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arxiv: 2606.04988 · v1 · pith:AK3DMGZVnew · submitted 2026-06-03 · 🌀 gr-qc

Interior marginally outer trapped surfaces in Hayward black holes

Abbas M. Sherif , Yen-Kheng Lim This is my paper

Pith reviewed 2026-06-28 05:19 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Hayward black holemarginally outer trapped surfacesMOTSinner horizonSturm-Liouville problemhypergeometric functionsregular black holestrapped surfaces
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The pith

Near the inner horizon of certain Hayward black holes, MOTS locations are given by hypergeometric functions after reduction to a singular Sturm-Liouville problem whose eigenspace is complete, not discrete, and discontinuous.

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

The paper locates interior marginally outer trapped surfaces and open surfaces in the Hayward regular black hole with parameter b, where a critical value b_c separates cases with no horizons from those with inner and outer horizons. Self-intersecting MOTS appear in pairs, but for b near the critical value they can be absent, allowing fine-tuning so that only near-spherical MOTS remain in the interior. In a neighborhood of the inner horizon for selected b, the locating equation reduces to a singular Sturm-Liouville problem solved by hypergeometric functions. A sympathetic reader would care because these surfaces control the causal structure inside regular black hole models that avoid singularities.

Core claim

In a neighborhood of the inner horizon for certain values of b, upon reduction of the problem to a singular Sturm-Liouville problem, the locations of the MOTS are given by hypergeometric functions, the eigenspace of the operator for which is complete, not discrete, and discontinuous. Self-intersecting MOTS occur in pairs. For b close to the critical value there are no self-intersecting MOTS/MOTOS, and one can fine-tune b so that the interior contains only near-spherical MOTS.

What carries the argument

Reduction of the MOTS locating equation to a singular Sturm-Liouville problem solved by hypergeometric functions, with the operator's eigenspace being complete yet not discrete and discontinuous.

If this is right

  • Self-intersecting MOTS occur in pairs for some range of b.
  • For b sufficiently close to the critical value, self-intersecting MOTS and MOTOS are absent.
  • Fine-tuning of b produces an interior containing only near-spherical MOTS.
  • The completeness of the eigenspace permits representation of general solutions in the neighborhood of the inner horizon.

Where Pith is reading between the lines

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

  • The same reduction technique may apply to the locating equations for trapped surfaces in other spherically symmetric regular black hole metrics.
  • The discontinuous character of the eigenspace suggests that standard spectral theorems for Sturm-Liouville operators require modification inside these spacetimes.
  • Explicit hypergeometric expressions for MOTS could serve as initial data or test cases for numerical evolutions of black hole interiors.

Load-bearing premise

The equation that locates MOTS can be reduced to a singular Sturm-Liouville problem whose solutions are hypergeometric functions possessing the stated completeness and discontinuity properties, and this reduction holds in a neighborhood of the inner horizon for selected values of b.

What would settle it

A direct numerical integration or shooting method that computes the actual radial locations of MOTS near the inner horizon for the selected b values and finds they deviate from the values predicted by the hypergeometric solutions.

Figures

Figures reproduced from arXiv: 2606.04988 by Abbas M. Sherif, Yen-Kheng Lim.

Figure 1
Figure 1. Figure 1: The first six self-intersecting MOTS in the Schwarz [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: r0 = 1.8m, 1.4m, 1.2m, respectively. −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0 0.4 0.8 1.2 1.6 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0 0.4 0.8 1.2 1.6 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0 0.4 0.8 1.2 1.6 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: r0 = 1.09m, 1.0794m, 1.0m, respectively. Next, we explore the solutions starting from the inner horizon [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: r0 = 0.81m, 0.80125m, 0.78m. −0.2 −0.1 0 0.1 0.2 0 −0.2 −0.1 0 0.1 0.2 0 −0.2 −0.1 0 0.1 0.2 0 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: r0 = 0.11m, 0.16m, 0.19m, respectively. −0.2 −0.1 0 0.1 0.2 0 −0.2 −0.1 0 0.1 0.2 0 −0.2 −0.1 0 0.1 0.2 0 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: r0 = 0.22m, 0.2635m, 0.26m, respectively. In the preceding discussions, note that the closed MOTS occur in pairs; one associated to the outer horizon and the other associated to the inner horizon. This is similar to that of the Gauss–Bonnet black holes, also with two horizons [19]. In particular, a MOTS with n self￾intersections occur in pairs, one associated with the outer horizon and the other associated… view at source ↗
Figure 7
Figure 7. Figure 7: Self-intersecting MOTS in the Hayward spacetime o [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: b = 0.05m, 0.1m, 0.2m, respectively. 3.2 Perturbing the MOTSodesic equations MOTS stability, via an elliptic operator and its spectrum, plays a crucial role in understanding the dynamical evolution of a black hole [16, 17]. The operator characterizes the deformation of 9 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: b = 0.3m, 0.4m, 0.5m, respectively. a MOTS S along its normal direction in the embeddding slice. LS = −D2 + 1 2  R − |σ| 2 − 2Gabk au b  , (3.7) where the scalar curvature of the MOTS R, the term involving the Einstein tensor Gab, and the norm of the shear σ of k a , in the case of the Hayward metric, are R = 2κ  r˙ r cot θ − ˙θ  , Gabk au b = Ψ r˙ 2 − 1  , Ψ = − 12m2 b 2 (r 3 + 2mb2 ) 2 , |σ| 2 = Q2 … view at source ↗
Figure 10
Figure 10. Figure 10: Eigenvalues of the spherical MOTS at (a) the inner [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: MOTS/MOTOS for the extremal Hayward black hole wi [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: ˙r vs λ for self-intersecting MOTS nearby the outer horizon (left panel) and inner horizon (right panel) respectively. the number of self-intersecting MOTS/MOTOS will vanish well inside the outer horizon. It therefore follows that the Theorem 3.4 of [18], for counting the least number of negative eigen￾values is not applicable for a relatively large interval of the Hayward parameter space. This was alread… view at source ↗
Figure 13
Figure 13. Figure 13: Near-spherical and looping MOTS: b = 0.767m, and near-spherical MOTS: b = 0.7679m To formalize the discussions in the preceding paragraphs, we look for MOTS in the spherical near-horizon regime via a perturbative analysis. As discussed in the previous section, we know that a constant r = rd and θ = λ rd is a solution to the MOTSOdesic equations corresponding to spherical MOTS, provided that rd coincides w… view at source ↗
Figure 15
Figure 15. Figure 15: Sketch of a MOTS S generated as a surface of revolution of a curve γ about the vertical axis, where the generator of the revolution is ζ. We take the angular coordinate φ to be adapted to the integral curves of ζ a , and the metric on Σ shall be written as habdy a dy b = h¯ ijdy i dy j + R 2 dφ 2 , (A.1) for i, j ∈ {1, 2}, where R is some function of y i . We shall assume ζ a is normalised so its 16 [PIT… view at source ↗
read the original abstract

We locate interior marginally outer trapped and marginally outer trapped open surfaces (MOTS/MOTOS) in the regular Haward metric with parameter $b$ for which a critical (extremal) value $b=b_c$ demarcates when the spacetime admits no horizon and when it admits inner and outer horizons. We identify self-intersecting MOTS which occur in pairs. For $b$ close to the critical value, there are no self-intersecting MOTS/MOTOS, and one can fine-tune $b$ so that the interior contains only near-spherical MOTS. We also show that in a neighborhood of the inner horizon for certain values of $b$, upon reduction of the problem to a singular Sturm-Liouville problem, the locations of the MOTS are given by hypergeometric functions, the eigenspace of the operator for which is complete, not discrete, and discontinuous.

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 locates interior marginally outer trapped surfaces (MOTS) and marginally outer trapped open surfaces (MOTOS) in the regular Hayward metric parameterized by b, with a critical value b_c separating regimes with and without horizons. It identifies self-intersecting MOTS occurring in pairs, conditions with no self-intersecting surfaces, and fine-tuning of b yielding only near-spherical interior MOTS. The central claim is that, in a neighborhood of the inner horizon for selected b, reduction of the locating equation to a singular Sturm-Liouville problem gives MOTS locations via hypergeometric functions, with the eigenspace of the operator being complete, not discrete, and discontinuous.

Significance. If the reduction to a singular Sturm-Liouville problem and the stated spectral properties hold with explicit verification, the work would supply an analytic handle on MOTS locations in a regular black-hole spacetime, including the possibility of a continuous family of surfaces. This could inform studies of trapped surfaces and horizon structure in non-singular geometries. The manuscript does not supply machine-checked proofs or reproducible code, so the result remains dependent on the correctness of the reduction steps.

major comments (2)
  1. Abstract: the claim that MOTS locations are 'given by hypergeometric functions' while the eigenspace is 'complete, not discrete, and discontinuous' requires clarification; a continuous spectrum ordinarily parametrizes a continuum of solutions rather than isolated locations, and the manuscript must show explicitly (via the weight function, singularity classification, or self-adjoint extension) how the hypergeometric family maps onto concrete MOTS positions without extra selection rules.
  2. Abstract (and implied § on the reduction): the reduction of the MOTS locating equation to a singular Sturm-Liouville problem is asserted for a neighborhood of the inner horizon and selected b, yet no explicit operator, boundary conditions, or verification that the assumptions of the Sturm-Liouville theory are satisfied are provided; this step is load-bearing for the hypergeometric claim and must be derived in full.
minor comments (1)
  1. Abstract: the metric is referred to as 'Haward' on one line; correct to 'Hayward' for consistency with the title.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. Both points identify places where the manuscript asserts results without sufficient explicit derivation or clarification; we will address them by expanding the relevant sections.

read point-by-point responses

  1. Referee: Abstract: the claim that MOTS locations are 'given by hypergeometric functions' while the eigenspace is 'complete, not discrete, and discontinuous' requires clarification; a continuous spectrum ordinarily parametrizes a continuum of solutions rather than isolated locations, and the manuscript must show explicitly (via the weight function, singularity classification, or self-adjoint extension) how the hypergeometric family maps onto concrete MOTS positions without extra selection rules.

    Authors: We agree the abstract phrasing risks conflating the continuous spectrum with discrete surface locations. The hypergeometric solutions arise for a continuous range of the spectral parameter that labels distinct MOTS near the inner horizon; the claimed completeness means the family spans all admissible solutions in that neighborhood. We will revise the abstract to state this parameterization explicitly and add a paragraph in the reduction section that identifies the weight function, classifies the singularities, and shows how each value of the continuous parameter corresponds to a unique MOTS without additional selection rules. revision: yes

  2. Referee: Abstract (and implied § on the reduction): the reduction of the MOTS locating equation to a singular Sturm-Liouville problem is asserted for a neighborhood of the inner horizon and selected b, yet no explicit operator, boundary conditions, or verification that the assumptions of the Sturm-Liouville theory are satisfied are provided; this step is load-bearing for the hypergeometric claim and must be derived in full.

    Authors: The referee correctly notes that the reduction is asserted rather than derived in full. In the revised manuscript we will insert a self-contained subsection that starts from the MOTS equation, performs the change of variables that produces the Sturm-Liouville form, writes the explicit differential operator and weight function, states the boundary conditions at the singular points, and verifies the hypotheses of singular Sturm-Liouville theory (limit-point/limit-circle classification and self-adjointness) for the chosen range of b. This will make the appearance of hypergeometric functions and the spectral properties fully traceable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation reduces MOTS equation to independent Sturm-Liouville analysis

full rationale

The paper's central step reduces the MOTS locating equation in the Hayward metric to a singular Sturm-Liouville problem whose eigenfunctions are identified as hypergeometric functions with a claimed complete, non-discrete, discontinuous spectrum. This reduction is performed directly from the metric and the MOTS definition (null expansion vanishing) without any parameter fitting, self-referential definitions, or load-bearing self-citations. The spectral properties are presented as mathematical consequences of the resulting operator, not as inputs renamed as outputs. No equations or citations in the provided text exhibit the enumerated circularity patterns; the derivation remains self-contained against the spacetime geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the central claim rests on the validity of the Hayward metric as a regular black hole solution and on the legitimacy of reducing the MOTS locating equation to a singular Sturm-Liouville problem.

axioms (2)
  • domain assumption The Hayward metric with parameter b is a valid regular black hole spacetime in general relativity.
    The paper takes this metric as given and studies its interior properties.
  • domain assumption The MOTS locating equation admits a reduction to a singular Sturm-Liouville problem near the inner horizon.
    This reduction is the step that allows the hypergeometric solution and completeness claim.

pith-pipeline@v0.9.1-grok · 5678 in / 1487 out tokens · 36641 ms · 2026-06-28T05:19:17.719336+00:00 · methodology

discussion (0)

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Reference graph

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