The Gravitational Action in Asymptotically AdS and Flat Spacetimes

TL;DR

The paper develops a covariant, intrinsic counterterm framework to render the gravitational action finite in asymptotically AdS and asymptotically flat spacetimes, addressing shortcomings of reference-spacetime subtraction. It introduces an iterative Gauss-Codazzi-based algorithm to generate boundary counterterms in arbitrary dimensions and provides explicit leading-order results, including curvature-squared structures, with applications to AdS-Schwarzschild. It extends the method to flat space by constructing two counterterm prescriptions, analyzes their impact on black hole thermodynamics and ADM quantities, and discusses anomalies and scheme-dependence of finite parts. The work clarifies the holographic interpretation of renormalized gravity in AdS and offers practical tools for calculating renormalized actions and energies in diverse geometries, while acknowledging limitations and the need for more general AFS counterterms for highly deformed boundaries.

Abstract

The divergences of the gravitational action are analyzed for spacetimes that are asymptotically anti-de Sitter and asymptotically flat. The gravitational action is rendered finite using a local counterterm prescription, and the relation of this method to the traditional reference spacetime is discussed. For AdS, an iterative procedure is devised that determines the counterterms efficiently. For asymptotically flat space, we use a different method to derive counterterms which are sufficient to remove divergences in most cases.