How to make the gravitational action on non-compact space finite

TL;DR

This paper develops a polynomial boundary counterterm method to regularize gravitational actions on non-compact spaces, applicable to both asymptotically Anti-de Sitter and flat geometries, demonstrated on several known metrics.

Contribution

It introduces a systematic prescription for constructing boundary counterterms polynomial in boundary curvature, improving regularization of gravitational actions on non-compact spaces.

Findings

Counterterms effectively cancel divergences for known non-compact metrics.

The method works for deviations from round boundary geometries.

Higher order divergences may require infinite series of counterterms.

Abstract

The recently proposed technique to regularize the divergences of the gravitational action on non-compact space by adding boundary counterterms is studied. We propose prescription for constructing the boundary counterterms which are polynomial in the boundary curvature. This prescription is efficient for both asymptotically Anti-de Sitter and asymptotically flat spaces. Being mostly interested in the asymptotically flat case we demonstrate how our procedure works for known examples of non-compact spaces: Eguchi-Hanson metric, Kerr-Newman metric, Taub-NUT and Taub-bolt metrics and others. Analyzing the regularization procedure when boundary is not round sphere we observe that our counterterm helps to cancel large $r$ divergence of the action in the zero and first orders in small deviations of the geometry of the boundary from that of the round sphere. In order to cancel the divergence in the second order in deviations a new quadratic in boundary curvature counterterm is introduced. We argue that cancelation of the divergence for finite deviations possibly requires infinite series of (higher order in the boundary curvature) boundary counterterms.