Evolution of 3-Space (Geometrodynamics)
In order to understand Julian Barbour's view on general relativity, we have to know the so-called the "3+1" approach to general relativity. This is the view developed by Dirac, and Wheeler in particular; instead of taking the 4-dimensional spacetime as basic, this view regards general relativity as a dynamic theory of a 3-space (closed Riemann space) changing through time. For this, we have to have an arena for expressing a change of 3-spaces (3-geometries), and that is what Wheeler called "superspace", and this is quite similar to Barbour's "Platonia" (relative configuration space). In the following, we will present a superspace for an approximation of a continuous 3-space by a finite number of flat segments (polyhedrons), but this approximation can be refined to any desired degree.
Based on this approximation, we can construct a superspace in which this (approximate) 3-space can change.
The relationships between superspace, 3-spaces, and a 4-dimensional spacetime can be illustrated as follows (Figure adapted from Misner et al, 1183). Each slice of spacetime, as in the Figure, is a 3-space, and they form a history in acordance with the Einstein (field) equations. Thus, in terms of superspace or its appropriate modifications, the general relativity may be reconstructed in conformity with the Machian ideas (Barbour in fact argues that his "best matching" is applicable in superspace). Notice that the superspace has no time-axis.
For more of this (far richer contents can be expressed in superspace), see Misner, Thorne, and Wheeler, Gravitation, Freeman, 1973, ch. 43.
Last modified July 31, 2001.(c) Soshichi Uchii suchii@bun.kyoto-u.ac.jp
