On the Validity of the Background Subtraction Method for Black Hole Thermodynamics in Matter-Coupled Gravity Theories

TL;DR

This work examines the validity of the background subtraction (Euclidean action) method for black hole thermodynamics in gravity theories coupled to matter. By deriving background subtraction from the Iyer–Wald formalism and applying it to two representative models—a higher-derivative gravity with non-minimally coupled Maxwell field and gravity with a Kalb-Ramond field—the authors show that the Euclidean approach yields thermodynamics consistent with the IW framework, provided the full action including boundary terms is used in cases with nonzero boundary contributions. They identify two subtleties: non-integrable variations from matter parameters and possible entropy–Wald entropy discrepancies when horizon fields diverge, offering holographic renormalization as a potential resolution in some instances. The results clarify when the Euclidean method can be trusted in matter-coupled contexts and provide practical guidance for computing black hole thermodynamics in holographic and higher-derivative settings.

Abstract

The background subtraction method has long served as a practical tool for computing the Euclidean action and thermodynamic quantities of black holes. While its equivalence to the Iyer--Wald formalism is well understood in pure gravity theories, its validity in matter-coupled theories remains less clear and has even been questioned in the literature. In this work, we revisit this issue and demonstrate that the equivalence between the Euclidean action method and the Iyer--Wald formalism persists in matter-coupled scenarios. We apply the resulting formulation to two representative examples of such theories, and in both cases, the Euclidean approach performs smoothly. We further identify situations where the method may encounter subtleties due to the special properties of certain matter fields. Our results clarify when background subtraction remains reliable beyond pure gravity and when additional care is necessary.