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My understanding of the Israel junction conditions are as they are laid out in Poisson's "A Relativist's Toolkit", namely that if one wishes to join 2 different spacetimes across some timelike or spacelike hypersurface (also possibly null, haven't looked at this case much), then the following junction conditions apply:

  1. The induced metric is necessarily continuous across this hypersurface.

  2. The extrinsic curvature is either continuous or discontinuous. If it is discontinuous, the there is a surface stress-energy shell that depends on said discontinuity.

In several papers including https://doi.org/10.1103/PhysRevD.42.2626 (see text prior to eq 9), avoiding the surface stress-energy shell at the boundary is reduced to requiring the continuity of the 'first metric derivatives' with respect to some coordinate, in this case proper radial distance.

I assume that this is not a general result because if so, then why formulate the junction conditions in terms of the extrinsic curvature?

So my question is: In what cases does the second junction condition reduce simply to continuity of metric first derivatives? (Assuming one wants to avoid a surface stress-energy density)

Any answers would be appreciated, thanks!

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  • $\begingroup$ I will post a proper answer when I get the time but: 1) The continuity of the extrinsic curvature is equivalent (for non-null surfaces) that the metric is $C^1$, but is more convenient for calculations, because you don't need to find a coordinate system that behaves "nicely" at the junction surface to evaluate it. 2) for null hypersurfaces it is possible to have a situation when the surface energy tensor vanishes, but the metric is not $C^1$. You can interpret these as "singular" gravitational waves. $\endgroup$ Commented May 24, 2023 at 12:22

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In many cases the extrinsic curvature is proportional to the Christoffell symbols projected along the normal to the hypersurface and the tangent vectors to the hypersurface (see Poisson's book). If $x^{\mu}$ are the spacetime coordinates and $y^{a}$ the hypersurface coordinates it could happen that (especially in symmetric situations) \begin{equation} K_{ab} =n_{\alpha} \left( \frac{\partial^{2}x^{\alpha}}{\partial y^{a}\partial y^{b}} + \Gamma^{\alpha}_{\mu \nu} \frac{\partial x^{\mu}}{\partial y^{a}} \frac{\partial x^{\nu}}{\partial y^{b}}\right)= n_{\alpha}\Gamma^{\alpha}_{\mu \nu}e^{\mu}_{a}e^{\nu}_{a} \end{equation} where $n_{\alpha}$ normal to the hypersurface and $e^{\mu}_{a}$ tangent vectors. It follows that in such cases \begin{equation} [ K_{ab}] = n_{\alpha}[\Gamma^{\alpha}_{\mu \nu}]e^{\mu}_{a}e^{\nu}_{a}. \end{equation} Imposing $[K_{ab}]=0$ is then equivalent to impose the continuity of first metric derivatives.

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