The Liu-Yau mass as a good quasi-local energy in general relativity

TL;DR

The paper discusses the Liu-Yau mass as a quasi-local energy in general relativity, highlighting its interpretation as an energy measure, especially in spherical geometries, and its advantages over other mass definitions.

Contribution

It reinterprets the Liu-Yau mass as a quasi-local energy and explores its properties, particularly in spherical geometries, providing new insights into interior energy content.

Findings

Liu-Yau mass equals the maximum Brown-York energy for a surface.

In spherical geometries, it equals the minimum energy needed to generate the surface.

Provides new interpretation of Liu-Yau mass as an interior energy measure.

Abstract

A quasi-local mass has been a long sought after quantity in general relativity. A recent candidate has been the Liu-Yau mass. One can show that the Liu-Yau mass of any two-surface is the maximum of the Brown-York energy for that two-surface. This means that it has significant disadvantages as a mass. It is much better interpreted as an energy and I will show one way of doing so. The Liu-Yau mass is especially interesting in spherical geometries, where mass and energy are indistinguishable. For a spherical two-surface, it equals the minimum of the amount of energy at rest that one needs to put inside the two-surface to generate the given surface geometry. Thus it gives interesting information about the interior, something no other mass or energy function does.