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

254 Infinite-dimensional symmetries in gravityThe Lagrangian is given byÄ = 1 4 egmn Tr P m P n (14.66)and then the field equations for P m readD m ( √ gg mn P n ) = 0. (14.67)In the following section we will show how it is possible to reproduce theLagrangians obtained by dimensional reduction from this general constructionof nonlinear σ -models.14.2.1 Ehlers Lagrangian as a nonlinear σ -modelTo link these arguments to the previous discussion, let us consider the groupsG = SL(2, R), H = SO(2). (14.68)The quotient space has only two degrees of freedom. We enforce the triangulargauge choosing for v the following expression(1/2Bv =−1/2 )0 −1/2 (14.69)and then( 12v −1 −1 ∂ m −1 )∂ m B∂ m v =0 − 1 2 −1 ∂ m = Pm 1 Y 1 + Pm 2 Y 2 + Q m Y 3 (14.70)where the coefficients of the generators of the algebra are given byPm 1 = 1 2 −1 ∂ m , Pm 2 = Q m = 1 2 −1 ∂ m B. (14.71)The evaluation of the Lagrangian is straightforward, and we getÄ = 1 4 egmn Tr P m P n = 1 8 egmn −2 (∂ m ∂ n + ∂ m B∂ n B). (14.72)This result matches exactly with the matter part of the Einstein–HilbertLagrangian found in the previous section. We have found that this expression canbe directly reduced to two dimensions, and then, coupled to gravity, it becomessimply the Ehlers Lagrangian Ä E seen before.The Ehlers Lagrangian after dimensional reduction isÄ E = gravity + 1 4 ρe(2) g µν Tr(P µ P ν )=− 1 2 λ−1 ∂ µ λ∂ µ ρ + 1 8 ρe(2) −2 g µν (∂ µ ∂ ν + ∂ µ B∂ ν B). (14.73)We saw in the previous section that, by another type of dimensional reduction, wegot a different reduced Lagrangian, the Matzner–Misner one.This one can be constructed as a nonlinear σ -model too: we need only adifferent gauge choice, as we will see in the next section; before this, let us lookat the equations of motion derived from the Ehlers Lagrangian.