Kiefer C. Quantum gravity

118 HAMILTONIAN FORMULATION OF GENERAL RELATIVITY
so the transformations in the phase space Γ spanned by (h ab ,p cd ) cannot be reduced
to space–time transformations. What, then, is the relation between both
types of transformations? Let (M,g) be a globally hyperbolic space–time. We
shall denote by Riem M the space of all (pseudo-) Riemannian metrics on M.
Since the group of space–time diffeomorphisms, Diff M, does not act transitively,
there exist non-trivial orbits in Riem M. One can make a projection down to
the space of all four-geometries, Riem M/Diff M. By considering a particular
section,
σ :RiemM/Diff M ↦→ Riem M , (4.84)
one can choose a particular representative metric on M for each geometry. In
this way one can define formal points of the ‘background manifold’ M, whicha
priori have no meaning (in GR, points cannot be disentangled from the metric
fields). The map between different sections is not a single diffeomorphism, but a
more complicated transformation (an element of the ‘Bergmann–Komar group’,
see Bergmann and Komar 1972). Hájíček and Kijowski (2000) have shown (see
also Hájíček and Kiefer 2001a and Section 7.2) that there exists a map from
Riem M/Diff M×Emb(Σ, M) ,
where Emb(Σ, M) denotes the space of all embeddings of Σ into M, intothe
phase space Γ but excluding points where the constructed space–times admit an
isometry. Therefore, the identification between space–time diffeomorphisms and
the transformations in phase space proceeds via whole ‘histories’. The necessary
exclusion of points representing Cauchy data for space–times with Killing vectors
from Σ is one of the reasons why GR cannot be equivalent globally to a
deparametrized theory (Torre 1993).
One interesting limit for the Hamiltonian constraint (4.69) is the ‘strongcoupling
limit’ defined by setting formally G →∞. This is the limit opposite to
the weak-coupling expansion of Chapter 2. It also corresponds formally to the
limit c → 0, that is, the limit opposite to the Galileian case of infinite speed
of light. This can be seen by noting that the constant in front of the potential
term in (4.69) in fact reads c 4 /16πG. Therefore, in this limit, the lightcones
effectively collapse to the axes x = constant; different spatial points decouple
because all spatial derivative terms being present in (3) R have disappeared. One
can show that this situation corresponds to having a ‘Kasner universe’ at each
space point; see Pilati (1982, 1983) for details. Since the potential term also
carries the signature σ, this limit also corresponds formally to σ =0,thatis,
the Poisson bracket between the Hamiltonian constraint (3.90) becomes zero.
The decoupling of space points can also be recognized in the BKL-oscillations
that occur when one approaches the cosmological singularity; cf. Belinskii et al.
(1982).
4.2.4 The case of open spaces
Up to now we have neglected the presence of possible spatial boundary terms
in the Hamiltonian. In this subsection, we shall briefly discuss the necessary