Black Hole Thermodynamic Ensembles, Euclidean Action and Legendre Transformation

TL;DR

This work clarifies how Legendre transformations of black hole thermodynamics correspond to changes in boundary conditions for the on-shell action, distinguishing between ensembles via the variational principle. By comparing the Wald formalism with Euclidean action methods, the authors show that, in four-dimensional dyons, not all conjugate-variable pairs yield consistent ensembles; only two—those fixing $(\Phi_e, Q_m)$ or $(\Phi_m, Q_e)$—are allowed. The study then extends these ideas through dimensional reduction to geometric quantities: angular momentum and angular velocity can be Legendre-transformed via KK reductions that recast rotation as a $U(1)$ gauge sector, while in Chern–Simons theories, a careful gauge choice and KK analysis is required to define charges and perform Legendre transformations compatibly with the first law. Across 4D and 5D examples (Kerr, boosted strings, KK monopoles, and CS supergravity), the authors demonstrate how to construct the appropriate total-derivative Legendre terms, regularize divergences, and maintain consistency with the Wald formalism, thus providing a cohesive framework for black hole thermodynamic ensembles in gravity theories with nontrivial boundary conditions.

Abstract

In thermodynamics, a Legendre transformation of the free energy provides a mapping between different statistical ensembles. In this work, we demonstrate that performing a Legendre transformation of the black hole on-shell action is equivalent to imposing different boundary conditions on the fields. Consequently, the choice of ensemble must be consistent with, and cannot contradict, the imposed boundary conditions. From this perspective, it follows that for four-dimensional dyonic black holes, the on-shell action can only be expressed either as a function of the electric charge and the magnetic potential, or alternatively as a function of the magnetic charge and the electric potential. Inspired by the Legendre transformation of the Maxwell field, we argue that for purely gravitational theories whose metric geometries admit a \(U(1)\) fiber bundle structure, i.e.\ rotating, boosted, or Kaluza-Klein monopole configurations, one can similarly introduce appropriate Legendre terms, in the sense of dimensional reduction, to modify the thermodynamic ensemble of the black hole. Within the dimensional reduction framework, we study the on-shell action of black holes in five-dimensional minimal supergravity with a Chern-Simons term, analyze the corresponding Legendre transformation procedure, and show how the resulting formulation remains consistent with the Wald formalism.