A note on the Penrose junction conditions

TL;DR

This paper investigates the transformation between distributional and continuous descriptions of impulsive pp-waves, clarifying their physical equivalence through generalized functions and regularization techniques.

Contribution

It provides a rigorous interpretation of the transformation as a generalized coordinate change using Colombeau's theory, resolving classical distribution theory issues.

Findings

Calculated the transformation T and Rosen form of the metric for arbitrary wave profiles.

Defined physical equivalence as a limit of regularized diffeomorphisms.

Showed T as an example of a generalized coordinate transformation.

Abstract

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 $T$ als 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.