On algebraically special vacuum spacetimes in five dimensions

Abstract

Vacuum solutions admitting a hypersurface-orthogonal repeated principal null direction are an important class of 4D algebraically special spacetimes. We investigate the 5D analogues of such solutions: vacuum spacetimes admitting a hypersurface-orthogonal multiple Weyl aligned null direction (WAND). Such spacetimes fall into four families determined by the rank of the 3 × 3 matrix that defines the expansion and shear of the multiple WAND. The rank 3 and rank 0 cases have been studied previously. We investigate the two remaining families. We show how to define coordinates which lead to a considerable simplification of the Einstein equation with cosmological constant. The rank 2 case gives warped product and Kaluza–Klein versions of the 4D Robinson–Trautman solutions as well as some new solutions. The rank 1 case gives product, or analytically continued Schwarzschild, spacetimes.