A new conformal quasi-local energy in general relativity

TL;DR

This paper introduces a novel class of conserved quasi-local energies in general relativity, leveraging optimal isometric embeddings and conformal Killing fields to achieve finiteness and establish conservation laws.

Contribution

It develops a new conformal quasi-local energy definition in general relativity using isometric embeddings, extending previous formalisms and proving their finiteness and conservation properties.

Findings

Energies are finite for asymptotically flat spacetimes of order 1.

Limits of these energies correspond to total energies of isolated systems.

A conservation law under Einsteinian evolution is established.

Abstract

We construct new conserved quasi-local energies in general relativity using the formalism developed by \cite{CWY}. In particular, we use the optimal isometric embedding defined in \cite{yau,yau1} to transplant the conformal Killing fields of the Minkowski space back to the $ 2-$ surface of interest in the physical spacetime. For an asymptotically flat spacetime of order $1$, we show that these energies are always finite. Their limit as the total energies of an isolated system is evaluated and a conservation law under Einsteinian evolution is deduced.