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arxiv: 2508.12646 · v2 · submitted 2025-08-18 · 🌀 gr-qc · hep-th

Black holes of multiple horizons without mass inflation

Changjun Gao , Toktarbay Saken This is my paper

Pith reviewed 2026-05-18 23:27 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords black holesmass inflationinner horizonssurface gravitynonlinear Maxwell fieldmultiple horizonsgeneral relativity
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The pith

Black holes with multiple horizons avoid mass inflation when inner horizons coincide and surface gravities vanish.

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

The paper constructs black hole solutions sourced by a nonlinear Maxwell field that possess several horizons. It arranges for the inner horizons to coincide exactly, which forces the surface gravity at each inner horizon to zero. Prior linear analysis indicates that zero surface gravity removes the exponential growth of mass inflation near those horizons. A sympathetic reader cares because this offers a route to stable multi-horizon spacetimes without the usual instability that plagues inner horizons in standard two-horizon geometries.

Core claim

By using nonlinear Maxwell field sources and making the inner horizons coincide so that the surface gravities of every inner horizon vanish, black hole solutions with multiple horizons are obtained that do not exhibit mass inflation.

What carries the argument

Coincidence of inner horizons in the metric, which sets surface gravities to zero and removes the exponential amplification of perturbations.

If this is right

  • The constructed solutions remain free of mass inflation because the surface-gravity factor driving exponential growth is absent.
  • Nonlinear Maxwell fields allow explicit metric solutions realizing these degenerate multi-horizon configurations.
  • Multiple-horizon black holes can be stable against perturbations that would otherwise trigger mass inflation.

Where Pith is reading between the lines

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

  • The same degeneracy trick could be tested in other nonlinear electrodynamics models to see whether stable inner horizons appear more generally.
  • If these solutions are regular, they might provide concrete examples for studying the causal structure of spacetimes with more than two horizons.
  • Perturbation analysis beyond linear order in these metrics would directly test whether the absence of mass inflation survives in the nonlinear regime.

Load-bearing premise

Vanishing surface gravity at a degenerate inner horizon, as seen in linear analysis, is enough to eliminate mass inflation under the full nonlinear dynamics.

What would settle it

Numerical evolution of small perturbations in the constructed spacetime that shows no exponential growth of mass near the inner horizon.

Figures

Figures reproduced from arXiv: 2508.12646 by Changjun Gao, Toktarbay Saken.

Figure 1
Figure 1. Figure 1: FIG. 1: A schech of effective potential [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: There are potential barriers (on the order of 10 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: There are potential wells (on the order of 10 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Potential barriers coexist with potential wells (on [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Potential barriers coexist with potential wells (on [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Potential barriers coexist with potential wells (on [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: There are infinite deep potential wells (on the order [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
read the original abstract

Mass inflation is a phenomenon happened in the vicinity of inner horizon in two-horizon spacetime. It is shown that the mass of initially small perturbations will grow exponentially as they approach the inner horizon. This implies that black hole inner horizon is unstable to the perturbations. In view of the fact that the mass inflation is determined by the surface gravity of inner horizon, Carballo-Rubio et al. showed that if one makes the surface gravity of inner horizon vanish, then the exponential growth character of mass inflation is not present. Basing on this conclusion, we look for the black hole solutions of multiple horizons with nonlinear Maxwell field. Then we make the inner horizons to coincide with each other such that the surface gravities of every inner horizon vanish. By this way, black holes of multiple horizons without mass inflation are constructed.

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 / 1 minor

Summary. The paper claims to construct black hole solutions with multiple horizons in nonlinear Maxwell theory by arranging for inner horizons to coincide, thereby setting their surface gravities to zero. Invoking the result of Carballo-Rubio et al. that vanishing surface gravity eliminates exponential growth of mass inflation, the authors conclude that these solutions are free of mass inflation.

Significance. If the construction is valid and the linear result extends to the nonlinear regime, the work would provide explicit examples of multi-horizon black holes whose inner horizons are stable against mass inflation. This could be relevant for understanding horizon stability in nonlinear electrodynamics and for constructing regular or stable interior geometries.

major comments (2)
  1. The abstract states the construction but supplies no explicit metric, field equations, or perturbation analysis; without these details the central claim cannot be checked for derivation gaps or post-hoc choices.
  2. The paper relies on the prior result that zero surface gravity removes exponential growth (from Carballo-Rubio et al.); however, that reference establishes the result only for linear perturbations. The manuscript provides no nonlinear perturbation analysis, no numerical evolution of the Einstein-nonlinear-Maxwell system, and no explicit check that the mass function remains bounded when the full nonlinear stress-energy is retained near the degenerate horizon.
minor comments (1)
  1. Notation for the nonlinear Maxwell coupling constants and the explicit form of the stress-energy tensor should be introduced earlier and used consistently when describing the field equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment point by point below, indicating the revisions we will make.

read point-by-point responses

  1. Referee: The abstract states the construction but supplies no explicit metric, field equations, or perturbation analysis; without these details the central claim cannot be checked for derivation gaps or post-hoc choices.

    Authors: The abstract is intentionally concise as a summary of the main result. The full manuscript contains the explicit metric ansatz for the multi-horizon spacetime, the nonlinear Maxwell field equations, the procedure for arranging coinciding inner horizons, and the resulting vanishing surface gravities. These are derived in the sections following the introduction. We will revise the abstract to briefly reference the key elements of the construction and the reliance on the surface-gravity condition. revision: yes

  2. Referee: The paper relies on the prior result that zero surface gravity removes exponential growth (from Carballo-Rubio et al.); however, that reference establishes the result only for linear perturbations. The manuscript provides no nonlinear perturbation analysis, no numerical evolution of the Einstein-nonlinear-Maxwell system, and no explicit check that the mass function remains bounded when the full nonlinear stress-energy is retained near the degenerate horizon.

    Authors: We agree that Carballo-Rubio et al. treat linear perturbations and that our manuscript does not include a dedicated nonlinear perturbation analysis or numerical evolution of the full Einstein-nonlinear-Maxwell system. The central construction ensures that the surface gravity vanishes at the degenerate inner horizons by design, which removes the exponential growth mechanism identified in the linear analysis. We will add a discussion paragraph noting this scope limitation and indicating that a full nonlinear check would be a natural extension for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; explicit solutions constructed to satisfy external zero-surface-gravity condition

full rationale

The paper solves the Einstein-nonlinear-Maxwell equations to obtain explicit metrics admitting multiple horizons, then tunes parameters so that inner horizons coincide and surface gravities vanish at those surfaces. This construction step is independent of the target claim and does not reduce any derived quantity to a fitted input or self-referential definition. The conclusion that mass inflation is absent rests on an external citation to Carballo-Rubio et al. for the linear-perturbation result; that citation is treated as independent support rather than a self-citation chain or ansatz smuggled from the authors' prior work. No load-bearing step equates a prediction to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The construction rests on the nonlinear Maxwell Lagrangian, the assumption that surface gravity controls mass inflation, and the ability to tune parameters so that multiple horizons coincide exactly.

free parameters (1)
  • nonlinear Maxwell coupling constants
    Parameters in the nonlinear electrodynamics Lagrangian that are chosen to allow degenerate horizons.
axioms (1)
  • domain assumption Vanishing surface gravity at a degenerate inner horizon eliminates exponential mass inflation
    Invoked from Carballo-Rubio et al. to justify the construction.

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Forward citations

Cited by 1 Pith paper

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

  1. Taming the Aretakis instability: extremal black holes with multi-degenerate horizons

    gr-qc 2026-04 unverdicted novelty 6.0

    Black holes with infinitely degenerate horizons are proposed to be stable against Aretakis instability, potentially serving as end states.

Reference graph

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