The solution space of the Einstein’s vacuum field equations for the case of five-dimensional Bianchi Type I (Type 4A1)

Abstract

We consider the 4+1 Einstein’s field equations (EFE’s) in vacuum, simplified by the assumption that there is a 4D sub-manifold on which an isometry group of dimension four acts simply transitive. In particular, we consider the Abelian group Type 4A1; and thus the emerging homogeneous sub-space is flat. Through the use of coordinate transformations that preserve the sub-manifold’s manifest homogeneity, a coordinate system is chosen in which the shift vector is zero. The resulting equations remain form invariant under the action of the constant Automorphisms group. This group is used in order to simplify the equations and obtain their complete solution space which consists of seven families corresponding to 21 distinct solutions. Apart form the Kasner type all the other solutions found are, to the best of our knowledge, new. Some of them correspond to cosmological solutions, others seem to depend on some spatial coordinate and there are also pp-wave solutions.