A rigorous solution concept for geodesic and geodesic deviation equations in impulsive gravitational waves

TL;DR

This paper introduces a rigorous mathematical framework using Colombeau algebras to solve the highly singular geodesic and geodesic deviation equations in impulsive gravitational waves, ensuring well-defined solutions and physical consistency.

Contribution

It develops a solution concept based on Colombeau algebras for impulsive gravitational wave equations, proving existence, uniqueness, and regularization-independent limits.

Findings

Solutions are well-defined within the Colombeau algebra framework.

Explicit distributional limits match expected physical behavior.

The approach handles highly singular products of distributions effectively.

Abstract

The geodesic as well as the geodesic deviation equation for impulsive gravitational waves involve highly singular products of distributions $(\theta\de$, $\theta^2\de$, $\de^2$). A solution concept for these equations based on embedding the distributional metric into the Colombeau algebra of generalized functions is presented. Using a universal regularization procedure we prove existence and uniqueness results and calculate the distributional limits of these solutions explicitly. The obtained limits are regularization independent and display the physically expected behavior.