Affine-null metric formulation of General Relativity at two intersecting null hypersurfaces

TL;DR

This paper revisits and extends Winicour's affine-null metric formulation of General Relativity, focusing on null hypersurfaces as boundaries, and demonstrates its application through the derivation of the Israel black hole solution.

Contribution

It introduces a null hypersurface boundary condition into the affine-null metric formulation and shows how to hierarchically solve Einstein equations in this setting.

Findings

Hierarchical solution of Einstein equations on null hypersurfaces.

Derivation of the Israel black hole solution in spherical symmetry.

Discussion of Penrose conformal compactification of the resulting spacetime.

Abstract

We revisit Winicour's affine-null metric initial value formulation of General Relativity, where the characteristic initial value formulation is set up with a null metric having two affine parameters. In comparison to past work, where the application of the formulation was aimed for the timelike-null initial value problem, we consider here a boundary surface that is a null hypersurface. All of the initial data are either metric functions or first derivatives of the metric. Given such a set of initial data, Einstein equations can be integrated in a hierarchical manner, where first a set of equations is solved hierarchically on the null hypersurface serving as a boundary. Second, with the obtained boundary values, a set of differential equations, similar to the equations of the Bondi-Sachs formalism, comprising of hypersurface and evolution equations is solved hierarchically to find the entire space-time metric. An example is shown how the double null Israel black hole solution arises after specification to spherical symmetry and vacuum. This black hole solution is then discussed to with respect to Penrose conformal compactification of spacetime.