Abstract
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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Hickethier, D., Dray, T. Covariant derivatives on null submanifolds. Gen Relativ Gravit 44, 225–238 (2012). https://doi.org/10.1007/s10714-011-1275-6
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DOI: https://doi.org/10.1007/s10714-011-1275-6