Junction conditions for LRS spacetimes in the $1+1+2$ covariant formalism

We use the distribution formalism to derive the complete set of junction conditions for general Local Rotationally Symmetric (LRS) spacetimes in the $1+1+2$ covariant formalism. We start by developing a parametric framework encompassing timelike, spacelike, or null hypersurfaces. We then introduce the distribution formalism in the $1+1+2$ framework and obtain the necessary conditions to preserve the regularity of the $1+1+2$ equations at the separation hypersurface. Using these results, we can deduce some general prescriptions on the junction of LRS spacetimes and the properties of the shell in the non-smooth cases. As examples of the application of the junction conditions, we use this formalism to perform the matching necessary to obtain well-known solutions, e.g., the Martinez thin-shell, the Schwarzschild constant-density fluid star, and the Oppenheimer-Snyder collapse.