Affine projection tensor geometry: Decomposing the curvature tensor when the connection is arbitrary and the projection is tilted
| Autor: | Robert H. Gowdy |
|---|---|
| Rok vydání: | 1994 |
| Předmět: |
Physics
Riemann curvature tensor Pure mathematics Spacetime FOS: Physical sciences Statistical and Nonlinear Physics General Relativity and Quantum Cosmology (gr-qc) Affine connection Curvature General Relativity and Quantum Cosmology Tensor field Connection (mathematics) symbols.namesake Projection (mathematics) symbols Mathematics::Differential Geometry Tensor Mathematical Physics |
| Zdroj: | Journal of Mathematical Physics. 35:1274-1301 |
| ISSN: | 1089-7658 0022-2488 |
| DOI: | 10.1063/1.530589 |
| Popis: | This paper constructs the geometrically natural objects which are associated with any projection tensor field on a manifold with any affine connection. The approaches to projection tensor fields which have been used in general relativity and related theories assume normal projection tensors of co-dimension one and connections which are metric compatible and torsion-free. These assumptions fail for projections onto lightlike curves or surfaces and other situations where degenerate metrics occur as well as projections onto two-surfaces and projections onto spacetime in the higher dimensional manifolds of unified field theories. This paper removes these restrictive assumptions. One key idea is to define two different ''extrinsic curvature tensors'' which become equal for normal projections. In addition, a new family of geometrical tensors is introduced: the cross-projected curvature tensors. In terms of these objects, projection decompositions of covariant derivatives, the full Riemann curvature tensor and the Bianchi identities are obtained and applied to perfect fluids, timelike curve congruences, string congruences, and the familiar 3+1 analysis of the spacelike initial value problem of general relativity. Comment: 49 pages, RevTeX 3.0 |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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