The Minkowski formula and the quasi-local mass

TL;DR

This paper develops estimates for quasi-local energy in various spacetimes using the Minkowski formula and conformal Killing-Yano 2-forms, leading to rigidity theorems that characterize Minkowski and hyperbolic spaces.

Contribution

It introduces a new approach to estimate quasi-local energy using conformal Killing-Yano 2-forms in different reference spacetimes, extending previous positive mass theorems.

Findings

Derived quasi-local energy estimates in Minkowski, anti-de Sitter, and Schwarzschild spacetimes.

Established rigidity theorems characterizing Minkowski spacetime and hyperbolic space.

Connected energy estimates with positive mass theorems to identify spacetime geometries.

Abstract

In this article, we estimate the quasi-local energy with reference to the Minkowski spacetime [16,17], the anti-de Sitter spacetime [4], or the Schwarzschild spacetime [3]. In each case, the reference spacetime admits a conformal Killing-Yano 2-form which facilitates the application of the Minkowski formula in [15] to estimate the quasi-local energy. As a consequence of the positive mass theorems in [9,13] and the above estimate, we obtain rigidity theorems which characterize the Minkowski spacetime and the hyperbolic space.