Thermodynamic analysis of shift-symmetric black-hole spacetimes in Horndeski gravity

TL;DR

The paper develops a Euclidean-action framework to extract black hole thermodynamics in shift-symmetric Horndeski and beyond-Horndeski theories, deriving a general entropy-variation formula for homogeneous spacetimes with $g_{tt}=1/g_{rr}$ and showing that both parity-preserving and parity-violating couplings contribute to $\mathcal{S}$. It analyzes two representative models—one parity-symmetric with analytic solutions and another parity-violating—finding that entropy can be driven by boundary terms and, in some cases, may vanish, yielding fixed-mass, zero-temperature black holes. Across concrete examples (GR limit, ESGB, Lovelock-related parity breaking, non-homogeneous spacetimes, and 3D Horndeski), the study reveals when the usual Bekenstein area law holds and when logarithmic or other corrections arise due to nonminimal scalar-curvature couplings. The approach clarifies ambiguities of Wald entropy in scalar-tensor theories by emphasizing boundary-term consistency and extends thermodynamic understanding of black holes in modified gravity with potential implications for stability and holography.

Abstract

In this study, we investigate the thermodynamic properties of shift symmetric Horndeski and beyond Horndeski theories (theories in which only derivatives of the scalar field appear in the action). Utilizing Euclidean methods, we first analyze two specific cases that serve as foundational examples for the broader framework. We derive the general expression for entropy variation within this setting, for homogeneous spacetimes (with $g_{tt}=1/g_{rr}$), demonstrating that both parity-preserving and parity violating terms contribute to the entropy formula. Given the functional form of the coupling terms, the black hole entropy can be determined by integrating this relation with respect to the event horizon. Our general result is consistent with previously reported special cases in the literature. Notably, for particular constraints on the coupling functions, the entropy identically vanishes. Furthermore, we establish that homogeneous spacetimes within shift and parity symmetric beyond Horndeski theories universally obey the Bekenstein Area Law for entropy, irrespective of the specific form of the coupling functions.

Figures (1)

• Figure 1: The temperature as a function of the horizon radius. Here we have set $\delta\zeta/\alpha=4$,$\delta/\varepsilon=2$ for three different ratios of $\mu/\zeta$.