Quasi-Local Conservation Equations in General Relativity

TL;DR

This paper derives exact quasi-local conservation equations in general relativity using a first-order Kaluza-Klein formalism, connecting them to known energy and momentum concepts in various spacetime scenarios.

Contribution

It introduces a novel set of quasi-local conservation equations derived from Einstein's equations, applicable to different spacetime geometries and linked to established energy measures.

Findings

Quasi-local energy reduces to Bondi energy in asymptotic flat regions.

Quasi-local energy becomes Misner-Sharp energy in spherical symmetry.

On black hole horizons, the conservation equation aligns with Thorne et al.'s results.

Abstract

A set of exact quasi-local conservation equations is derived from the Einstein's equations using the first-order Kaluza-Klein formalism of general relativity in the (2,2)-splitting of 4-dimensional spacetime. These equations are interpreted as quasi-local energy, momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasi-local energy and energy-flux integral reduce to the Bondi energy and energy-flux, respectively. In spherically symmetric spacetimes, the quasi-local energy becomes the Misner-Sharp energy. Moreover, on the event horizon of a general dynamical black hole, the quasi-local energy conservation equation coincides with the conservation equation studied by Thorne {\it et al}. We discuss the remaining quasi-local conservation equations briefly.