Symmetries of vacuum type-N metrics

Abstract

The main part of the first half of this paper is a review of symmetries of metrics which are solutions of Einstein's vacuum. The symmetries dealt with are curvature collineations, (special) conformal motions, homothetic motions and motions in the two cases when the symmetry vector fields are null and non-null and the results are listed in a table. Many references are given to papers where such symmetries have been discussed and further results are proved in this paper to complete the table. Reasons are given for why a vacuum metric with a non-trivial conformal motion must be a pp-wave metric. In the case of twisting type-N vacuum metrics, the equations are examined when the metrics admit a two-parameter group of a homothetic motion and an isometric motion. The Hauser metric, the only known exact solution of this class, admits such a two-parameter group. The general metric with such a group is given in terms of a solution of a sixth-order ordinary differential equation, provided that a constraint equation is satisfied.