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

TL;DR

This paper uncovers a universal relation for Euclidean integrals in higher-derivative gravity theories, applicable to various black hole spacetimes, and identifies conditions under which matter fields violate this relation.

Contribution

It establishes a universal equality for Euclidean gravitational integrals in higher-derivative theories and extends it to AdS spacetimes, providing new insights into gravitational thermodynamics.

Findings

Universal relation holds for asymptotically flat black holes

Relation extends to AdS spacetimes with regularization

Violations occur with pathological matter fields

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.