plane waves by a compact body

    T. Stratton and S. R. Dolan, Rainbow scattering of grav- itational plane waves by a compact body, Phys. Rev. D100, 024007 (2019), arXiv:1903.00025 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  24. [24]

    S. R. Dolan and T. Stratton, Rainbow scattering in the gravitational field of a compact object, Phys. Rev.D95, 124055 (2017), arXiv:1702.06127 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  25. [25]

    Ould El Hadj, T

    M. Ould El Hadj, T. Stratton, and S. R. Dolan, Scatter- ing from compact objects: Regge poles and the complex angular momentum method, Phys. Rev. D101, 104035 (2020), arXiv:1912.11348 [gr-qc]

    work page arXiv 2020

  26. [26]

    G. N. Watson, The diffraction of electric waves by the Earth, Proc. R. Soc. London A95, 83 (1918)

    work page 1918

  27. [27]

    Sommerfeld,Partial Differential Equations of Physics (Academic Press, New York, 1949)

    A. Sommerfeld,Partial Differential Equations of Physics (Academic Press, New York, 1949)

    work page 1949

  28. [28]

    de Alfaro and T

    V. de Alfaro and T. Regge,Potential Scattering(North- Holland Publishing Company, Amsterdam, 1965)

    work page 1965

  29. [29]

    R. G. Newton,Scattering Theory of Waves and Particles, 2nd ed. (Springer-Verlag, New York, 1982)

    work page 1982

  30. [30]

    H. M. Nussenzveig,Diffraction effects in semiclassical scattering, Vol. 1 (Cambridge University Press, 2006)

    work page 2006

  31. [31]

    W. T. Grandy,Scattering of Waves from Large Spheres (Cambridge University Press, Cambridge, 2000)

    work page 2000

  32. [32]

    Überall,Acoustic Resonance Scattering(Gordon and Breach, New York, 1992)

    H. Überall,Acoustic Resonance Scattering(Gordon and Breach, New York, 1992). 23

    work page 1992

  33. [33]

    Aki and P

    K. Aki and P. Richards,Quantitative Seismology, 2nd ed. (University Science Book, Sausalito, 2002)

    work page 2002

  34. [34]

    V. N. Gribov,The Theory of Complex Angular Momenta: Gribov Lectures on Theoretical Physics(Cambridge Uni- versity Press, Cambridge, 2003)

    work page 2003

  35. [35]

    P. D. B. Collins,An Introduction to Regge Theory and High-Energy Physics(Cambridge University Press, Cam- bridge, 1977)

    work page 1977

  36. [36]

    V.BaroneandE.Predazzi,High-Energy Particle Diffrac- tion(Springer-Verlag, Berlin, 2002)

    work page 2002

  37. [37]

    Donnachie, G

    S. Donnachie, G. Dosch, P. V. Landshoff, and O. Nacht- mann,Pomeron Physics and QCD(Cambridge Univer- sity Press, Cambridge, 2005)

    work page 2005

  38. [38]

    Andersson and K

    N. Andersson and K. E. Thylwe, Complex angular mo- mentum approach to black hole scattering, Class. Quant. Grav.11, 2991 (1994)

    work page 1994

  39. [39]

    Andersson, Complex angular momenta and the black hole glory, Class

    N. Andersson, Complex angular momenta and the black hole glory, Class. Quant. Grav.11, 3003 (1994)

    work page 1994

  40. [40]

    Decanini, A

    Y. Decanini, A. Folacci, and B. Jensen, Complex angu- lar momentum in black hole physics and the quasinor- mal modes, Phys. Rev.D67, 124017 (2003), arXiv:gr- qc/0212093 [gr-qc]

    work page arXiv 2003

  41. [41]

    Regge pole description of scattering of scalar and electromagnetic waves by a Schwarzschild black hole

    A. Folacci and M. Ould El Hadj, Regge pole descrip- tion of scattering of scalar and electromagnetic waves by a Schwarzschild black hole, Phys. Rev.D99, 104079 (2019), arXiv:1901.03965 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  42. [42]

    Rev.D100, 064009 (2019), arXiv:1906.01441 [gr-qc]

    A.FolacciandM.OuldElHadj,Reggepoledescriptionof scatteringofgravitationalwavesbyaSchwarzschildblack hole, Phys. Rev.D100, 064009 (2019), arXiv:1906.01441 [gr-qc]

    work page arXiv 2019

  43. [43]

    Regge poles of the Schwarzschild black hole: a WKB approach

    Y. Decanini and A. Folacci, Regge poles of the Schwarzschild black hole: A WKB approach, Phys. Rev. D81, 024031 (2010), arXiv:0906.2601 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  44. [44]

    Universality of high-energy absorption cross sections for black holes

    Y. Decanini, G. Esposito-Farese, and A. Folacci, Univer- sality of high-energy absorption cross sections for black holes, Phys. Rev.D83, 044032 (2011), arXiv:1101.0781 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  45. [45]

    Fine structure of high-energy absorption cross sections for black holes

    Y. Decanini, A. Folacci, and B. Raffaelli, Fine structure of high-energy absorption cross sections for black holes, Class. Quant. Grav.28, 175021 (2011), arXiv:1104.3285 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  46. [46]

    Folacci and M

    A. Folacci and M. Ould El Hadj, Alternative description of gravitational radiation from black holes based on the Regge poles of theS-matrix and the associated residues, Phys. Rev. D98, 064052 (2018), arXiv:1807.09056 [gr- qc]

    work page arXiv 2018

  47. [47]

    Torres, M

    T. Torres, M. O. E. Hadj, S.-Q. Hu, and R. Gregory, Regge pole description of scattering by dirty black holes, Phys. Rev. D107, 064028 (2023), arXiv:2211.17147 [gr- qc]

    work page arXiv 2023

  48. [48]

    Ould El Hadj, Black hole absorption cross sections: Spin and Regge poles, Phys

    M. Ould El Hadj, Black hole absorption cross sections: Spin and Regge poles, Phys. Rev. D111, 124041 (2025), arXiv:2504.19324 [gr-qc]

    work page arXiv 2025

  49. [49]

    Chandrasekhar andV.Ferrari,On the non-radialoscil- lations of a star iii

    S. Chandrasekhar andV.Ferrari,On the non-radialoscil- lations of a star iii. a reconsideration of the axial modes, Proc. Roy. Soc. Lond. A434, 449 (1991)

    work page 1991

  50. [50]

    Cardoso, V

    V. Cardoso, V. F. Foit, and M. Kleban, Gravitational wave echoes from black hole area quantization, JCAP 08, 006, arXiv:1902.10164 [hep-th]

    work page arXiv 1902

  51. [51]

    Maggio, P

    E. Maggio, P. Pani, and G. Raposo, Testing the nature of dark compact objects with gravitational waves (2021), arXiv:2105.06410 [gr-qc]

    work page arXiv 2021

  52. [52]

    De Laurentis and P

    M. De Laurentis and P. Pani, Testing the nature of com- pact objects and the black hole paradigm, Gen. Rel. Grav.57, 39 (2025)

    work page 2025

  53. [53]

    Nambu, S

    Y. Nambu, S. Noda, and Y. Sakai, Wave Optics in Space- times with Compact Gravitating Object, Phys. Rev. D 100, 064037 (2019), arXiv:1905.01793 [gr-qc]

    work page arXiv 2019

  54. [54]

    He, Simulating the scattering of low-frequency grav- itational waves by compact objects using the finite ele- ment method (2019), arXiv:1912.00325 [gr-qc]

    J.-h. He, Simulating the scattering of low-frequency grav- itational waves by compact objects using the finite ele- ment method (2019), arXiv:1912.00325 [gr-qc]

    work page arXiv 2019

  55. [55]

    Black hole interference patterns in flavour oscillations

    J. Alexandre and K. Clough, Black hole interference patterns in flavor oscillations, Phys. Rev.D98, 043004 (2018), arXiv:1805.01874 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  56. [56]

    Marchant, K

    P. Marchant, K. Breivik, C. P. L. Berry, I. Mandel, and S. L. Larson, Eclipses of continuous gravitational waves as a probe of stellar structure, Phys. Rev. D101, 024039 (2020), arXiv:1912.04268 [astro-ph.SR]

    work page arXiv 2020

  57. [57]

    A. N. Seenivasan and S. R. Dolan, Self-force and the Schwarzschild star, Class. Quant. Grav.42, 185002 (2025), arXiv:2501.13176 [gr-qc]

    work page arXiv 2025

  58. [58]

    On the discovery of Birkhoff's theorem

    N. Voje Johansen and F. Ravndal, On the discovery of Birkhoff’s theorem, Gen. Rel. Grav.38, 537 (2006), arXiv:physics/0508163 [physics]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  59. [59]

    S. L. Shapiro and S. A. Teukolsky,Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Ob- jects(Wiley, New-York, 1983)

    work page 1983

  60. [60]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun,Handbook of Mathe- matical Functions(Dover, New-York, 1965)

    work page 1965

  61. [61]

    Wolfram Research, Inc., Mathematica, version 10.0 (2014), champaign, IL

    work page 2014