Role of Mann Counterterm in Gravitational Energy

TL;DR

This paper explores the Mann counterterm's role in gravitational energy, demonstrating its effectiveness in removing divergences and setting ground state energy to zero across various spacetime geometries.

Contribution

It extends the application of Mann's counterterm beyond divergence removal to setting the ground state energy to zero for diverse boundary geometries.

Findings

Counterterm eliminates divergence in asymptotically AdS and flat spacetimes.

Counterterm sets ground state energy to zero for spacetimes with $S^{d-2} imes \, \mathbb{R}$ boundary.

Speculation on broader applicability to other boundary geometries.

Abstract

In 1999, R. B. Mann proposed a counterterm that is some sort of generalization of the well-known Holographic counterterm and that can eliminate the divergence of the gravitational action of asymptotically AdS and flat spacetimes (Phys. Rev. D $\boldsymbol{60}$ (1999) 104047 [1]). I show it is not only for eliminating the divergence of such spacetimes but also for setting the ground state energy to zero for any $d$-dimensional spacetimes with an $S^{d-2} \times \mathbb{R} $ boundary geometry, and speculate it is also true for spacetimes with any (suitable) boundary geometry and topology.