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

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.

• The paper establishes a unified framework embedding static black holes with the g_tt g_rr = -1 condition into extended quasi-topological gravity.
• It employs dimensional reduction to an effective 2D dilaton-Horndeski theory, ensuring integrable, second-order field equations that avoid Ostrogradsky ghosts.
• The study derives explicit generating function expressions for black hole mass, temperature, and entropy, clarifying corrections to the Bekenstein-Hawking area law.

Black Hole Thermodynamics in Extended Quasi-Topological Gravity with $g_{tt} g_{rr} = -1$

Introduction

The paper " $g_{tt} g_{rr} =-1$ black hole thermodynamics in extended quasi-topological gravity" (2604.24101) develops a unified framework for analyzing the thermodynamics of $d$-dimensional static black holes with arbitrary compact horizon topology, subject to the metric condition $g_{tt} g_{rr} = -1$ (Schwarzschild gauge). The key insight is that such spacetimes can be systematically realized as vacuum solutions within an "extended quasi-topological gravity" (QTG), reducible to integrable $2$D effective dilaton-Horndeski theories. The author rigorously establishes a generalized first law of black hole thermodynamics, with the thermodynamic mass appearing as an integration constant in a universal generating function formulation and the entropy evaluated through Wald's formalism.

Framework: Extended Quasi-Topological Gravities and Dimensional Reduction

The analysis proceeds from $d$-dimensional static metrics

$$ds^2 = -f(r) dt^2 + \frac{dr^2}{f(r)} + r^2 d\Sigma_{d-2}^2,$$

with $d\Sigma_{d-2}^2$ the metric on a maximally symmetric $(d-2)$-dimensional surface of curvature $k = 0, \pm1$. Any such geometry parameterized by a mass term is shown to be reproducible as a solution to a (potentially non-polynomial) generally covariant QTG. Critically, these admit second-order field equations (akin to Lovelock, general quasi-topological, or Horndeski theories when reduced), ensuring avoidance of Ostrogradsky ghosts.

Dimensional reduction on the isometry group leads to an effective 2D dilaton-Horndeski theory for the field $g_{tt} g_{rr} =-1$0 and a reduced 2-metric $g_{tt} g_{rr} =-1$1. The reduced Lagrangian,

$g_{tt} g_{rr} =-1$2

involves functions $g_{tt} g_{rr} =-1$3 of $g_{tt} g_{rr} =-1$4 and its kinetic term $g_{tt} g_{rr} =-1$5. Notably, integrability in this sector—a necessary and sufficient condition for all black hole solutions of interest—is encoded in a specific partial differential constraint upon $g_{tt} g_{rr} =-1$6, $g_{tt} g_{rr} =-1$7, $g_{tt} g_{rr} =-1$8 analogous to a generalized Frobenius condition. This ensures that the field equation for $g_{tt} g_{rr} =-1$9 can be integrated to yield an algebraic equation.

Generating Function Formalism and Thermodynamic Identification

Central to the discussion is the definition of a generating function $d$0 that encodes the integrated equations of motion for the metric function $d$1. The black hole solution corresponds to the level set

$d$2

where $d$3 can be interpreted as the thermodynamic mass (identical to the ADM mass in Einstein gravity).

For the class of (generalized) polynomial QTGs—such as Lovelock or higher-order quasi-topological gravities—a universal form for $d$4 is

$d$5

with $d$6 a power series in its argument. This allows for the systematic generation of solutions, including regular and singular black holes with AdS, flat, or de Sitter asymptotics.

Thermodynamics: Mass, Temperature, and Wald Entropy

The Hawking temperature is derived from the surface gravity at the event horizon $d$7: $d$8 The entropy is computed using the Wald formalism, with the entropy functional generically expressed as

$d$9

This formula makes manifest the corrections to the Bekenstein-Hawking area law present in QTG, including higher-curvature and non-polynomial contributions. In the Einstein gravity limit, the area law $g_{tt} g_{rr} = -1$0 is recovered, while in generic QTG, explicit analytic expressions for $g_{tt} g_{rr} = -1$1 involve derivatives of the generating function with respect to $g_{tt} g_{rr} = -1$2.

Combining the temperature and entropy expressions yields the generalized first law,

$g_{tt} g_{rr} = -1$3

where $g_{tt} g_{rr} = -1$4 is interpreted as the thermodynamic (ADM) mass.

Extended First Law and Smarr Relations

For spacetimes where the black hole parameters depend on multiple dimensionful couplings (such as the cosmological constant $g_{tt} g_{rr} = -1$5 and higher-order parameters $g_{tt} g_{rr} = -1$6), the mass function is

$g_{tt} g_{rr} = -1$7

The extended first law generalizes to

$g_{tt} g_{rr} = -1$8

with pressure $g_{tt} g_{rr} = -1$9 and conjugate volume $2$0, in line with the "black hole chemistry" framework. Smarr-like relations, derived via Eulerian scaling arguments, are established for theories with homogeneous mass functions, providing closed-form expressions for the enthalpy structure.

Explicit Example: Regular AdS Bardeen Black Hole

The general constructions are illustrated with the regular Bardeen black hole in $2$1: $2$2 demonstrating the explicit construction of the generating function, the mass, temperature, and entropy. The entropy formula exhibits non-trivial functions of the regularization parameter $2$3, deviating from the area law and illustrating the efficacy of the unified framework even for regular black holes beyond the Einstein sector.

Theoretical and Practical Implications

This work systematically bridges $2$4-dimensional black hole thermodynamics and the theory of integrable dilaton gravities. Any black hole metric of Schwarzschild type with arbitrary compact horizon and invertible $2$5 dependence can be promoted to a vacuum solution of some generally covariant extended QTG, with consistent thermodynamics governed by the above formal structure. The framework is manifestly constructive, enabling both the top-down generation of new models and the systematic embedding of phenomenological or quantum-corrected geometries in a covariant, thermodynamically consistent action principle.

From the practical perspective, this general framework allows for the calculation of thermodynamic quantities, phase structure, and criticality for a vast landscape of higher-derivative and non-polynomial gravitational models. Furthermore, the method applies equally to regular black holes—of interest in quantum gravity phenomenology and singularity resolution programs—and to their singular limits. Explicit formulas are provided for the computation of entropy corrections and chemical potentials associated with higher-curvature couplings, ensuring the accessibility of the analysis for practical calculations in arbitrary dimensions.

Theoretically, the generality of the result—constructing all such black holes as vacuum configurations of a QTG—addresses a persistent issue in black hole model building: the lack of action-level realization for regular and phenomenological geometries and their thermodynamic ambiguity. By reducing the black hole physics to dilaton-Horndeski theory and recasting mass and entropy relations via generating function methods, the formalism provides a structurally unifying picture, amenable to further extensions (e.g., to include matter, charges, non-spherical topologies, and non-trivial asymptotics).

Conclusion

The paper delivers a systematic and universal approach to black hole thermodynamics in extended quasi-topological gravities for all static, spherically/toroidally/hyperbolically symmetric spacetimes satisfying $2$6. The explicit construction of the generating function $2$7, integrability conditions, entropy correction formulas, and the identification of mass parameters render the thermodynamics of such black holes manifestly covariant and algorithmically tractable. This approach clarifies the embedding of a wide class of regular and quantum-inspired metrics within generally covariant Lagrangian frameworks and lays the groundwork for further developments in the thermodynamics and dynamics of modified gravity and quantum gravity motivated black holes.