General Relativity: Thin Shell (Israel/Darmois) & Differentiability

Question

This may be a naive question, but...

In applying the Israel/Darmois thin shell formalism, delta function matter distributions are considered acceptable.

How is the fact that this typically (necessarily?) leads to a curvature discontinuity - which suggests to me that differentiability of the manifold is lost - accommodated, given that differentiability lies at the heart of the concept of global hyperbolicity (before any considerations of causal structure) per Leray (1953) "Hyperbolic Differential Equations"?

How can one do physics on a manifold with such a thin shell if the geodesic equation (the essential hyperbolic differential equation) no longer applies (and thus the existence & uniqueness of geodesics is indeterminate)?

Isn't such a manifold singular?

EDIT: Is part of the answer that the geodesic equation can still be applied locally where the manifold is differentiable, so that as long as a geodesic does not intersect the thin shell it's OK?

NB Links to copies of the relevant Israel/Darmois papers would also be welcomed.

Answer 1

While the curvature fails to be differentiable, it is still weakly differentiable (indeed that's the point behind using distributions). As for the geodesic equation, it is still well defined, even though in the Geroch-Traschen class of metrics, this will imply that the connection is at best a square-integrable distribution (it will generally be a piecewise-continuous function).

While there's no specific problem with this, the issue of the uniqueness of geodesics is more tricky to show, in particular it may not be trivial to prove that there exists convex normal neighbourhoods around such points (as far as I know, the usual proof assumes at least a $C^2$ metric for this to use the Picard–Lindelöf theorem). I haven't seen any proof for it in any lower differentiability class but I assume it might be possible to do using the general Carathéodory theorem for piecewise continuous functions.