Isometric embeddings into the Minkowski space and new quasi-local mass

TL;DR

This paper develops a new approach to defining quasi-local mass in general relativity using isometric embeddings into Minkowski space, proving existence and uniqueness, and deriving a positive mass expression with desirable properties.

Contribution

It introduces a novel method for defining quasi-local mass via Minkowski isometric embeddings, extending previous Euclidean-based approaches.

Findings

Established existence and uniqueness of Minkowski isometric embeddings.

Derived a new positive quasi-local mass expression.

Mass vanishes in Minkowski space and behaves well at infinity.

Abstract

The definition of quasi-local mass for a bounded space-like region in space-time is essential in several major unsettled problems in general relativity. The quasi-local mass is expected to be a type of flux integral on the boundary two-surface and should be independent of whichever space-like region it bounds. An important idea which is related to the Hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi-local mass from the difference of the extrinsic geometries. This approach has been taken by Brown-York and Liu-Yau (see also related works) to define such notions using the isometric embedding theorem into the Euclidean three-space. However, there exist surfaces in the Minkowski space whose quasilocal mass is strictly positive. It appears that the momentum information needs to be accounted for to reconcile the difference. In order to fully capture this information, we use isometric embeddings into the Minkowski space as references. In this article, we first prove an existence and uniqueness theorem for such isometric embeddings. We then solve the boundary value problem for Jang's equation as a procedure to recognize such a surface in the Minkowski space. In doing so, we discover new expression of quasi-local mass. The new mass is positive when the ambient space-time satisfies the dominant energy condition and vanishes on surfaces in the Minkowski space. It also has the nice asymptotic behavior at spatial and null infinity. Some of these results were announced in [29].