A winding number analysis of Schwarzschild black hole stability in light of Planck-scale modified kinematics

Mohsen Khodadi; Nosratolla Jafari; Shahin Mamedov

Planck-scale cubic entropy corrections leave Schwarzschild black holes thermodynamically unstable.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-14 10:33 UTC pith:YNHORZ6Q

load-bearing objection Clean negative result: this cubic MDR entropy leaves only the unstable Schwarzschild branch once physical cuts are enforced. the 2 major comments →

arxiv 2607.10600 v2 pith:YNHORZ6Q submitted 2026-07-12 gr-qc hep-th

A winding number analysis of Schwarzschild black hole stability in light of Planck-scale modified kinematics

Mohsen Khodadi , Nosratolla Jafari , Shahin Mamedov This is my paper

classification gr-qc hep-th

keywords winding numbersthermodynamic topologySchwarzschild black holemodified dispersion relationsPlanck-scale kinematicsentropy-geometry correspondencephase stabilityblack-hole thermodynamics

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.

This paper asks whether a common class of Planck-scale modifications to particle kinematics can turn an evaporating Schwarzschild black hole into a thermodynamically stable remnant. Starting from a modified dispersion relation with a cubic energy correction, the authors derive a cubic correction to the Bekenstein-Hawking entropy and then apply thermodynamic topology: they map black-hole equilibria to zeros of a vector field and read stability from winding numbers. Once physical requirements of positive mass and positive temperature are imposed through the entropy-geometry map, only a single unstable branch survives, with winding number -1 for both signs of the correction. An apparent second root that would look stable is discarded because it has negative mass or temperature and lies outside the regime where the entropy formula is valid. The result is that this particular family of modified kinematics does not stabilize Schwarzschild black holes, though other leading corrections might.

Core claim

When a cubic entropy correction S = π r_h² - α r_h³ arising from the modified dispersion relation η E³/E_P is analyzed with thermodynamic topology, physical constraints S'(r_h) > 0 and T > 0 leave only one equilibrium branch. That branch always carries winding number w = -1 and total topological charge W = -1, for both signs of the correction parameter. The mathematical root that would give w = +1 is unphysical (negative ADM mass or temperature) and lies outside the perturbative domain of the entropy formula. Hence this class of Planck-scale kinematics does not produce stable Schwarzschild black holes.

What carries the argument

Thermodynamic topology: equilibrium states are zeros of a two-component vector field built from the off-shell free energy; each zero is assigned a winding number w = ±1 fixed by the convexity of free energy, and the sum of winding numbers gives the global topological charge W that classifies stability once the physical domain S' > 0, T > 0 is enforced via the entropy-geometry correspondence M = S'(r_h)/(4π).

Load-bearing premise

The claim rests on the entropy-geometry correspondence that converts any entropy function into a unique metric and ADM mass; if that map is not how entropy corrections actually back-react, the cuts that discard the would-be stable root no longer hold.

What would settle it

Derive the same cubic entropy from a different first-principles route (or from a fully consistent DSR multiparticle model) and recompute the winding numbers without imposing the entropy-geometry map; if a physical w = +1 branch then appears inside the perturbative regime, the paper's negative conclusion is overturned.

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

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

Clean negative result: this cubic MDR entropy leaves only the unstable Schwarzschild branch once physical cuts are enforced.

read the letter

This is a careful negative result. The authors take a standard phenomenological MDR (leading ηE³/E_P), derive the cubic entropy S=πr_h²−αr_h³ via Hamilton–Jacobi tunneling, feed it into the existing thermodynamic-topology machinery, and show that once you enforce S′>0 and T>0 the only surviving physical root has w=−1 and W=−1 for both signs of α. The would-be stable root is discarded because it has negative mass/temperature and sits outside the perturbative regime. That combination of derivation plus constrained classification is the actual new piece; the topology formalism and the entropy-geometry map are imported.

What they do well is the bookkeeping. The free-energy Hessian, the always-positive discriminant of the quadratic equilibrium condition, the domain cuts r_h<π/(3α), and the tables that separate physical from mathematical roots are all checkable from the text and hold. They also flag their own limitations (perturbative control near the boundary, no full DSR implementation, reliance on the Anand et al. map) rather than burying them.

The softest external premise is exactly the one the stress-test notes: the entropy-geometry correspondence M=S′(r_h)/4π is taken as given. If that map is not the right back-reaction, the physical-domain cuts that kill the w=+1 root no longer apply. Inside the framework they adopt, however, the algebra is solid and the negative claim is honest. No circularity, no hidden free parameters beyond the single phenomenological η, and the citation pattern is appropriate.

This is for people who work on thermodynamic topology of black holes or on remnant/stabilization scenarios from modified kinematics. It will not reorganize the field, but it cleanly rules out one concrete class of MDRs. I would send it to peer review; a referee can argue about the imported map, but the paper deserves that conversation rather than a desk reject.

Referee Report

2 major / 5 minor

Summary. The paper derives a cubic entropy correction S=πr_h²−αr_h³ from a phenomenological MDR E²=p²c²+m²c⁴+ηE³/E_P via the Hamilton–Jacobi tunneling method, then applies the thermodynamic-topology formalism of Wei–Liu–Mann together with the entropy–geometry correspondence of Anand et al. to classify Schwarzschild equilibria. After imposing the physical cuts S′(r_h)>0 and T>0, only a single branch with winding number w=−1 (total charge W=−1) survives for both signs of the deformation parameter α; a second mathematical root that would carry w=+1 is discarded because it yields negative ADM mass and temperature and lies outside the perturbative regime of the MDR expansion. The authors conclude that this particular class of MDRs does not stabilize Schwarzschild black holes, while leaving open the possibility that other leading-order corrections might.

Significance. The central negative result is a clean, falsifiable check on whether a widely used cubic MDR correction can produce thermodynamically stable Schwarzschild remnants. The derivation of the entropy from the MDR (Sec. II), the exhaustive root-counting of the free-energy equilibrium condition (Eq. 4.18), and the explicit enforcement of positivity constraints (Tables I–II, Figs. 1–2) are carefully executed and tabulated. The work usefully underscores that winding numbers alone are insufficient without physical-domain cuts, and it correctly flags the imported entropy–geometry map and the perturbative limitation as open issues. Within the stated framework the claim is sound and of moderate but genuine interest for the black-hole thermodynamics and quantum-gravity phenomenology communities.

major comments (2)

Sec. III C and Eqs. (3.21)–(3.22): the entire physical-domain cut that discards the w=+1 root rests on the entropy–geometry map M=S′(r_h)/(4π). While the authors acknowledge this premise in Sec. VI, the manuscript would be strengthened by a short explicit discussion of how sensitive the W=−1 conclusion is to alternative back-reaction prescriptions (e.g., fixed-background generalized-entropy analyses). A one-paragraph robustness check or a clear statement that the result is conditional on this map would make the load-bearing assumption transparent to readers.
Sec. II, after Eq. (2.13) and Sec. V: for α>0 the physical window extends up to r_h→r_c where the expansion parameter η r_h/κ reaches O(1). The authors correctly note that the discarded stable root lies outside this regime, yet the physical root itself approaches the boundary of perturbative control. A quantitative estimate of the size of the neglected O(1/E_P²) terms near r_max, or an explicit statement that the W=−1 result is reliable only well below r_c, would tighten the claim.

minor comments (5)

Abstract and Sec. I: the phrase “Planck-scale modified kinematics” is used interchangeably for both LIV and DSR; a single clarifying sentence early on that the conclusions apply only to the entropy correction, not to a full DSR multiparticle structure, would prevent misreading.
Figs. 1–2: the captions already state which root is physical, but adding the numerical values of S′ and T at each plotted zero would make the exclusion of the second root immediately visible without consulting the tables.
Eq. (4.1) and the definition α≡2η/(3κ): a brief parenthetical reminder that the sign of α tracks the sign of η would help readers who jump directly to Sec. IV.
Appendix A is helpful but lengthy; a one-sentence pointer in the main text (near Eq. 3.18) that the sign convention for ϕ_rh produces the apparent reversal between thermodynamic and phase-space stability would improve readability.
References [61,62] are central; ensuring they are cited with full arXiv identifiers (or journal details if available) will aid readers tracking the entropy–geometry correspondence.

Circularity Check

0 steps flagged

No significant circularity: winding numbers and physical-domain cuts follow by direct calculation from the derived cubic entropy under an externally imported (non-self) entropy-geometry map.

full rationale

The load-bearing chain is: phenomenological MDR (2.1) o Hamilton-Jacobi tunneling derivation of cubic entropy S=πrh^{2}-αrh^{3} (2.13, 4.1) o free-energy Hessian and winding numbers via the standard topological formalism of Wei et al. [69,70] o physical cuts S'>0, T>0 that discard the mathematical w=+1 root. The entropy-geometry map M=S'(rh)/4π (3.21–3.22) is taken from independent Refs. [61,62] (Anand/Jusufi/Saridakis et al.; no author overlap with the present paper) and is not derived from, nor justified by, the stability conclusion itself. The algebraic fact that every physical root has wi=-1 follows immediately from Q(r)>0 for all r (4.31) together with (4.30)/(4.32) once D(ri)>0 is imposed; this is a calculation on the cubic form, not a definitional identity with the input MDR or with the winding-number definition. Author self-citations appear only as background on minimum-length phenomenology and do not underwrite the topological classification or the domain cuts. No parameters are fitted to data and re-presented as predictions, no uniqueness theorem is self-imported, and no ansatz is smuggled via self-citation. The paper is therefore self-contained against its stated external premises and exhibits no circular reduction of the claimed result.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

The central negative claim rests on three imported ingredients (the phenomenological MDR, the entropy-geometry map, and the thermodynamic-topology dictionary) plus one free phenomenological coefficient. No new particles or forces are postulated; the cubic entropy is derived rather than invented. The free parameter α (or η) is never fitted to data inside the paper; its sign is scanned and both signs yield the same topological conclusion.

free parameters (1)

η (or equivalently α≡2η/(3κ))
Dimensionless strength of the leading cubic MDR correction; treated as a free phenomenological parameter whose sign is scanned. Not fitted to any data set in this work.

axioms (4)

domain assumption Phenomenological MDR E²=p²c²+m²c⁴+ηE³/EP+O(1/EP²) is a valid low-energy parametrization of Planck-scale kinematics (both LIV and DSR).
Stated in Sec. II and used to derive the cubic entropy; the paper explicitly declines to break the LIV/DSR degeneracy.
domain assumption Entropy-geometry correspondence: any entropy function S(r) determines the metric via f(r)=1−4πM/S′(r) and the ADM mass via M=S′(rh)/4π.
Adopted wholesale from Refs. [61,62] (Sec. III C); all physical-domain cuts (S′>0, T>0) rest on this map.
domain assumption Winding number wi=sgn(∂²F/∂rh²) classifies thermodynamic stability (wi=+1⇔C>0) if and only if M′>0 and S′>0.
Taken from the thermodynamic-topology literature (Wei et al.) and re-derived under the entropy-geometry map (Sec. III B).
domain assumption Standard Hamilton-Jacobi tunneling probability and WKB approximation remain valid once the group velocity is corrected by the MDR.
Used in Sec. II to convert the MDR into an effective temperature and thence into the cubic entropy.

pith-pipeline@v1.1.0-grok45 · 26563 in / 2726 out tokens · 35150 ms · 2026-07-14T10:33:04.957395+00:00 · methodology

0 comments

read the original abstract

Determining whether Planck-scale effects can stabilize black holes addresses fundamental questions about black hole evaporation and quantum gravity consistency. Here, we analyze the thermodynamic topology of Schwarzschild black holes under Planck-scale modified kinematics, using a cubic entropy correction derived from a well-known phenomenological MDR with leading correction \(\eta E^3/E_P\). Enforcing physical constraints (\(S'(r_h) > 0\), \(T > 0\)) via the entropy-geometry correspondence, we find a single unstable branch with \(w = -1\) and \(W = -1\) for both signs of the correction parameter. A second root suggesting stability (\(w = +1\)) is excluded due to negative mass/temperature and lies outside the perturbative regime. Thus, this class of MDRs does not yield stable Schwarzschild black holes. However, MDRs with different leading-order corrections may behave otherwise, leaving the search for Planck-scale stabilization an open endeavor.

Figures

Figures reproduced from arXiv: 2607.10600 by Mohsen Khodadi, Nosratolla Jafari, Shahin Mamedov.

A winding number analysis of Schwarzschild black hole stability in light of Planck-scale modified kinematics

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)