Double Null Data and the Characteristic Problem in General Relativity

TL;DR

This paper develops an abstract formalism for studying hypersurfaces of any causal character in General Relativity, specifically solving the characteristic Cauchy problem for Einstein vacuum equations in a gauge-invariant manner.

Contribution

It introduces a formalism that describes null hypersurfaces and solves the characteristic problem in a fully diffeomorphism and gauge covariant way, unifying it with the standard Cauchy problem.

Findings

Formulated hypersurface data formalism for null and non-null hypersurfaces.

Solved the characteristic Cauchy problem in an abstract, gauge-invariant manner.

Established the characteristic problem as comparable to the standard Cauchy problem.

Abstract

General hypersurfaces of any causal character can be studied abstractly using the hypersurface data formalism. In the null case, we write down all tangential components of the ambient Ricci tensor in terms of the abstract data. Using this formalism, we formulate and solve in a completely abstract way the characteristic Cauchy problem of the Einstein vacuum field equations. The initial data is detached from any spacetime notion, and it is fully diffeomorphism and gauge covariant. The results of this paper put the characteristic problem on a similar footing as the standard Cauchy problem in General Relativity.