Entropy release from Minkowski breaking in regular Schwarzschild black holes

Francisco S. N. Lobo; Manuel E. Rodrigues

REVIEW 4 major objections 5 minor 3 cited by

This paper claims that regular black holes with a de Sitter core carry a hidden entropy in their inner Cauchy horizon, and that this entropy must be released as radiation when the horizon vanishes during the transition to a singular Schwarz

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2026-08-02 02:49 UTC pith:DS72VHB6

load-bearing objection A clean, internally consistent thermodynamic add-on to the OCK regular black holes, but its central 'must be released' claim is explicitly built on an unproven additive-entropy rule and should be framed as conditional. the 4 major comments →

arxiv 2607.14079 v1 pith:DS72VHB6 submitted 2026-07-15 gr-qc astro-ph.HEhep-th

Entropy release from Minkowski breaking in regular Schwarzschild black holes

Francisco S. N. Lobo , Manuel E. Rodrigues This is my paper

classification gr-qc astro-ph.HEhep-th PACS 04.70.-s

keywords regular black holesinner Cauchy horizonBekenstein–Hawking entropyMinkowski breakinggeneralized second lawarea quantizationde Sitter coreentropy spectrum

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper argues that the classically forbidden transition from a regular black hole (a de Sitter core) to the singular Schwarzschild geometry leaves a thermodynamic footprint: the inner Cauchy horizon carries an extra Bekenstein–Hawking entropy S_inner = π h_c^2, hidden from outside observers in equilibrium but required to be released by the generalized second law once the horizon disappears. As the integer collapse parameter n decreases, the inner horizon shrinks and this hidden entropy is shed, culminating in a quantum event at n = 0 where 'Minkowski breaking' eliminates the inner horizon. Because n labels a discrete family of static solutions, the total entropy S_reg(n) = A/4 + π h_c^2 forms a quantized spectrum, with the Schwarzschild black hole as the ground state. The paper quantifies the stored entropy (about 59% of A/4 for n = 3, approaching a full A/4 for large n) and interprets regular black holes as excited geometric configurations of the Schwarzschild black hole.

Core claim

For the one-parameter OCK family of regular Schwarzschild interiors, the metric function f(r) has a second zero at an inner horizon radius h_c < h (the event horizon). Since this inner horizon is a Killing horizon, the paper assigns it a formal Bekenstein–Hawking entropy S_inner = A_inner/4 = π h_c^2 and asserts that the total gravitational entropy of the regular state is the sum S_reg = A/4 + π h_c^2. Invoking the generalized second law, the paper concludes that the difference ΔS = π h_c^2 cannot simply vanish when the inner horizon collapses; it must be released as radiation. The paper shows that as n → 0+ the inner horizon radius vanishes like (2/3)^{1/n}, the surface gravity diverges as

What carries the argument

The load-bearing object is the inner Killing (Cauchy) horizon with radius h_c(n), defined by f(h_c) = 0, carrying the formal entropy S_inner = A_inner/4 = π h_c^2. The argument runs on the additive bookkeeping rule S_reg = A/4 + π h_c^2 — the working hypothesis that the two horizons have independent microstates — together with the OCK mass function m(r) = (r/2(n−2))[(n+1)r²/h² − 3(r/h)ⁿ], whose polynomial form makes n a discrete label. The inner horizon's progressive shrinking and the Minkowski-breaking discontinuity at n = 0 are what convert formal horizon area into released entropy, while the exponentially growing surface gravity near n = 0 marks the quantum threshold where the semiclassic

Load-bearing premise

The central claim stands on the assumption that the inner Cauchy horizon's area is genuine, additive entropy (S_reg = A/4 + π h_c²) that must be released under the generalized second law; if the two horizons' microstates are not independent, the 'entropy release' is only a bookkeeping device.

What would settle it

A direct calculation of the Euclidean action or transition amplitude between a regular state (say n = 3) and the singular Schwarzschild state would settle the claim: if the emitted entropy is not at least π h_c² ≈ 0.59 A/4, the generalized-second-law release argument fails. Alternatively, a classical evolution that connects the regular interior to Schwarzschild without the n = 0 discontinuity (so that the inner horizon never vanishes abruptly) would remove the need for the predicted quantum jump.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The transition from a regular black hole to the singular Schwarzschild black hole is a discrete quantum event, not a smooth classical collapse.
The total entropy of a regular black hole always exceeds A/4, reaching up to 2(A/4) for highly excited states, and the excess must be emitted to satisfy the generalized second law.
For the n = 3 state the released entropy is approximately 0.59 A/4; for n ≫ 1 it approaches a full A/4, so the most excited states liberate an entropy equal to the entire event-horizon entropy.
The integer parameter n quantizes the entropy spectrum, offering a concrete mechanism for horizon-area quantization via discrete internal geometric states.
In the N > 1 families, collapse produces a multi-stage entropy cascade, with both n_1- and n_2-transitions contributing discrete pulses of entropy release.

Where Pith is reading between the lines

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

We infer that if this picture is correct, the final moments of such a collapse would emit a burst of radiation (likely gravitational waves or Hawking-like quanta) whose total entropy equals π h_c² — a potentially observable signature that could distinguish regular-black-hole models from alternatives.
We infer that the statistical interpretation (S = ln Ω) is only consistent if a maximum allowed value of n exists; otherwise the infinite tower of states would imply an unbounded density of microstates for a fixed mass, so a quantum-gravity cutoff is implicitly required.
We infer that the 59% figure for n = 3 is a sharp, testable prediction: any quantum-gravity calculation that finds significant correlations between the two horizons would change the released entropy, so this number provides a clean quantitative target for future computations.
We infer that the logarithmic threshold n ~ 1/ln(h/ℓ_P) means the quantum transition takes place while the black hole is still macroscopic, suggesting that stochastic-semiclassical methods could be used to probe the transition without a full theory of quantum gravity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

A clean, internally consistent thermodynamic add-on to the OCK regular black holes, but its central 'must be released' claim is explicitly built on an unproven additive-entropy rule and should be framed as conditional.

read the letter

Short version: this paper reads a discrete entropy spectrum off the OCK regular black holes, and it does so carefully. The arithmetic checks out, the assumptions are stated, and the N=2 extension is concrete. But the load-bearing physical claim—that hidden inner-horizon entropy 'must be released'—is exactly as conditional as the paper's own 'working hypothesis' label. The abstract and conclusions state it more firmly than the body justifies.

What is actually new: the interpretation of S_inner = πh_c^2 as a potential gravitational entropy liberated when the Cauchy horizon disappears, the quantized entropy ladder labelled by integer n, the ~59% figure for n=3, and the N=2 tables showing algebraic approach to the N=1 limit. Those are genuine extensions of the OCK construction, not contained in the cited papers. The authors also deserve credit for flagging the infinite-tower problem and suggesting a cutoff, and for explicitly saying no instanton has been constructed.

The soft spot is the bridge. Eq. (14) just assumes additivity of event-horizon and inner-horizon entropies. Wald's Noether charge gives an area term for any Killing horizon, but it does not make the inner horizon a thermodynamic subsystem that participates in the generalized second law when it is destroyed. The GSL is proven for event horizons; extending it to disappearing Cauchy horizons is an assumption, not a theorem. The paper says this in Section III, but the title and abstract don't. If additivity fails, the 'entropy release' is bookkeeping, not physics. The transition probability Γ ∝ exp[ΔS] is similarly an expectation, not a derived result—again stated, but easy to miss.

The mass-inflation discussion is honest and appropriately hedged: the quantum disappearance of the Cauchy horizon is a plausible way to bypass the instability, but it is speculative. The N>1 cascade is a nice concrete prediction—a train of pulses if the transitions are quantum jumps—but it inherits the same additivity assumption.

Who this is for: people working on regular black holes, the OCK construction, and horizon thermodynamics. It deserves referee time because it is internally consistent, builds on a concrete prior framework, and makes a testable-in-principle prediction. But it should be accepted only after the unconditional language is softened and the additive rule is either derived from a model or explicitly framed as a bookkeeping consequence. My own verdict would be conditional, not accept as is.

Referee Report

4 major / 5 minor

Summary. The paper studies the OCK family of regular Schwarzschild black holes with a de Sitter core, whose metric depends on an integer parameter n>2. It observes that the inner Killing horizon carries a formal Bekenstein–Hawking entropy S_inner = A_inner/4 = π h_c^2, and proposes that this entropy is a hidden contribution to the total gravitational entropy, S_reg = A/4 + π h_c^2 (Eq. 14). Because the OCK collapse dynamics forces n to decrease and the inner horizon to shrink and vanish at n=0 (Minkowski breaking), the paper argues that the generalized second law requires this hidden entropy to be released. It computes h_c(n), the released fraction ΔS/(A/4), the divergence of κ_inner as n→0^+, and the resulting discrete entropy spectrum. It also discusses N>1 families and a conjectured statistical interpretation in which black hole entropy counts distinct interior geometries. The arithmetic and asymptotic limits are internally consistent and reproduce the tables.

Significance. If the central thermodynamic identification were justified, the paper would provide a concrete, testable link between singularity avoidance, Minkowski breaking, and entropy quantization. Its strengths are the explicit calculations with no fitted parameters, the reproducible numerical values in Tables I and II, the clean asymptotic analysis (e.g., h_c/h ≃ (2/3)^{1/n}, κ_inner divergence), and the unusually candid discussion of its own assumptions and limitations. The paper is not a derivation from quantum gravity; it is a proposal for how the OCK geometry would behave if the additive inner-horizon entropy rule and an extended second law hold. That conditional nature is explicitly acknowledged in Sec. III, but the abstract and conclusions sometimes state the release as unconditional. The significance is therefore real but conditional: the paper offers a well-defined geometric model whose physical interpretation depends on an unproven premise.

major comments (4)

The load-bearing premise is the additive rule S_reg = A/4 + π h_c^2. The paper itself labels it a 'working assumption' and 'working hypothesis' in Sec. III, and notes that alternative rules would require correlations between horizons. However, the abstract, Sec. IV.A, and Sec. VI state that the inner-horizon entropy 'must be released' when the inner horizon disappears. The Wald–Noether theorem quoted in Sec. III shows that an area term can be assigned to any Killing horizon, but it does not establish that the inner horizon contributes additively to the thermodynamic entropy of the external black hole, nor that the generalized second law applies to a disappearing Cauchy horizon. The GSL is established for event horizons, not for destroyed inner horizons. Since the entire entropy-release prediction hinges on this identification, the central claim is currently conditional. The authors shoul
The transition probability Γ ∝ exp(ΔS) is presented as supporting the 'thermodynamically favoured' decay to the Schwarzschild state. The paper explicitly says no instanton has been constructed and that Γ ∝ exp(ΔS) is 'an expectation based on the standard semiclassical framework, not an established result.' Yet this unproven relation is later used to assert that the system is 'overwhelmingly likely' to tunnel to the ground state and that the entropy release is irreversible. This is another load-bearing link. If the additive-entropy premise is accepted, ΔS>0 alone does not determine the transition rate without a dynamical model; the Euclidean-action relation I_E = βM − S is itself known to require care for non-stationary or multiple-horizon spacetimes. The claim should be labeled as conjecture in the conclusions as well as in the body.
The claim that mass inflation is circumvented because 'the inner horizon shrinks and vanishes before the full nonlinear development of the mass inflation can occur' is not demonstrated. The paper does not provide a dynamical timescale comparison between the collapse of n(v) and the e-folding time of the mass-inflation instability. This is explicitly acknowledged as 'plausible' and 'beyond the scope,' but it is used in the conclusions to support the quantum nature of the final transition. As it stands, this is an unresolved physical question; the paper should either present evidence for the timescale hierarchy or describe this as an open possibility, not a consequence of the model.
The statistical-mechanics proposal S_BH = ln Ω(M), with Ω(M) counting ordered integer tuples subject to a cutoff n_max, is an interesting conjecture but is not derived. The appendix correctly limits itself to counting; however, the text in Sec. V.D and the conclusions suggests that 'black hole entropy emerges as the logarithm of the number of such states.' For a fixed ADM mass, the event horizon area is fixed, so all interior states have the same exterior, but the entropy of the interior is not independently computed; the additive rule is simply imported. The paper should make clear that this is a combinatorial analogy, not a derivation, unless a concrete gravitational path-integral or microstate count is supplied.

minor comments (5)

The mass function (3) is stated for n>2, but the paper later refers to 'n=−1' as the Schwarzschild ground state (Sec. V, Fig. 1). For n=−1 the expression does reduce to m(r)=M, which is a useful observation, but it should be stated explicitly; otherwise the ground state appears to lie outside the claimed domain of the family.
The surface gravity formula has a sign choice: the derivative f'(h_c) is negative for the Cauchy horizon, but Eq. (17) writes κ_inner = −(1/2)f'(h_c). This should be flagged as a convention and compared with the usual definition κ = |(1/2)f'| at Killing horizons, to avoid confusion with the event-horizon sign convention.
Table I gives values for n=3,...,10 and ∞, but Fig. 1 mentions the ground state at n=−1. The horizontal axis label in the figure would benefit from including the mapping n>2 → regular states and n=−1 → Schwarzschild, since the reader may not infer this from the text.
The relation I_E = βM − S is stated without derivation and with reference [13] only. For a spacetime with multiple horizons, the identification of β and S is not automatic; a one-sentence clarification of the regime in which this is expected to hold would improve precision.
The coefficient C is written as 3(n_1+1)/[(n_2−2)(n_2−n_1)]; for n_2>n_1 this is positive. The sign of the x^{n_2} term in Eq. (A2) is then negative, consistent with the polynomial structure. This is correct, but the text could note that B and C are both positive for the ordered range.

Circularity Check

0 steps flagged

No significant circularity: the entropy-release conclusion is a transparent conditional consequence of explicitly labeled working hypotheses, with independent OCK/Wald inputs and no fitted parameters or self-citations.

full rationale

The derivation chain is: (i) the OCK regular metric and the Minkowski-breaking obstruction are taken from refs. [8,9], which are external to the present authors; (ii) the inner-horizon entropy S_inner=A_inner/4=πh_c^2 is obtained by applying the Wald–Noether area law for Killing horizons, with the paper explicitly saying 'We adopt this identification as a working assumption' (Sec. III); (iii) the additivity S_reg=A/4+πh_c^2 is introduced as 'a natural bookkeeping device' and then 'adopt[ed] ... as a working hypothesis throughout this work' (Sec. III, Eq. 14), not as a derived result; (iv) the claim that this entropy 'must be released' is conditional on that working hypothesis plus the externally assumed generalized second law, and the abstract/conclusions state the condition; (v) the quantized spectrum follows directly from the input integer parameter n and is presented as such: 'The integer character of the parameter n ... has a direct thermodynamic consequence.' No parameter is fitted and then renamed a prediction, no load-bearing step is justified by a self-citation, and the paper itself flags the main physical caveats: the inner-horizon entropy is 'formal,' the additive rule is a working hypothesis, and 'no instanton describing the transition has been constructed' (Sec. IV-B). These are limitations and conditionalities, not circular reductions.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 1 invented entities

Once the OCK metric and S=A/4 are accepted, the arithmetic is deterministic and no free parameters are fitted. The speculative weight is concentrated in four axioms: the formal inner-horizon entropy is treated as physical and additive; GSL forces its release; the OCK collapse kinematics (integer n, n-dot<0, Minkowski breaking) are taken from prior work; and a Planck-scale breakdown licenses the quantum-transition picture. The integer n is inherited, not derived.

free parameters (2)

Integer exponent n = 3 (used for the headline 59% figure); arbitrary integer > 2
Labels the OCK polynomial interior. Its integrality is the entire source of the claimed quantized entropy spectrum; it is inherited from the prior construction, not derived in this paper.
N=2 exponents n1, n2 = e.g. (3,4), (3,10), (5,100) in Table II
Same role in the Appendix A extension; the ordering 2 < n1 < n2 is assumed and affects the multi-stage entropy cascade.

axioms (5)

domain assumption Wald-Noether area law S=A/4 applies to the inner Cauchy horizon as a Killing horizon.
Standard for Killing horizons, but the paper explicitly calls the inner-horizon identification a working assumption (Sec. III), and the physical meaning differs from the event horizon.
ad hoc to paper Horizon entropies add: S_reg = A/4 + π h_c^2.
Paper states no proof; additive rule assumes independent microstates and factorizing partition function (Sec. III).
domain assumption Generalized second law requires hidden inner-horizon entropy to be released when the inner horizon disappears.
Central logical bridge from geometry to radiation; assumed, not derived.
domain assumption OCK result: continuous transition to Schwarzschild point mass is impossible; collapse has n-dot<0.
Taken from refs [8,9]; used to justify that entropy release accompanies a discrete/quantum event.
ad hoc to paper Semiclassical breakdown κ_inner ~ 1/ℓ_P implies a quantum transition that can bypass mass inflation.
Plausibility argument; the paper says a full quantum/semiclassical analysis lies beyond scope.

invented entities (1)

Hidden inner-horizon entropy S_inner = π h_c^2 as 'potential gravitational entropy' no independent evidence
purpose: Bookkeeping device for internal geometric microstates that become observable when the inner horizon is destroyed
No independent observable signature is derived; it is a formal assignment of S=A/4 to the Cauchy horizon plus GSL. The paper itself labels it a working hypothesis.

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0 comments

read the original abstract

The classical formation of a Schwarzschild black hole from a regular, non-singular configuration has recently been shown to be impossible within general relativity: the geometry inevitably develops a discontinuity at the origin, a phenomenon termed Minkowski breaking by Ovalle, Casadio, and Kamenshchik [PRD 113 (2026), 064042]. This obstruction signals that the transition to the Schwarzschild point mass must be a discrete, quantum event. We uncover the thermodynamic footprint of this transition. Using the explicit family of regular Schwarzschild black holes with a de Sitter core, we show that the inner Killing horizon carries a formal Bekenstein-Hawking entropy $S_{\rm inner} = A_{\rm inner}/4$ that is absent in the singular Schwarzschild state. This entropy is hidden from external observers in equilibrium but, assuming the generalized second law, must be released when the inner horizon disappears. As the collapse parameter $n$ decreases, the inner horizon shrinks and its entropy is gradually released during classical evolution, until the horizon finally vanishes at $n=0$ with the Minkowski breaking. The surface gravity diverges as $n\to0^+$, with the semiclassical description breaking down at $n \sim 1/\ln(h/\ell_P)$; the final disappearance is therefore a deep quantum process. For the $n=3$ regular black hole, the stored entropy is approximately $59\%$ of $A/4$; in the semiclassical limit $n\gg1$, it approaches the full $A/4$. The integer nature of $n$ implies a quantized entropy spectrum, with the Schwarzschild black hole as the ground state within the OCK family. We discuss how the classical mass-inflation instability may be circumvented by the quantum disappearance of the Cauchy horizon, and clarify the continuous vs. discrete nature of the collapse.

Figures

Figures reproduced from arXiv: 2607.14079 by Francisco S. N. Lobo, Manuel E. Rodrigues.

**Figure 1.** Figure 1: gives a schematic representation of these energy levels. As n decreases, the inner horizon shrinks and the total entropy Sreg drops by finite amounts, until it reaches the ground state within the OCK family (n = −1, Schwarzschild) with entropy S = A/4. A. Physical interpretation: regular black holes as excited states The discrete entropy spectrum suggests that the regular black holes are excited geometr… view at source ↗

discussion (0)

Forward citations

Cited by 3 Pith papers

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

Kerr black holes without primary hairs
gr-qc 2026-07 conditional novelty 6.0

Explicit infinite family of axisymmetric black holes with a Kerr exterior, a regular or mildly singular interior controlled by free exponents n_i, and no additional asymptotic charges.
From gravastar to central singularity
gr-qc 2026-07 conditional novelty 5.0

A simple thermodynamic model with negative entropy for the core finds gravastars unstable and always decaying to singular Schwarzschild black holes.
Entropy dynamics in gravitational collapse: From Minkowski breaking to de Sitter thermodynamics
gr-qc 2026-07 conditional novelty 4.0

The sign of the Hubble parameter unifies OCK entropy release and Volovik de Sitter thermodynamics as complementary pictures of entropy flow in collapse, without a classical bridge between them.

Reference graph

Works this paper leans on

14 extracted references · 5 linked inside Pith · cited by 3 Pith papers

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