pith. machine review for the scientific record. sign in

arxiv: 2604.21219 · v1 · submitted 2026-04-23 · 🌀 gr-qc · hep-th

Recognition: unknown

Calculation of a regularized Teukolsky Green function in Schwarzschild spacetime

David Q. Aruquipa , Marc Casals , Brien C. Nolan
Authors on Pith no claims yet

Pith reviewed 2026-05-09 21:47 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords Teukolsky equationretarded Green functionHadamard formSchwarzschild spacetimeconformal transformationvan Vleck determinantregularizationself-force
0
0 comments X

The pith

Exact expressions for factors in the Hadamard form of the Teukolsky retarded Green function are derived on Schwarzschild spacetime.

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

The paper shows how a conformal transformation to a spacetime with direct-product metric allows the singular part of the retarded Green function for the Teukolsky equation to be written in a separable form. The angular factor is calculated explicitly in terms of spin-weighted spherical harmonics and geodesics on the sphere. The factor from the two-dimensional subspace is given exactly for constant-radius orbits using the van Vleck determinant and a transport equation. These results permit the multipolar modes of the direct part to be found for electromagnetic and gravitational perturbations and improve the practical calculation of the regularized Green function by subtracting the singular part. This matters for accurate modeling of fields near black holes where coincidence points arise in self-force problems.

Core claim

We obtain exact expressions for various factors involved in the Hadamard form of the retarded Green function for the Teukolsky equation on Schwarzschild spacetime by working in a conformally related direct-product spacetime. The direct part separates into an angular factor computed explicitly and an M2 factor involving the square root of the van Vleck determinant times a spin-dependent transport term, both exact for constant radius orbits. This separability yields the ℓ-modes of the direct part and allows computation of the retarded Green function minus its direct part for gravitational perturbations.

What carries the argument

The separable Hadamard direct part of the retarded Green function, obtained from the conformal transformation to the M2 × S² metric, with its angular component from spin-weighted spherical harmonics and its M2 component from the van Vleck determinant and transport equation.

If this is right

  • The angular factor can be computed for arbitrary points using the interplay of geodesics, harmonics, and Euler angles.
  • Exact M2 factors are available for static and circular orbits in Schwarzschild.
  • Multipolar ℓ-modes of the direct part are available for electromagnetic and gravitational perturbations.
  • The regularized retarded Green function is obtained by subtracting the direct part, providing a better representation near coincidence.

Where Pith is reading between the lines

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

  • The exact factors could support higher-precision self-force calculations for extreme mass ratio inspirals around black holes.
  • The separability might suggest similar conformal simplifications for other wave equations on spherically symmetric backgrounds.
  • Direct comparison of the derived ℓ-modes against independent numerical integrations of the Teukolsky equation near coincidence would test the regularization procedure.

Load-bearing premise

That the Hadamard direct part becomes separable under the conformal transformation to the direct-product metric, allowing exact computation of its factors.

What would settle it

Numerical evaluation of the Teukolsky Green function near coincidence points that deviates from the predicted regularized form after subtracting the computed direct part would show the expressions to be incorrect.

Figures

Figures reproduced from arXiv: 2604.21219 by Brien C. Nolan, David Q. Aruquipa, Marc Casals.

Figure 1
Figure 1. Figure 1: FIG. 1: The rotations and Euler angles of Proposition 1 and the preceding paragraphs. Left image: A rotation [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Plot of the van Vleck determinant [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Log-plot of the relative error in the setting of Fig. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The coefficient [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Left image: Plot of the proper time [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Plots of the [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Plots of the [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Log-plot of the BPT GF [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Log-plot of the BPT GF [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
read the original abstract

We obtain exact expressions for various factors involved in the Hadamard form of the retarded Green function for the (Bardeen-Press-)Teukolsky equation on Schwarzschild spacetime. We use these to improve on previous results for the calculation of this Green function. We work in a spacetime $\mathcal{M}_2\times\mathbb{S}^2$ conformal to Schwarzschild, in which the metric takes a direct product form. This allows us to derive a separable form for the direct (i.e., singular) part of the Hadamard form of the retarded Green function. The angular factor in this quantity is calculated explicitly. This shows an interesting interplay between geodesics of $\mathbb{S}^2$, spin-weighted spherical harmonics, and Euler angles. The $\mathcal{M}_2$ factor equates to a spin-dependent factor that satisfies a transport equation along geodesics, times the square root of the van Vleck determinant. Both terms are calculated in an exact form for constant radius orbits (which includes the cases of circular timelike geodesics and static worldlines of Schwarzschild spacetime). This separable form also allows us to obtain the multipolar $\ell$-modes of the direct part for electromagnetic and gravitational field perturbations. We then use these $\ell$-modes to calculate, in the gravitational case, the retarded Green function minus its direct part: this is a better representation in practise of the retarded Green function for points near coincidence.

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 paper claims to obtain exact expressions for factors in the Hadamard form of the retarded Green function for the Teukolsky equation on Schwarzschild spacetime. By conformally transforming to a direct-product spacetime M2 × S², the direct (singular) part of the Green function is made separable; the angular factor is computed explicitly using spin-weighted spherical harmonics and geodesics on S², while the M2 factor (involving a spin-dependent transport equation and van Vleck determinant) is obtained exactly for constant-radius orbits. These are used to derive multipolar ℓ-modes of the direct part and, for gravitational perturbations, the retarded Green function minus its direct part as a practical representation near coincidence.

Significance. If the central derivations hold, the work supplies exact, closed-form expressions for the singular part of the Teukolsky Green function, which improves regularization techniques for black-hole perturbation theory and self-force calculations. The separable structure arising from the conformal product metric and the explicit angular calculation represent a technical advance over prior numerical or approximate approaches.

major comments (1)
  1. The mapping between the Hadamard direct part computed in the conformally rescaled metric and the physical retarded Green function for the original Teukolsky operator is not shown explicitly. The Teukolsky operator is not conformally invariant, and the Green function transforms with additional factors from the volume element (Ω⁴) and spin-connection terms; without the precise transformation law for the bitensor U and the full parametrix, it is unclear whether the separable direct part subtracts the correct singularity from the physical retarded solution. This relation is load-bearing for the claim that the computed ℓ-modes and regularized Green function are valid for Schwarzschild.
minor comments (2)
  1. The abstract states that the M2 factor 'equates to a spin-dependent factor that satisfies a transport equation... times the square root of the van Vleck determinant' but does not indicate the explicit form of the transport equation or the boundary conditions used for its solution along constant-r geodesics.
  2. The manuscript would benefit from a short table or explicit list of the 'various factors' for which exact expressions are obtained, cross-referenced to the relevant equations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising this important point concerning the conformal transformation. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [—] The mapping between the Hadamard direct part computed in the conformally rescaled metric and the physical retarded Green function for the original Teukolsky operator is not shown explicitly. The Teukolsky operator is not conformally invariant, and the Green function transforms with additional factors from the volume element (Ω⁴) and spin-connection terms; without the precise transformation law for the bitensor U and the full parametrix, it is unclear whether the separable direct part subtracts the correct singularity from the physical retarded solution. This relation is load-bearing for the claim that the computed ℓ-modes and regularized Green function are valid for Schwarzschild.

    Authors: We agree that the explicit mapping requires clearer exposition. The Teukolsky operator is indeed not conformally invariant, so the Green function in the physical Schwarzschild metric relates to that in the conformally rescaled M₂ × S² spacetime through additional factors arising from the volume element (scaling as Ω⁴) and the spin-connection terms in the covariant derivatives. In the manuscript we compute the direct (singular) part of the Hadamard parametrix in the product spacetime to obtain separability, but we will add a new subsection (or appendix) that derives the precise transformation law for the bitensor U and the full parametrix, including the appropriate powers of the conformal factor Ω and the spin-dependent adjustments. This will explicitly confirm that the separable expression subtracts the correct singularity from the physical retarded Green function, thereby validating the ℓ-modes and the regularized representation for Schwarzschild. The core derivations remain unchanged; only the presentation of the mapping will be strengthened. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation proceeds via explicit conformal transformation and transport equations

full rationale

The paper constructs the Hadamard direct part in the conformally rescaled product spacetime M2 × S², derives its separable form, computes the angular factor from spin-weighted harmonics and geodesics, and solves the M2 transport equation plus van Vleck factor exactly for constant-r orbits. These steps are presented as direct calculations rather than fits or self-referential definitions. The subsequent subtraction to obtain the regularized retarded Green function follows from the same construction without reducing the final result to its inputs by construction. No load-bearing self-citation or ansatz smuggling is indicated in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard domain assumptions of general relativity and wave propagation in curved spacetime; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The Hadamard parametrix supplies the correct singular structure for the retarded Green function of the Teukolsky operator.
    Invoked to define the direct part whose factors are computed.
  • domain assumption The chosen conformal transformation to M2 × S² preserves the causal structure and allows separation of the direct Hadamard term.
    Central step stated in the abstract that enables the subsequent exact calculations.

pith-pipeline@v0.9.0 · 5564 in / 1433 out tokens · 52480 ms · 2026-05-09T21:47:54.162287+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 1 Pith paper

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

  1. Pole Structure of Kerr Green's Function

    gr-qc 2026-05 unverdicted novelty 6.0

    Homogeneous solutions and connection coefficients in the radial Teukolsky equation for Kerr black holes exhibit simple poles at Matsubara frequencies that cancel in the Green's function, along with canceling zero-freq...

Reference graph

Works this paper leans on

32 extracted references · 10 canonical work pages · cited by 1 Pith paper

  1. [1]

    Hence, Eq

    for general null geodesics of Schwarzschild spacetime. Hence, Eq. (2.20) yields sU(x, x ′) = r′ r s(3ϵ+1)/2 r′ −2M r−2M s(1−ϵ)/2 along radial null geodesics.(2.21) Further progress taking this 4-dimensional approach seems difficult, and as we have seen, the results above are limited by the assumption thatθ=θ ′ =π/2. Thus we take an alternative approach th...

  2. [2]

    There is an interesting corollary of the calculation above which we state as the following proposition

    An aside on Euler angles. There is an interesting corollary of the calculation above which we state as the following proposition. This relates geodesics and Euler angles onS 2. The result follows immediately from the fact thatκis constant along geodesics, and from the formula (4.33). Proposition 1LetCbe a great circle ofS 2 and fixP ′ ∈C. LetPbe any other...

  3. [3]

    TwodLocalExpan- sions arxiv.nb

    fors= 0 (even if the values ofℓare also different). B. Calculation of the non-direct GF In the previous subsection we have calculated theℓ-modes −2Gd ℓ of the direct part of the BPT GF. In [13], two of the authors calculated the BPT GF −2Gret and itsℓ-modes −2Gℓ; we refer the reader to that paper for details of those calculations. Equipped with these, we ...

  4. [4]

    J. J. Duistermaat and L. H¨ ormander, Acta Mathematica128, 183 (1972), URLhttps://doi.org/10.1007/ BF02392165

    1972

  5. [5]

    Poisson, A

    E. Poisson, A. Pound, and I. Vega, Living Reviews in Relativity14, 7 (2011)

    2011

  6. [6]

    Casals, S

    M. Casals, S. Dolan, A. C. Ottewill, and B. Wardell, Phys. Rev. D88, 044022 (2013), URLhttp://link.aps.org/ doi/10.1103/PhysRevD.88.044022

    work page doi:10.1103/physrevd.88.044022 2013

  7. [7]

    Wardell, C

    B. Wardell, C. R. Galley, A. Zengino˘ glu, M. Casals, S. R. Dolan, and A. C. Ottewill, Phys. Rev. D89, 084021 (2014), URLhttp://link.aps.org/doi/10.1103/PhysRevD.89.084021

    work page doi:10.1103/physrevd.89.084021 2014

  8. [8]

    Hadamard,Lectures on Cauchy’s Problem in Linear Partial Differential Equations(Dover Publications, 1923), ISBN 978-0486495491

    J. Hadamard,Lectures on Cauchy’s Problem in Linear Partial Differential Equations(Dover Publications, 1923), ISBN 978-0486495491

    1923

  9. [9]

    F. G. Friedlander,The wave equation on a curved space-time, vol. 2 (Cambridge university press, 1975)

    1975

  10. [10]

    R. H. Jonsson, D. Q. Aruquipa, M. Casals, A. Kempf, and E. Mart´ ın-Mart´ ınez, Phys. Rev. D101, 125005 (2020), URLhttps://link.aps.org/doi/10.1103/PhysRevD.101.125005

    work page doi:10.1103/physrevd.101.125005 2020

  11. [11]

    Casals, B

    M. Casals, B. C. Nolan, A. C. Ottewill, and B. Wardell, Physical Review D100, 104037 (2019)

    2019

  12. [12]

    Casals and B

    M. Casals and B. C. Nolan, Physical Review D108, 044033 (2023)

    2023

  13. [13]

    J. M. Bardeen and W. H. Press, Journal of Mathematical Physics14, 7 (1973), https://doi.org/10.1063/1.1666175, URLhttps://doi.org/10.1063/1.1666175

    work page doi:10.1063/1.1666175 1973

  14. [14]

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

    1972

  15. [15]

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

    1973

  16. [16]

    D. Q. Aruquipa and M. Casals,Green functions of the Regge-Wheeler and Teukolsky equations in Schwarzschild spacetime(2026), 2603.07747, URLhttps://arxiv.org/abs/2603.07747

    work page arXiv 2026

  17. [17]

    Wald,General Relativity(Chicago, University of Chicago Press, 1984)

    R. Wald,General Relativity(Chicago, University of Chicago Press, 1984)

    1984

  18. [18]

    Iuliano and J

    C. Iuliano and J. Zahn, Phys. Rev. D108, 125017 (2023), 2307.07467

    work page arXiv 2023

  19. [19]

    Goldberg, A

    J. Goldberg, A. Macfarlane, E. T. Newman, F. Rohrlich, and E. Sudarshan, Journal of Mathematical Physics8, 2155 (1967)

    1967

  20. [20]

    T. J. Hollowood, G. M. Shore, and R. J. Stanley, Journal of High Energy Physics2009, 089 (2009)

    2009

  21. [21]

    Casals and B

    M. Casals and B. C. Nolan, Physical Review D92, 104030 (2015)

    2015

  22. [22]

    Casals and B

    M. Casals and B. C. Nolan, Physical Review D—Particles, Fields, Gravitation, and Cosmology86, 024038 (2012)

    2012

  23. [23]

    D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii,Quantum theory of angular momentum(World Scientific, 1988)

    1988

  24. [24]

    Monteverdi and E

    A. Monteverdi and E. Winstanley, arXiv preprint arXiv:2410.23201 (2024)

    work page arXiv 2024

  25. [25]

    Michel and K

    V. Michel and K. Seibert, inMathematische Geod¨ asie/Mathematical Geodesy: Handbuch der Geod¨ asie, herausgegeben von Willi Freeden und Reiner Rummel(Springer, 2020), pp. 195–307

    2020

  26. [26]

    Allen and T

    B. Allen and T. Jacobson, Communications in Mathematical Physics103, 669 (1986). 28

    1986

  27. [27]

    B. C. Nolan,Coordinates on the geodesic spray and two-point functions on conformal Schwarzschild spacetime, In preparation. (2026)

    2026

  28. [28]

    Casals, S

    M. Casals, S. Dolan, A. C. Ottewill, and B. Wardell, Phys. Rev.D79, 124044 (2009), 0903.5319

    work page arXiv 2009

  29. [29]

    Probing Direct Waves in Black Hole Ringdowns,

    N. Oshita, S. Ma, Y. Chen, and H. Yang,Probing direct waves in black hole ringdowns(2025), 2509.09165, URL https://arxiv.org/abs/2509.09165

    work page arXiv 2025

  30. [30]

    J. Su, N. Khera, M. Casals, S. Ma, A. Chowdhuri, and H. Yang, arXiv e-prints arXiv:2601.22015 (2026), 2601.22015

    work page arXiv 2026

  31. [31]

    P. F. Byrd and M. D. Friedman,Handbook of elliptic integrals for engineers and physicists(Springer, 2013)

    2013

  32. [32]

    Q. D. Aruquipa,https://github.com/DSSystems/M2BPTHadamardU