A universal relation among Euclidean integrals for black holes in higher-derivative gravity theories

TL;DR

The paper establishes a universal relation among Euclidean integrals of higher-derivative gravity actions, showing that for asymptotically-flat black holes the combination $\sum_m (2m - D) \alpha_m I_m$ vanishes, and extends this to asymptotically-AdS spacetimes via a regularized definition of $I_m$. The proof relies on the trace form of the field equations and the vanishing of the total-derivative flux $\int_{\mathcal{M}} \nabla_\mu K^\mu$ under the chosen regularization; matter couplings preserve the relation only if the fields are well-behaved, with explicit counterexamples illustrating violations. The results connect the distribution of Euclidean action among higher-order terms to black hole thermodynamics and the Smarr relation, while offering a diagnostic for ill-behaved matter configurations such as Kalb-Ramond backgrounds or scalar hair with horizon divergences. Overall, the work provides a unifying constraint on Euclidean gravitational integrals across a broad class of theories and spacetime asymptotics, and clarifies when Euclidean methods reliably capture black hole thermodynamics.

Abstract

In this paper, we establish a universal equality governing Euclidean integrals of gravitational actions in higher-derivative theories. This relation is shown to hold universally for asymptotically flat black holes in pure gravity, and is generalized to asymptotically anti-de Sitter (AdS) spacetimes through appropriate regularization. We further examine its validity in systems with matter-gravity coupling, identifying that violations occur only when matter fields exhibit pathological behaviors: divergence at the horizon or non-decaying profiles at infinity. These findings reveal fundamental constraints on gravitational thermodynamics and provide diagnostic tools for identifying ill-behaved matter configurations.