Which action for brane worlds?

In his pioneering work on singular shells in general relativity, Lanczos had derived jump conditions across energy-momentum carrying hypersurfaces from the Einstein equation with codimension 1 sources. However, on the level of the action, the discontinuity of the connection arising from a codimension 1 energy-momentum source requires to take into account two adjacent space–time regions separated by the hypersurface. The purpose of the present note is to draw attention to the fact that Lanczosʹ jump conditions can be derived from an Einstein action but not from an Einstein–Hilbert action.