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

In the Bondi–Sachs formulation of general relativity, the spacetime 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 regular boundary conditions at the vertices. We explore the general requirements of these regularity conditions when the world line is a time-like geodesic, and use axisymmetric vacuum spacetimes to demonstrate these requirements. We correct and complete vertex-boundary conditions that were used in the past. We derive nonlinear boundary conditions, which approximate locally a metric up to fourth order in normal coordinates, and linear boundary conditions of arbitrary order. It is shown that if the metric is represented by power series expansions of the metric variables, the coefficients of such series have a very rigid angular structure and a rigid hierarchical dependence of time derivatives of the expansion coefficients. (paper)