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.