Quantum Gravity

THE 3+1 DECOMPOSITION OF GENERAL RELATIVITY 111This can be seen as follows. Defining the three-dimensional volume elementas (see e.g. Wald 1984)(3) e µνλ = e ρµνλ t ρ ,with t ρ according to (4.40) and e ρµνλ denoting the time-independent four-dimensionalvolume element, one has by using ɛ ρµνλ = √ −ge ρµνλ ,ɛ ρµνλ t ρ = √ √−ge µνλ = − g h ɛ µνλ ,from which (4.58) follows after using (4.40) and taking purely spatial components.Equation (4.58) can also be found from (4.42).We shall now assume in the following that Σ is compact without boundary;the boundary terms for the non-compact case will be discussed separately inSection 4.2.4. In order to rewrite the curvature scalar, we use first (4.56) in thefollowing form,R = (3) R + K 2 − K ab K ab − 2R µν n µ n ν . (4.59)Using the definition of the Riemann tensor in terms of second covariant derivatives,R ρ µρνn µ = ∇ ρ ∇ ν n ρ −∇ ν ∇ ρ n ρ ,the second term on the right-hand side can be written as−2R µν n µ n ν =2(∇ ρ n ν )(∇ ν n ρ ) − 2∇ ρ (n ν ∇ ν n ρ )−2(∇ ν n ν )(∇ ρ n ρ )+2∇ ν (n ν ∇ ρ n ρ ) . (4.60)The second and fourth term are total divergences. They can thus be cast into surfaceterms at the temporal boundaries. The first surface term yields −2(n ν ∇ ν n ρ )n ρ = 0, while the second one gives 2∇ µ n µ = −2K (recall (4.45)). The two remainingterms in (4.60) can be written as 2K ab K ab and −2K 2 , respectively.Inspecting the Einstein–Hilbert action (1.1), one recognizes that the temporalsurface term is cancelled, and that the action now reads∫16πG S EH = dtd 3 xN √ h(K ab K ab − K 2 + (3) R − 2Λ)∫≡MMdtd 3 xN(G abcd K ab K cd + √ )h[ (3) R − 2Λ], (4.61)where in the second line DeWitt’s metric (4.25) was introduced. The action (4.61)is also called the ‘ADM action’ in recognition of the work by Arnowitt, Deser,and Misner, see Arnowitt et al. (1962). It has the classic form of kinetic energyminus potential energy, since the extrinsic curvature contains the ‘velocities’ ḣab,see (4.48). Writing∫S EH ≡ dtd 3 x L g ,one gets for the canonical momenta the following expressions. First,M