(1+1)-dimensional formalism and quasi-local conservation equations

TL;DR

This paper derives exact quasi-local conservation equations in a (1+1)-dimensional framework of Einstein's equations, linking them to known conservation laws at infinity and on horizons, with invariant surface integral expressions.

Contribution

It introduces a novel (1+1)-dimensional formalism for Einstein's equations that yields invariant quasi-local conservation laws applicable at different spacetime regions.

Findings

Reduces to Bondi conservation laws at infinity

Coincides with horizon conservation equations

Expressed as invariant two-surface integrals

Abstract

A set of exact quasi-local conservation equations is obtained in the (1+1)-dimensional description of the Einstein's equations of (3+1)-dimensional spacetimes. These equations are interpreted as quasi-local energy, linear momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that these quasi-local conservation equations reduce to the conservation equations of Bondi energy, linear momentum, and angular momentum, respectively. When restricted to the quasi-local horizon of a generic spacetime, which is defined without referring to the infinity, the quasi-local conservation equations coincide with the conservation equations on the stretched horizon studied by Price and Thorne. All of these quasi-local quantities are expressed as invariant two-surface integrals, and geometrical interpretations in terms of the area of a given two-surface and a pair of null vector fields orthogonal to that surface are given.