Revisiting the characteristic initial value problem for the vacuum Einstein field equations

TL;DR

This paper investigates the local existence of solutions to the vacuum Einstein field equations using the Newman-Penrose formalism and characteristic initial data on intersecting null hypersurfaces, extending previous results with a bootstrap approach.

Contribution

It demonstrates local existence of solutions in vacuum General Relativity for data on intersecting null hypersurfaces using a gauge and methods inspired by Stewart, Luk, and Rendall.

Findings

Proves local existence near the intersection of null hypersurfaces.

Extends the understanding of characteristic initial value problems in vacuum GR.

Employs a bootstrap argument with Newman-Penrose variables.

Abstract

Using the Newman-Penrose formalism we study the characteristic initial value problem in vacuum General Relativity. We work in a gauge suggested by Stewart, and following the strategy taken in the work of Luk, demonstrate local existence of solutions in a neighbourhood of the set on which data are given. These data are given on intersecting null hypersurfaces. Existence near their intersection is achieved by combining the observation that the field equations are symmetric hyperbolic in this gauge with the results of Rendall. To obtain existence all the way along the null-hypersurfaces themselves, a bootstrap argument involving the Newman-Penrose variables is performed.