A simple expression for the ADM mass

TL;DR

This paper presents a straightforward method to compute the ADM mass of an asymptotically flat space using a limit involving the rate of change of area of a closed surface, aligning with Brown and York's formulation.

Contribution

It introduces a simple, nearly elementary calculation for the ADM mass, connecting it to surface area change rates and providing two different proofs.

Findings

The ADM mass can be expressed as a limit involving area change rates.

The new formula aligns with the Brown and York approach.

Two proofs demonstrate the validity of the formula.

Abstract

We show by an almost elementary calculation that the ADM mass of an asymptotically flat space can be computed as a limit involving a rate of change of area of a closed 2-surface. The result is essentially the same as that given by Brown and York. We will prove this result in two ways, first by direct calculation from the original formula as given by Arnowitt, Deser and Misner and second as a corollary of an earlier result by Brewin for the case of simplicial spaces.