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

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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_<μν>= σg_<μν>, 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 u σ ≠ 0 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 σ, 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.
理論物理学刊行会の論文
2002-08-25