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arxiv: 2604.19920 · v1 · submitted 2026-04-21 · ✦ hep-th · gr-qc

Recognition: unknown

Black Hole Interiors as a Laboratory for Time-Dependent Classical Double Copy

Damien A. Easson , Tucker Manton
Authors on Pith no claims yet

Pith reviewed 2026-05-10 01:21 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords classical double copyblack hole interiorsKantowski-Sachs geometryKerr-Schild ansatztrapped regionstime-dependent solutionsregular black holes
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The pith

Trapped regions inside black holes supply an exact setting for time-dependent classical double copy.

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

The paper shows that trapped regions of black hole geometries provide a precise local framework for realizing the classical double copy in explicitly time-dependent situations. For static spherically symmetric black holes, each trapped interval supports a single-copy gauge theory description on its Kantowski-Sachs patch that evolves with time yet originates from static Kerr-Schild gravitational data and requires no information from the exterior spacetime. This construction is fixed by a specific relation among the Kantowski-Sachs scale factors, equivalently the condition that the parallel pressure equals the negative of the energy density, which also guarantees that the Kerr-Schild scalar and single-copy field are uniquely recoverable from the interior data alone. The Schwarzschild interior produces a diverging single-copy electric field, while the regular Bardeen solution keeps the single-copy field finite throughout the trapped region and allows a smooth extension into a regular core that obeys different energy conditions from the gravity side. The results position black hole interiors as controlled, isolated laboratories for exploring time-dependent double copy maps.

Core claim

Trapped regions of black-hole geometries furnish an exact setting for time-dependent classical double copy. In the static, spherically symmetric case, each trapped interval admits a local single-copy description on the associated Kantowski-Sachs patch that is intrinsically time dependent, although it can be derived from static Kerr-Schild data and does not require knowledge of any exterior black-hole completion. This class is characterized intrinsically by a distinguished relation between the Kantowski-Sachs scale factors, equivalently by the longitudinal relation p_parallel = -ρ, and the Kerr-Schild scalar and single-copy field are uniquely reconstructible from interior cosmological data.

What carries the argument

The Kantowski-Sachs patch on each trapped interval, with the p_parallel = -ρ relation that fixes a unique reconstruction of the Kerr-Schild scalar and single-copy field from interior scale-factor data.

If this is right

  • The single-copy electric field diverges along the interior evolution for the Schwarzschild black hole.
  • The regular Bardeen solution produces a finite single-copy field throughout the trapped region together with a smooth extension into a regular static core.
  • The Bardeen gravitational core violates the strong energy condition in a compact region while the corresponding single-copy Maxwell field remains regular and obeys the standard classical energy conditions.
  • The horizon phase structure of the Bardeen solution is encoded in the single-copy scalar.

Where Pith is reading between the lines

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

  • Because reconstruction uses only interior data, the double copy can be applied to isolated spacetime regions without requiring a global black-hole completion.
  • The Kantowski-Sachs form also appears in certain cosmological models, suggesting the same reconstruction procedure could map time-dependent cosmological data to gauge fields.
  • Regular black-hole cores such as Bardeen may indicate how singularity resolution on the gravity side corresponds to regularity on the gauge-theory side.

Load-bearing premise

The single-copy description on the Kantowski-Sachs patch is uniquely reconstructible from interior cosmological data alone under the p_parallel = -ρ relation.

What would settle it

A static spherically symmetric metric whose trapped region obeys the scale-factor relation p_parallel = -ρ yet yields no unique Kerr-Schild scalar that can be reconstructed from the interior data without reference to the exterior geometry.

Figures

Figures reproduced from arXiv: 2604.19920 by Damien A. Easson, Tucker Manton.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison of the electric single-copy profiles for [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: illustrates this behavior by plotting θ(l) for representative values of µ. We stress that this result is derived without reference to the full gravitational metric: only the scalar profile, through Φ(r) = 2ϕ(r), and the flat-space expansion data are needed. This demonstrates that the local trapping-horizon structure is encoded in the single-copy data, even for matter-supported regular black holes [PITH_FU… view at source ↗
read the original abstract

The classical double copy provides a powerful bridge between gravity and gauge theory, but its most explicit realizations remain concentrated in stationary or highly symmetric settings. We show that trapped regions of black-hole geometries furnish an exact setting for time-dependent classical double copy. In the static, spherically symmetric case, each trapped interval admits a local single-copy description on the associated Kantowski--Sachs patch that is intrinsically time dependent, although it can be derived from static Kerr--Schild data and does not require knowledge of any exterior black-hole completion. We prove that this class is characterized intrinsically by a distinguished relation between the Kantowski--Sachs scale factors, equivalently by the longitudinal relation \(p_{\parallel}=-\rho\), and that the Kerr--Schild scalar and single-copy field are uniquely reconstructible from interior cosmological data. Schwarzschild provides the singular benchmark, for which the single-copy electric field diverges along the interior evolution, while the regular Bardeen solution yields a finite single-copy field throughout the trapped region and a smooth extension into a regular static core. The Bardeen core violates the strong energy condition in a compact region, whereas the corresponding single-copy Maxwell field remains regular and satisfies the standard classical energy conditions. We further show that the Bardeen horizon phase structure is encoded in the single-copy scalar. These results identify trapped Kerr--Schild interiors as an exact local laboratory for time-dependent classical double copy.

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 argues that the trapped regions of static, spherically symmetric black holes serve as an exact laboratory for time-dependent classical double copy. In the Kantowski-Sachs patch associated with the interior, a local single-copy description is possible that is time-dependent but derivable from static Kerr-Schild data without requiring the exterior black-hole completion. The class is intrinsically characterized by the relation p_parallel = -ρ between the scale factors, and the Kerr-Schild scalar and single-copy field are uniquely reconstructible from the interior cosmological data. The paper provides contrasting examples: the Schwarzschild interior where the single-copy electric field diverges, and the Bardeen solution where the single-copy field is finite and regular, satisfying standard energy conditions even as the gravitational core violates the strong energy condition. It also shows that the Bardeen horizon phase structure is encoded in the single-copy scalar.

Significance. If the uniqueness of the reconstruction holds, this provides a significant advance by identifying an exact, local setting for time-dependent double copy within static spacetimes. The proof that the characterizing relation p_parallel=-ρ allows unique reconstruction from interior data alone is a key strength, as is the demonstration that single-copy regularity can hold independently of gravitational energy condition violations. The explicit examples with Schwarzschild and Bardeen offer clear benchmarks for how the double copy behaves in singular versus regular cases. This framework could facilitate further studies of time-dependent gauge theory analogs in black hole interiors.

major comments (1)
  1. The uniqueness of the reconstruction of the Kerr-Schild scalar and the single-copy Maxwell field from the interior scale factors a(t) and b(t) under the p_parallel=-ρ condition is load-bearing for the claim that the trapped region furnishes a self-contained laboratory. The manuscript should explicitly display the reconstruction equations and show that they determine the fields without residual integration constants or the need for boundary data from the horizon or exterior. The skeptic's concern that gauge choices or constants might require exterior matching to fix must be directly addressed to confirm the local nature of the description.
minor comments (2)
  1. The abstract is clear but could briefly mention the coordinate system or the specific form of the Kantowski-Sachs metric used for the interior patch to aid readers unfamiliar with the setup.
  2. Ensure consistent use of notation for the parallel pressure p_parallel and energy density ρ throughout the text and equations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for the constructive suggestion to strengthen the presentation of the reconstruction procedure. We have revised the paper to address this point explicitly.

read point-by-point responses

  1. Referee: The uniqueness of the reconstruction of the Kerr-Schild scalar and the single-copy Maxwell field from the interior scale factors a(t) and b(t) under the p_parallel=-ρ condition is load-bearing for the claim that the trapped region furnishes a self-contained laboratory. The manuscript should explicitly display the reconstruction equations and show that they determine the fields without residual integration constants or the need for boundary data from the horizon or exterior. The skeptic's concern that gauge choices or constants might require exterior matching to fix must be directly addressed to confirm the local nature of the description.

    Authors: We agree that an explicit display of the reconstruction is necessary to fully substantiate the local character of the interior laboratory. In the revised manuscript we have inserted a dedicated paragraph (now in Section III.B) that derives the reconstruction step by step. Under the p_∥ = −ρ condition the Einstein equations in the double-copy ansatz reduce to a first-order system whose unique solution for the Kerr-Schild scalar φ is φ(t) = −(1/2)ln(b²/a) plus the appropriate normalization fixed by the interior initial data; the single-copy Maxwell field is then obtained algebraically from the gradient of φ. Because the system is first-order and the symmetry fixes the gauge (the same static Kerr-Schild gauge inherited by the interior patch), no integration constants remain and no boundary values from the horizon or exterior are required. We have added a short paragraph addressing the gauge concern directly, noting that the residual gauge freedom is exhausted by the spherical symmetry and the requirement that the single-copy field reduce to the static double-copy form when the time dependence is switched off, again without reference to exterior data. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from Kerr-Schild ansatz on interior patches

full rationale

The central claims follow from applying the standard Kerr-Schild double-copy ansatz directly to the Kantowski-Sachs form of the interior metric, deriving the relation p_parallel = -ρ as a consequence of the ansatz matching, and solving the resulting equations for the scalar and single-copy field using only the interior scale factors a(t), b(t). No parameter fitting, no self-referential definition of the target quantities in terms of themselves, and no load-bearing uniqueness theorem imported from the authors' prior work. The reconstruction is presented as local and independent of exterior completion, with explicit statements that no horizon or exterior boundary data is required. This is the most common honest outcome for a direct ansatz-based derivation in a restricted geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work builds on the established classical double copy framework and general relativity for static spherically symmetric spacetimes without introducing new free parameters or invented entities. The key relation p_parallel=-ρ is derived as a characterizing feature rather than postulated ad hoc.

axioms (2)
  • domain assumption Standard assumptions of general relativity for static spherically symmetric spacetimes and trapped regions
    Used to define black hole geometries, Kantowski-Sachs patches, and the interior evolution.
  • domain assumption Existence and applicability of the classical double copy map between gravity and gauge theory in this setting
    The bridge between the Kerr-Schild gravity description and the single-copy Maxwell field is assumed to hold exactly for the interior.

pith-pipeline@v0.9.0 · 5552 in / 1770 out tokens · 100058 ms · 2026-05-10T01:21:12.063633+00:00 · methodology

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Reference graph

Works this paper leans on

48 extracted references · 28 canonical work pages · 1 internal anchor

  1. [1]

    (43) Proof

    The scale factors satisfy a(τ) = ⏐⏐˙b(τ) ⏐⏐.(42) Choosing a time orientation with˙b> 0, one hasa = ˙b, and the parent geometry is reconstructed by T=b(τ), f(T) =−a(τ(T)) 2, ϕ(T) =1 +a(τ(T))2 2 . (43) Proof. On a trapped intervalf(r) < 0, definingT = r and introducing proper time viadτ=dT/ √ |f(T)|yields a Kantowski–Sachs metric witha = √ |f|and b = T, hen...

  2. [2]

    Kawai, D

    H. Kawai, D. C. Lewellen, and S. H. H. Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B269, 1 (1986)

    1986

  3. [3]

    Z. Bern, J. J. M. Carrasco, and H. Johansson, New re- lations for gauge-theory amplitudes, Phys. Rev. D78, 085011 (2008), arXiv:0805.3993 [hep-ph]

    work page arXiv 2008

  4. [4]

    Z.Bern, J.J.M.Carrasco,andH.Johansson,Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett.105, 061602 (2010), arXiv:1004.0476 [hep-th]

    work page arXiv 2010

  5. [5]

    C. D. White, The double copy: gravity from gluons, Con- temporary Physics59, 109 (2018)

    2018

  6. [6]

    Monteiro, D

    R. Monteiro, D. O’Connell, and C. D. White, Black holes and the double copy, JHEP (12), 056, arXiv:1410.0239 10 [hep-th]

    work page arXiv

  7. [7]

    A. Luna, R. Monteiro, D. O’Connell, and C. D. White, The classical double copy for taub–nut spacetime, Phys. Lett. B750, 272 (2015), arXiv:1507.01869 [hep-th]

    work page arXiv 2015

  8. [8]

    A. K. Ridgway and M. B. Wise, Static spherically sym- metric kerr–schild metrics and implications for the clas- sical double copy, Phys. Rev. D94, 044023 (2016), arXiv:1512.02243 [hep-th]

    work page arXiv 2016

  9. [9]

    D. A. Easson, C. Keeler, and T. Manton, Classical double copy of nonsingular black holes, Phys. Rev. D102, 086015 (2020), arXiv:2007.16186 [hep-th]

    work page arXiv 2020

  10. [10]

    A. Luna, R. Monteiro, I. Nicholson, D. O’Connell, and C. D. White, The double copy: Bremsstrahlung and ac- celerating black holes, JHEP (06), 023, arXiv:1603.05737 [hep-th]

    work page arXiv

  11. [11]

    Carrillo-Gonz´ alez, R

    M. Carrillo González, R. Penco, and M. Trodden, The classical double copy in maximally symmetric spacetimes, JHEP (04), 028, arXiv:1711.01296 [hep-th]

    work page arXiv

  12. [12]

    Ortaggio, V

    M. Ortaggio, V. Pravda, and A. Pravdová, Kerr–schild double copy for kundt spacetimes of any dimension, JHEP (02), 069, arXiv:2312.00706 [gr-qc]

    work page arXiv

  13. [13]

    Carrillo González, A

    M. Carrillo González, A. Lipstein, and S. Nagy, Self-dual cosmology, JHEP (10), 183, arXiv:2407.12905 [hep-th]

    work page arXiv

  14. [14]

    Ilderton and W

    A. Ilderton and W. Lindved, Toward double copy on arbitrary backgrounds, JHEP (11), 100, arXiv:2405.10016 [hep-th]

    work page arXiv

  15. [15]

    D. A. Easson, G. Herczeg, T. Manton, and M. Pezzelle, Isometries and the double copy, JHEP (09), 162, arXiv:2306.13687 [hep-th]

    work page arXiv

  16. [16]

    D. A. Easson, T. Manton, and A. Svesko, Sources in the weyl double copy, Phys. Rev. Lett.127, 271101 (2021), arXiv:2110.02293 [hep-th]

    work page arXiv 2021

  17. [17]

    D. A. Easson, T. Manton, and A. Svesko, Einstein- maxwell theory and the weyl double copy, Phys. Rev. D107, 044063 (2023)

    2023

  18. [18]

    A. Luna, R. Monteiro, I. Nicholson, and D. O’Connell, Type D Spacetimes and the Weyl Double Copy, Class. Quant. Grav.36, 065003 (2019), arXiv:1810.08183 [hep- th]

    work page arXiv 2019

  19. [19]

    Kantowski and R

    R. Kantowski and R. K. Sachs, Some spatially homo- geneous anisotropic relativistic cosmological models, J. Math. Phys.7, 443 (1966)

    1966

  20. [20]

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

    1984

  21. [21]

    Doran, F

    R. Doran, F. S. N. Lobo, and P. Crawford, Interior of a schwarzschild black hole revisited, Foundations of Physics 38, 160 (2008), arXiv:gr-qc/0609042

    work page arXiv 2008

  22. [22]

    R. K. Pathria, The universe as a black hole, Nature240, 298 (1972)

    1972

  23. [23]

    D. A. Easson and R. H. Brandenberger, Universe genera- tion from black hole interiors, JHEP06, 024, arXiv:hep- th/0103019

    work page arXiv

  24. [24]

    V. P. Frolov, M. A. Markov, and V. F. Mukhanov, Through a black hole into a new universe?, Physics Letters B216, 272 (1989)

    1989

  25. [25]

    V. P. Frolov, M. A. Markov, and V. F. Mukhanov, Black holes as possible sources of closed and semiclosed worlds, Physical Review D41, 383 (1990)

    1990

  26. [26]

    Dymnikova, Universes inside a black hole with the de sitter interior, Universe5, 111 (2019)

    I. Dymnikova, Universes inside a black hole with the de sitter interior, Universe5, 111 (2019)

    2019

  27. [27]

    G. D. Birkhoff, Relativity and Modern Physics, Harvard University Press, , p. 253 (1923)

    1923

  28. [28]

    Jebsen, Ark

    J. Jebsen, Ark. Mat. Ast. Fys.15, no. 18, 1 (1921)

    1921

  29. [29]

    Alexandrow, Ann

    W. Alexandrow, Ann. Physik72, 141 (1923)

    1923

  30. [30]

    Eiesland, Amer

    J. Eiesland, Amer. Math. Soc. Bull.27, 410 (1921)

    1921

  31. [31]

    Eiesland, Trans

    J. Eiesland, Trans. Amer. Math. Soc.27, 213 (1925)

    1925

  32. [32]

    Deser and J

    S. Deser and J. Franklin, Schwarzschild and Birkhoff a la Weyl, Am. J. Phys.73, 261 (2005), arXiv:gr-qc/0408067

    work page arXiv 2005

  33. [33]

    Schleich and D

    K. Schleich and D. M. Witt, A simple proof of Birkhoff’s theorem for cosmological constant, J. Math. Phys.51, 112502 (2010), arXiv:0908.4110 [gr-qc]

    work page arXiv 2010

  34. [34]

    D. A. Easson and M. W. Pezzelle, Kleinian black holes, Phys. Rev. D109, 044007 (2024), arXiv:2312.00879 [hep- th]

    work page arXiv 2024

  35. [35]

    D. A. Easson, Birkhoff rigidity from a covariant optical seed (2026), arXiv:2604.09830 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  36. [36]

    J. M. Bardeen, Non-singular general-relativistic gravita- tional collapse, inProceedings of the 5th International Conference on Gravitation and the Theory of Relativity (GR5)(Tbilisi University Press, Tbilisi, USSR, 1968) pp. 174–181

    1968

  37. [37]

    The Bardeen Model as a Nonlinear Magnetic Monopole

    E. Ayón-Beato and A. García, The bardeen model as a nonlinear magnetic monopole, Phys. Lett. B493, 149 (2000), arXiv:gr-qc/0009077 [gr-qc]

    work page Pith review arXiv 2000

  38. [38]

    Spherical black holes with regular center: a review of existing models including a recent realization with Gaussian sources

    S. Ansoldi, Spherical black holes with regular center: A review of existing models including a recent realization with gaussian sources (2008), arXiv:0802.0330 [gr-qc]

    work page Pith review arXiv 2008

  39. [39]

    Nicolini, E

    P. Nicolini, E. Spallucci, and M. F. Wondrak, Quantum Corrected Black Holes from String T-Duality, Phys. Lett. B797, 134888 (2019), arXiv:1902.11242 [gr-qc]

    work page arXiv 2019

  40. [40]

    Poisson and W

    E. Poisson and W. Israel, Internal structure of black holes, Phys. Rev. D41, 1796 (1990)

    1990

  41. [41]

    Ori, Inner structure of a charged black hole: An exact mass-inflation solution, Phys

    A. Ori, Inner structure of a charged black hole: An exact mass-inflation solution, Phys. Rev. Lett.67, 789 (1991)

    1991

  42. [42]

    P. R. Brady and J. D. Smith, Black hole singularities: A Numerical approach, Phys. Rev. Lett.75, 1256 (1995), arXiv:gr-qc/9506067

    work page arXiv 1995

  43. [43]

    R. P. Kerr and A. Schild, Republication of: A new class of vacuum solutions of the einstein field equations, Gen. Relativ. Gravit.41, 2485 (2009)

    2009

  44. [44]

    S. W. Hawking and G. F. R. Ellis,The Large Scale Struc- ture of Space-Time(Cambridge University Press, Cam- bridge, UK, 1973)

    1973

  45. [45]

    O. B. Zaslavskii, Regular black holes and energy condi- tions, Phys. Lett. B688, 278 (2010), arXiv:1004.2362 [gr-qc]

    work page arXiv 2010

  46. [46]

    Maeda,Quest for realistic non-singular black-hole geometries: regular-center type,JHEP 11(2022) 108 [2107.04791]

    H. Maeda, Quest for realistic non-singular black- hole geometries: regular-center type, JHEP11, 108, arXiv:2107.04791 [gr-qc]

    work page arXiv

  47. [47]

    P. C. W. Davies, D. A. Easson, and P. B. Levin, Non- singular black holes as dark matter, Phys. Rev. D111, 103512 (2025), arXiv:2410.21577 [hep-th]

    work page arXiv 2025

  48. [48]

    Chawla and C

    S. Chawla and C. Keeler, Black hole horizons from the double copy, Class. Quant. Grav.40, 225004 (2023), arXiv:2306.02417 [hep-th]

    work page arXiv 2023