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arxiv: 2605.19829 · v1 · pith:UJBFZ5ZMnew · submitted 2026-05-19 · 🌀 gr-qc

Hamiltonian formalism, master functions and Darboux transformations for perturbed (interiors and exteriors of) nonrotating black holes

Michele Lenzi , Guillermo A. Mena Marug\'an , Andr\'es M\'inguez-S\'anchez , Carlos F. Sopuerta This is my paper

Pith reviewed 2026-05-20 04:08 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Hamiltonian formalismDarboux transformationsmaster functionsblack hole perturbationsgauge invariantsnonrotating black holespolar perturbationsaxial perturbations
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The pith

Canonical transformations bijectively correspond to Darboux transformations for polar perturbations of nonrotating black holes and can mix axial and polar sectors.

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

The paper builds a Hamiltonian formalism for perturbations of nonrotating black holes that works for both interiors and exteriors. It interprets Darboux transformations geometrically as generalized canonical transformations that keep the perturbations as harmonic oscillators with potentials. This establishes a bijective correspondence for polar perturbations, extending the axial case, and shows transformations that mix the two types of master functions. A reader would care because this unifies the treatment of gauge invariants and supports classical and quantum studies of black hole perturbations.

Core claim

Adopting the Hamiltonian formalism, Darboux transformations between pairs of master functions are characterized as generalized canonical transformations that preserve the Hamiltonian structure of the perturbations as harmonic oscillators subject to certain potentials. The bijective correspondence between such canonical transformations and Darboux transformations, previously proved for axial perturbations, is extended to polar perturbations. In addition, canonical transformations that mix axial and polar master functions are demonstrated to exist.

What carries the argument

The arrangement of perturbative gauge invariants into canonical pairs of master functions that behave as harmonic oscillators with effective potentials, with Darboux transformations acting as generalized canonical transformations.

If this is right

  • The formalism describes background physical degrees of freedom as well as perturbations.
  • It applies to classical and quantum aspects of perturbed black holes.
  • The geometric interpretation aids in relating different master functions.
  • All physical perturbative degrees of freedom are captured in this canonical structure.

Where Pith is reading between the lines

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

  • This could enable derivation of new relations between quasinormal mode spectra for different potentials.
  • The mixing of axial and polar sectors might have implications for parity-violating effects or coupled dynamics.
  • Extensions to time-dependent backgrounds or other cosmologies could follow from the same Hamiltonian approach.

Load-bearing premise

All physical perturbative degrees of freedom can be arranged into canonical pairs associated with master functions that behave as harmonic oscillators with effective potentials.

What would settle it

A counterexample where a Darboux transformation for a polar master function fails to preserve the canonical Hamiltonian structure of the oscillator would disprove the bijective correspondence.

read the original abstract

Motivated by their relevance to the interior of nonrotating black holes, classical and quantum Kantowski-Sachs cosmologies have recently attracted increasing attention. This interest has led to the development of a Hamiltonian formalism for axial and polar perturbations, which can be extended to applications in the exterior region. The formalism provides also a description of the background physical degrees of freedom. Moreover, it allows for the construction of all physical perturbative gauge invariants, which can be arranged into canonical pairs associated with master functions. In this work, we review the basis of this Hamiltonian formalism, putting the emphasis on its foundations and fundamental steps rather than on details of the involved calculations. Our discussion focuses on classical and effective aspects, although we also briefly comment on its natural role in the quantization of perturbed black holes. Adopting this formalism we present a geometric interpretation of Darboux transformations between pairs of master functions, characterizing them as generalized canonical transformations that preserve the Hamiltonian structure of the perturbations as harmonic oscillators subject to certain potentials. This bijective correspondence between such canonical transformations and Darboux transformations, which was recently proved to hold for axial perturbations, is here extended to the case of polar perturbations. In addition, we demonstrate the existence of canonical transformations that, similarly to Darboux transformations, mix axial and polar master functions.

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

2 major / 2 minor

Summary. The manuscript reviews the foundations of a Hamiltonian formalism for axial and polar perturbations of nonrotating black holes (interiors and exteriors), including the background degrees of freedom. It constructs all physical perturbative gauge invariants and arranges them into canonical pairs associated with master functions that behave as harmonic oscillators subject to effective potentials. Adopting this formalism, the paper gives a geometric interpretation of Darboux transformations as generalized canonical transformations preserving the Hamiltonian structure, extends the previously established bijective correspondence between canonical and Darboux transformations from axial to polar perturbations, and demonstrates the existence of canonical transformations that mix axial and polar master functions.

Significance. If the derivations and extensions hold, the work supplies a unified Hamiltonian framework for black-hole perturbations that links gauge-invariant master functions to harmonic-oscillator dynamics. The extension of the bijective correspondence to polar perturbations and the construction of mixing transformations could furnish new analytic tools for solving the perturbation equations and for quantizing the system, with direct relevance to interior geometries such as Kantowski-Sachs cosmologies.

major comments (2)
  1. [§4] §4 (polar perturbations): the extension of the bijective correspondence relies on the construction of gauge-invariant canonical pairs; the manuscript should supply an explicit verification that the resulting master functions reduce to the known Regge-Wheeler and Zerilli equations in the Schwarzschild exterior limit, including a check that no post-hoc redefinitions are introduced.
  2. [§5] §5 (mixing transformations): the claim that canonical transformations mixing axial and polar master functions preserve the harmonic-oscillator Hamiltonian structure is load-bearing for the central result; an explicit transformation matrix or generating function should be displayed together with the verification that the effective potentials remain unchanged under the mixing.
minor comments (2)
  1. [Abstract] The abstract states the extensions without error estimates or limit checks; a short paragraph summarizing the verification steps performed in the body would improve readability.
  2. [Notation] Notation for the master functions (e.g., the symbols used for axial versus polar cases) should be collected in a single table or glossary to avoid ambiguity when discussing the mixing transformations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments, which we believe will improve the clarity of our presentation. We address each major comment below and will incorporate the requested explicit verifications into the revised version.

read point-by-point responses

  1. Referee: [§4] §4 (polar perturbations): the extension of the bijective correspondence relies on the construction of gauge-invariant canonical pairs; the manuscript should supply an explicit verification that the resulting master functions reduce to the known Regge-Wheeler and Zerilli equations in the Schwarzschild exterior limit, including a check that no post-hoc redefinitions are introduced.

    Authors: We agree that an explicit check in the Schwarzschild exterior limit will strengthen the presentation. In the revised manuscript we will add a short subsection at the close of §4 that takes the exterior limit of the gauge-invariant canonical pairs constructed for polar perturbations. We will show step by step that the resulting master functions satisfy the Zerilli equation (and recover the Regge-Wheeler equation for the axial sector) with effective potentials that match the standard expressions exactly. Because the canonical pairs are defined directly from the Hamiltonian formalism without additional redefinitions, the reduction proceeds without post-hoc adjustments; we will display the intermediate steps to make this manifest. revision: yes

  2. Referee: [§5] §5 (mixing transformations): the claim that canonical transformations mixing axial and polar master functions preserve the harmonic-oscillator Hamiltonian structure is load-bearing for the central result; an explicit transformation matrix or generating function should be displayed together with the verification that the effective potentials remain unchanged under the mixing.

    Authors: We thank the referee for underscoring the importance of this verification. In the revised version we will insert an explicit example in §5, presenting both the transformation matrix and the associated generating function that mixes one axial and one polar master function while preserving the canonical commutation relations. Direct substitution into the Hamiltonian will then be carried out to confirm that the effective potentials are left invariant and that the harmonic-oscillator form is retained. This calculation will be included as a self-contained paragraph so that the preservation of the structure is fully transparent. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for axial case; polar extension and mixing are independent

full rationale

The paper reviews its Hamiltonian formalism for perturbations and extends a bijective correspondence between canonical and Darboux transformations from axial to polar perturbations while constructing mixing transformations. It references prior proof of the axial case but presents the polar extension and mixing results as new derivations within this work, using gauge-invariant canonical pairs and the oscillator structure. This constitutes at most one minor self-citation that is not load-bearing for the central claims, with the new steps appearing self-contained and independent of fitted inputs or definitional reductions. No other circular patterns are exhibited by the provided abstract and claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of a Hamiltonian formulation for perturbations that reduces to harmonic-oscillator form with effective potentials, plus the assumption that Darboux transformations can be realized as canonical transformations within that structure. No explicit free parameters or invented entities are stated in the abstract.

axioms (1)
  • domain assumption Perturbations of nonrotating black holes admit a Hamiltonian formulation in which physical degrees of freedom are captured by gauge-invariant master functions behaving as harmonic oscillators subject to effective potentials.
    Invoked throughout the abstract when describing the formalism and when linking Darboux transformations to canonical transformations.

pith-pipeline@v0.9.0 · 5793 in / 1490 out tokens · 33482 ms · 2026-05-20T04:08:19.632035+00:00 · methodology

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Works this paper leans on

60 extracted references · 60 canonical work pages · 1 internal anchor

  1. [1]

    ℓ(ℓ+ 1) 2πn L0 |pc| 4 L2 0 p2 b Ωbh−n 5 − p−n 5 ℓ(ℓ+ 1) −2 L2 0 p2 b |pc|Ωbhn 6 − |pc|pn 3 + 2|pc|pn 4 − (ℓ+ 2)! (ℓ−2)!|p c|(Ωb + Ωc)hn 4 # , C4[kn 4] = X n kn 4

    ifm <0,(7) where cY m ℓ stands for the standard complex spherical harmonics, and the star symbol denotes complex conjugation. Similarly, for the Fourier expansion, we employ the real modes F0 = 1/ p L0, F n = p 2/L0 cos (ωnζ) ifn >0, F n = p 2/L0 sin (ωnζ) ifn <0.(8) Alternative, one can instead use complex spherical harmonics and Fourier plane waves, but...

    work page doi:10.13019/501100011033 2021

  2. [2]

    Genzel, R., Eisenhauer, F., Gillessen, S.: Experimental studies of black holes: status and future prospects. Astron. Astrophys. Rev.32, 3 (2024)

    work page 2024

  3. [3]

    Oxford University Press, Oxford, UK (1983)

    Chandrasekhar, S.:The Mathematical Theory of Black Holes. Oxford University Press, Oxford, UK (1983)

    work page 1983

  4. [4]

    Cambridge University Press, Cambridge, UK (1973)

    Hawking, S.W., Ellis, G.F.R.:The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge, UK (1973)

    work page 1973

  5. [5]

    Barceló, C., Liberati, S., Sonego, S., Visser, M.: Fate of gravitational collapse in semiclassical gravity. Phys. Rev. D77, 044032 (2008)

    work page 2008

  6. [6]

    Edited by Wald R.M

    Hawking, S.W.: Is information lost in black holes?, inBlack Holes and Relativistic Stars. Edited by Wald R.M. University of Chicago Press, Chicago (1998)

    work page 1998

  7. [7]

    Regge, T., Wheeler, J.A.: Stability of a Schwarzschild singularity. Phys. Rev.108, 1063 (1957)

    work page 1957

  8. [8]

    Teukolsky, S.A.: Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations. Astrophys. J.185, 635 (1973)

    work page 1973

  9. [9]

    Nollert, H.-P.: Quasinormal modes: the characteristic “sound” of black holes and neutron stars. Class. Quantum Grav.16, R159 (1999)

    work page 1999

  10. [10]

    Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys.43, 199 (1975)

    work page 1975

  11. [11]

    Black hole spectroscopy: from theory to experiment

    Berti, E.,et al.: Black hole spectroscopy: from theory to experiment. arXiv:2505.23895 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  12. [12]

    Berti, E.,et al.: Testing general relativity with present and future astrophysical observations. Class. Quantum Grav.32 243001 (2015)

    work page 2015

  13. [13]

    Sitzungsber

    Einstein, A.: Über Gravitationswellen. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math.Phys.)1918, 154 (1918)

    work page 1918

  14. [14]

    Bardeen, J.M.: Gauge-invariant cosmological perturbations. Phys. Rev. D22, 1882 (1980)

    work page 1980

  15. [15]

    Kodama, H., Sasaki, M.: Cosmological perturbation theory. Prog. Theor. Phys. Suppl.78, 1 (1984)

    work page 1984

  16. [16]

    I - Linearized odd-parity radiation

    Cunningham, C.T., Price, R.H., Moncrief, V.: Radiation from collapsing relativistic stars. I - Linearized odd-parity radiation. Astrophys. J.224, 643 (1978)

    work page 1978

  17. [17]

    Zerilli, F.J.: Effective potential for even parity Regge-Wheeler gravitational perturbation equations. Phys. Rev. Lett.24, 737 (1970)

    work page 1970

  18. [18]

    Moncrief, V.: Gravitational perturbations of spherically symmetric systems. I. The exterior problem. Ann. Phys. (N.Y.) 88, 323 (1974)

    work page 1974

  19. [19]

    Darboux, G.: Sur une proposition relative aux équations linéaires. C. R. Acad. Sci. (Paris)94, 1456 (1882)

    work page

  20. [20]

    Lenzi, M., Sopuerta, C.F.: Darboux covariance: a hidden symmetry of perturbed Schwarzschild black holes. Phys. Rev. D 104, 124068 (2021). 19

    work page 2021

  21. [21]

    Lenzi, M., Sopuerta, C.F.:Master functions and equations for perturbations of vacuum spherically symmetric spacetimes. Phys. Rev. D104, 084053 (2021)

    work page 2021

  22. [22]

    Lenzi, M., Sopuerta, C.F.: Gauge-independent metric reconstruction of perturbations of vacuum spherically-symmetric spacetimes. Phys. Rev. D109, 084030 (2024)

    work page 2024

  23. [23]

    Chandrasekar, S.: On one-dimensional potential barriers having equal reflexion and transmission coefficients. Proc. Roy. Soc. Lond. A369, 425 (1980)

    work page 1980

  24. [24]

    Yurov, A., Yurov, V.: A look at the generalized Darboux transformations for the quasinormal spectra in Schwarzschild black hole perturbation theory: just how general should it be? Phys. Lett. A383, 2571 (2019)

    work page 2019

  25. [25]

    Glampedakis, K., Johnson, A.D., Kennefick, D.: The Darboux transformation in black hole perturbation theory. Phys. Rev. D96, 024036 (2017)

    work page 2017

  26. [26]

    University of Chicago Press, Chicago (1984)

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

    work page 1984

  27. [27]

    Springer, Berlin (1991)

    Matveev, V.B., Salle, M.A.:Darboux Transformation and Solitons. Springer, Berlin (1991)

    work page 1991

  28. [28]

    Brizuela, D., Martín-García, J.M.: Hamiltonian theory for the axial perturbations of a dynamical spherical background. Class. Quantum Grav.26, 015003 (2009)

    work page 2009

  29. [29]

    http://hdl.handle.net/10486/121, Ph.D thesis, Univ

    Brizuela, D.: High-order perturbation theory of spherical spacetimes with application to vacuum and perfect fluid matter. http://hdl.handle.net/10486/121, Ph.D thesis, Univ. Autónoma de Madrid, Madrid (2008). Accessed 29 December 2025

    work page 2008

  30. [30]

    Brizuela, D., Martín-García, J.M., Mena Marugán, G.A.: Second and higher-order perturbations of a spherical spacetime. Phys. Rev. D74, 044039 (2006)

    work page 2006

  31. [31]

    Brizuela, D., Martín-García, J.M., Mena Marugán, G.A.: High-order gauge-invariant perturbations of a spherical space- time. Phys. Rev. D76, 024004 (2007)

    work page 2007

  32. [32]

    Kantowski, R., Sachs, R.K.: Some spatially inhomogeneous dust models. J. Math. Phys.7, 443 (1966)

    work page 1966

  33. [33]

    Weber, E.: Kantowski-Sachs cosmological models as big-bang models. J. Math. Phys.26, 1308 (1985)

    work page 1985

  34. [34]

    Adhav, K.S., Mete, V.G., Nimkar, A.S., Pund, A.M.: Kantowski-Sachs cosmological model in general theory of relativity. Int. J. Theor. Phys.47, 2314 (2008)

    work page 2008

  35. [35]

    Ashtekar, A., Olmedo, J., Singh, P.: Quantum transfiguration of Kruskal black holes. Phys. Rev. Lett.121, 241301 (2018)

    work page 2018

  36. [36]

    Ashtekar, A., Olmedo, J., Singh, P.: Quantum extension of the Kruskal spacetime. Phys. Rev. D98, 126003 (2018)

    work page 2018

  37. [37]

    Ashtekar, A., Olmedo, J.: Properties of a recent quantum extension of the Kruskal geometry. Int. J. Mod. Phys. D29, 2050076 (2020)

    work page 2020

  38. [38]

    Elizaga Navascués, B., García-Quismondo, A., Mena Marugán, G.A.: Space of solutions of the Ashtekar-Olmedo-Singh effective black hole model. Phys. Rev. D106, 063516 (2022)

    work page 2022

  39. [39]

    Elizaga Navacués, B., Mena Marugán, G.A., Mínguez-Sánchez, A.: Extended phase space quantization of a black hole interior model in loop quantum cosmology. Phys. Rev. D108, 106001 (2023)

    work page 2023

  40. [40]

    Mena Marugán, G.A., Mínguez-Sánchez, A.: Axial perturbations in Kantowski-Sachs spacetimes and hybrid quantum cosmology. Phys. Rev. D109, 106009 (2024)

    work page 2024

  41. [41]

    Mena Marugán, G.A., Mínguez-Sánchez, A.: Polar perturbations in Kantowski-Sachs spacetimes and hybrid quantum cosmology. Phys. Rev. D111, 086024 (2025)

    work page 2025

  42. [42]

    Lenzi, M., Mena Marugán, G.A., Mínguez-Sánchez, A., Sopuerta, C.F.: Master variables and Darboux symmetry for axial perturbations of the exterior and interior of black hole spacetimes. Phys. Rev. D113, 064039 (2026)

    work page 2026

  43. [43]

    arXiv:2512.10692 (2025)

    Lenzi, M., Mena Marugán, G.A., Mínguez-Sánchez, A.: Master functions and hybrid quantization of perturbed nonrotating black hole interiors. arXiv:2512.10692 (2025)

    work page arXiv 2025

  44. [44]

    Arnowitt, R., Deser, S., Misner, C.W.: Dynamical structure and definition of energy in general relativity. Phys. Rev. D 116, 1322 (1959)

    work page 1959

  45. [45]

    Cunningham, C.T., Price, R.H., Moncrief, V.: Radiation from collapsing relativistic stars. II. Linearized even parity radiation. Astrophys. J.230, 870 (1979)

    work page 1979

  46. [46]

    Castelló Gomar, L., Martín-Benito, M., Mena Marugán, G.A.: Gauge-invariant perturbations in hybrid quantum cosmol- ogy. J. Cosmol. Astropart. Phys.06, 045 (2015)

    work page 2015

  47. [47]

    Phys Rev

    Livine, E.R., Yokokura, Y.: Effective dynamics of spherically symmetric static spacetime. Phys Rev. D112, 104034 (2025)

    work page 2025

  48. [48]

    Universe12, 59 (2026)

    Koch, B., Riahinia, A.: Quantum uncertainties of static spherically symmetric spacetimes. Universe12, 59 (2026)

    work page 2026

  49. [49]

    Endlich, S., Gorbenko, V., Huang, J., Senatore, L.: An effective formalism for testing extensions to general relativity with gravitational waves. J. High Energ. Phys.2017, 122 (2017)

    work page 2017

  50. [50]

    Cardoso, V., Kimura, M., Maselli, A., Senatore, L.: Black holes in an effective field theory extension of general relativity. Phys. Rev. Lett.121, 251105 (2018); Erratum: Phys. Rev. Lett.131, 109903 (2023)

    work page 2018

  51. [51]

    Silva, H.O., Tambalo, G., Glampedakis, K., Yagi, K., Steinhoff, J.: Quasinormal modes and their excitation beyond general relativity. Phys. Rev. D110, 024042 (2024)

    work page 2024

  52. [52]

    Pezzella, L., Destounis, K., Maselli, A., Cardoso, V.: Quasinormal modes of black holes embedded in halos of matter. Phys. Rev. D111, 064026 (2025)

    work page 2025

  53. [53]

    Ashtekar, A., Singh, P.: Loop quantum cosmology: a status report. Class. Quantum Grav.28, 213001 (2011)

    work page 2011

  54. [54]

    Ashtekar, A., Bianchi, E.: A short review of loop quantum gravity. Rep. Prog. Phys.84, 042001 (2021)

    work page 2021

  55. [55]

    Ashtekar, A., Lewandowski, J.: Background independent quantum gravity: a status report. Class. Quantum Grav.21, R53 (2004)

    work page 2004

  56. [56]

    Cambridge University Press, Cambridge, UK (2007)

    Thiemann, T.:Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge, UK (2007)

    work page 2007

  57. [57]

    Velhinho, J.M.: The quantum configuration space of loop quantum cosmology. Class. Quantum Grav.24, 3745 (2007)

    work page 2007

  58. [58]

    Elizaga Navascués, B., Mena Marugán, G.A.: Hybrid loop quantum cosmology: an overview. Front. Astron. Space Sci.8, 624824 (2021). 20

    work page 2021

  59. [59]

    Mathematics11, 3922 (2023)

    Cortez, J., Elizaga Navacués, B., Mena Marugán, G.A., Torres-Caballeros, A., Velhinho, J.M.: Fock quantization of a Klein-Gordon field in the interior geometry of a non-rotating black hole. Mathematics11, 3922 (2023)

    work page 2023

  60. [60]

    Cortez, J., Mena Marugán, G.A., Torres-Caballeros, A., Velhinho, J.M.: Time-dependent scalings and Fock quantization of a massless scalar field in Kantowski-Sachs. Class. Quantum Grav.41, 195002 (2024)

    work page 2024