Covariant phase space analysis of Lanczos-Lovelock gravity with boundaries

TL;DR

This paper develops a covariant phase space method to define finite thermodynamic potentials in Lanczos-Lovelock gravity with boundaries, allowing for a consistent first law and Smarr formula for black holes without background dependence.

Contribution

It introduces a boundary-inclusive covariant phase space approach to derive finite thermodynamic potentials in Lanczos-Lovelock gravity with multiple couplings.

Findings

Derived a new prescription for thermodynamic potentials in Lanczos-Lovelock gravity.

Established a version of the first law and Smarr formula for static black holes with arbitrary asymptotics.

Showed boundary and corner terms are essential for relaxing integrability conditions.

Abstract

This work introduces a novel prescription for the expression of the thermodynamic potentials associated with the couplings of a Lanczos-Lovelock theory. These potentials emerge in theories with multiple couplings, where the ratio between them provide intrinsic length scales that break scale invariance. Our prescription, derived from the covariant phase space formalism, differs from previous approaches by enabling the construction of finite potentials without reference to any background. To do so, we consistently work with finite-size systems with Dirichlet boundary conditions and rigorously take into account boundary and corner terms: including these terms is found to be crucial for relaxing the integrability conditions for phase space quantities that were required in previous works. We apply this prescription to the first law of (extended) thermodynamics for stationary black holes, and derive a version of the Smarr formula that holds for static black holes with arbitrary asymptotic behaviour.