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Imagine a metric space to be discontinuous: a Schwarzschild outer metric that changes to a flat FLRW metric once inside the boundary of a $2$-sphere of fixed $r$. If an object (or ray) just hitting the boundary at $r=R$ from the outside has its own four velocity, then in order to continue it, one would have to find the FLRW version of the vector at $r=$R. So the trajectory would be discontinuous. What I want to know is how to find that corresponding vector.

All I know is that it involves the tensor $h_{ab}$, which I don't know the name of. But I would appreciate any help.

Miyase
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user345249
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  • What reference talks about $h_{ab}$? – Ghoster Aug 27 '23 at 17:44
  • I don’t think “discontinuous spacetimes” are mainstream physics. – Ghoster Aug 27 '23 at 17:45
  • The question seems underspecified since the FLRW metric is time-dependent and you don’t specify what time it’s supposed to be when you cross the boundary. – Ghoster Aug 27 '23 at 17:56
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    @Ghoster: These are known as "thin-shell" metrics and are well-studied in GR. The idea is that you can model the boundary between two "smooth" metrics as containing a "thin shell" of stress-energy, similarly to how surface charges & surface currents create discontinuous EM fields. For more details, see Poisson's A Relativist's Toolkit among other sources. – Michael Seifert Aug 27 '23 at 18:43
  • $h_{ab}$ is usually called the "induced metric" on the boundary hypersurface. – Michael Seifert Aug 27 '23 at 18:45
  • @Ghoster: You're right, I didn't specify the $t$ value at the boundary. Also, the other thing is the scale factor, which I don't want to be just $t$ because I want to avoid singularities. So then I guess $e^t$. – user345249 Aug 27 '23 at 19:20
  • @Michael Seifert: One thing that confuses me is: when it is written with Greek indices, doesn't that undermine the inducedness? Anyway, I think that for the purpose of extending the geodesic in this case it would be the induced metric on the 2-sphere surface only, with $r$ as the ambient space. – user345249 Aug 27 '23 at 19:24
  • @Ghoster: Here is the reference. https://arxiv.org/pdf/1108.3793.pdf – user345249 Aug 27 '23 at 19:32
  • This answer might be interesting to you: https://physics.stackexchange.com/a/480194/27732. A keyword you might find useful is the "Israel Junction Conditions". – Andrew Aug 27 '23 at 22:36
  • @MichaelSeifert Thanks for the info about this topic. – Ghoster Aug 27 '23 at 22:41
  • @Andrew: The second answer says that the two conditions are that the extrinsic curvature and induced metric on the boundary must be the same on both sides of it. However, I still don't understand how to get the induced metric. – user345249 Aug 30 '23 at 02:07

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