Doubly covariant action principle of singular hypersurfaces in general relativity and scalar-tensor theories

An action principle of singular hypersurfaces in general relativity and scalar-tensor type theories of gravity in the Einstein frame is presented without assuming any symmetry. The action principle is manifestly doubly covariant in the sense that coordinate systems on and off a hypersurface are disentangled and can be independently specified. It is shown that, including variation of the metric, the position of the hypersurface, and the matter fields, the variational principle gives the correct set of equations of motion: the Einstein equation off the hypersurface, Israel’s junction condition in a doubly covariant form, and the equations of motion of the matter fields including the scalar fields. The position of the hypersurface measured from one side of the hypersurface and that measured from the other side can be independently varied as required by the double covariance.