Stability of spatially homogeneous plane wave spacetimes. I

Abstract

The problem of ascertaining the stability of a given spatially homogeneous solution of Einstein's equations to small metric perturbations was examined by Barrow and Sonoda (1986), and their method was discussed by Siklos (1988). In this paper, one of the points mentioned by Siklos is investigated further: the role of the constraint equation. It is shown that the constraint equation can lead to problems aside from those (e.g. linearization stability) usually considered. A particularly simple example is used to illustrate these points, namely the homogeneous vacuum plane wave metric, which for a certain range of values of its parameters admits spatially homogeneous hypersurfaces. One way of avoiding the problems associated with the constraint equation is simply to ignore it. In most cases, this will not affect the outcome.