The characteristic initial value problem for colliding plane waves: The linear case

TL;DR

This paper explores solving the characteristic initial value problem for colliding plane gravitational waves in a Minkowski background using the Abel transform method, focusing on the linear case with constant polarization.

Contribution

It demonstrates how the Abel transform method can be applied to the linear case of colliding plane waves and discusses solutions for known and more complex initial data scenarios.

Findings

Method works for known solutions

Problems arise with certain initial data

Infinite series solutions are possible

Abstract

The physical situation of the collision and subsequent interaction of plane gravitational waves in a Minkowski background gives rise to a well-posed characteristic initial value problem in which initial data are specified on the two null characteristics that define the wavefronts. In this paper, we analyse how the Abel transform method can be used in practice to solve this problem for the linear case in which the polarization of the two gravitational waves is constant and aligned. We show how the method works for some known solutions, where problems arise in other cases, and how the problem can always be solved in terms of an infinite series if the spectral functions for the initial data can be evaluated explicitly.