Feynman Path Integral Formulation

4.3 Arnowitt-Deser-Misner (ADM) Formalism 107In this formalism the pure gravitational action can be written asI = 1 ∫16πG()d 4 x|det e| ea μ eb ν Rab μν − 2λ, (4.26)where we have added, for later reference, a cosmological constant term. The aboveaction can be shown to reproduce, after varying with respect to eaμ and ω abμ ,theEinstein field equations. By construction it is invariant under both local Lorentz andgeneral coordinate transformations.A closely related approach is the starting point for the loop quantization of gravity.There one introduces a self-dual connection defined as()A abμ = 1 2ω abμ − 1 2 i εab cd ωcd μ , (4.27)where the dual ( ∗ ) of an antisymmetric two index object is defined here as f ∗ab ≡− 1 2 iεab cd f cd with f ∗∗ = f . From the field A abμ one can define a curvature Fμν, ab andfrom it an actionI = 1 ∫ ()d 4 x|det e| ea μ eb ν 8πGFab μν − λ , (4.28)which can also be shown to reproduce correctly the classical field equations(Ashtekar, 1986a,b; 2004). These variables then provide the basis for the so-calledloop quantum gravity method, for which recent reviews can be found in (Smolin,2003; Thiemann, 2007a,b).4.3 Arnowitt-Deser-Misner (ADM) FormalismThe next step in developing a Hamiltonian formulation for gravity is to introducea time-slicing of space-time, by introducing a sequence of spacelike hypersurfaces,labeled by a continuous time coordinate t (Arnowitt, Deser and Misner, 1960; 1962).The invariant distance ds 2 = −dτ 2 is then written as−dτ 2 = g μν dx μ dx ν= g ij (dx i + N i dt)(dx j + N j dt) − N 2 dt 2= g ij dx i dx j + 2g ij N i dx j dt − (N 2 − g ij N i N j )dt 2 , (4.29)where x i (i = 1,2,3) are coordinates on a three-dimensional manifold. The relationshipbetween the quantities dτ, dt, dx i , N and N i here basically expresses theLorentzian version of Pythagoras’ theorem (see Fig. 4.1).Indices are raised and lowered with g ij (x)(i, j = 1,2,3), which denotes the threemetricon the given spacelike hypersurface, and N(x) and N i (x) are the lapse andshift functions, respectively. These last two quantities describe the lapse of propertime (N) between two infinitesimally close hypersurfaces, and the corresponding