Geometry of general hypersurfaces in spacetime: junction conditions

Abstract

The authors study embedded general hypersurfaces in spacetime, i.e. hypersurfaces whose timelike, spacelike or null character can change from point to point. Inherited geometrical structures on these hypersurfaces are defined by two distinct methods: the first one, in which a rigging vector (a vector not tangent to the hypersurface anywhere) induces the standard rigged connection; and the other one, more adapted to physical aspects, where each observer in spacetime induces a completely new type of connection that is called the rigged metric connection which is volume preserving. The generalizations of the Gauss and Codazzi equations are also given. With the above machinery, they attack the problem of matching two spacetimes across a general hypersurface. It is seen that the preliminary junction conditions allowing for the correct definition of Einstein's equations in the distributional sense reduce to the requirement that the first fundamental form of the hypersurface be continuous, because then there exists a maximal C1 atlas in which the metric is continuous. The Bianchi identities are then proven to hold in the distributional sense. Next, they find the proper junction conditions which forbid the appearance of singular parts in the curvature. These are shown to be equivalent to the existence of coordinate systems where the metric is C1. Finally, they derive the physical implications of the junction conditions: only six independent discontinuities of the Riemann tensor are allowed. These are six matter discontinuities at non-null points of the hypersurface. For null points, the existence of two arbitrary discontinuities of the Weyl tensor (together with four in the matter tensor) are also allowed. The classical results for timelike, spacelike or null hypersurfaces are trivially recovered.