Quantum Gravity

108 HAMILTONIAN FORMULATION OF GENERAL RELATIVITYÒ ÉÒ È Ò È¦ÉFig. 4.2. Geometric interpretation of the extrinsic curvature.ÈIn order to introduce the corresponding ‘velocity’ for h ab ,onecanstartbyconsidering the following tensor field, 10K µν = h ρµ ∇ ρn ν . (4.45)Since K µν n µ =0=K µν n ν , this tensor field is a purely spatial quantity andcan be mapped to its spatial version K ab (with indices being moved by thethree-metric). One can prove, using Frobenius’ theorem for the hypersurfaceorthogonalvector field n µ , that this tensor field is symmetric, K µν = K νµ .Its geometric interpretation can be inferred from Fig. 4.2. Consider the normalvectors at two different points P and Q of a hypersurface. Be ñ µ the vectorat P resulting from parallel transporting n µ along a geodesic from Q to P. Thedifference between n µ and ñ µ is a measure for the embedding curvature of Σinto M at P. One therefore recognizes that the tensor field (4.45) can be used todescribe this embedding curvature, since it vanishes for n µ =ñ µ .Onecanalsorewrite K µν in terms of a Lie derivative,K µν = 1 2 L nh µν , (4.46)where n denotes the normal vector field. Therefore, K ab can be interpreted asthe ‘velocity’ associated with h ab . It is called ‘extrinsic curvature’ or ‘secondfundamental form’. Its trace,K ≡ K aa = hab K ab ≡ θ (4.47)can be interpreted as the ‘expansion’ of a geodesic congruence orthogonal to Σ. 11In terms of lapse and shift, the extrinsic curvature can be written asK ab = 1)(ḣab − D a N b − D b N a , (4.48)2Nand the two terms involving the spatial covariant derivative are together equivalentto −L N h ab . As we shall see in the next subsection, the components of thecanonical momentum are obtained by a linear combination of the K ab .10 Sometimes a different sign is used in this definition.11 For a Friedmann universe (cf. Section 8.1.2) K is three times the Hubble parameter.