Explicit form of the Mann-Marolf surface term in (3+1) dimensions

TL;DR

This paper derives an explicit expression for the Mann-Marolf surface term in four-dimensional gravity, expressing it in terms of the boundary Einstein tensor, which simplifies its application in asymptotically flat spacetimes.

Contribution

The paper provides the first explicit formula for the Mann-Marolf surface term in (3+1) dimensions, replacing its previous implicit Ricci tensor form with an explicit Einstein tensor expression.

Findings

Explicit form of the Mann-Marolf surface term derived

Simplifies calculations of gravitational actions in asymptotically flat spacetimes

Facilitates more straightforward application in theoretical physics

Abstract

The Mann-Marolf surface term is a specific candidate for the "reference background term" that is to be subtracted from the Gibbons-Hawking surface term in order make the total gravitational action of asymptotically flat spacetimes finite. That is, the total gravitational action is taken to be: (Einstein-Hilbert bulk term) + (Gibbons-Hawking surface term) - (Mann-Marolf surface term). As presented by Mann and Marolf, their surface term is specified implicitly in terms of the Ricci tensor of the boundary. Herein I demonstrate that for the physically interesting case of a (3+1) dimensional bulk spacetime, the Mann-Marolf surface term can be specified explicitly in terms of the Einstein tensor of the (2+1) dimensional boundary.