arxiv: 2604.24101 · v1 · submitted 2026-04-27 · 🌀 gr-qc · hep-th

Recognition: unknown

g_{tt}g_{rr} =-1 black hole thermodynamics in extended quasi-topological gravity

Johanna Borissova

Authors on Pith no claims yet

Pith reviewed 2026-05-08 02:19 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords black hole thermodynamicsquasi-topological gravityWald entropydilaton gravitygeneralized first lawstatic black holeshigher-curvature gravity

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The pith

For black holes with g_tt g_rr = -1 the generating function of f(r) supplies the thermodynamic mass in a generalized first law using Wald entropy.

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

The paper develops a unified framework for the thermodynamics of static d-dimensional black holes obeying g_tt g_rr = -1 in Schwarzschild gauge, regardless of horizon topology. It treats every such solution as arising from an integrable two-dimensional effective dilaton theory, which in turn counts as a vacuum solution of an extended notion of quasi-topological gravity. Under this identification the generating function that determines f(r) = -g_tt directly furnishes the mass parameter that enters the generalized first law, while entropy is obtained from the Wald formula. The construction covers both singular and regular black holes with flat or anti-de Sitter asymptotics and all three standard horizon topologies.

Core claim

Static black holes with g_tt g_rr = -1 are solutions of an integrable 2D effective dilaton theory and therefore vacuum solutions of d-dimensional extended quasi-topological gravity. The generating function that fixes f(r) = -g_tt in the integrated equation of motion therefore supplies the thermodynamic mass appearing in the generalized first law, with entropy given by the Wald prescription. The framework applies uniformly to spherical, toroidal and hyperbolic horizons and to both singular and regular solutions with flat or anti-de Sitter asymptotics.

What carries the argument

The generating function determining f(r) = -g_tt in the integrated equation of motion, which directly supplies the thermodynamic mass.

If this is right

The thermodynamic mass is read off directly from the generating function without separate integration constants or asymptotic matching.
Entropy follows automatically from the Wald formula for the underlying higher-derivative theory.
The same first-law relation holds for spherical, toroidal and hyperbolic horizons alike.
Both singular and regular black holes with flat or anti-de Sitter asymptotics are covered by the identical construction.

Where Pith is reading between the lines

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

The reduction to an integrable 2D dilaton model may simplify thermodynamic calculations in other higher-curvature theories that admit similar static metrics.
The approach suggests that many features of d-dimensional black-hole thermodynamics are already encoded in the 2D dilaton description for this restricted class of metrics.
Similar generating-function techniques could be tested on time-dependent or non-static solutions that still obey g_tt g_rr = -1 in a suitable gauge.

Load-bearing premise

Any black hole satisfying g_tt g_rr = -1 can be treated as a solution of an integrable 2-dimensional effective dilaton theory and thereby as a vacuum solution of extended quasi-topological gravity.

What would settle it

An explicit black-hole solution with g_tt g_rr = -1 for which the value of the generating function fails to equal the mass required by the first law when entropy is computed via the Wald formula.

read the original abstract

We present a unified framework for the discussion of black hole thermodynamics of $d$-dimensional static black holes with spherical, toroidal or compact hyperbolic horizon topology satisfying $g_{tt}g_{rr}=-1$ in Schwarzschild gauge. To that end, we consider any such black hole as a solution to an integrable $2$-dimensional effective dilaton theory and thereby as a vacuum solution to an extended notion of $d$-dimensional quasi-topological gravity. We show that the generating function determining $f(r) = -g_{tt} $ in the integrated equation of motion provides the thermodynamic mass in a generalised first law with entropy computed as the Wald entropy. The framework presented here can be applied to singular and regular black holes with flat or anti-de Sitter asymptotics.

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

The paper unifies thermodynamics for g_tt g_rr=-1 black holes by mapping them to integrable 2D dilaton models from an extended quasi-topological gravity, letting the generating function serve as the mass in a generalized first law.

read the letter

The main point is that any static black hole with g_tt g_rr = -1 can be treated as a solution to a 2D dilaton theory whose integrability gives a generating function for f(r). That function is then the thermodynamic mass, paired with Wald entropy, and the same setup covers spherical, toroidal, and hyperbolic topologies plus both singular and regular solutions with flat or AdS asymptotics. The framework avoids re-deriving the first law for each new metric in this class. The reduction itself is clean once the 2D model is accepted, and the identification of mass follows directly from the integrated equations without extra fitting. This is a practical step for quick thermodynamic analysis in higher-curvature settings. The softer element is the higher-dimensional lift. The paper works with an extended notion of quasi-topological gravity chosen so the spherical reduction yields the desired dilaton theory. This works by construction for the metrics at hand, but it leaves open whether a single fixed Lagrangian covers arbitrary f(r) or whether the extension must be adjusted per solution. The stress-test concern about guaranteeing a consistent lift for every such metric is reasonable to check in the explicit action, though the paper presents the extension as sufficient for the cases considered. Readers working on black hole thermodynamics in modified gravity will get the most from it as a calculational tool. I would send the paper to peer review. The core construction is direct and the results are new enough to merit expert scrutiny on the scope of the extension.

Referee Report

1 major / 0 minor

Summary. The paper presents a unified framework for black hole thermodynamics of d-dimensional static black holes (spherical, toroidal or hyperbolic horizons) obeying g_tt g_rr = -1 in Schwarzschild gauge. Any such metric is treated as a solution of an integrable 2D effective dilaton theory obtained by spherical reduction, which is asserted to be a vacuum solution of some extended d-dimensional quasi-topological gravity. The generating function that determines f(r) = -g_tt after integrating the equation of motion is identified as the thermodynamic mass conjugate to the Wald entropy in a generalized first law. The construction is claimed to apply uniformly to singular and regular solutions with flat or AdS asymptotics.

Significance. If the mapping from arbitrary f(r) to an extended quasi-topological Lagrangian is rigorously established, the result supplies a general, generating-function-based route to the first law that bypasses the need to fix the higher-dimensional action in advance and automatically incorporates the Wald entropy. This would unify thermodynamic calculations across a wide class of higher-curvature theories while remaining parameter-free once the 2D integrability is granted.

major comments (1)

[framework construction (around the effective 2D theory and its lift)] The central identification of the generating function with thermodynamic mass rests on the claim that every metric with g_tt g_rr = -1 is a vacuum solution of some extended quasi-topological gravity. The manuscript must supply an explicit construction (or existence proof) of the d-dimensional Lagrangian whose spherical reduction reproduces an arbitrary integrable 2D dilaton model with the given f(r). Without this step, the 'thereby' step in the abstract remains an assumption rather than a derivation, and the thermodynamic interpretation does not follow for arbitrary f(r).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for identifying the key point requiring clarification in the framework construction. We address the major comment below and will revise the manuscript accordingly to strengthen the derivation.

read point-by-point responses

Referee: The central identification of the generating function with thermodynamic mass rests on the claim that every metric with g_tt g_rr = -1 is a vacuum solution of some extended quasi-topological gravity. The manuscript must supply an explicit construction (or existence proof) of the d-dimensional Lagrangian whose spherical reduction reproduces an arbitrary integrable 2D dilaton model with the given f(r). Without this step, the 'thereby' step in the abstract remains an assumption rather than a derivation, and the thermodynamic interpretation does not follow for arbitrary f(r).

Authors: We agree that the manuscript would benefit from an explicit construction or existence proof to make the lift from the 2D effective dilaton theory to the d-dimensional extended quasi-topological gravity fully rigorous rather than asserted. In the revised version, we will add a dedicated subsection that constructs the d-dimensional Lagrangian explicitly. For any given integrable 2D dilaton model with arbitrary f(r) satisfying the g_tt g_rr = -1 condition, the spherical reduction of the extended quasi-topological action (built from the standard Lovelock terms plus the quasi-topological invariants that reduce to total derivatives or zero in spherical symmetry) can be matched term-by-term to the 2D action by solving for the coefficients of the higher-curvature polynomials. Because the quasi-topological terms provide sufficient freedom in their curvature invariants while preserving the integrability of the 2D equations, an existence proof follows from the invertibility of this coefficient-matching procedure for any smooth f(r) with the required asymptotic behavior. This will turn the 'thereby' statement into a derived result and confirm that the generating function is indeed the thermodynamic mass conjugate to the Wald entropy for arbitrary such solutions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework assumption is explicit and derivation proceeds independently within it.

full rationale

The paper explicitly adopts the assumption that any g_tt g_rr=-1 black hole can be viewed as an integrable 2D dilaton solution and hence a vacuum solution in an extended quasi-topological gravity. Within this setup the integration of the effective equations yields a generating function that is then shown to enter the first law as mass when paired with Wald entropy. This identification follows from the structure of the 2D theory and the standard Wald formula rather than from re-labeling fitted parameters or self-referential definitions. No load-bearing self-citations, ansatzes smuggled via prior work, or uniqueness theorems imported from the same authors appear in the abstract or described chain. The central result is therefore self-contained once the initial mapping is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that black holes satisfying g_tt g_rr=-1 are integrable solutions of a 2D dilaton theory that also solve the extended quasi-topological equations; no explicit free parameters or new entities are mentioned in the abstract.

axioms (2)

domain assumption Black holes with g_tt g_rr=-1 are solutions to an integrable 2D effective dilaton theory
Invoked in the abstract to unify the d-dimensional solutions.
domain assumption These are vacuum solutions to extended quasi-topological gravity
Stated as the higher-dimensional embedding of the effective theory.

pith-pipeline@v0.9.0 · 5428 in / 1389 out tokens · 37175 ms · 2026-05-08T02:19:16.526513+00:00 · methodology

discussion (0)

Reference graph

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