Wormhole Construction & "Jump Conditions"

Question

Throughout the literature Wormholes are typically constructed by "Minkowski" or "Schwarzschild Surgery" (see e.g. Visser, Lorentzian Wormholes...), i.e. under quite simple and/or highly symmetric circumstances.

In the former case, regions are excised from a single manifold and their boundaries identified, and in the latter, two manifolds are joined.

In the Minkowski case, the joining is trivial (modulo orientational considerations) since the metric is constant (-1,1,1,1), and in the Schwarzschild case shell Jump Conditions (ref, Israel, Darmois, Lichnerowicz, O'Brien & Synge - of which I have only been able to track down a copy of Darmois online) are applied.

Question:

If one were to attempt similar surgery on a more general spacetime foliated by Cauchy hypersurfaces, adopting the approach that two regions are excised and identified on each slice, what conditions (metric, derivatives, etc.) should be imposed on the spacelike identifications within surfaces and what conditions on the timelike identifications between surfaces?

Slereah · Answer

Well general relativity doesn't really have any standard condition on the smoothness of the metric tensor, but generally the metric is assumed to be at least $mathcal C^2$, so that the Levi Civitta connection will be $mathcal C^1$ and the Riemann tensor $mathcal C^0$. Minkowski surgery usually involves a bit of a weaker condition in the thin shell formalism, with the metric being $mathcal C^0$, the connexion with some step discontinuities and the Riemann tensor some Dirac distributions. Since the computations done on the Riemann tensor are generally linear, this works out alright.

So the surgery should have the metric at least have the same value along the stitching, and perhaps better if you could identify up to second derivatives as well.

Wormhole Construction & "Jump Conditions"

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