Quantum Gravity

106 HAMILTONIAN FORMULATION OF GENERAL RELATIVITY4.2.1 The canonical variablesThe Hamiltonian formalism starts from the choice of a configuration variableand 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)intoaset of three-dimensional space-like hypersurfaces Σ t ; cf. also Section 3.3. Thecovariance of GR is preserved by allowing for the possibility to consider allfeasible 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, numericalrelativity needs a description in terms of foliations in order to describe thedynamical evolution of events, for example, the coalescence of black holes andtheir 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 whichinitial data can be described to determine uniquely the whole space–time, seefor example, Wald (1984) or Hawking and Ellis (1973) for details. In such cases,the classical initial value formulation makes sense, and the Hamiltonian form ofGR can be constructed. The occurrence of naked singularities is prohibited bythis 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 aCauchy surface; therefore, M can be foliated into Cauchy hypersurfaces, and itstopology is a direct product,M ∼ = R × Σ . (4.38)The topology of space–time is thus fixed. This may be a reasonable assumption inthe classical theory, since topology change is usually connected with singularitiesor closed time-like curves. In the quantum theory, topology change may be aviable option and its absence in the formalism could be a possible weakness ofthe canonical approach. 8 Nevertheless, the resulting quantum theory is generalenough to cope with many of the interesting situations.One therefore starts with performing a foliation of space–time into Cauchysurfaces Σ t ,witht denoting the global time function (‘3+1 decomposition’). Thecorresponding vector field (‘flow of time’) is denoted by t µ ,obeyingt µ ∇ µ t =1.The relation between infinitesimally neighboured hypersurfaces is the same asshown in Fig. 3.1. 9 The space–time metric g µν induces a three-dimensional metricon each Σ t according toh 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 pathintegralapproach of Section 2.2.9 The vector field t µ was called Ẋ µ in Fig. 3.1 and the relation (3.74).