Inversion of a "discontinuous coordinate transformation" in general relativity

TL;DR

This paper demonstrates that a historically introduced discontinuous coordinate transformation in general relativity is mathematically invertible within the framework of generalized functions, clarifying its formal properties.

Contribution

It establishes the invertibility of Penrose's discontinuous coordinate transformation using modern generalized function theory.

Findings

The transformation is shown to be an invertible generalized function.

It provides a rigorous mathematical foundation for Penrose's formal transformation.

Clarifies the mathematical structure of impulsive pp-wave metrics.

Abstract

As early as 1972, Penrose - in a purely formal way - introduced a "discontinuous coordinate transformation", which relates a continuous representation of the metric of impulsive pp-waves to a discontinuous one. On the basis of the invertibility concept for generalized functions developed recently by the first author, we show that this discontinuous coordinate transformation indeed represents an invertible generalized function in the appropriate sense.