Thermodynamics of Asymptotically Locally AdS Spacetimes

TL;DR

This paper develops a comprehensive thermodynamic framework for asymptotically locally AdS spacetimes, establishing well-posed variational principles, deriving conserved charges, and verifying the first law of black hole mechanics.

Contribution

It introduces covariant counterterms for AdS gravity, rederives holographic charges via Noether's theorem, and proves the first law for general asymptotically locally AdS black holes.

Findings

Validated the first law for Kerr-Newman-AdS black holes

Computed conserved charges for specific AdS black holes

Established the necessity of covariant counterterms in variational problems

Abstract

We formulate the variational problem for AdS gravity with Dirichlet boundary conditions and demonstrate that the covariant counterterms are necessary to make the variational problem well-posed. The holographic charges associated with asymptotic symmetries are then rederived via Noether's theorem and `covariant phase space' techniques. This allows us to prove the first law of black hole mechanics for general asymptotically locally AdS black hole spacetimes. We illustrate our discussion by computing the conserved charges and verifying the first law for the four dimensional Kerr-Newman-AdS and the five dimensional Kerr-AdS black holes.