Kiefer C. Quantum gravity

THE 3+1 DECOMPOSITION OF GENERAL RELATIVITY 115
constraints in phase space, 4×∞ 3 variables have to be subtracted. The remaining
8 ×∞ 3 variables define the constraint hypersurface Γ c . Since the constraints
generate a four-parameter set of gauge transformations on Γ c (see Section 3.1.2),
4 ×∞ 3 degrees of freedom must be subtracted in order to ‘fix the gauge’. The
remaining 4 ×∞ 3 variables define the reduced phase space Γ r and correspond
to 2 ×∞ 3 degrees of freedom in configuration space—in accordance with the
counting above. It must be emphasized that this counting always holds modulo
a finite number of degrees of freedom—an example is gravity in 2+1 dimensions
(Section 8.1.3).
Does a three-dimensional geometry indeed contain information about time?
Consider a situation in non-gravitational physics, for example, electrodynamics.
There the specification of the, say, magnetic field on two hypersurfaces does not
suffice to determine the field everywhere. In addition, the time parameters of the
two surfaces must be specified for an appropriate boundary-value problem. In
contrast to the gravitational case, the background space–time is fixed here (i.e.,
it is non-dynamical). As we have seen in Section 3.4, two configurations (e.g. of a
clock) in classical mechanics do not suffice to determine the motion—one needs
in addition the two times of the clock configurations or its speed.
The situation in the gravitational case is related to the ‘sandwich conjecture’.
This conjecture states that two three-geometries do (in the generic case) determine
the temporal separation (the proper times) along each time-like worldline
connecting them in the resulting space–time. Whereas not much is known about
the finite version of this conjecture, results are available for the infinitesimal
case. In this ‘thin-sandwich conjecture’, one specifies on one hypersurface the
three-metric h ab and its ‘time derivative’ ∂h ab /∂t—the latter is only required
up to a numerical factor, since the ‘speed’ itself is meaningless; only the ‘direction’
in configuration space is of significance. The thin-sandwich conjecture
holds if one can determine from these initial conditions lapse and shift from the
constraints. 16 It has been shown that this can be done locally for ‘generic’ situations;
see Bartnik and Fodor (1993) for pure gravity and Giulini (1999) for
gravity plus matter. 17
The ‘temporal’ degree of freedom of the three-geometry cannot in general be
separated from other variables, that is, all three degrees of freedom contained in
h ab (after the diffeomorphism constraints have been considered) should be interpreted
as physical variables, and be treated on equal footing. In the special case
of linear gravity (Section 2), a background structure is present. This enables one
to separate a distinguished time and to regard the remaining variables, the two
degrees of freedom of the graviton, as the only physical variables. The identification
of one variable contained in the h ab (x) as ‘time’ thus seems only possible in
situations where the hypersurface is already embedded in a space–time satisfying
16 This boundary-value problem must be distinguished from the one above where h ab and
p cd are specified on Σ and a space–time can be chosen after lapse and shift are freely chosen.
17 The condition is that the initial speed must have at each space point a negative square
with respect to the DeWitt metric.