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I'm confused about the Israel Junction Conditions. I've seen them written several different ways so far, but here I'll use: $$K^-_{ij}-K^+_{ij}=8\pi(S_{ij}-\frac{1}{2}g_{ij}S).$$

My understanding is that $K^-_{ij}$ and $K^+_{ij}$ are the extrinsic curvatures of the shell measured from the inside and outside respectively, $S_{ij}$ is the energy-momentum tensor of the induced metric on the shell, $S$ is its trace, and $g_{ij}$ is the metric of the resulting manifold after the two are combined.

  1. But I've seen it written in other ways too. This is another way I’ve seen it written. Do those other ways mean the same thing as this?

  2. Also, does this apply for any case of two Lorentzian manifolds being glued?

Here is the image I got the first equation from:

enter image description here

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  • $\begingroup$ I wanted to answer but the option isn't there anymore as the question is closed. The conditions do apply for Lorentzian manifolds. In AdS/CFT, when taking the CPT-conjugation and conformally completing a coarse-grained wedge, one uses these Barrabes-Isreal conditions to identify the codimension-2 extremal surface to find the conformal completion of the coarse-graining. As of the first part, I am not sure what the other ways are, but technically the idea is to identify conditions to glue objects across some surface without discontinuities. I hope that answers your question. $\endgroup$
    – VaibhavK
    Commented Mar 18 at 18:02
  • $\begingroup$ But did I get the terms right? Also, aren’t there cases where a continuous metric isn’t realizable? $\endgroup$
    – user345249
    Commented Mar 18 at 18:17

1 Answer 1

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  1. This is just a trace-reversed version of the EFE on the hypersurface, i.e. the 2nd Israel junction condition. This is equivalent to the forward version. (Recall there is also a trace-reversed version of the EFE in the bulk.)

  2. The gluing of 2 Lorentzian manifolds along a common hypersurface satisfying the EFE is given by Israel's 2 junction conditions, cf. e.g. Ref. 1.

References:

  1. Eric Poisson, A Relativist's Toolkit, 2004; space-like/time-like hypersurface: section 3.7 + null-like hypersurface: section 3.11.
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  • $\begingroup$ The Israel conditions are only valid for a timelike or spacelike hypersurface though. The mentioned book also contains a separate discussion for the null case. It is also possible to treat all hypersurfaces simultaneously (including those whose signature is non-constant), e.g. iopscience.iop.org/article/10.1088/0264-9381/10/9/026 . But then the equations are more complicated. $\endgroup$ Commented Mar 19 at 9:37
  • $\begingroup$ $\uparrow$ Good point. $\endgroup$
    – Qmechanic
    Commented Mar 19 at 9:47
  • $\begingroup$ Thank you, these responses are really helpful. $\endgroup$
    – user345249
    Commented Mar 22 at 19:36

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