How to make the gravitational action on noncompact space finite

The recently proposed technique to regularize the divergences of the gravitational action on noncompact space by adding boundary counterterms is studied. We propose a prescription for constructing 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 noncompact spaces: Eguchi-Hanson metric, Kerr-Newman metric, Taub-NUT, Taub-bolt metrics, and others. Analyzing the regularization procedure when the boundary is not a round sphere we observe that our counterterm helps to cancel the 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 the cancellation of the divergence for finite deviations possibly requires an infinite series of (higher order in the boundary curvature) boundary counterterms.