Geometry of General Hypersurfaces in Spacetime: Junction Conditions

TL;DR

This work extends junction conditions to general hypersurfaces in spacetime whose causal character can change pointwise, by developing two induced connections—the rigged connection and a physically motivated rigged metric connection—and deriving a generalized Gauss–Codazzi formalism. The authors show that continuity of the first fundamental form provides the preliminary condition for well-defined Einstein equations in the distributional sense, and they establish distributional Bianchi identities and a precise six-parameter set of allowable curvature discontinuities, partitioned into matter versus Weyl components. The central result is the junction conditions {[}{\cal H}_{\mu\nu}{]}=0, which eliminate curvature singularities and yield a unified, covariant treatment of non-null and null junctions, including explicit expressions for the six independent discontinuities via vectors and Weyl/Ricci decompositions. The framework has implications for physical scenarios involving surface layers, shock gravitational waves, or phase transitions, and clarifies how matter and gravitational fields may discontinuously align across arbitrarily shaped, causally variable interfaces in general relativity.

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.

Theorems & Definitions (21)

• Lemma 1
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• Theorem 1
• Proposition 1
• Proposition 2
• Proposition 3
• Theorem 2
• Theorem 3
• Lemma 3
• Proposition 4