Covariant Derivatives on Null Submanifolds

Details Covariant Derivatives on Null Submanifolds Covariant Derivatives on Null Submanifolds

2011-08-11

arxiv.org/abs/1108.2332v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

13 pages, no figures

Scientific paper

The degenerate nature of the metric on null hypersurfaces makes it difficult to define a covariant derivative on null submanifolds. Recent approaches using decomposition to define a covariant derivative on null hypersurfaces are investigated, with examples demonstrating the limitations of the methods. Motivated by Geroch's work on asymptotically flat spacetimes, conformal transformations are used to construct a covariant derivative on null hypersurfaces, and a condition on the Ricci tensor is given to determine when this construction can be used. Several examples are given, including the construction of a covariant derivative operator for the class of spherically symmetric hypersurfaces.

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