The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions

TL;DR

This paper develops explicit constraints and solutions for the Einstein equations on a null hypersurface in any dimension, establishing conditions for the existence and uniqueness of vacuum solutions in a characteristic Cauchy problem.

Contribution

It provides explicit formulas for constraints on null hypersurfaces and proves a geometric uniqueness theorem for the vacuum Einstein equations in arbitrary dimensions.

Findings

Derived explicit constraint formulas for Einstein equations on null hypersurfaces.

Solved the constraints to characterize solutions satisfying full Einstein equations.

Proved a geometric uniqueness theorem for the vacuum case.

Abstract

We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary dimensions. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.