Quantum Gravity

140 QUANTUM GEOMETRODYNAMICS5. ‘Space–time problem’: Writing X A = (T,X i ), the ‘time’ T must be aspace–time scalar. This means that, although it is constructed from thecanonical data h ab and p cd on Σ, it must weakly vanish with the Hamiltonianconstraint, { ∫}T (x), d 3 y H ⊥ (y)N(y) ≈ 0 (5.15)Σfor all lapse functions N(y) withN(x) = 0. Otherwise, one would getfrom two hypersurfaces Σ and Σ ′ crossing at x, twodifferentvaluesofT ,depending on whether the canonical data of Σ or of Σ ′ are used. The variableT would in this case have no use as a time variable. This problem isrelated to the fact that the algebra of hypersurface transformations doesnot coincide with space–time diffeomorphisms. A possible solution to thespace–time problem can be obtained by using matter variables, for example,the ‘reference fluid’ used by Brown and Kuchař (1995). The ‘space–timeproblem’ already anticipates a space–time picture which, however, is absentin quantum gravity. The space–time problem therefore refers mainlyto the classical theory.6. ‘Anomalies’: Quantum anomalies may spoil the consistency of this approach;cf. Section 5.3.7. ‘Problem of construction’: The actual transformation (5.9) has been performedonly in very special cases, for example, linearized gravity, cylindricalgravitational waves, black holes, and dust shells (Chapter 7), andhomogeneous cosmological models (Chapter 8).In the full theory, concrete proposals for the canonical transformation (5.9)are rare. A possibility that was developed to a certain extent makes use of ‘York’stime’ or ‘extrinsic time’, which is defined byT ( x; h ab ,p cd] = 23 √ h pcd h cd , P T = − √ h ; (5.16)cf. Al’tshuler and Barvinsky (1996) and the references therein. Since p cd h cd =− √ hK/8πG, cf. (4.63), T is proportional to the trace of the extrinsic curvatureK. It is canonically conjugated to P T .NotethatT does not obey (5.15) and isthus not a space–time scalar. It has been shown 5 that the Hamiltonian constraintcan be written in the form (5.10), that is, written as P T + h T ≈ 0, where h A isknown to exist, but not known in explicit form, that is, not known as an explicitfunction of T and the remaining variables. From (5.16) it is clear that the ‘true’Hamiltonian contains the three-dimensional volume as its dynamical part, thatis,∫H true = d 3 x √ ∫h + d 3 xN a H a . (5.17)5 This involves a detailed study of the ‘Lichnerowicz equation’, a non-linear (but quasi-linear)elliptical equation for P T , which under appropriate conditions possesses a unique solution (cf.Choquet-Bruhat and York 1980).