Vacuum Branes in D-Dimensional Static Spacetimes with Spatial Symmetry IO(D - 2), O(D - 1) or O+(D - 2,1)

Abstract

In this paper, we give a complete classification of vacuum branes, i.e., everywhere umbilical time-like hypersurfaces whose extrinsic curvature is a constant multiple of the induced metric, K_mn=k g_mn, in D-dimensional static spacetimes with spatial symmetry G(D-2,K), where G(n,K) is the isometry group of an n-dimensional space with constant sectional curvature K. D>=4 is assumed. It is shown that all possible configurations of a brane are invariant under an isometry subgroup G(D-3,K') for some K'>= K. In particular, configurations of a brane with non-zero k are always G(D-2,K) invariant, except for those in five special one-parameter families of spacetimes. Further, such G(D-2,K)-invariant configurations are allowed only in spacetimes whose Ricci tensors are isotropic in the two planes orthogonal to each G(D-2,K)-orbit, or for special values of k, which do not exist in generic cases. On the basis of these results, we prove the non-existence of a vacuum brane with black hole geometry in static bulk spacetimes with spatial symmetry G(D-2,K). We also discuss mathematical implications of these results.