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arxiv: 2606.03045 · v1 · pith:JGUU3VWPnew · submitted 2026-06-02 · 🌀 gr-qc · hep-th

Off-shell Hessian thermodynamic stability of higher-curvature black holes

Chen-Hao Hao , Jieci Wang This is my paper

Pith reviewed 2026-06-28 09:20 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords black hole thermodynamicshigher-curvature gravityWald entropyoff-shell free energythermodynamic stabilityLovelock gravityAdS black holescritical phenomena
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The pith

Local stability of higher-curvature black holes is governed by the off-shell Hessian H = S'_W T' rather than temperature slope alone.

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

The paper builds a thermodynamic framework based on the off-shell Gibbs free energy whose stationary points are the equilibrium black holes. Local stability on fixed-parameter slices is read from the sign of the Hessian formed by the product of the Wald-entropy derivative and the temperature derivative. In the five-dimensional charged regular AdS black hole the entropy derivative stays positive, so the familiar temperature-slope criterion is recovered as a special case. In Lovelock theories the entropy derivative can flip sign on non-planar branches and thereby reverse the stability assignment, yet it remains positive on ghost-free physical branches. The same off-shell construction also reproduces the A3 cusp normal form and the associated mean-field branch-separation exponent near criticality.

Core claim

Equilibrium black holes are stationary points of the off-shell Gibbs free energy G_off. Their local stability is governed by the Hessian H = S'_W(r_h) T'(r_h) rather than by the temperature slope alone. The same off-shell structure yields the local A3 cusp normal form near criticality with mean-field 1/2 branch separation exponent. In Lovelock black holes S'_W can change sign on non-planar branches, reversing the temperature-slope stability assignment, but S'_W stays positive on ghost-free and branch-regular exteriors.

What carries the argument

The off-shell Hessian H = S'_W(r_h) T'(r_h) obtained from the off-shell Gibbs free energy G_off together with the Wald entropy S_W.

If this is right

  • In the five-dimensional charged regular AdS black hole in quasi-topological gravity, monotonic S_W recovers the ordinary temperature-slope stability rule.
  • In Lovelock black holes S'_W can change sign on non-planar branches and thereby reverse the stability assignment given by temperature slope.
  • On ghost-free and branch-regular Lovelock exteriors S'_W remains positive, protecting the ordinary slope rule.
  • The off-shell Hessian produces the local A3 cusp normal form and explains the mean-field 1/2 exponent that smooth observables inherit near criticality.

Where Pith is reading between the lines

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

  • Smooth nondegenerate observables such as the Lyapunov exponent inherit the same 1/2 scaling because they are regular functions on the branches.
  • The framework supplies a diagnostic that can be applied to any higher-curvature theory once the Wald entropy and temperature functions are known.
  • Numerical evolution of small perturbations on a branch where S'_W changes sign would directly test whether the Hessian correctly predicts the onset of instability.

Load-bearing premise

Equilibrium black holes are stationary points of the off-shell Gibbs free energy and the Wald entropy together with temperature supply the complete data needed for the Hessian stability test on fixed-parameter slices.

What would settle it

A concrete counter-example would be a physically admissible black-hole branch on which the sign of S'_W T' disagrees with the sign of the specific heat obtained from linear perturbation analysis.

Figures

Figures reproduced from arXiv: 2606.03045 by Chen-Hao Hao, Jieci Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. Parameters are chosen as [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Off-shell Gibbs free energy at [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
read the original abstract

We develop a branch-sensitive thermodynamic framework for higher-curvature black holes using the off-shell Gibbs free energy $G_{\rm off}$ and the Wald entropy$S_W$ as the basic data. On fixed-parameter slices, equilibrium black holes are stationary points of $G_{\rm off}$, and their local stability is governed by the Hessian $H=S'_W(r_h)T'(r_h)$, rather than by the temperature slope alone. For the five-dimensional charged regular AdS black hole in quasi-topological gravity, $S_W$ remains monotonic on the physical branch, so the usual temperature-slope rule is recovered only as a special consequence. The same off-shell structure also gives the local $A_3$ cusp normal form near criticality, yielding the mean-field $1/2$ branch separation exponent and explaining why smooth nondegenerate observables, such as the Lyapunov exponent, inherit the same scaling. In Lovelock black holes, $S'_W$ can change sign on non-planar branches, reversing the temperature slope stability assignment. However, on ghost-free and branch-regular Lovelock exteriors $S'_W$ remains positive. Thus the off-shell Hessian criterion also diagnoses why the ordinary slope rule is protected on physically admissible black holes branches.

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

0 major / 3 minor

Summary. The manuscript develops an off-shell thermodynamic framework for higher-curvature black holes, taking the off-shell Gibbs free energy G_off = M(r_h) - T S_W(r_h) and Wald entropy S_W as basic data. Equilibrium black holes are stationary points of G_off on fixed-parameter slices, with local stability governed by the Hessian H = S'_W(r_h) T'(r_h) rather than the temperature slope T' alone. For the five-dimensional charged regular AdS black hole in quasi-topological gravity, S_W is monotonic on the physical branch so the standard temperature-slope rule is recovered; in Lovelock gravity S'_W can change sign on non-planar branches (reversing the stability assignment) but remains positive on ghost-free, branch-regular exteriors. The same off-shell structure yields the local A3 cusp normal form near criticality, producing the mean-field 1/2 branch-separation exponent for smooth nondegenerate observables such as the Lyapunov exponent.

Significance. If the central claims hold, the work supplies a general, branch-sensitive stability criterion that explains why the conventional temperature-slope rule is protected on physically admissible branches while diagnosing exceptions in higher-curvature theories. The derivation of the A3 cusp normal form directly from the off-shell potential is a notable strength, furnishing a model-independent mechanism for the 1/2 exponent without additional assumptions. The framework is logically economical, relying only on the first law to reduce the second-derivative test, and applies uniformly to the quasi-topological and Lovelock examples.

minor comments (3)
  1. [Abstract] Abstract: the explicit functional form of G_off and the precise definition of the off-shell Hessian H are introduced without an accompanying equation; adding the relation G_off'' = S'_W T' (which follows from the first law at equilibrium) would improve immediate readability.
  2. [Abstract] The statement that 'the same off-shell structure also gives the local A3 cusp normal form' would be clearer if the leading terms of the normal-form expansion (e.g., G_off ~ a (r-r_c)^3 + b (T-T_c)(r-r_c)) were written explicitly, even if only to recall the standard A3 form.
  3. The claim that S_W remains monotonic on the physical branch of the quasi-topological black hole is central to recovering the temperature-slope rule; an explicit derivative plot or inequality establishing S'_W > 0 throughout the physical range would strengthen the presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recommendation for minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The central result follows from the standard second-derivative test applied to the off-shell Gibbs free energy G_off = M(r_h) - T S_W(r_h). With the first law dM = T dS_W, the second derivative at equilibrium points reduces identically to S'_W T' by direct differentiation, yielding the Hessian criterion as a mathematical identity rather than an independent prediction or fitted input. The A3 cusp normal form is likewise the generic local Taylor expansion of this G_off when T'(r_c) = 0. No self-citations, ansatzes smuggled via prior work, or uniqueness theorems are invoked to support the load-bearing steps. The derivation is self-contained against external thermodynamic benchmarks and does not reduce any claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Framework rests on two domain assumptions extracted from the abstract; no free parameters, invented entities, or ad-hoc axioms are stated.

axioms (2)
  • domain assumption Equilibrium black holes are stationary points of the off-shell Gibbs free energy G_off
    Stated as the starting point of the branch-sensitive framework.
  • domain assumption Wald entropy S_W is the appropriate entropy for constructing the stability Hessian in higher-curvature theories
    Used together with G_off as the basic data.

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