Quantum Gravity

126 HAMILTONIAN FORMULATION OF GENERAL RELATIVITYInserting (4.109) and (4.110) intoΓ i kj = ed k ef j ei c Γc df − ed k ef j ∂ de i f . (4.110)D a e i b = ∂ a e i b − Γ c abe i c + ω iaje j b ,one finds the covariant constancy of the triads,in analogy to D a h bc = 0. Parallel transport is defined byD a e i b =0, (4.111)dv i = −ω iajv j dx a .Defining 22 Γ i a = − 1 2 ω ajkɛ ijk , (4.112)this parallel transport corresponds to the infinitesimal rotation of the vector v iby an angleδω i =Γ i a dxa , (4.113)that is,dv i = ɛ i jk vj δω k .(Recall that for an orthonormal frame we have ω ajk = −ω akj .) From (4.111)one finds∂ [a e i b] = −ω [a ij ej b] = −ɛi jk Γj [a ek b] . (4.114)Parallel transport around a closed loop yieldsdv i = −R i jabv j dx a dx b ≡ ɛ i jkv j δω k ,where Rjab i are the components of the curvature two-form. The angle δωk canbe written asδω k = −Rabdx k a dx b , (4.115)with Rab k ɛi jk ≡ Ri jab . The curvature components Ri abobey (from Cartan’s secondequation)Rab i =2∂ [aΓ i b] + ɛi jk Γj a Γk b (4.116)and the ‘cyclic identity’Rabe i b i =0. (4.117)The curvature scalar is given byR[e] =−R i abɛ jki e a j e b k = −R j kab ea j e bk . (4.118)The triad e a i (and similarly Ki a) contains nine variables instead of the six variablesof h ab . The Gauss constraints (4.106) reduce the number again from nine to six.22 ɛ ijk is here always the invariant tensor density, that is, ɛ 123 =1,etc.