Kiefer C. Quantum gravity

106 HAMILTONIAN FORMULATION OF GENERAL RELATIVITY
4.2.1 The canonical variables
The Hamiltonian formalism starts from the choice of a configuration variable
and the definition of its momentum. Since the latter requires a time coordinate
(‘p = ∂L/∂ ˙q’), one must cast GR in a form where it exhibits a ‘distinguished’
time.Thisisachievedbyfoliating the space–time described by (M,g)intoa
set of three-dimensional space-like hypersurfaces Σ t ; cf. also Section 3.3. The
covariance of GR is preserved by allowing for the possibility to consider all
feasible foliations of this type.
This is not only of relevance for quantization (which is our motivation here),
but also for important applications in the classical theory. For example, numerical
relativity needs a description in terms of foliations in order to describe the
dynamical evolution of events, for example, the coalescence of black holes and
their emission of gravitational waves (Baumgarte and Shapiro 2003).
As a necessary condition we want to demand that (M,g) be globally hyperbolic,
that is, it possesses a Cauchy surface Σ (an ‘instant of time’) on which
initial data can be described to determine uniquely the whole space–time, see
for example, Wald (1984) or Hawking and Ellis (1973) for details. In such cases,
the classical initial value formulation makes sense, and the Hamiltonian form of
GR can be constructed. The occurrence of naked singularities is prohibited by
this assumption.
An important theorem states that for a globally hyperbolic space–time (M,g)
there exists a global ‘time function’ f such that each surface f = constant is a
Cauchy surface; therefore, M can be foliated into Cauchy hypersurfaces, and its
topology is a direct product,
M ∼ = R × Σ . (4.38)
The topology of space–time is thus fixed. This may be a reasonable assumption in
the classical theory, since topology change is usually connected with singularities
or closed time-like curves. In the quantum theory, topology change may be a
viable option and its absence in the formalism could be a possible weakness of
the canonical approach. 8 Nevertheless, the resulting quantum theory is general
enough to cope with many of the interesting situations.
One therefore starts with performing a foliation of space–time into Cauchy
surfaces Σ t ,witht denoting the global time function (‘3+1 decomposition’). The
corresponding vector field (‘flow of time’) is denoted by t µ ,obeyingt µ ∇ µ t =1.
The relation between infinitesimally neighboured hypersurfaces is the same as
shown in Fig. 3.1. 9 The space–time metric g µν induces a three-dimensional metric
on each Σ t according to
h µν = g µν + n µ n ν , (4.39)
where n µ denotes again the unit normal to Σ t ,withn µ n µ = −1.
8 A more general formulation allowing topology change to occur in principle is the pathintegral
approach of Section 2.2.
9 The vector field t µ was called Ẋ µ in Fig. 3.1 and the relation (3.74).