16 - Values and Arguments in Homogeneous Spaces

Abstract

Curved homogeneous spaces have played important roles in relativity and cosmology. A number of well-known applications are listed with Taub's contributions noted. In these familiar cases the homogeneous space was space–time or a slice of it, and served as the domain (or argument manifold) for the physical functions or fields. Another use of homogeneous spaces is then proposed, using them as the range (or manifold of values) of fields defined on flat (or curved) space–time. Few physical examples are yet known of physical fields taking values in a curved manifold, but the mathematics of harmonic maps, their aptitude for expressing symmetry breaking, and suggestive relationships to gauge theories make exploration of this idea attractive. (A brief introduction to harmonic maps is given, summarizing Misner [16] and Misner, Schild Memorial Lectures, 1978.) The style or technique of symmetry breaking that harmonic maps implement is a style that is implicit, but not often recognized, in general relativity. The main new result of this paper is a simple proof, by analogy to the geodesic equations, that if a field ϕ satisfying a harmonic mapping field equation takes its values in a homogeneous space, then for every Killing vector ξA in that space, the quantity is a conserved current Jμ:μ = 0 even if space–time itself has no Killing vectors. (A third section of the lecture concerning harmonic connections is omitted from the text, as it is given briefly in Misner [16] and more fully in a paper in draft.)