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

14.3 Symmetries of nonlinear σ -modelsSymmetries of nonlinear σ -models 257We have seen that for preserving the gauge choice, in particular the triangulargauge, the symmetryv → v ′ = g −1 v, g ∈ G (14.87)must be realized in a nonlinear way, namelyv → v ′ (x) = g −1 v(x)h(x), g ∈ G, h ∈ H. (14.88)Now consider the infinitesimal form of (14.88). The infinitesimal variation of v isδv(x) =−δg −1 v(x) + v(x)δh(x) (14.89)applying now this linearized transformation to the two σ -models seen before.Considering in particular the Chevalley–Serre generators for the SL(2, R)Lie algebrae ≡ T + =( )0 1, f ≡ T − =0 0endowed with the following commutation rules( )(0 0, h ≡ T 3 1 0=1 00 −1)(14.90)[h, e] = 2e, [h, f ] =−2 f, [e, f ] = h (14.91)one can check that this nonlinear transformation has been introduced to preservethe gauge. Let us now analyse the action of the Ehlers and Matzner–Misnergroups in turn.14.3.1 Nonlinear realization of SL(2, R) EWe use the Chevalley–Serre generators for the algebra. Considering the triangulargauge(1/2Bv =−1/2 )0 −1/2 (14.92)we now linearize the transformation (14.87). The variation of v is only due to thealgebra element a:δv = v ′ − v =−av. (14.93)Then, given the triangular form of v, it follows also( 12 −1/2 δ − 1δv =2 −3/2 Bδ + −1/2 )δB0 − 1 . (14.94)2 −3/2 δIn the following we will refer to the variation δ by, for example, the generator ewith the compact notation e() or e(B) for B. Now, we realize the transformationusing the Chevalley–Serre algebra generators, e, h and f .Fore we have( )(0 1 1/2Be 1 : −−1/2 ) ( )0 −−1/20 0 0 −1/2 =(14.95)0 0