(ed.). Gravitational waves (IOP, 2001)(422s).

246 Infinite-dimensional symmetries in gravityx M = (x 0 ,...,x D−1 ). The metric can be expressed in terms of the vielbein asg MN = E A M E B N η AB (14.1)with the flat metric η AB ≡ (+, −,...,−). For the following it will be importantthat the vielbein can be viewed as an element of a coset space according toE A MThe metric must be covariantly conserved∈ GL(D, R)/SO(1, D − 1). (14.2)D N (Ɣ)g MP = 0 (14.3)where Ɣ is the Christoffel symbol of the metric g MN . We next introduce a spinconnection one-form, with coefficients ω MA B . The vielbein postulate, that is thecovariant constancy of the vielbein, which agrees exactly with Cartan’s structureequation for the torsion two-form, isWriting out this equation, we haveD M (ω, Ɣ)E N A = 0. (14.4)∂ [M E N] A + ω MA B E N B = Ɣ [MN] P E P A . (14.5)We assume there is no torsion, so the Christoffel symbols are symmetric inspacetime indices, henceThe coefficients of the anholonomy are∂ [M E N] A + ω [MA N] = 0. (14.6) AB C = 2E [A M E B] N ∂ M E N C . (14.7)Using the torsion-free condition for the spin connection and permuting the indicesof the coefficients of the anholonomy we obtain the following equations ABC + ω AC B − ω BC A = 0− BC A − ω BAC + ω CAB = 0 (14.8) CAB + ω CBA − ω ABC = 0.Employing then the property of the spin-connection ω ABC =−ω AC B , due tothe fact that the generators of the algebra of the D-Lorentz Group are totallyantisymmetric matrices, we have the expression of the spin connection as afunction of ABCω ABC = 1 2 ( ABC − BC A + CAB ). (14.9)