Vacuum branes in D-dimensional static spacetimes with spatial symmetry IO(D-2), O(D-1) or O+(D-2,1)

Details Vacuum branes in D-dimensional static spacetimes with spatial symmetry IO(D-2), O(D-1) or O+(D-2,1) Vacuum branes in D-dimensional static spacetimes with spatial symmetry IO(D-2), O(D-1) or O+(D-2,1)

2002-04-12

arxiv.org/abs/gr-qc/0204042v2

Prog.Theor.Phys. 108 (2002) 253-295

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

42 pages in LaTeX with the PTP style. No figure. Minor errors are corrected, and notations are partially changed

Scientific paper

10.1143/PTP.108.253

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.

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