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arxiv: 2601.01861 · v2 · submitted 2026-01-05 · 🌀 gr-qc

Regular Black Holes in Quasitopological Gravity: Null Shells and Mass Inflation

Valeri P. Frolov , Andrei Zelnikov This is my paper

Pith reviewed 2026-05-16 18:22 UTC · model grok-4.3

classification 🌀 gr-qc
keywords regular black holesquasitopological gravitymass inflationnull shellsinner horizonCauchy horizonjunction conditions
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The pith

In quasitopological gravity regular black holes, significant mass inflation near the inner horizon requires null shells to collide at radial distances much smaller than the fundamental scale ℓ.

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

The paper models ingoing and outgoing perturbations inside regular black holes in quasitopological gravity as the collision of two spherical null shells. It shows that substantial amplification of the mass function and curvature invariants occurs only when the shells intersect at distances from the inner horizon satisfying r minus r star less than or equal to ℓ times (ℓ over r sub g) to the power 2n times (D minus 3). For macroscopic black holes where r sub g greatly exceeds ℓ this distance is tiny compared to the fundamental scale. A sympathetic reader would care because the result indicates that the classical mass inflation instability may be strongly suppressed in these geometries with bounded curvature cores.

Core claim

Unlike the Reissner-Nordström or Kerr cases, mass inflation in quasitopological gravity regular black holes with an inner horizon near scale ℓ requires the two null shells to intersect at r minus r star less than or equal to ℓ (ℓ over r sub g) to the power 2n(D minus 3). This bound follows from applying the Dray-t Hooft-Barrabes-Israel junction condition to the shell collision and tracking the resulting discontinuity in the metric function and the growth of curvature invariants.

What carries the argument

Collision of two spherical null shells inside the regular black hole spacetime, governed by the Dray-t Hooft-Barrabes-Israel junction condition that determines the jump in the metric function and the condition for curvature amplification.

If this is right

  • Significant mass inflation is unlikely for generic perturbations because the required intersection distances are much smaller than ℓ for macroscopic black holes.
  • Curvature invariants remain bounded near the inner horizon under typical conditions.
  • The bounded curvature core of these regular black holes is protected against classical inflation effects.
  • The suppression strengthens with increasing black hole size r_g relative to ℓ and depends on spacetime dimension D and model parameter n.

Where Pith is reading between the lines

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

  • These regular black holes may furnish stable interiors that avoid the classical inner-horizon instabilities without extra mechanisms.
  • Numerical evolution of finite-width wave packets rather than thin shells could test whether the required closeness persists for more realistic perturbations.
  • Similar shell-collision analyses in other families of regular black holes with bounded cores could reveal if the suppression is generic.
  • The result suggests that the fundamental scale ℓ controls the interior dynamics for astrophysical black holes, potentially altering expectations for singularity resolution.

Load-bearing premise

The collision of two spherical null shells via the junction condition accurately captures the interaction of generic ingoing and outgoing perturbations.

What would settle it

A calculation showing substantial mass inflation for shell intersections at radial separations larger than ℓ (ℓ/r_g) to the power 2n(D-3) would falsify the requirement that such extreme closeness is needed.

Figures

Figures reproduced from arXiv: 2601.01861 by Andrei Zelnikov, Valeri P. Frolov.

Figure 1
Figure 1. Figure 1: FIG. 1. Colliding spherical thin null shells near the inner [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The typical shape of functions [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

We investigate the phenomenon of mass inflation in the interior of regular black holes arising in quasitopological gravity (QTG). These geometries are characterized by a bounded curvature core and the presence of an inner (Cauchy) horizon located near the fundamental scale $\ell$. To examine whether mass inflation persists in this setting, we model the interaction of ingoing and outgoing perturbations by considering the collision of two spherical null shells inside the black hole. Using the Dray-'t\,Hooft-Barrabes-Israel junction condition, we derive conditions under which the metric function and curvature invariants may experience significant amplification near the inner horizon. Our analysis shows that, unlike in classical Reissner--Nordstr\"om or Kerr geometries, significant mass inflation requires shell intersection at radii very close to the horizon, with radial separations from it of the order $r-r_* \lesssim \ell \big(\ell/r_g\big)^{2n(D-3)}$, where $r_g$ is the gravitational radius of the black hole, $D$ is the number of spacetime dimensions and $n\ge 1$ is a parameter depending on a concrete QTG model. For macroscopic black holes with $r_g\gg \ell$ this distance is much smaller than the fundamental scale $\ell$. We discuss possible consequences of this effect.

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 / 1 minor

Summary. The paper investigates mass inflation inside regular black holes in quasitopological gravity by modeling collisions of ingoing and outgoing spherical null shells. Using the Dray-'t Hooft-Barrabes-Israel junction condition on the assumed metric family, it derives that significant amplification of the metric function and curvature invariants near the inner horizon requires shell intersections at radial separations r-r_* ≲ ℓ(ℓ/r_g)^{2n(D-3)}, a distance much smaller than the fundamental scale ℓ for macroscopic black holes with r_g ≫ ℓ.

Significance. If the result holds, it suggests that mass inflation is strongly suppressed in these regular black hole models unless perturbations intersect extremely close to the Cauchy horizon. This points to potentially greater stability of the inner horizon compared to classical RN or Kerr geometries, with implications for singularity resolution and the dynamics of black hole interiors in higher-curvature gravity theories.

major comments (1)
  1. [main derivation of the junction matching and near-horizon expansion] The central scaling r-r_* ≲ ℓ(ℓ/r_g)^{2n(D-3)} is obtained by substituting the GR Dray-'t Hooft-Barrabes-Israel junction condition into the near-horizon expansion of f(r). Because quasitopological gravity introduces higher-order curvature invariants (controlled by n and ℓ), the field equations contain fourth-order derivatives; the correct junction conditions across a null hypersurface must include additional surface terms from jumps in second derivatives and contractions of the higher-curvature tensors. These terms are absent from the GR matching used here and would directly modify the amplification factor and the required proximity threshold.
minor comments (1)
  1. [Abstract and opening paragraphs] The abstract states that the analysis uses 'the given metric family' but does not display the explicit form of f(r) or the regularizing function; including the concrete expression for the metric function (with the parameters n and D) would make the substitution into the junction condition fully transparent.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for raising this important point about the junction conditions. We address the major comment below and will revise the manuscript to derive the matching conditions directly from the quasitopological action.

read point-by-point responses

  1. Referee: [main derivation of the junction matching and near-horizon expansion] The central scaling r-r_* ≲ ℓ(ℓ/r_g)^{2n(D-3)} is obtained by substituting the GR Dray-'t Hooft-Barrabes-Israel junction condition into the near-horizon expansion of f(r). Because quasitopological gravity introduces higher-order curvature invariants (controlled by n and ℓ), the field equations contain fourth-order derivatives; the correct junction conditions across a null hypersurface must include additional surface terms from jumps in second derivatives and contractions of the higher-curvature tensors. These terms are absent from the GR matching used here and would directly modify the amplification factor and the required proximity threshold.

    Authors: We agree that the junction conditions cannot be imported directly from general relativity. Although the quasitopological field equations reduce to second order for the static spherically symmetric ansatz, the underlying action contains higher-curvature invariants whose variation produces additional surface contributions when the metric is discontinuous across a null shell. In the original manuscript we applied the standard Dray-'t Hooft-Barrabes-Israel conditions to the assumed metric family; this was an approximation whose validity must be checked. In the revised version we will integrate the quasitopological equations of motion across the shell, extract the correct jump relations (including the extra terms generated by the higher-curvature pieces), and recompute the near-horizon expansion of the metric function. We will then determine whether the scaling r-r_* ≲ ℓ(ℓ/r_g)^{2n(D-3)} for significant mass inflation survives or is modified by these additional contributions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is algebraically self-contained from metric ansatz and external junction rules

full rationale

The paper obtains the reported threshold r-r_* ≲ ℓ(ℓ/r_g)^{2n(D-3)} by substituting the near-horizon series expansion of the regular black-hole metric function f(r) (taken from the quasitopological gravity solution) into the standard Dray-'t Hooft-Barrabes-Israel null-shell matching conditions. This produces the required proximity scaling directly from the functional form of f(r) and the jump relations; no parameters are fitted to the target quantity, no self-citation supplies the load-bearing step, and the result is not equivalent to the input by definition. The derivation therefore remains independent of the final claim once the metric and the external junction condition are granted.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of null-shell junction conditions to the quasitopological gravity metric family and on the existence of an inner horizon at scale ℓ; no new entities are postulated.

free parameters (2)
  • n
    Model-dependent integer parameter appearing in the quasitopological gravity action
  • D
    Spacetime dimension
axioms (2)
  • domain assumption Dray-'t Hooft-Barrabes-Israel junction conditions hold for null shells in this geometry
    Invoked to relate metric jumps across the colliding shells
  • domain assumption The regular black hole solution in quasitopological gravity possesses an inner horizon located near the fundamental scale ℓ
    Taken from the known properties of quasitopological gravity black hole solutions

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Forward citations

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Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · cited by 7 Pith papers · 6 internal anchors

  1. [1]

    explosive

    We call such a solution with boundedp(r) satisfying condition (2.8) aregular black hole. A characteristic property of regular black holes is the existence of the mass gapm 0, such that for mass param- eterm < m 0 the metric does not have horizons. For m > m 0 there exist the external (event) and internal (Cauchy) horizons (see Fig. 2). *r r ()fr gr FIG. 2...

    work page

  2. [3]

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

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

    work page 1991

  3. [4]

    Classical stability and quantum instability of black-hole cauchy horizons,

    D. Markovi´ c and E. Poisson, “Classical stability and quantum instability of black-hole cauchy horizons,” Phys. Rev. Lett.74, 1280 (1995)

    work page 1995

  4. [5]

    Black hole singularities: a numerical approach,

    P. R. Brady and J. D. Smith, “Black hole singularities: a numerical approach,” Phys. Rev. Lett.75, 1256 (1995)

    work page 1995

  5. [6]

    Structure of the black hole's Cauchy horizon singularity

    L. M. Burko, “Structure of the black hole’s Cauchy horizon singularity,” Phys. Rev. Lett.79, 4958 (1997), arXiv:gr-qc/9710112

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  6. [7]

    Strength of the null singularity inside black holes,

    L. M. Burko, “Strength of the null singularity inside black holes,” Phys. Rev. D60, 104033 (1999), arXiv:gr- qc/9907061

    work page arXiv 1999

  7. [8]

    Collapse of charged scalar fields,

    Y. Oren and T. Piran, “Collapse of charged scalar fields,” Phys. Rev. D68, 044013 (2003)

    work page 2003

  8. [9]

    The interior of charged black holes and the problem of uniqueness in general relativity,

    M. Dafermos, “The interior of charged black holes and the problem of uniqueness in general relativity,” Com- munications on Pure and Applied Mathematics58, 445 (2005)

    work page 2005

  9. [10]

    Inside charged black holes I. Baryons

    A. J. S. Hamilton and S. E. Pollack, “Inside charged black holes. I. Baryons,” Phys. Rev. D71, 084031 (2005), arXiv:gr-qc/0411061

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  10. [11]

    The physics of the relativistic counter-streaming instability inside black holes,

    A. J. S. Hamilton and P. P. Avelino, “The physics of the relativistic counter-streaming instability inside black holes,” Phys. Rept.495, 1 (2010)

    work page 2010

  11. [12]

    Phenomenological aspects of black holes beyond general relativity

    R. Carballo-Rubio, F. Di Filippo, S. Liberati, and M. Visser, “Phenomenological aspects of black holes be- yond general relativity,” Phys. Rev. D98, 124009 (2018), arXiv:1809.08238 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  12. [13]

    V. P. Frolov and I. D. Novikov,Black Hole Physics: Ba- sic Concepts and New Developments(Kluwer Academic Publishers, Dordrecht, 1998)

    work page 1998

  13. [14]

    Poisson,A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics(Cambridge University Press, Cambridge, 2004)

    E. Poisson,A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics(Cambridge University Press, Cambridge, 2004)

    work page 2004

  14. [15]

    The effect of spherical shells of matter on the Schwarzschild black hole,

    T. Dray and G. ’t Hooft, “The effect of spherical shells of matter on the Schwarzschild black hole,” Communica- tions in Mathematical Physics99, 613 (1985)

    work page 1985

  15. [16]

    Thin shells in general relativ- ity and cosmology: The lightlike limit,

    C. Barrab` es and W. Israel, “Thin shells in general relativ- ity and cosmology: The lightlike limit,” Physical Review D43, 1129 (1991)

    work page 1991

  16. [17]

    Inner-horizon instability and mass inflation in black holes,

    E. Poisson and W. Israel, “Inner-horizon instability and mass inflation in black holes,” Phys. Rev. Lett.63, 1663 (1989)

    work page 1989

  17. [18]

    Internal structure of black holes,

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

    work page 1990

  18. [19]

    A new cubic theory of gravity in five dimensions: black hole, Birkhoff’s theorem and C- function,

    J. Oliva and S. Ray, “A new cubic theory of gravity in five dimensions: black hole, Birkhoff’s theorem and C- function,” Classical and Quantum Gravity27, 225002 (2010)

    work page 2010

  19. [20]

    Black Holes in Quasi-topological Gravity

    R. C. Myers and B. Robinson, “Black holes in quasi-topological gravity,” JHEP08, 067 (2010), arXiv:1003.5357 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  20. [21]

    Four-dimensional black holes in Einsteinian cubic gravity

    P. Bueno and P. A. Cano, “Four-dimensional black holes in einsteinian cubic gravity,” Phys. Rev. D94, 124051 (2016), arXiv:1610.08019 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  21. [22]

    Generalized quasi-topological gravity

    R. A. Hennigar, D. Kubiznak, and R. B. Mann, “Gener- alized quasitopological gravity,” Phys. Rev. D95, 104042 (2017), arXiv:1703.01631 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  22. [23]

    Bueno, P

    P. Bueno, P. A. Cano, J. Moreno, and A. Mur- cia, “All higher-curvature gravities as generalized quasi-topological gravities,” JHEP11, 062 (2019), arXiv:1906.00987 [hep-th]

    work page arXiv 2019

  23. [24]

    Bueno, P

    P. e. a. Bueno, “Generalized quasi-topological gravities: the whole shebang,” Class. Quant. Grav.40, 015004 (2023), arXiv:2203.05589 [hep-th]

    work page arXiv 2023

  24. [25]

    Moreno and ´A

    J. Moreno and A. J. Murcia, “Classification of generalized quasitopological gravities,” Phys. Rev. D108, 044016 (2023), arXiv:2304.08510 [gr-qc]

    work page arXiv 2023

  25. [26]

    Charged Regular Black Holes From Quasi-topological Gravities inD≥5,

    C.-H. Hao, J. Jing, and J. Wang, “Charged Regular Black Holes From Quasi-topological Gravities inD≥5,” (2025), arXiv:2512.04604 [gr-qc]

    work page arXiv 2025

  26. [27]

    Regular black holes from pure gravity,

    P. Bueno, P. A. Cano, and R. A. Hennigar, “Regular black holes from pure gravity,” Phys. Lett. B861, 139260 (2025)

    work page 2025

  27. [28]

    Regular black holes inspired by quasitopological grav- ity,

    V. P. Frolov, A. Koek, J. P. Soto, and A. Zelnikov, “Regular black holes inspired by quasitopological grav- ity,” Phys. Rev. D111, 044034 (2025), arXiv:2411.16050 [gr-qc]

    work page arXiv 2025

  28. [29]

    Birkhoff implies quasi-topological,

    P. Bueno, R. A. Hennigar, and A. J. Murcia, “Birkhoff implies quasi-topological,” (2025), arXiv:2510.25823 [gr- qc]

    work page arXiv 2025

  29. [30]

    Bueno, P.A

    P. Bueno, P. A. Cano, R. A. Hennigar, and ´A. J. Murcia, “Regular black hole formation in four-dimensional non- polynomial gravities,” Phys. Rev. D113, 024019 (2026), arXiv:2509.19016 [gr-qc]

    work page arXiv 2026

  30. [31]

    Ling and Z

    Y. Ling and Z. Yu, “Big bounce and black bounce in quasi-topological gravity,” JCAP03, 004 (2026), arXiv:2509.00137 [gr-qc]

    work page arXiv 2026

  31. [32]

    Classification of global struc- tures of evaporating regular black holes with infinite pe- riods of time,

    K. Sueto and H. Yoshino, “Classification of global struc- tures of evaporating regular black holes with infinite pe- riods of time,” (2025), arXiv:2508.21315 [gr-qc]

    work page arXiv 2025

  32. [33]

    Regular black holes from Oppenheimer-Snyder collapse,

    P. Bueno, P. A. Cano, R. A. Hennigar, A. J. Mur- cia, and A. Vicente-Cano, “Regular black holes from Oppenheimer-Snyder collapse,” Phys. Rev. D112, 064039 (2025), arXiv:2505.09680 [gr-qc]

    work page arXiv 2025

  33. [34]

    J. A. Pinedo Soto,Modified Gravity and Regular Black Hole Models, Ph.D. thesis, University of Alberta (2025), arXiv:2511.12902 [gr-qc]

    work page arXiv 2025

  34. [35]

    Di Filippo, I

    F. Di Filippo, I. Kol´ aˇ r, and D. Kubiznak, “Inner- extremal regular black holes from pure gravity,” Phys. Rev. D111, L041505 (2025), arXiv:2404.07058 [gr-qc]

    work page arXiv 2025

  35. [36]

    C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravi- tation(W. H. Freeman, San Francisco, 1973)

    work page 1973