Covariant Derivatives on Null Submanifolds

TL;DR

This paper explores methods to define covariant derivatives on null submanifolds with degenerate metrics, proposing a conformal transformation approach inspired by Geroch's work, and provides examples including spherically symmetric hypersurfaces.

Contribution

It introduces a conformal transformation method to construct covariant derivatives on null hypersurfaces, addressing limitations of previous decomposition approaches.

Findings

Conformal transformations enable covariant derivative construction on null hypersurfaces.

A Ricci tensor condition determines when the method applies.

Examples include covariant derivatives for spherically symmetric hypersurfaces.

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.