Thermodynamics of Asymptotically Locally AdS Spacetimes
Ioannis Papadimitriou, Kostas Skenderis
TL;DR
The paper develops a covariant, background-independent framework for asymptotically locally AdS spacetimes, showing that boundary counterterms are essential for a well-posed variational problem and finite holographic charges. It unifies holographic, Noether, and Wald approaches to define charges associated with asymptotic symmetries and proves a general first law for AlAdS black holes, including the role of the conformal anomaly. The authors illustrate the formalism with explicit 4D Kerr–Newman–AdS and 5D Kerr–AdS black holes, highlighting the appearance of Casimir energy in odd dimensions and the necessity of appropriate Weyl/PBH transformations to maintain a fixed conformal representative. Overall, the work clarifies how boundary data, anomalies, and regulator choices influence thermodynamics and conserved quantities in AdS/CFT contexts, providing a robust toolkit for holographic black hole thermodynamics.
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.