Kiefer C. Quantum gravity

122 HAMILTONIAN FORMULATION OF GENERAL RELATIVITY
for β1/3 it is indefinite (this
includes the GR-case).
At each space point, G abcd
β
canbeconsideredasametricinthespaceof
symmetric positive definite 3 × 3 matrices, which is isomorphic to R 6 .Thus,
R 6 ∼ = GL(3, R)/SO(3) ∼ = SL(3, R)/SO(3) × R + . (4.98)
The ˜h ab are coordinates on SL(3, R)/SO(3), and τ is the coordinate on R + .The
relation (4.98) corresponds to the form (4.97) of the line element. All structures
on SL(3, R)/SO(3) × R + can be transferred to Riem Σ, since G abcd
β
is ultralocal.
One can give an interpretation of one consequence of the signature change
that occurs for β =1/3 in (4.97). For this one calculates the acceleration of
the three-volume V = ∫ d 3 x √ h (assuming it is finite) for N =1.Aftersome
calculation, one finds the expression (Giulini and Kiefer 1994)
d 2 ∫
dt 2
d 3 x √ ∫
h = −3(3α − 1)
−16πG
d 3 x √ ( 2
h
(3) R − 2Λ
3
[
H m − 1 ∂H ])
3 hab m
∂h ab
. (4.99)
We call gravity ‘attractive’ if the sign in front of the integral on the right-hand
side is negative. This is because then
1. a positive (3) R contributes with a negative sign and leads to a deceleration
of the three-volume;
2. a positive cosmological constant acts repulsively;
3. in the coupling to matter, an overall sign change corresponds to a sign
change in G. 20
In the Hamiltonian constraint (4.69), the inverse metric (4.94) enters. The
critical value separating the positive definite from the indefinite case is thus
α =1/3. One, therefore, recognizes that there is an intimate relation between
the signature of the DeWitt metric and the attractivity of gravity: only for an
indefinite signature is gravity attractive. From observations (primordial Helium
abundance) one can estimate (Giulini and Kiefer 1994) that
0.4 α 0.55 . (4.100)
This is, of course, in accordance with the GR-value α =0.5.
We return now to the case of GR. The discussion so far concerns the metric
on Riem Σ, given by the DeWitt metric (4.25). Does this metric also induce a
metric on superspace (Giulini 1995b)? In Riem Σ, one can distinguish between
‘vertical’ and ‘horizontal’ directions. The vertical directions are the directions
along the orbits generated by the three-dimensional diffeomorphisms. Metrics
20 This does not, of course, necessarily mean that G changes sign in all relations.