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

The Riemann tensor (the curvature two-form) can be defined byEinstein theory 247[D M (ω), D N (ω)]V A = R MNA B V B . (14.10)The explicit expression in terms of the spin connection isR MNA B = 2∂ [M ω N]A B + 2ω [MA C ω N]C B . (14.11)From the Riemann tensor the Ricci tensor and the curvature scalar are obtained inthe usual way: R MN := R MPN P and R := g MN R MN . The metric determinant is14.1.3 The Einstein–Hilbert actionE = det E M A = √ −g. (14.12)Now we have all the elements to define Einstein theory. The Einstein–Hilbertaction is∫S = d 4 x Ä (14.13)and the Lagrangian Ä can be expressed in function of the spin connectionÄ =− 1 4 ER =−1 4 EE A M E B N R MNAB=− 1 2 EE A M E B N ∂ M ω N AB − 1 4 Eω A AC ω BC B + 1 4 ω BACω AC B . (14.14)Substituting now the expression ω = ω ( ) , integrating by parts and droppingtotal derivatives, we arrive at− 1 4 ER = 16 1 E( ABC ABC − 2 ABC BC A − 4 C AC A D D ). (14.15)This is the expression best suited for dimensional reduction of Einstein’s theory.In the remainder, we will now set D = 4, i.e. work in four spacetime dimensions.14.1.4 Dimensional reduction D = 4 → D = 3‘Dimensional reduction’ is equivalent to searching for solutions of Einstein’sequations with one Killing vector, which we take to be ξ M ∂ M ≡ ∂ 3 . For thispurpose, we proceed from the ‘Kaluza–Klein ansatz’ for the vierbein.(E A M =−1/2 ea m 1/2 )(B m0 1/2 , E M A =1/2 e m a −e n a B n 1/2 )0 −1/2 .(14.16)The matrix e a m is the three-bein; B m is called the Kaluza–Klein vector and theKaluza–Klein scalar. The ansatz fixes a part of the SO(1, 3) Lorentz symmetry.The residual symmetry group preserving the gauge condition E a 3 = 0 is thegauge group SO(1, 2).