arxiv: 2605.06602 · v1 · submitted 2026-05-07 · 🌀 gr-qc

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Quasi-homogeneous black hole geometrothermodynamics in Einstein-Maxwell theory

Hernando Quevedo

Authors on Pith no claims yet

Pith reviewed 2026-05-08 06:58 UTC · model grok-4.3

classification 🌀 gr-qc

keywords geometrothermodynamicsblack holesphase transitionscurvature singularitiesquasi-homogeneous functionsEinstein-Maxwell theoryReissner-NordstromKerr-Newman

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The pith

Quasi-homogeneous equations for black holes produce Legendre-invariant metrics whose curvature singularities mark the divergences of their heat capacities.

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

The review sets out geometrothermodynamics as a geometric description suited to non-extensive, self-gravitating systems. It notes that black-hole fundamental equations are quasi-homogeneous, so the usual Euler relation of laboratory thermodynamics does not apply. Legendre-invariant metrics are constructed on the equilibrium manifold for the Reissner-Nordström, Kerr and Kerr-Newman solutions. Curvature singularities of these metrics are shown to occur at the same parameter values where the heat capacities diverge, thereby locating the phase transitions.

Core claim

Because the mass and entropy of black holes are quasi-homogeneous functions of the extensive variables, the standard thermodynamic Euler identity fails. The resulting Legendre-invariant metric on the space of equilibrium states has scalar curvature whose singularities coincide exactly with the poles of the heat capacities for the Reissner-Nordström, Kerr and Kerr-Newman black holes.

What carries the argument

Legendre-invariant metric on the equilibrium manifold whose curvature singularities locate the phase transitions of the black-hole heat capacities.

If this is right

Phase transitions of these black holes can be identified by locating curvature singularities rather than by direct computation of heat capacities.
The same geometric construction applies to any other quasi-homogeneous, non-extensive system.
Standard thermodynamic relations that assume extensivity must be replaced by quasi-homogeneous versions when treating black holes.
Stability and critical behavior of gravitational systems can be read from the curvature of a single metric on the equilibrium space.

Where Pith is reading between the lines

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

The same singularity correspondence may hold for black holes in higher-dimensional or modified gravity theories.
Geometrothermodynamics could supply a geometric criterion for stability in other non-extensive systems such as neutron-star interiors.
If the link is universal, curvature analysis might reveal previously unnoticed phase transitions in black-hole thermodynamics.

Load-bearing premise

The fundamental thermodynamic equations of black holes are quasi-homogeneous functions of their extensive variables.

What would settle it

A calculation for any of the three black-hole families in which a heat-capacity divergence fails to coincide with a curvature singularity of the GTD metric, or the converse.

read the original abstract

In this review, we establish the mathematical framework of geometrothermodynamics (GTD) as a formalism capable of describing non-extensive, quasi-homogeneous, self-gravitating systems in a Legendre-invariant manner. We argue that the fundamental equations of black holes are quasi-homogeneous functions, a property that invalidates the standard Euler identity of laboratory thermodynamics. We derive the metrics for the equilibrium manifold and analyze their curvature singularities for the Reissner-Nordstr\"om, Kerr, and Kerr-Newman black holes. Furthermore, we establish a direct correspondence between the curvature singularities of the equilibrium space and phase transitions, as determined by the divergences of the corresponding heat capacities.

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.

Desk Editor's Note private letter to a colleague

This review applies GTD to quasi-homogeneous black holes and shows curvature singularities align with heat capacity divergences for the RN, Kerr, and Kerr-Newman cases.

read the letter

This paper is a review that sets up geometrothermodynamics for quasi-homogeneous black hole systems and verifies that the curvature singularities in the equilibrium manifold correspond to the divergences in heat capacity for the Reissner-Nordström, Kerr, and Kerr-Newman black holes. It does a solid job of deriving the metrics explicitly under the Legendre-invariant condition and analyzing their singularities case by case. The quasi-homogeneous property is used to justify why the standard Euler relation doesn't hold, which is a fair point for self-gravitating systems. The correspondence is shown through direct calculation rather than just asserted. The main limitation is that much of this builds on prior GTD work, so the novelty is in the application and the confirmation for these solutions rather than a fundamental advance. The circularity concern is minor because the paper does the explicit check, but the metrics are constructed with the invariance in mind, so the match is partly by design. No major gaps in the logic for the cases presented. This is useful for specialists in black hole thermodynamics who are interested in geometric approaches to phase transitions. Readers looking for new predictions or broader applications might find it thin, but those wanting the detailed GTD metrics for these spacetimes will appreciate the calculations. It deserves a serious referee because the explicit derivations and analysis provide something concrete to evaluate. I would recommend peer review; the work is grounded enough in the calculations to warrant feedback from the community.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to establish the framework of geometrothermodynamics (GTD) for non-extensive, quasi-homogeneous self-gravitating systems in a Legendre-invariant manner. It argues that the fundamental equations of black holes are quasi-homogeneous functions, invalidating the standard Euler identity. The authors derive the GTD metrics on the equilibrium manifold, analyze curvature singularities, and establish a direct correspondence between these singularities and phase transitions (identified via divergences in heat capacities) for the Reissner-Nordström, Kerr, and Kerr-Newman black holes in Einstein-Maxwell theory.

Significance. If the explicit constructions hold, the work is significant as it adapts GTD to black hole thermodynamics by incorporating the quasi-homogeneous scaling that underlies the Smarr relation, providing a geometric tool for identifying phase transitions that avoids issues with extensivity. The case-by-case metric derivations and singularity analyses for the three standard solutions offer concrete, falsifiable checks that could advance geometric methods in gravitational thermodynamics.

major comments (2)

[Derivation of metrics and curvature analysis] The section deriving the GTD metrics and analyzing curvature singularities: the claimed direct correspondence between curvature singularities and heat-capacity divergences is verified explicitly for RN, Kerr, and KN; the manuscript should clarify whether this alignment follows generally from the quasi-homogeneous Legendre-invariant construction or is verified only case-by-case, to address potential circularity in the central claim.
[Discussion of quasi-homogeneous property] The argument that quasi-homogeneity invalidates the standard Euler identity: an explicit calculation showing the discrepancy (e.g., the scaling-induced deviation from the Euler relation) is needed to make this foundational point load-bearing and rigorous for the subsequent metric derivations.

minor comments (2)

The abstract refers to 'this review' while the content includes original derivations and analyses; adjust the language for consistency if the work is primarily original research.
Standardize notation for thermodynamic variables and potentials across all sections and equations to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which have helped us improve the clarity and rigor of the presentation. We address each major comment in detail below.

read point-by-point responses

Referee: The section deriving the GTD metrics and analyzing curvature singularities: the claimed direct correspondence between curvature singularities and heat-capacity divergences is verified explicitly for RN, Kerr, and KN; the manuscript should clarify whether this alignment follows generally from the quasi-homogeneous Legendre-invariant construction or is verified only case-by-case, to address potential circularity in the central claim.

Authors: The quasi-homogeneous Legendre-invariant construction is developed as a general framework applicable to any non-extensive self-gravitating system whose fundamental equation satisfies the quasi-homogeneity condition. This general structure ensures that curvature singularities of the derived metrics correspond to phase transitions identified by heat capacity divergences. The explicit calculations for the Reissner-Nordström, Kerr, and Kerr-Newman black holes then serve as concrete verifications of this general prediction. To eliminate any potential ambiguity regarding circularity, we have revised the manuscript (in the introduction and the concluding section) to state explicitly that the alignment follows from the general properties of the construction, with the three cases providing illustrative confirmation rather than the sole foundation of the claim. revision: yes
Referee: The argument that quasi-homogeneity invalidates the standard Euler identity: an explicit calculation showing the discrepancy (e.g., the scaling-induced deviation from the Euler relation) is needed to make this foundational point load-bearing and rigorous for the subsequent metric derivations.

Authors: We agree that an explicit demonstration of the discrepancy strengthens the foundational argument. In the revised manuscript we have inserted a dedicated paragraph (immediately preceding the metric derivations) that performs the explicit calculation for a general quasi-homogeneous function of degree k. The computation shows that the standard Euler relation is replaced by an identity containing an additional term proportional to (k-1) times the extensive variable, thereby quantifying the deviation induced by the non-extensive scaling. This addition renders the point fully rigorous and directly supports the subsequent construction of the Legendre-invariant GTD metrics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit case-by-case verification

full rationale

The paper argues that black-hole fundamental equations are quasi-homogeneous (invalidating the standard Euler relation), constructs Legendre-invariant GTD metrics on the equilibrium manifold by design, and then performs explicit curvature calculations for the RN, Kerr, and Kerr-Newman cases. The claimed direct correspondence between curvature singularities and heat-capacity divergences is obtained by inspecting the resulting scalar curvature for each metric and comparing the locations of divergences with the known thermodynamic phase-transition loci; this matching is a derived result of the calculation rather than an identity imposed by the metric ansatz or by any self-citation chain. No step reduces the target claim to a fitted parameter, a renamed known result, or an unverified uniqueness theorem imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that black hole thermodynamic equations are quasi-homogeneous and on the validity of the geometrothermodynamics construction for non-extensive systems; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption The fundamental equations of black holes are quasi-homogeneous functions
Explicitly stated in the abstract as the property that invalidates the standard Euler identity.

pith-pipeline@v0.9.0 · 5402 in / 1270 out tokens · 68542 ms · 2026-05-08T06:58:18.366053+00:00 · methodology

discussion (0)

Reference graph

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