Collision of plane gravitational and electromagnetic waves in a Minkowski background: solution of the characteristic initial value problem

TL;DR

This paper develops a geometric and analytical framework for solving the characteristic initial value problem of colliding plane gravitational and electromagnetic waves in Minkowski space, enabling explicit construction of interaction solutions.

Contribution

It introduces a new geometric formulation and a linear spectral problem approach for solving wave collision equations with arbitrary polarizations.

Findings

Explicit solutions for colliding wave space-times are constructed from initial data.

A linear integral evolution system is developed for the nonlinear wave interaction equations.

The approach generalizes previous methods to include electromagnetic waves with arbitrary polarization.

Abstract

We consider the collisions of plane gravitational and electromagnetic waves with distinct wavefronts and of arbitrary polarizations in a Minkowski background. We first present a new, completely geometric formulation of the characteristic initial value problem for solutions in the wave interaction region for which initial data are those associated with the approaching waves. We present also a general approach to the solution of this problem which enables us in principle to construct solutions in terms of the specified initial data. This is achieved by re-formulating the nonlinear dynamical equations for waves in terms of an associated linear problem on the spectral plane. A system of linear integral ``evolution'' equations which solve this spectral problem for specified initial data is constructed. It is then demonstrated explicitly how various colliding plane wave space-times can be constructed from given characteristic initial data.