Junction Conditions: In what cases is matching the extrinsic curvature at a boundary tantamount to matching metric derivatives at the boundary?

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!

• 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. Commented May 24, 2023 at 12:22

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) $$$$K_{ab} =n_{\alpha} \left( \frac{\partial^{2}x^{\alpha}}{\partial y^{a}\partial y^{b}} + \Gamma^{\alpha}_{\mu \nu} \frac{\partial x^{\mu}}{\partial y^{a}} \frac{\partial x^{\nu}}{\partial y^{b}}\right)= n_{\alpha}\Gamma^{\alpha}_{\mu \nu}e^{\mu}_{a}e^{\nu}_{a}$$$$ where $$n_{\alpha}$$ normal to the hypersurface and $$e^{\mu}_{a}$$ tangent vectors. It follows that in such cases $$$$[ K_{ab}] = n_{\alpha}[\Gamma^{\alpha}_{\mu \nu}]e^{\mu}_{a}e^{\nu}_{a}.$$$$ Imposing $$[K_{ab}]=0$$ is then equivalent to impose the continuity of first metric derivatives.