Black hole thermodynamics from a variational principle: Asymptotically conical backgrounds

TL;DR

This paper extends the variational principle approach to black hole thermodynamics beyond asymptotically AdS spaces, specifically for subtracted geometries in supergravity, establishing finite conserved charges and thermodynamic laws.

Contribution

It formulates a covariant variational principle for subtracted geometries in four-dimensional supergravity and connects their thermodynamics to BTZ black holes via dimensional uplift and reduction.

Findings

Finite conserved charges satisfying the first law of thermodynamics.

Covariant boundary conditions for subtracted geometries.

Mapping of thermodynamic variables to BTZ black holes.

Abstract

The variational problem of gravity theories is directly related to black hole thermodynamics. For asymptotically locally AdS backgrounds it is known that holographic renormalization results in a variational principle in terms of equivalence classes of boundary data under the local asymptotic symmetries of the theory, which automatically leads to finite conserved charges satisfying the first law of thermodynamics. We show that this connection holds well beyond asymptotically AdS black holes. In particular, we formulate the variational problem for $\mathcal{N}=2$ STU supergravity in four dimensions with boundary conditions corresponding to those obeyed by the so called `subtracted geometries'. We show that such boundary conditions can be imposed covariantly in terms of a set of asymptotic second class constraints, and we derive the appropriate boundary terms that render the variational problem well posed in two different duality frames of the STU model. This allows us to define finite conserved charges associated with any asymptotic Killing vector and to demonstrate that these charges satisfy the Smarr formula and the first law of thermodynamics. Moreover, by uplifting the theory to five dimensions and then reducing on a 2-sphere, we provide a precise map between the thermodynamic observables of the subtracted geometries and those of the BTZ black hole. Surface terms play a crucial role in this identification.