Kantowski–Sachs metrics with source: A massless scalar field

Abstract

Solutions of the coupled Einstein massless scalar field equation are discussed under the assumption that the metric belongs to the ‘‘Kantowski–Sachs’’ class of metrics. More precisely, the general solution of the coupled Einstein massless scalar field equations is derived under the following assumptions: (a) The scalar field is massless minimally coupled to gravity and obeys the Klein–Gordon equation; and (b) the metric belongs to the spherical Kantowski–Sachs class of metrics. The main feature of the results is a tractable, very simple, compact form of the space-time metric. This is accomplished by expressing the solution in a special coordinate gauge. An analysis of the geometry shows the presence of curvature singularities which consist of a spacelike part and a null part. The spacelike singularity appears to ‘‘squeeze’’ the SO(3) orbits to zero proper area while simultaneously stretching their proper separation to infinity length. The null part appears to squeeze to zero values both SO(3) orbits and their proper separation as well. There also exists a subclass of metrics for which the singularity structure is different. The null singularity is replaced by a spacelike part. Both ‘‘initial’’ and ‘‘final’’ singularities exhibit the same qualitative features and they are of finite proper time away from each other.