Geodesics and geodesic deviation for impulsive gravitational waves

TL;DR

This paper rigorously analyzes the geometry of impulsive gravitational waves using regularization techniques to handle distributional singularities in geodesic equations, confirming previous results with mathematical rigor.

Contribution

It introduces a regularization approach to define and analyze geodesic and geodesic deviation equations in impulsive pp-waves, ensuring distributional limits are well-defined and independent of regularization.

Findings

Regularization yields well-defined solutions for geodesic equations.

Distributional limits are independent of regularization method.

The geometric structure of impulsive pp-waves is consistent within distribution theory.

Abstract

The geometry of impulsive pp-waves is explored via the analysis of the geodesic and geodesic deviation equation using the distributional form of the metric. The geodesic equation involves formally ill-defined products of distributions due to the nonlinearity of the equations and the presence of the Dirac $\de$-distribution in the space time metric. Thus, strictly speaking, it cannot be treated within Schwartz's linear theory of distributions. To cope with this difficulty we proceed by first regularizing the $\de$-singularity,then solving the regularized equation within classical smooth functions and, finally, obtaining a distributional limit as solution to the original problem. Furthermore it is shown that this limit is independent of the regularization without requiring any additional condition, thereby confirming earlier results in a mathematical rigorous fashion. We also treat the Jacobi equation which, despite being linear in the deviation vector field, involves even more delicate singular expressions, like the ``square'' of the Dirac $\de$-distribution. Again the same regularization procedure provides us with a perfectly well behaved smooth regularization and a regularization-independent distributional limit. Hence it is concluded that the geometry of impulsive pp-waves can be described consistently using distributions as long as careful regularization procedures are used to handle the ill-defined products.