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

258 Infinite-dimensional symmetries in gravitywhere the subscript 1 refers to the Ehlers group. From (14.94), one deducese 1 () = 0, e 1 (B) =−1 (14.96)and the triangular gauge is preserved. The calculation is analogous for thegenerator h( )(1 0 1/2Bh 1 : −−1/2 ) (−1/2−B0 −1 0 −1/2 =−1/2 )0 −1/2 (14.97)with h 1 () =−2 and h 1 (B) =−2B. The triangular gauge is still preserved.This is not so for the third generator, f . Repeating the above steps we find( )(0 0 1/2Bf 1 : −−1/2 ) ( )0 01 0 0 −1/2 =− 1/2 −B −1/2 (14.98)namely the triangular gauge is not preserved. Therefore, we have to introduce acompensating term, i.e. we need the transformation rule (14.89). We introducea local H transformation parametrized by a function ω, which is determined insuch a way as to preserve the gauge. Remember that the H generator is Y 3 :( ) (f 1 : − f 1 v + v(−ωY 3 0 0B−1/2−) =− 1/2 −B −1/2 + ω1/2 ) −1/2 .0(14.99)The triangular gauge is defined by the condition− √ +and so the transformation readsf 1 : δv =Hence the variations of the fields and B areω √= 0 → ω = (14.100)(B1/2− 3/2 )0 −B −1/2 . (14.101)f 1 () = 2B, f 1 (B) = B 2 − 2 (14.102)clearly not linear in the fields.Note that the SL(2, R) transformations leave the fields ρ and λ unchanged,i.e.δλ = 0, δρ = 0.14.3.2 Nonlinear realization of SL(2, R) MMOn the other side, identical calculations can be done to evaluate the action ofSL(2, R) MM on the fields (, B 2 ). Also in this case the symmetry is realized in