The mass of the static extension of small spheres

TL;DR

This paper provides a straightforward proof for calculating the ADM mass of static extensions of small spheres, utilizing a specific mass formula for asymptotically flat static manifolds with boundary.

Contribution

It introduces a simplified proof method for the ADM mass computation of static extensions of small spheres, building on previous work by Wiygul.

Findings

Simplified proof of ADM mass calculation

Application of mass formula to static manifolds

Clarification of static extension properties

Abstract

We give a simple proof to the computation of ADM mass of the static extensions of small spheres in Wiygul \cite{W1, W2}. It makes use of the mass formula $m = \frac{1}{4\pi} \int_{\partial M} \frac{\partial V}{\partial \nu}$ for an asymptotically flat static manifold with boundary.