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Is there anything like Rankine-Hugoniot jump conditions for general relativity?

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Yes, to glue together in GR two Lorentzian manifolds (possibly with matter) along a common boundary (which could contain a brane), the Israel junction conditions should be satisfied, cf. e.g. this related Phys.SE post.

This is by the way similar to the interface conditions between two media in E&M.

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