A note on the Penrose junction conditions

Impulsive pp-waves are commonly described either by a distributional spacetime metric or, alternatively, by a continuous one. The transformation T relating these forms clearly has to be discontinuous, which causes two basic problems. First, it changes the manifold structure and second, the pullback of the distributional form of the metric under T is not well defined within classical distribution theory. Nevertheless, from a physical point of view both pictures are equivalent. In this work, after calculating T as well as the 'Rosen' form of the metric in the general case of a pp-wave with arbitrary wave profile we give a precise meaning to the term 'physically equivalent' by interpreting T as the distributional limit of a suitably regularized sequence of diffeomorphisms. Moreover, it is shown that T provides an example of a generalized coordinate transformation in the sense of Colombeau's generalized functions.