Quantum Field Theory and Gravity: Conceptual and Mathematical ...

278 J. Barbour
they leave all length ratios unchanged, and are conceptually distinct from the
general transformations (18), which change the ratios of the geodesic lengths
d(a, b) andd(c, d) between point pairs a, b and c, d. As a result, general conformal
transformations open up a vastly richer field for study than similarity
transformations. Another subgroup consists of the volume-preserving conformal
transformations (18). They leave the total volume V = ∫ √ gd 3 x of the
universe unchanged. We shall see that these transformations play an important
role in cosmology. The seemingly minor restriction of the transformations
(18) to be volume preserving is the mysterious last vestige of absolute space
that I mentioned in the introduction.
The idea of geometrodynamics is nearly 150 years old. Clifford, the
translator of Riemann’s 1854 paper on the foundations of geometry, conjectured
in 1870 that material bodies in motion might be nothing more than
regions of empty but differently curved three-dimensional space moving relative
to each other [24], p. 1202. This idea is realized in Einstein’s general
relativity in the vacuum (matter-free) case in the geometrodynamic interpretation
advocated by Wheeler [24]. I shall briefly describe his superspace-based
picture, before taking it further to conformal superspace.
Consider a matter-free spacetime that is globally hyperbolic. This means
that one can slice it by nowhere intersecting spacelike hypersurfaces identified
by a monotonic time label t (Fig. 7). Each hypersurface carries a 3-geometry,
which can be represented by many different 3-metrics g ij .Atanypointx on
one hypersurface labelled by t one can move in spacetime orthogonally to
the t + δt hypersurface, reaching it after the proper time δτ = Nδt,where
N is called the lapse. If the time labelling is changed, N is rescaled in such
awaythatNδt is invariant. In general, the coordinates on successive 3-
geometries will be chosen arbitrarily, so that the point with coordinate x on
hypersurface t + δt will not lie at the point at which the normal erected at
point x on hypersurface t pierces hypersurface t + δt. There will be a lateral
displacement of magnitude δx i = N i δt. ThevectorN i is called the shift.
The lapse and shift encode the g 00 and g 0i components respectively of the
4-metric: g 00 = N i N i − N 2 , g 0i = N i .
Each 3-metric g ij on the successive hypersurfaces is a point in Riem, and
the one-parameter family of successive g ij ’s is represented as a curve in Riem
parametrized by t. This is just one representation of the spacetime. First, one
can change the time label freely on the curve (respecting monotonicity). This
leaves the curve in Riem unchanged and merely changes its parametrization.
Second, by changing the spatial coordinates on each hypersurface one can
change the successive 3-metrics and move the curve around to a considerable
degree in Riem. However, each of these curves corresponds to one and
thesamecurveinsuperspace.But,third,oneandthesamespacetimecan
be sliced in many different ways because the definition of simultaneity in
general relativity is to a high degree arbitrary (Fig. 8). Thus, an infinity of
curves in superspace, and an even greater infinity of curves in Riem, represent
the same spacetime. In addition, they can all carry infinitely many different