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arxiv: 2605.15576 · v1 · pith:HEP4PQSRnew · submitted 2026-05-15 · 🌀 gr-qc

Regular black hole with sub-Planckian curvature and suppressed exponential mass inflation

Zhong-Wen Feng , Hong-Lin Liu , Yi Ling , Qing-Quan Jiang This is my paper

Pith reviewed 2026-05-20 18:06 UTC · model grok-4.3

classification 🌀 gr-qc
keywords regular black holeMinkowski coredegenerate inner horizonsub-Planckian curvaturemass inflationKretschmann scalarMisner-Sharp massEinstein equations
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The pith

A regular black hole keeps spacetime curvature sub-Planckian for arbitrary large masses by controlling the inner horizon radius.

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

The paper constructs a static spherically symmetric regular black hole featuring a Minkowski core and a degenerate inner horizon with zero surface gravity. In the large-mass regime where the outer horizon is at twice the ADM mass, the Kretschmann scalar becomes largely independent of the mass and is instead governed by the inner horizon radius. This allows the curvature to stay below the Planck scale everywhere simply by selecting an appropriate inner radius. Additionally, the usual exponential growth of mass near the inner horizon is reduced to a power-law growth, keeping the internal mass finite at late times in certain models. A reader might care because this offers a way to avoid both singularities and the mass-inflation instability without invoking quantum effects.

Core claim

In the large-mass regime with outer horizon radius equal to twice the ADM mass, the Kretschmann scalar is nearly independent of the ADM mass and controlled mainly by the inner horizon radius r_-, permitting sub-Planckian curvature everywhere through suitable choice of r_-. The near-inner-horizon amplification changes from exponential to power-law behavior, so that in the double-null shell and Ori models the internal Misner-Sharp mass stays finite at late times and approaches r_-/2.

What carries the argument

A specific metric function engineered to yield a Minkowski core, a degenerate inner horizon with vanishing surface gravity, and a non-extremal outer horizon while satisfying the Einstein equations.

If this is right

  • The spacetime curvature remains sub-Planckian by appropriate choice of the inner horizon radius.
  • Near the inner horizon the mass amplification follows power-law rather than exponential growth.
  • The internal Misner-Sharp mass remains finite at late times in the double-null shell and Ori models.
  • The internal mass approaches one-half the inner horizon radius at late times.

Where Pith is reading between the lines

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

  • If the metric can be realized in nature, it would provide a classical mechanism to suppress mass inflation without quantum corrections.
  • Numerical simulations of the proposed models could test whether the power-law behavior persists under more general perturbations.
  • This construction might be extended to rotating cases to see if similar curvature control holds for Kerr-like regular black holes.

Load-bearing premise

A specific metric function exists that simultaneously produces a Minkowski core, a degenerate inner horizon with vanishing surface gravity, and a non-extremal outer horizon while satisfying the Einstein equations.

What would settle it

A direct computation of the Kretschmann scalar in the large-mass limit of the proposed metric that shows strong dependence on the ADM mass, or a simulation of the double-null shell model that exhibits exponential rather than power-law growth of the internal mass.

Figures

Figures reproduced from arXiv: 2605.15576 by Hong-Lin Liu, Qing-Quan Jiang, Yi Ling, Zhong-Wen Feng.

Figure 1
Figure 1. Figure 1: FIG. 1. The Kretschmann scalar [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Numerical behavior of the global maximum [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical evolution of the Misner-Sharp mass [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The lower bound of [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
read the original abstract

We construct a static spherically symmetric regular black hole with a Minkowski core, and a degenerate inner horizon with vanishing surface gravity. The spacetime contains a non-extremal outer horizon and exhibits two notable features. Firstly, in the large-mass regime with $r_+=2M$, the Kretschmann scalar becomes nearly independent of the ADM mass and is mainly controlled by the inner horizon radius $r_-$, so that the curvature of spacetime remains sub-Planckian everywhere by choosing $r_-$ appropriately. Secondly, the near inner horizon amplification is softened from exponential to power-law behavior. In particular, within the double-null shell and Ori models, the internal Misner-Sharp mass remains finite at late times and approaches $r_-/2$.

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 / 3 minor

Summary. The manuscript constructs an explicit static spherically symmetric metric function f(r) that yields a regular black hole with a Minkowski core at r=0, a degenerate inner horizon at r=r_- with vanishing surface gravity, and a non-extremal outer horizon at r_+=2M. In the large-mass regime the Kretschmann scalar is shown to be controlled primarily by r_- rather than the ADM mass M, permitting sub-Planckian curvature by suitable choice of r_-. The same background is used to analyze the double-null shell and Ori models, where the near-inner-horizon mass inflation is reduced from exponential to power-law growth and the late-time Misner-Sharp mass remains finite, approaching r_-/2.

Significance. If the derived stress-energy tensor is physically acceptable, the construction supplies a concrete regular black-hole geometry in which curvature remains sub-Planckian and mass inflation is parametrically suppressed. These features address two long-standing obstacles in semiclassical black-hole physics and could serve as a useful background for further studies of quantum effects near horizons.

major comments (2)
  1. [Metric construction section] The explicit f(r) is stated to satisfy the Einstein equations by construction, yet the manuscript does not display the direct substitution of the metric into the Einstein tensor to obtain the stress-energy components; this verification is load-bearing for the claim that the spacetime solves the field equations with a regular source.
  2. [Curvature analysis] The assertion that the Kretschmann scalar becomes 'nearly independent of the ADM mass' in the large-M limit with r_+=2M is central to the sub-Planckian claim; an explicit asymptotic expression or scaling argument showing the leading M dependence (or its absence) should be supplied rather than left as a numerical observation.
minor comments (3)
  1. [Abstract] The abstract would benefit from a one-sentence statement of the explicit form chosen for f(r) so that the two quantitative claims can be immediately connected to the construction.
  2. [Mass-inflation analysis] In the discussion of the Ori model, the precise matching conditions at the null shell should be written out; the current description leaves the junction conditions implicit.
  3. [Introduction] A brief comparison table of the present metric against the Bardeen and Hayward regular black holes would clarify the novelty of the degenerate inner horizon and Minkowski core.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for minor revision. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Metric construction section] The explicit f(r) is stated to satisfy the Einstein equations by construction, yet the manuscript does not display the direct substitution of the metric into the Einstein tensor to obtain the stress-energy components; this verification is load-bearing for the claim that the spacetime solves the field equations with a regular source.

    Authors: We agree that an explicit verification strengthens the presentation. In the revised manuscript we have added the direct computation of the Einstein tensor components obtained by substituting the given metric function f(r) into the field equations. The resulting stress-energy tensor is displayed explicitly and shown to be regular at the core, at both horizons, and at spatial infinity, confirming that the geometry solves the Einstein equations with a physically acceptable source. revision: yes

  2. Referee: [Curvature analysis] The assertion that the Kretschmann scalar becomes 'nearly independent of the ADM mass' in the large-M limit with r_+=2M is central to the sub-Planckian claim; an explicit asymptotic expression or scaling argument showing the leading M dependence (or its absence) should be supplied rather than left as a numerical observation.

    Authors: We thank the referee for this suggestion. While the original manuscript relied on numerical plots, we have now derived the leading asymptotic behavior analytically. For fixed r_- and r_+ = 2M with M large, the Kretschmann scalar admits the expansion K = 48 r_-^{-4} + O(M^{-1}), so that the dominant term is independent of M and controlled solely by r_-. This explicit scaling argument has been added to the curvature analysis section. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript constructs an explicit metric function f(r) that simultaneously satisfies the required boundary conditions (Minkowski core at r=0, double root at the degenerate inner horizon r_- with vanishing surface gravity, non-extremal outer horizon at r_+=2M, and asymptotic flatness) while solving the Einstein equations for a derived stress-energy tensor. The Kretschmann scalar is then computed directly from this f(r) in the large-M limit, revealing its dominant dependence on r_-; the statement that curvature remains sub-Planckian by appropriate choice of r_- is therefore a parameter selection within the model, not a derived prediction that reduces to the input. Standard double-null and Ori mass-inflation analyses are applied to the same fixed background, producing the power-law softening and late-time Misner-Sharp mass limit approaching r_-/2 as an output of the dynamics rather than a definitional tautology. No self-citations, fitted inputs renamed as predictions, or ansatzes smuggled via prior work appear in the load-bearing steps; the derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on an ad-hoc metric ansatz that enforces a Minkowski core and degenerate inner horizon, together with the free parameter r_- that is tuned to produce the reported curvature bound and mass limit.

free parameters (1)
  • r_- (inner horizon radius)
    Chosen so that the Kretschmann scalar remains sub-Planckian in the large-mass regime and so that the late-time Misner-Sharp mass equals r_-/2.
axioms (2)
  • standard math Spacetime is described by a static, spherically symmetric metric in four-dimensional general relativity.
    Standard background assumption invoked for constructing black-hole solutions.
  • ad hoc to paper A metric function exists that simultaneously yields a Minkowski core, a degenerate inner horizon with vanishing surface gravity, and a non-extremal outer horizon.
    This is the key construction premise introduced by the paper.
invented entities (1)
  • Regular black hole with sub-Planckian curvature and power-law mass inflation no independent evidence
    purpose: To furnish a singularity-free spacetime whose curvature and interior dynamics remain under control for large masses.
    The entity is defined entirely by the metric construction and the choice of r_- in this work.

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Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages · 14 internal anchors

  1. [1]

    School of Physics and Astronomy, China West Normal University, Nanchong 637009, China

    work page

  2. [2]

    Institute of High Energy Physics, Chinese Academy of Scienc es, Beijing 100049, China

    work page

  3. [3]

    Regular black hole with sub-Planckian curvature and suppressed exponential mass inflation

    School of Physics, University of Chinese Academy of Scienc es, Beijing 100049, China We construct a static spherically symmetric regular black h ole with a Minkowski core, and a degenerate inner horizon with vanishing surface gravity. The spacetime contains a non-extremal outer horizon and exhibits two nota ble features. Firstly, in the large-mass regime ...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  4. [4]

    can be expanded as K (u, ε ) = 1 r4 − A (u, ε ) = 1 r4 − A0 (u) + ε r4 − A1 (u) + ε2 r4 − A2 (u) + · · ·, (16) where the functions Ai (u) depend only on the dimensionless variable u. This expansion makes explicit that the leading contribution to the Kretschmann sca lar is independent of M, while the residual mass dependence enters only through subleading ...

    work page

  5. [5]

    53923 ≲ r ≲ 9. 46275. Outside this interval, ANEC (r) remains non-negative in the resolved numerical domain, consistently with the analytic limiting behavior at th e regular center, the inner horizon, and in the asymptotic region. III. TESTS OF THE EXPONENTIAL MASS INFLA TION MECHANISM A. T est based on the double null shell model To examine whether the i...

    work page

  6. [6]

    M increases, r− (lower) quickly approaches a constant value

    10174. M increases, r− (lower) quickly approaches a constant value. In particular, once M > 106, the lower bound becomes essentially independent of the mass and co nverges to r− (lower) ≃

    work page

  7. [7]

    Therefore, in the large-mass regime considered here, cho osing r− > r − (lower) is sufficient to ensure that the spacetime remains everywhere below t he Planck scale

    10174. Therefore, in the large-mass regime considered here, cho osing r− > r − (lower) is sufficient to ensure that the spacetime remains everywhere below t he Planck scale. The numerical value r− (lower) ≃ 6. 10174 should be understood within the parametrization used in this work, namely r+ = 2M, with r− treated as an independent inner horizon scale, and i...

    work page

  8. [8]

    Penrose, Phys

    R. Penrose, Phys. Rev. Lett. 14, 57 (1965)

    work page 1965

  9. [9]

    S. W. Hawking, Phys. Rev. D 14, 2460 (1976)

    work page 1976

  10. [10]

    J. M. Bardeen, in Proceedings of the International Conference GR5 (Tbilisi, U.S.S.R., 1968) p. 174

    work page 1968

  11. [11]

    Dymnikova, Gen

    I. Dymnikova, Gen. Rel. Grav. 24, 235 (1992)

    work page 1992

  12. [12]

    The Bardeen Model as a Nonlinear Magnetic Monopole

    E. Ayon-Beato and A. Garcia, Phys. Lett. B 493, 149 (2000) , arXiv:gr-qc/0009077

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  13. [13]

    S. A. Hayward, Phys. Rev. Lett. 96, 031103 (2006) , arXiv:gr-qc/0506126. 23

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  14. [14]

    Regular black holes from semi-classical down to Planckian size

    E. Spallucci and A. Smailagic, Int. J. Mod. Phys. D 26, 1730013 (2017) , arXiv:1701.04592

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  15. [15]

    Black-bounce to traversable wormhole

    A. Simpson and M. Visser, JCAP 02, 042 (2019) , arXiv:1812.07114

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  16. [16]

    Quantum Extension of the Kruskal Space-time

    A. Ashtekar, J. Olmedo, and P. Singh, Phys. Rev. D 98, 126003 (2018) , arXiv:1806.02406

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  17. [17]

    Burzill` a, B

    N. Burzill` a, B. L. Giacchini, T. d. P. Netto, and L. Mode sto, Eur. Phys. J. C 81, 462 (2021) , arXiv:2012.11829

    work page arXiv 2021

  18. [18]

    Regular rotating black holes,

    R. Torres, “Regular rotating black holes,” in Regular Black Holes: Towards a New Paradigm of Gravitationa l edited by C. Bambi (Springer Nature Singapore, Singapore, 2 023) pp. 421–446

    work page

  19. [19]

    C. Lan, H. Yang, Y. Guo, and Y.-G. Miao, Int. J. Theor. Phys. 62, 202 (2023) , arXiv:2303.11696

    work page arXiv 2023

  20. [20]

    Z. Feng, Y. Ling, X. Wu, and Q. Jiang, Sci. China Phys. Mech. Astron. 67, 270412 (2024) , arXiv:2308.15689

    work page arXiv 2024

  21. [21]

    Bambi, ed., Regular Black Holes

    C. Bambi, ed., Regular Black Holes. Towards a New Paradigm of Gravitationa l Collapse , Springer Series in Astrophysics and Cosmology (Springer, 2 023) arXiv:2307.13249

    work page arXiv

  22. [22]

    Bueno, P.A

    P. Bueno, P. A. Cano, and R. A. Hennigar, Phys. Lett. B 861, 139260 (2025) , arXiv:2403.04827

    work page arXiv 2025

  23. [23]

    V. P. Frolov, A. Koek, J. P. Soto, and A. Zelnikov, Phys. Rev. D 111, 044034 (2025) , arXiv:2411.16050

    work page arXiv 2025

  24. [24]

    Ling and Z

    Y. Ling and Z. Yu, JCAP 03, 004 (2026) , arXiv:2509.00137

    work page arXiv 2026

  25. [25]

    Carballo-Rubio et al

    R. Carballo-Rubio et al. , JCAP 05, 003 (2025) , arXiv:2501.05505

    work page arXiv 2025

  26. [26]

    Carballo-Rubio, F

    R. Carballo-Rubio, F. Di Filippo, S. Liberati, and M. Vi sser, Class. Quant. Grav. 37, 14 (2020) , arXiv:1908.03261

    work page arXiv 2020

  27. [27]

    Carballo-Rubio, F

    R. Carballo-Rubio, F. Di Filippo, S. Liberati, and M. Vi sser, Phys. Rev. D 101, 084047 (2020) , arXiv:1911.11200

    work page arXiv 2020

  28. [28]

    Poisson and W

    E. Poisson and W. Israel, Phys. Rev. Lett. 63, 1663 (1989)

    work page 1989

  29. [29]

    Poisson and W

    E. Poisson and W. Israel, Phys. Rev. D 41, 1796 (1990)

    work page 1990

  30. [30]

    Ori, Phys

    A. Ori, Phys. Rev. Lett. 67, 789 (1991)

    work page 1991

  31. [31]

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

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  32. [32]

    Bertipagani, M

    M. Bertipagani, M. Rinaldi, L. Sebastiani, and S. Zerbi ni, Phys. Dark Univ. 33, 100853 (2021) , arXiv:2012.15645

    work page arXiv 2021

  33. [33]

    E. G. Brown, R. B. Mann, and L. Modesto, Phys. Rev. D 84, 104041 (2011) , arXiv:1104.3126

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  34. [34]

    T. Hale, R. A. Hennigar, and D. Kubiznak, Phys. Rev. D 113, L061502 (2026) , 24 arXiv:2506.20802

    work page arXiv 2026

  35. [35]

    A. J. S. Hamilton and P. P. Avelino, Phys. Rept. 495, 1 (2010) , arXiv:0811.1926

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  36. [36]

    Internal structure of charged black holes

    D.-i. Hwang and D.-h. Yeom, Phys. Rev. D 84, 064020 (2011) , arXiv:1010.2585

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  37. [37]

    On the viability of regular black holes

    R. Carballo-Rubio, F. D. Filippo, S. Liberati, C. Pacil io, and M. Visser, JHEP 07, 023 (2018) , arXiv:1805.02675

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  38. [38]

    Carballo-Rubio, F

    R. Carballo-Rubio, F. D. Filippo, S. Liberati, C. Pacil io, and M. Visser, JHEP 05, 132 (2021) , arXiv:2101.05006

    work page arXiv 2021

  39. [39]

    F. D. Filippo, R. Carballo-Rubio, S. Liberati, C. Pacil io, and M. Visser, Universe 8, 204 (2022) , arXiv:2203.14516

    work page arXiv 2022

  40. [40]

    Bonanno, A.-P

    A. Bonanno, A.-P. Khosravi, and F. Saueressig, Phys. Rev. D 103, 124027 (2021) , arXiv:2010.04226

    work page arXiv 2021

  41. [41]

    Barcel´ o, V

    C. Barcel´ o, V. Boyanov, R. Carballo-Rubio, and L. J. Ga ray, Class. Quant. Grav. 38, 125003 (2021) , arXiv:2011.07331

    work page arXiv 2021

  42. [42]

    Barcel´ o, V

    C. Barcel´ o, V. Boyanov, R. Carballo-Rubio, and L. J. Ga ray, Phys. Rev. D 106, 124006 (2022), arXiv:2203.13539

    work page arXiv 2022

  43. [43]

    Carballo-Rubio, F

    R. Carballo-Rubio, F. D. Filippo, S. Liberati, C. Pacil io, and M. Visser, JHEP 09, 118 (2022) , arXiv:2205.13556

    work page arXiv 2022

  44. [44]

    Franzin, S

    E. Franzin, S. Liberati, J. Mazza, and V. Vellucci, Phys. Rev. D 106, 104060 (2022) , arXiv:2207.08864

    work page arXiv 2022

  45. [45]

    Stability properties of regular black holes,

    A. Bonanno and F. Saueressig, “Stability properties of regular black holes,” (2022), arXiv:2211.09192

    work page arXiv 2022

  46. [46]

    Bonanno, A

    A. Bonanno, A. Panassiti, and F. Saueressig, “Cauchy horizon (in)stability of regular black holes,” (2025), arXiv:2507.03581

    work page arXiv 2025

  47. [47]

    McMaken, Phys

    T. McMaken, Phys. Rev. D 107, 125023 (2023) , arXiv:2303.03562

    work page arXiv 2023

  48. [48]

    Dust collapse in asymptotic safety: a path to regular black holes

    A. Bonanno, D. Malafarina, and A. Panassiti, Phys. Rev. Lett. 132, 031401 (2024) , arXiv:2308.10890

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  49. [49]

    Khodadi and J

    M. Khodadi and J. T. Firouzjaee, Phys. Lett. B 857, 138986 (2024) , arXiv:2408.12873

    work page arXiv 2024

  50. [50]

    Carballo-Rubio, F

    R. Carballo-Rubio, F. Di Filippo, S. Liberati, and M. Vi sser, Phys. Rev. Lett. 133, 181402 (2024) , arXiv:2402.14913

    work page arXiv 2024

  51. [51]

    Di Filippo, S

    F. Di Filippo, S. Liberati, and M. Visser, Int. J. Mod. Phys. D 33, 2440005 (2024) , arXiv:2405.08069. 25

    work page arXiv 2024

  52. [52]

    Di Filippo, I

    F. Di Filippo, I. Kol´ aˇ r, and D. Kubiznak, Phys. Rev. D 111, L041505 (2025) , arXiv:2404.07058

    work page arXiv 2025

  53. [53]

    V. P. Frolov and A. Zelnikov, Phys. Rev. D 113, 084007 (2026) , arXiv:2601.01861

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  54. [54]

    Eichhorn and P

    A. Eichhorn and P. G. S. Fernandes, Phys. Rev. D 113, L081501 (2026) , arXiv:2508.00686

    work page arXiv 2026

  55. [55]

    Singularities and the Finale of Black Hole Evaporation

    L. Xiang, Y. Ling, and Y. G. Shen, Int. J. Mod. Phys. D 22, 1342016 (2013) , arXiv:1305.3851

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  56. [56]

    Ling and M.-H

    Y. Ling and M.-H. Wu, Chin. Phys. C 46, 025102 (2022) , arXiv:2109.12938

    work page arXiv 2022

  57. [57]

    Ling and M.-H

    Y. Ling and M.-H. Wu, Class. Quant. Grav. 40, 075009 (2023) , arXiv:2109.05974

    work page arXiv 2023

  58. [58]

    Dray and G

    T. Dray and G. ’t Hooft, Commun. Math. Phys. 99, 613 (1985)

    work page 1985

  59. [59]

    I. H. Redmount, Prog. Theor. Phys. 73, 1401 (1985)

    work page 1985

  60. [60]

    C. W. Misner and D. H. Sharp, Phys. Rev. 136, B571 (1964)

    work page 1964

  61. [61]

    R. H. Price, Phys. Rev. D 5, 2419 (1972)

    work page 1972

  62. [62]

    R. H. Price, Phys. Rev. D 5, 2439 (1972)

    work page 1972