An Example of Indeterminacy in the Time-Development of “Already Unified Field Theory”: A Collision between Electomagnetic Plane Waves

Summary

Abstract

An example is presented which points to a certain basic difficulty in the “already unified” approach to unified field theory. It is shown that one can construct a pair of solutions of the combined Einstein-Maxwell equations for which the two space-times are identical in the neighbourhood of an initial spacelike hypersurface (and in fact they may also be identical at all earlier times), but the time-development of the equations leads to space-times which are essentially different in their futures. The construction of such examples requires the electromagnetic field to be null (or zero) in some regions. The example given here represents a collision between two gravitational-electromagnetic waves.

Introductory preamble

This paper was written in late 1959 or early 1960, while I was at Princeton University in the early part of my research career in general relativity. It was at a time when I knew Charlie Misner best, since he was also in Princeton then, and I learnt a great deal from him about issues of general relativity, such as the initial value problem etc. As far as I can recall, it was discussions with him, and also with John Wheeler, that led to the ideas described in this paper.

I had completed the paper, and gave it to John Wheeler for his comments. Unfortunately, unforseen circumstances intervened, and it was not until several months later that the paper resurfaced, at which time my own interests had moved elsewhere. The celebration of Charlie's 60th birthday seemed an ideal occasion on which to resurrect the paper, and I searched through old files in order to locate it.