The Bondi-Sachs metric at the vertex of a null cone: axially symmetric vacuum solutions

TL;DR

This paper analyzes the regularity conditions of the Bondi-Sachs metric at the vertex of null cones in axially symmetric vacuum spacetimes, deriving boundary conditions and solving Einstein equations near the vertex.

Contribution

It provides explicit solutions and boundary conditions for the Bondi-Sachs metric near the vertex in axially symmetric vacuum spacetimes, including higher-order corrections.

Findings

Regularity at the vertex requires specific angular structure in initial data.

Initial data depend only on the geodesic observer’s time.

Boundary conditions are derived for arbitrary order corrections.

Abstract

In the Bondi-Sachs formulation of General Relativity space-time is foliated via a family of null cones. If these null cones are defined such that their vertices are traced by a regular world-line then the metric tensor has to obey regularity conditions at the vertices. We explore these regularity conditions when the world line is a time-like geodesic. In particular, we solve the Einstein equations for the Bondi-Sachs metric near the vertices for axially symmetric vacuum space- times. The metric is calculated up to third order corrections with respect to a flat metric along the time-like geodesic, as this is the lowest order where non- linear coupling of the metric coefficients occurs. We also determine the boundary conditions of the metric to arbitrary order of these corrections when a linearized and axially symmetric vacuum space-time is assumed. In both cases we find that (i) the initial data on the null cone must have a very rigid angular structure for the vertex to be a regular point, and (ii) the initial data are determined by functions depending only on the time of a geodesic observer tracing the vertex. The latter functions can be prescribed freely, but if the vertex is assumed to be regular they must be finite and have finite derivatives along the geodesic.