Covariant double-null dynamics: $(2+2)$-splitting of the Einstein equations

TL;DR

This paper introduces a covariant $(2+2)$-splitting formalism for Einstein's equations, simplifying their structure and aiding the analysis of characteristic initial-value problems in general relativity.

Contribution

It develops a novel $(2+2)$-covariant formalism for Einstein equations, providing transparent and tractable expressions applicable to characteristic initial-value problems.

Findings

Formalism is two-dimensionally covariant and geometrically transparent.

Simplifies Einstein equations and Einstein-Hilbert action expressions.

Facilitates analysis of characteristic initial-value problems.

Abstract

The paper develops a $(2+2)$-imbedding formalism adapted to a double foliation of spacetime by a net of two intersecting families of lightlike hypersurfaces. The formalism is two-dimensionally covariant, and leads to simple, geometrically transparent and tractable expressions for the Einstein field equations and the Einstein-Hilbert action, and it should find a variety of applications. It is applied here to elucidate the structure of the characteristic initial-value problem of general relativity.