Harmonic Map Formulation of Colliding Electrovac Plane Waves

Summary

The formulation of the Einstein field equations admitting two Killing vectors in terms of harmonic mappings of Riemannian manifolds is a subject in which Charlie Misner has played a pioneering role. We shall consider the hyperbolic case of the Einstein-Maxwell equations admitting two hypersurface orthogonal Killing vectors which physically describes the interaction of two electrovac plane waves. Following Penrose's discussion of the Cauchy problem we shall present the initial data appropriate to this collision problem. We shall also present three different ways in which the Einstein-Maxwell equations for colliding plane wave spacetimes can be recognized as a harmonic map. The goal is to cast the Einstein-Maxwell equations into a form adopted to the initial data for colliding impulsive gravitational and electromagnetic shock waves in such a way that a simple harmonic map will directly yield the metric and the Maxwell potential 1-form of physical interest.

*for Charles W. Misner on his 60th birthday

Introduction

Charlie Misner was the first to recognize that the subject of harmonic mappings of Riemannian manifolds finds an important application in general relativity. In a pioneering paper with Richard Matzner [1] he found that stationary, axially symmetric Einstein field equations can be formulated as a harmonic map. Eells and Sampson's theory of harmonic mappings of Riemannian manifolds [2] provides a geometrical framework for thinking of a set of pde's, in the same spirit as “mini-superspace” that Charlie was to introduce [3] for ode Einstein equations a little later.