I have recently learned about the Israel junction conditions in GR (as explained in for example Gravitation by MTW). I then tried to generalize it when including Electromagnetism, i.e. matching two spacetimes over a junction but when there is some non-zero electromagnetic content in, but I ran into difficulties. Does anyone know where this has been studied in the literature?
Basically the problem I ran into was that the Einstein equations are schematically on the form
Here $T$ is a matter stress energy tensor for some delta function matter at the junction, $F$ is the electromagnetic field tensor and $G$ is the Einstein tensor. When deriving the Israel junction conditions we would integrate over a small slice over the junction and it turns out that the Einstein tensor has a delta function which comes from a discontinuity in the extrinsic curvature so we obtain relations between the jump of the extrinsic curvature and the stress energy tensor of the shell. However, no such delta function can appear in F^2 if we assume a discontinuity in e.g. the four potential. Differentiating A then gives a delta function in $E$ and/or $B$ but $F^2$ is then a delta function squared and contributes nothing.
If we argue that it is really $B$ and $E$ that are the physical quantities of interest and they can not possess a delta function, then $F^2$ would contribute zero to the junction integral since there are not derivatives wrt E or B. Thus the Israel junction conditions should be unchanged by adding electromagnetism, but I dont think this is the right answer?