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 “physicially 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.