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REVIEW 3 major objections 5 minor 57 references

Entropy bound predicts remnant for evaporating rotating black hole

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2026-07-10 03:26 UTC pith:UE2EZWTD

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arxiv 2607.08661 v1 pith:UE2EZWTD submitted 2026-07-09 hep-th gr-qc

The Remnant of an Evaporating Rotating Regular Black Hole from the Generalized Entropy in the Final Stage of Evaporation

Shingo Takeuchi This is my paper
classification hep-th gr-qc
keywords alphaevaporationfinalrotatingstageconsideredentropymass
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 an evaporating rotating regular black hole cannot shrink all the way to its extremal limit. The author writes the black hole mass as m_ext + α, where m_ext is the extremal mass and α parameterizes how far the hole is from extremality. The generalized entropy of Hawking radiation in the final evaporation stage splits into two pieces: an area term Λ₁ that stays positive, and a correction term Λ₂ that behaves as -ln(α) and grows negative as α shrinks. Because entropy is generally non-negative, the author assumes these two contributions should not change sign during evaporation. The correction term Λ₂ vanishes at a finite value α₂ ≈ √(2/(3π)) · m_ext. This finite vanishing point is taken as a lower bound on α, meaning the black hole mass cannot drop below m_ext + α₂. The author interprets this as the emergence of a remnant — a stable end-state object that halts evaporation before the extremal limit is reached. The black hole studied is a rotating regular black hole, where regular means the central singularity is replaced by a de Sitter core at Planck scale, motivated by the idea that the fine structure of the central region matters in the final evaporation stage.

Core claim

The generalized entropy of Hawking radiation for a near-extremal rotating regular black hole decomposes into an area contribution Λ₁ ~ 4πa²/G_N (positive, bounded away from zero) and a correction contribution Λ₂ ~ (1/6)ln(a²G²_N/(α²ε⁴)) that diverges to -∞ as α → 0. The correction term crosses zero at α₂ ≈ √(2/(3π)) · m_ext, which is finite and of order m_ext. Under the assumption that neither contribution should change sign during evaporation, this crossing point sets a hard lower bound on how far the black hole can evaporate. The remnant mass is m_ext + α₂ ≈ (1 + √(2/(3π))) m_ext, and the regularization parameter ℓ_p enters only as a subleading correction, so the remnant formation does not

What carries the argument

island formula

Load-bearing premise

The argument depends on assuming that the area term and the correction term in the generalized entropy each keep a fixed sign throughout the entire evaporation process. The lower bound α₂ is identified as the point where the correction term hits zero, and the claim that the black hole cannot evaporate past this point rests entirely on the correction term not flipping to positive below α₂. If the correction term were to change sign at some smaller α, the lower-bound argument

What would settle it

A demonstration that the correction term Λ₂ in the generalized entropy changes sign at some α < α₂, or that the sign-constancy assumption is violated by a more complete calculation of the field-theory contribution to the entropy, would remove the lower bound and invalidate the remnant conclusion.

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

Where Pith is reading between the lines

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

  • If remnants generically form at a mass floor of order m_ext for rotating black holes, this could provide a concrete mechanism for preserving unitarity: information is not destroyed but trapped in a stable Planck-scale object rather than radiated away.
  • The fact that the regularization parameter ℓ_p enters only as a correction to α₂ suggests the remnant mass floor is robust to the specific choice of singularity regularization, which would strengthen the result's generality if confirmed.
  • The exponentially large upper bound α₁ ~ e^{a²/G_N} where the total entropy vanishes implies the Page curve transition would occur extremely early in the evaporation process, which has implications for how quickly information begins to be recovered from Hawking radiation.
  • If the sign-constancy assumption on Λ₁ and Λ₂ could be derived from first principles rather than postulated, the remnant conclusion would become a theorem rather than an argument from plausibility.

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 / 5 minor

Summary. This paper studies the generalized entropy of Hawking radiation in the final stage of evaporation of a rotating regular black hole (RBH). The author parametrizes the BH mass as m_0 = m_ext + alpha, where m_ext is the extremal mass, and expands the entanglement entropy (EE) of the Hawking radiation in alpha using the island formula. The EE is decomposed into an area term (Lambda_1) and a correction term (Lambda_2). The author assumes that the signs of these two contributions remain unchanged throughout evaporation, identifies the value alpha_2 at which Lambda_2 vanishes as a lower bound on alpha, and finds alpha_2 ~ sqrt(2/(3*pi)) * m_ext. This finite lower bound is interpreted as evidence for a remnant. The paper also provides a detailed perturbative solution of the quintic horizon equation for the rotating RBH metric, demonstrating smooth connection between non-extremal and extremal horizon radii.

Significance. The question of whether black hole evaporation terminates in a remnant is of broad interest for the information paradox and unitarity. The paper provides an explicit quantitative estimate of a remnant mass in a rotating regular BH setting, which is a novel contribution. The perturbative solution of the quintic horizon equation (Sec. 3) and the demonstration of smooth connection between non-extremal and extremal cases is a technically useful result that, to the author's knowledge, has not appeared elsewhere. The falsifiable prediction alpha_2 ~ sqrt(2/(3*pi)) * m_ext is a concrete, testable result.

major comments (3)
  1. Sec. 6, point ii: The sign-stability assumption is the sole logical bridge from 'Lambda_2 crosses zero at alpha_2' to 'alpha_2 is a lower bound on alpha.' Entropy non-negativity constrains only the sum Lambda_1 + Lambda_2 >= 0, not the individual signs of Lambda_1 and Lambda_2. Without additional justification, Lambda_2 could change sign at some alpha < alpha_2 while Lambda_1 keeps the total non-negative, and no remnant would be implied. The paper should either derive this assumption from more fundamental principles or provide a physical argument for why sign changes are disallowed in this system. As stated, the central claim depends on an undemonstrated premise.
  2. Eq. (56) vs. Eq. (49) and Appendix E, Eq. (78): The constant C_{kappa r_b} is dropped from Lambda_2 in Eq. (56) when computing alpha_2. However, Appendix E (Eq. 78) shows that C_{kappa r_b} contains a constant piece 2*ln[tanh(sqrt(3)/2)] ~ -0.40, which is the same order as the retained 1/6 * ln(...) term at alpha ~ alpha_2 (which is ~0.63 by construction). Including this constant would shift the zero-crossing of Lambda_2 by a factor of order e^{6*0.4} ~ 11, potentially violating the assumption alpha << m_ext. The author should either justify dropping C_{kappa r_b} quantitatively or include it in the computation of alpha_2.
  3. Abstract vs. body notation: The abstract states the lower bound is alpha_1, while the body (Sec. 6.2, Eq. 58) identifies alpha_2 as the lower bound and alpha_1 as the upper bound (Sec. 6.1, Eq. 57). This inconsistency between abstract and body should be corrected, as it creates confusion about which quantity is the main result.
minor comments (5)
  1. Sec. 5.3: The treatment of r_b involves choosing it by hand to avoid divergences (case 1) and to avoid unphysical largeness (case 3), settling on case 2 where kappa*r_b is finite. The author acknowledges in a footnote that the origin of some stationary points is unclear. While this does not block the main result, the sensitivity of the final answer to this choice should be discussed more transparently.
  2. Eq. (52): The result r_a ~ r_+ - G_N/(12*a*pi) places the extremal surface inside the horizon, contrary to typical results in the island formula literature where r_a > r_+. The author attributes this to the approximation in Eq. (42), but this is a notable deviation that warrants further discussion, as it may affect the validity of the island formula application.
  3. Sec. 5.2, Eq. (43): The derivation of kappa*t << 1 for t >> 1 is performed in Appendix D using Schwarzschild greybody factors as an approximation for the Kerr case. The author acknowledges this is an approximation, but the validity of using non-rotating greybody factors for a near-extremal rotating BH should be briefly justified, as it affects the relevant expression.
  4. The paper would benefit from a summary table of the key quantities (r_ext, m_ext, alpha_1, alpha_2) and their leading-order expressions, as the reader must track many expansions across many equations.
  5. Reference [42] is cited twice (also as [31]) for the same review on black hole remnants. This should be consolidated.

Circularity Check

0 steps flagged

No significant circularity: the derivation of α₂ from the island formula is self-contained, but the load-bearing sign-stability assumption is an unverified ansatz, not a circular definition.

full rationale

The paper's central result is the lower bound α₂ ~ √(2/(3π)) · m_ext (Eq. 58), obtained by setting the correction term Λ₂ to zero. Walking the derivation chain: the generalized entropy S_rad (Eq. 39) is constructed from the island formula (Eq. 1-2), the area term A (Eq. 37) from the rotating regular BH metric (Eq. 3), and the correction term S_field (Eq. 38) from the two-point function on the near-horizon geometry (Eq. 27). The extremal surface position r_a is solved from ∂S_rad/∂r_a = 0 (Eq. 50-52), and r_b is fixed by ∂S_rad/∂r_b = 0 (Appendix E, Eq. 76). The resulting Λ₂ (Eq. 53, fourth line) is a logarithmic function of α whose zero-crossing yields α₂. None of these steps reduce to a self-referential definition: α₂ is not defined in terms of itself, and no self-citation chain is load-bearing for the computation. The key logical step — interpreting α₂ as a physical lower bound — rests on the assumption (Sec. 6, point ii) that the signs of Λ₁ and Λ₂ remain unchanged throughout evaporation. This is explicitly stated as an assumption ('we assume that their signs should respectively remain unchanged'), motivated by the general non-negativity of entropy. While this assumption is the weakest link in the argument (entropy non-negativity constrains only the sum Λ₁+Λ₂ ≥ 0, not individual signs), it is an unverified physical ansatz rather than a circular definition. The result α₂ is computed from the geometry and the island formula, not assumed. The reader's concern about the dropped constant C_{κr_b} (Eq. 49, 56) affecting the quantitative value of α₂ is a correctness/approximation issue, not a circularity issue. No self-definitional, fitted-input-as-prediction, or self-citation-load-bearing circularity is present. The derivation, while resting on a strong assumption, is self-contained against the island formula framework and the BH geometry specified in the paper. Score 2 reflects the presence of a load-bearing unverified assumption that is not independently justified but is also not circularly defined.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 1 invented entities

The paper introduces one ad-hoc assumption (sign stability) that is load-bearing for the central claim, treats the angular momentum parameter as free, and uses a regular BH metric that is not a solution to any known field equation. The cutoff surface position is chosen by hand to avoid divergences. These are the main elements the paper contributes vs. pulls from prior literature.

free parameters (3)
  • a (angular momentum parameter) = treated as a parameter, ~ m_ext
    The author treats a as a parameter (Eq. 11) because the system has two equations for three unknowns (a, m0, rext). It is held fixed during the near-extremal analysis.
  • rb (cutoff surface position) = chosen so that kappa*rb is finite (Eq. 48)
    rb is a free parameter chosen by hand. The author exploits this freedom to select case 2 (kappa*rb = finite) to avoid divergences in the entropy expression.
  • c_kappa_rb = sqrt(6) (from extremization, Eq. 76-77)
    A finite constant arising from the choice of rb. The author assumes it is large enough for Eq. 42 to hold, achievable by rescaling rb.
axioms (4)
  • ad hoc to paper The signs of the area term and correction term in the generalized entropy remain unchanged throughout the entire evaporation process.
    Stated in Sec. 6, point ii. This is the load-bearing assumption: entropy non-negativity is used to argue that signs cannot change, but this does not logically follow—non-negativity of the total entropy does not require each term to maintain its sign.
  • domain assumption The island formula (Eq. 2) applies to a shrinking black hole in asymptotically flat spacetime.
    The island formula has been derived/verified primarily for eternal black holes. The author acknowledges this difference (Sec. 1) and applies it to a shrinking BH without independent justification.
  • domain assumption The adiabatic approximation holds up to the extremal limit.
    Stated in the bullet points under Eq. 18: the expansion forms are assumed valid up to alpha = 0, equivalent to assuming adiabatic evaporation throughout.
  • domain assumption The near-horizon effective 2D theory captures the relevant physics of the QES extremization.
    The reduction to 2D (Sec. 4.1) and the use of the 2D geodesic distance formula (Eq. 38) assume the near-horizon approximation is sufficient for computing S_field.
invented entities (1)
  • Regular black hole metric (Eq. 3) with de Sitter core no independent evidence
    purpose: Regularizes the central singularity via Planck-length parameter lp
    The metric does not satisfy any known gravitational field equation (acknowledged in Appendix A). No Lagrangian admitting rotating regular BH solutions exists. The regularization is a phenomenological model of quantum gravity effects.

pith-pipeline@v1.1.0-glm · 34883 in / 2843 out tokens · 670002 ms · 2026-07-10T03:26:31.575769+00:00 · methodology

0 comments
read the original abstract

We express the generalized entropy (GE) of the Hawking radiation in the final stage of an evaporating rotating regular black hole (BH) by writing the mass of the BH as $m_{\rm ext}+\alpha$, where $m_{\rm ext}$ represents the mass at the extremal limit and $\alpha$ is a parameter. Generally, entropy is non-negative. Based on this fact, we assume that, in the GE considered in this study, the signs of the contributions from the area term and the correction term remain unchanged throughout the entire evaporation process of the BH. Therefore, we regard $\alpha$ at which the correction term vanishes as its lower bound and determine it. As a result, we find that such a value of $\alpha$ is finite. Denoting this value by $\alpha_1$, this result indicates that the mass of the BH cannot become smaller than $m_{\rm ext}+\alpha_1$, which can be interpreted as the emergence of a remnant at the final stage of BH evaporation. The BH considered in this study is a rotating regular BH. The regularization is motivated by the fact that the fine structure of the central region becomes relevant in the final stage of evaporation. A rotating BH is considered from the viewpoint of generality.

Figures

Figures reproduced from arXiv: 2607.08661 by Shingo Takeuchi.

Figure 1
Figure 1. Figure 1: Plots of ∆ ( ˜ r 3 + ℓ 3 p ) = 0 in the non-extremal case with parameters (a, m, ℓp) = (2, 4, 0.1). The right figure shows a magnified view of the left one. It can be seen that ∆ admits two positive and one ˜ negative real solutions, while the remaining two solutions are complex [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plots of ∆ ( ˜ r 3 + ℓ 3 p ) = 0 in the extremal case with parameters (a, m0, ℓp) = (2, 2.000282, 0.1). The middle and right plots show the magnified views of the left one. It can be seen that ∆ admits one ˜ positive (double root) and one negative solutions [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plots of ∆ ( ˜ r 3 + ℓ 3 p ) = 0 in the case beyond the extremal, where (a, m0, ℓp) = (2, 1, 0.1). The right plot is magnified view of the left plot. It can be seen that ∆ admits one negative solution. ˜ 3.4 The horizon radii in the extremal case We have plotted ∆ = 0 in the non-extremal, extremal and the beyond-extremal cas ˜ es in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plots of 2m0r and f(r) = (r 2 + a 2 )(1 + ℓ 3 p/r3 ), which are the functions constituting ∆ = 0 in ˜ (3), where (m0, a, ℓp) are chosen as (2, 1, 0.1), (2, 2, 0.1) and (2, 3, 0.1) in the left, middle and right figures, respectively. The middle plot corresponds to the extremal case. Since a tangent line of a function f(r) at r = rext can be written as f ′ (rext)(r − rext) + f(rext), we obtain 2m0r = f ′ (re… view at source ↗
Figure 5
Figure 5. Figure 5: The Penrose diagrams of the maximally extended spacetime [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The blue lines represent the complementary surfaces of t [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: “No expansion” is obtained from (71). On the other hand, [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: These are plots of ∂S(late) rad /∂rb in (75) for α. The left figure shows the overall behavior, while the middle and right figures show its small and large rb regions. It can be seen in the middle and right figures that there is a stationary point of S (late) rad in each of the small rb and large rb regions. Since the cutoff surface is generally assumed to be located far away, we focus on the solution in t… view at source ↗

discussion (0)

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