Foliations of lightlike hypersurfaces and their physical interpretation

Abstract

This paper deals with a family of lightlike (null) hypersurfaces (Hu) of a Lorentzian manifold M such that each null normal vector ℓ of Hu is not entirely in Hu, but, is defined in some open subset of M around Hu. Although the family (Hu) is not unique, we show, subject to some reasonable condition(s), that the involved induced objects are independent of the choice of (Hu) once evaluated at u = constant. We use (n+1)-splitting Lorentzian manifold to obtain a normalization of ℓ and a well-defined projector onto H, needed for Gauss, Weingarten, Gauss-Codazzi equations and calculate induced metrics on proper totally umbilical and totally geodesic Hu. Finally, we establish a link between the geometry and physics of lightlike hypersurfaces and a variety of black hole horizons.