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The Simpson–Visser regular black hole undergoes a Davies phase transition at a = √2 m and its quantum-corrected entropy ends in a purely logarithmic remnant.

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2026-08-03 17:54 UTC pith:L5YJIDPB

load-bearing objection Semiclassical half is correct but routine; the quantum corrections have a sign error and a remnant formula that contradicts the paper's own Eq. (27). the 3 major comments →

arxiv 2512.07786 v3 pith:L5YJIDPB submitted 2025-12-08 gr-qc

Thermodynamic Phase Transitions and Quantum Entropy Corrections in the Simpson-Visser Regular Black Hole

Vinayak Joshi , Ashok B. Joshi This is my paper
classification gr-qc PACS 04.70.-s04.60.-m
keywords regular black holeSimpson-Visserblack-bounceheat capacity divergenceDavies phase transitionquantum entropy correctiontunneling formalismblack hole remnant
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 tries to show that resolving the central singularity—as the Simpson–Visser black-bounce metric does with a length scale a—changes black-hole thermodynamics in two measurable ways. First, the heat capacity C = −8πm³√(4m²−a²)/(2m²−a²) diverges at a = √2 m, a Davies-type second-order phase transition where the heat capacity changes sign between an unstable Schwarzschild-like phase and a stable phase. Second, using the tunneling-expansion method for the temperature, the paper derives a quantum-corrected entropy whose leading term already depends on a, and whose value at the extremal remnant a = 2m is purely logarithmic, S_rem = 8πβ₁ ln a + const. The central message is that singularity resolution is not just a geometric detail: it imprints itself on the phase structure, stability, and final quantum state of the black hole.

Core claim

At the heart of the paper is the one-parameter Simpson–Visser metric, f(r) = 1 − 2m/√(r² + a²), whose outer horizon sits at r₊ = √(4m² − a²). From the surface gravity the authors obtain T_H = √(4m² − a²)/(16πm²), and therefore C = (∂T_H/∂m)⁻¹ = −8πm³√(4m² − a²)/(2m² − a²). The denominator vanishes at a = √2 m, which they identify as a Davies-type second-order phase transition separating an unstable phase (C < 0) from a stable phase (C > 0). For the quantum part, they use the corrected temperature T_corr = T_H (1 + β₁ℏ/m² + β₂ℏ²/m⁴)⁻¹ and integrate the first law to obtain S(m,a) = (2π/ℏ)[m√(4m²−a²) + (a²/2) ln((2m+√(4m²−a²))/a)] + 8πβ₁ ln((2m+√(4m²−a²))/a) − 16πβ₂ℏ√(4m²−a²)/(a²m) + S_const; a

What carries the argument

The central object is the Simpson–Visser geometry, a one-parameter family that interpolates between Schwarzschild (a = 0), a regular black hole with an inner and outer horizon (0 < a < 2m), an extremal black-bounce (a = 2m), and a traversable wormhole (a > 2m). The semiclassical analysis uses the horizon radius r₊ = √(4m² − a²), the Hawking temperature T_H = √(4m² − a²)/(16πm²), and the heat-capacity definition C = (∂T_H/∂m)⁻¹. The quantum analysis is carried by the Hamilton–Jacobi tunneling formalism: an ℏ-expansion of the particle action produces the corrected temperature T_corr, which is then integrated through the first law dS = dm/T_corr to obtain the corrected entropy. The named object

Load-bearing premise

The load-bearing premise is that the quantum correction coefficients β₁ and β₂ in the tunneling temperature are the same constants for the Simpson–Visser geometry as for Schwarzschild; the paper never computes the tunneling action for the a-dependent metric, so if β₁ or β₂ depend on a, the corrected entropy and the logarithmic-remnant conclusion change.

What would settle it

Compute the full Hamilton–Jacobi tunneling probability Γ ∼ exp(−2 Im I/ℏ) for the metric f(r) = 1 − 2m/√(r² + a²) with a > 0 and extract T_corr. If the coefficients multiplying ℏ/m² and ℏ²/m⁴ depend on a when a is not zero, then Eq. (27) is not the entropy of the Simpson–Visser black hole, and the remnant formula S = 8πβ₁ ln a is not its end-state entropy. A direct derivation of T_corr from the tunneling action, or a numerical integration for several values of a/m, would settle it.

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

If this is right

  • For 0 < a < √2 m the heat capacity is negative, so the black hole evaporates unstably like Schwarzschild; for √2 m < a < 2m it is positive, allowing stable equilibrium with a heat bath.
  • At a = √2 m the heat capacity diverges, marking a Davies-type second-order phase transition between the two phases.
  • The Bekenstein–Hawking entropy 4πm² is independent of a, but the corrected leading-order entropy depends on a, breaking the area-law degeneracy among equal-mass regular black holes at the quantum level.
  • In the extremal limit a = 2m, the temperature and area-law term vanish, leaving a stable remnant whose entropy is S = 8πβ₁ ln a + const, a purely logarithmic quantum-statistical end state.
  • In the evaporation narrative at fixed a, a black hole crosses into the stable phase as its mass decreases below a/√2, slowing evaporation and ending at a remnant.

Where Pith is reading between the lines

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

  • A direct Hamilton–Jacobi calculation for the a-dependent Simpson–Visser metric—rather than the Schwarzschild-derived ansatz for T_corr—would show whether β₁ and β₂ depend on a; the paper's entropy formula, its 1/a² term, and the remnant formula all stand or fall on that calculation.
  • The threshold a_c/m = √2 is scale-free; if it survives in rotating or charged versions of the model, it would provide a concrete prediction for when a regular black hole's evaporation end-state changes character.
  • The 1/a² divergence of the β₂ correction near the Schwarzschild limit suggests the perturbative ℏ-expansion is not uniformly valid, so non-perturbative or resummed treatments may be needed exactly where the model approaches the singular geometry.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper analyzes the thermodynamics of the Simpson–Visser regular black hole, a one-parameter (a) interpolation between Schwarzschild, a black-bounce, and a traversable wormhole. In the semiclassical section it derives the Hawking temperature T_H = sqrt(4m^2 − a^2)/(16πm^2), the Bekenstein–Hawking entropy S_BH = 4πm^2, the heat capacity C = −8πm^3 sqrt(4m^2−a^2)/(2m^2−a^2), and the free energy F = m − sqrt(4m^2−a^2)/4. It identifies a Davies-type divergence at a = √2 m, separating a thermodynamically unstable phase from a stable one, and interprets the extremal limit as a remnant. In the quantum section the paper assumes a corrected temperature T_corr = T_H(1 + β1ℏ/m^2 + β2ℏ^2/m^4)^{-1} and integrates dS = dm/T_corr to obtain a corrected entropy S(m,a) with logarithmic and 1/a^2 terms. It then claims that at the extremal endpoint m = a/2 the entropy reduces to a purely logarithmic remnant S_remnant = 8πβ1 ln a + S_const, which is presented as evidence for a singularity-free information-storing end state.

Significance. If the quantum claims were correct, the paper would establish a concrete example of singularity resolution leaving observable thermodynamic imprints: a stability transition controlled by the regularization scale and a modified entropy with a nonzero logarithmic remnant. The semiclassical heat-capacity calculation is clean, correct, and reproducible; the identification of the Davies point at a = √2 m is a valid result. However, the quantum half does not support its conclusions. The central bridge (Eq. 17) is assumed rather than derived for this geometry, and even granting that assumption there are internal algebraic inconsistencies: the sign of the β2 term is wrong, and the final entropy formula contradicts the claimed logarithmic remnant. These are not presentation issues but errors in the load-bearing derivation, so the paper's main advertised results — the quantum-corrected entropy and its remnant interpretation — are not established.

major comments (3)
  1. [IV.A, Eq. (17)] The quantum-corrected temperature T_corr = T_H(1 + Σ β_i ℏ^i/m^{2i})^{-1} is taken over from the Schwarzschild analyses of Refs. [44,45] without performing any Hamilton–Jacobi or WKB computation for the Simpson–Visser metric (1). The near-horizon geometry depends on a through r_+ = sqrt(4m^2−a^2), f(r) and its derivative; nothing in the manuscript rules out β_i = β_i(a,m). Since every a-dependent feature of Eq. (27) — the log coefficient, the 1/a^2 term, and the remnant — comes from this assumed form, Eq. (17) is the load-bearing bridge of the quantum section. The abstract's claim to 'derive' the corrections is therefore not supported by the presented calculation.
  2. [IV.C–D, Eqs. (24)–(28)] There is an internal inconsistency between the final entropy formula and the remnant claim. In Eqs. (24)–(25) the logarithmic argument is ln(2m + sqrt(4m^2−a^2)); in Eq. (27) it is silently replaced by ln((2m + sqrt(4m^2−a^2))/a). This is not a harmless integration-constant shift because the difference involves a, which is not constant in the evaporation scenario. At m = a/2, Eq. (27) gives sqrt = 0 and ln(1) = 0, so S(m=a/2,a) = S_const, not 8πβ1 ln a + S_const. Thus Eq. (28) follows only from the unnormalized logarithm of Eqs. (24)–(25), not from the paper's own final result. The pure-logarithmic-remnant narrative and its information-loss discussion are unsupported.
  3. [IV.C, Eq. (26)] The sign of the β2 contribution is incorrect. From Eq. (23), the β2 term is (16π/ℏ) β2ℏ^2 ∫ dm/(m^2 sqrt(4m^2−a^2)). Direct integration gives ∫ dm/(m^2 sqrt(4m^2−a^2)) = sqrt(4m^2−a^2)/(a^2 m) > 0, so the integrated contribution is +16πβ2ℏ sqrt(4m^2−a^2)/(a^2m). Eq. (26) and hence Eq. (27) carry a minus sign. This changes the sign structure of the subleading correction and the stated 1/a^2 behavior, and it also affects the figures that use Eq. (27).
minor comments (4)
  1. [III.B, after Eq. (15)] The text states that the free energy 'monotonically decreases with increasing a, vanishing in the extremal limit a→2m'. The formula F = m − sqrt(4m^2−a^2)/4 and Fig. 3 show the opposite: F increases from m/2 at a=0 to m at a=2m. The subsequent statement that the Schwarzschild configuration is thermodynamically preferred is correct, but the sentence is wrong and should be fixed.
  2. [Eq. (24)] Eq. (24) writes the antiderivative with ln(2m + sqrt(4m^2−a^2)), whose argument has dimensions of mass. The correct indefinite integral is m sqrt(4m^2−a^2)/8 + (a^2/16) ln((2m+sqrt(4m^2−a^2))/a). The /a appears only later in Eq. (27); the intermediate equation is dimensionally inconsistent as written.
  3. [Figures and references] Several figure captions and reference entries contain garbled encoding artifacts (e.g., '/uni00000014/uni00000011/uni00000013' in Fig. 1 and Fig. 5 captions and in Refs. [35,46]). These rendering artifacts must be cleaned before any resubmission.
  4. [General conventions] The paper declares G=c=ℏ=k_B=1, but Eqs. (17)–(27) explicitly keep factors of ℏ and k_B. Please state the convention clearly; otherwise the status of β_i and the dimensions of the logarithms are ambiguous.

Circularity Check

0 steps flagged

No circularity found; the quantum section rests on an explicit external ansatz and contains an internal inconsistency, but no result reduces to its own input.

full rationale

The classical phase-transition analysis (Eqs. 5-13) is self-contained calculus from the Simpson-Visser metric: the Hawking temperature is computed from surface gravity, the heat capacity is its reciprocal mass derivative, and the divergence at a=sqrt(2)m follows directly. Nothing in that chain is fitted or defined in terms of the claimed phase transition. The quantum-entropy section is not circular in the rubric's sense: Eq. (17) is openly presented as a phenomenological corrected-temperature ansatz imported from external references [44,45] ('The dimensionless coefficients beta_i are phenomenological parameters... They are expected to be of order unity'), and the beta_i are not fitted to the S-V entropy nor renamed as predictions. The final S(m,a) is the integral of that assumed T_corr, so it is conditional on an ansatz, but a conditional derivation is not a by-construction equivalence. There is no load-bearing self-citation and no uniqueness theorem imported from the present authors' prior work. I do flag two non-circular weaknesses in the manuscript: (i) the applicability of Eq. (17) to the S-V geometry, with a-independent beta_i, is asserted rather than derived from a Hamilton-Jacobi tunneling calculation for S-V; and (ii) Eq. (28) does not follow from Eq. (27), because at m=a/2 the logarithm ln((2m+sqrt(4m^2-a^2))/a) vanishes, so Eq. (27) gives S_const rather than 8*pi*beta_1*ln(a)+S_const. These are correctness or support gaps, not circular reductions, and therefore do not raise the circularity score.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central results rest on zero new entities but on several imported or assumed inputs: the S-V metric itself, the canonical-ensemble identification, the Banerjee-Majhi temperature ansatz with a-independent coefficients (the most fragile), the fixed-a first law, and the literature value of β1. The phase-transition half of the paper needs only the first two; the quantum half needs all five, and the two free coefficients β1, β2 plus S_const carry the tunable content of the headline formulas.

free parameters (3)
  • β1 (leading quantum-correction coefficient) = -1/(8π), imported from refs [57,60-64] and used in Fig. 4
    Controls the logarithmic correction and the claimed remnant entropy; its value is assumed from prior literature, never derived for the S-V geometry.
  • β2 (next-to-leading quantum-correction coefficient) = not specified; 'enhanced, intermediate and strong coupling taken arbitrarily' in Fig. 4
    Controls the claimed 1/a² term (Eqs. 26-27); its sign and magnitude are arbitrary in the paper, yet the small-a divergence narrative depends on it.
  • S_const (entropy integration constant) = unspecified
    Absorbs ln(a) and a² ln(a) pieces depending on convention; the remnant-entropy value and the 'convergence to the area law' claims depend on it, so the physical content is unconstrained.
axioms (5)
  • domain assumption The Simpson-Visser metric (Eq. 1) is an adequate effective description of singularity resolution.
    Imported from ref [34]; all subsequent calculations inherit this modeling choice, including the NEC-violating matter discussed in Sec. II.B.
  • domain assumption Canonical-ensemble identification E = m and C = (∂T_H/∂m)^{-1} (Eq. 9).
    Standard in black-hole thermodynamics but an assumption about which ensemble and energy variable apply; enters at the start of Sec. III.A.
  • ad hoc to paper The quantum-corrected temperature has the a-independent form T_corr = T_H(1 + β1ℏ/m² + β2ℏ²/m⁴)^{-1} (Eq. 17).
    Imported from refs [44,45]; the Hamilton-Jacobi action for the S-V metric is never computed, and the a-independence of the β_i is assumed without evidence. This is the weakest structural premise of Section IV.
  • domain assumption First law dS = dm/T at fixed a applied across all mass scales (Eq. 18).
    Assumes no extra work term associated with the scale a; acceptable if a is fixed, but the microscopic status of a is left unstated.
  • domain assumption β1 = −1/8π from previous quantum-gravity/Cardy-type results.
    Taken from refs [57,60-64]; external input, not checked in this paper.

pith-pipeline@v1.3.0-alltime-deepseek · 16694 in / 45176 out tokens · 380198 ms · 2026-08-03T17:54:30.469507+00:00 · methodology

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read the original abstract

Regular black holes offer a compelling framework to explore the consequences of resolving the central singularity of standard black holes. Using the Simpson-Visser ``black-bounce" geometry as an elegant, analytically tractable framework, we explore the intricate thermodynamic behavior in such models. We demonstrate that this regular spacetime exhibits a critical instability, marked by a phase transition where the heat capacity is discontinuous. This transition signals a fundamental change in the black hole's evaporation state, which depends on the regularization parameter. Pushing beyond the semiclassical limit, we then derive the leading-order quantum corrections to the entropy via the Hamilton-Jacobi tunneling formalism. Our analysis provides a refined statistical basis for the entropy of non-singular spacetimes and offers a quantitative analysis of the nature of the black hole end-state. These results reveal that singularity resolution is not merely a geometric modification but a profound thermodynamic event, with direct implications for the stability and ultimate fate of evaporating black holes.

Figures

Figures reproduced from arXiv: 2512.07786 by Ashok B. Joshi, Vinayak Joshi.

Figure 1
Figure 1. Figure 1: FIG. 1. Metric structure and energy density for the Simpson– [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Normalized heat capacity as a function of the di [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The free energy indicates the thermodynamically [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The quantum-corrected entropy [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. A detailed decomposition and analysis of the quantum-corrected entropy for various phenomenological scenarios. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

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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. Optical appearance of regularized compact objects without an exterior photon sphere

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