Covariant Derivatives on Null Submanifolds

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.Comment: 13 pages, no figure

Covariant derivatives on null submanifolds

The degenerate nature of the metric on null hypersurfaces creates many difficulties when attempting to define a covariant derivative on null submanifolds. This dissertation investigates these challenges and provides a technique for defining a connection on null hypersurfaces in some cases. 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. In addition, a condition on the Ricci tensor is given to determine when this construction can be used. All of the results are motivated through a sequence of examples of null surfaces on which the covariant derivative is defined. Finally, a covariant derivative operator is given for the class of spherically symmetric hypersurfaces