Geometric Inequalities for Quasi-Local Masses

TL;DR

This paper establishes geometric inequalities for various quasi-local masses in general relativity, relating them to charge, angular momentum, and horizon area, and provides new proofs and generalizations of key positivity and inequality results.

Contribution

It introduces new lower bounds for quasi-local masses based on a Hamiltonian approach, extending positivity proofs and inequalities to higher dimensions and static cases.

Findings

Derived lower bounds for Brown-York, Liu-Yau, and Wang-Yau masses.

Provided a new proof of positivity for the Wang-Yau mass.

Generalized the Penrose inequality for static Wang-Yau mass.

Abstract

In this paper lower bounds are obtained for quasi-local masses in terms of charge, angular momentum, and horizon area. In particular we treat three quasi-local masses based on a Hamiltonian approach, namely the Brown-York, Liu-Yau, and Wang-Yau masses. The geometric inequalities are motivated by analogous results for the ADM mass. They may be interpreted as localized versions of these inequalities, and are also closely tied to the conjectured Bekenstein bounds for entropy of macroscopic bodies. In addition, we give a new proof of the positivity property for the Wang-Yau mass which is used to remove the spin condition in higher dimensions. Furthermore, we generalize a recent result of Lu and Miao to obtain a localized version of the Penrose inequality for the static Wang-Yau mass.