Geometry of General Hypersurfaces in Spacetime: Junction Conditions

TL;DR

This paper develops a geometric framework for hypersurfaces in spacetime with changing causal character, deriving junction conditions that ensure consistent Einstein equations and matter discontinuities across these hypersurfaces.

Contribution

It introduces new geometric structures and junction conditions for hypersurfaces with variable causal character, extending classical matching conditions in general relativity.

Findings

Continuity of the first fundamental form is necessary for Einstein equations in distributional form.

Proper junction conditions prevent singular curvature parts, ensuring physical consistency.

Six matter discontinuities are allowed at non-null points; additional Weyl tensor discontinuities at null points.

Abstract

We study imbedded hypersurfaces in spacetime whose causal character is allowed to change from point to point. Inherited geometrical structures on these hypersurfaces are defined by two methods: first, the standard rigged connection induced by a rigging vector (a vector not tangent to the hypersurface anywhere); and a second, more physically adapted, where each observer in spacetime induces a new type of connection that we call the rigged metric connection. The generalisation of the Gauss and Codazzi equations are also given. With the above machinery, we 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. The Bianchi identities are then proven to hold in the distributional sense. Next, we find the proper junction conditions which forbid the appearance of singular parts in the curvature. Finally, we 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.