Regularization of odd-dimensional AdS gravity: Kounterterms

TL;DR

This paper introduces Kounterterms as boundary terms involving extrinsic curvature to regularize odd-dimensional AdS gravity, providing finite actions and conserved charges without Dirichlet boundary conditions, and correctly reproducing black hole thermodynamics.

Contribution

It proposes a novel boundary term approach using Kounterterms for regularizing AdS gravity in odd dimensions, compatible with various asymptotic conditions.

Findings

Kounterterms regularize the Euclidean action in odd dimensions.

The method yields correct mass and angular momentum for AAdS black holes.

The approach reproduces known results for vacuum energy and black hole thermodynamics.

Abstract

As an alternative to the Dirichlet counterterms prescription, I introduce the concept of Kounterterms as the boundary terms with explicit dependence on the extrinsic curvature K_{ij} that regularize the AdS gravity action. Instead of a Dirichlet boundary condition on the metric, a suitable choice of the boundary conditions --compatible with any asymptotically AdS (AAdS) spacetime-- ensures a finite action principle for all odd dimensions. Background-independent conserved quantities are obtained as Noether charges associated to asymptotic symmetries and their general expression appears naturally split in two parts. The first one gives the correct mass and angular momentum for AAdS black holes and vanishes identically for globally AdS spacetimes. Thus, the second part is a covariant formula for the vacuum energy in AAdS spacetimes and reproduces the results obtained by the Dirichlet counterterms method in a number of cases. It is also shown that this Kounterterms series regularizes the Euclidean action and recovers the correct black hole thermodynamics in odd dimensions.