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

256 Infinite-dimensional symmetries in gravityThen, the trace is14 Tr ˜P µ ˜P µ = 1 4 ( ˜P µ 1 ˜P µ1 + ˜P µ 2 ˜P µ2 ) = 1 8 ρ−1 ∂ µ ρ(ρ −1 ∂ µ ρ − 2 −1 ∂ µ )) 2∂ µ B 2 ∂ µ B 2 . (14.80)+ 1 8 −2 ∂ µ ∂ µ + 1 ( 8 ρNow, the two Lagrangians coincide if ˜λ satisfies the condition− 1 2 ˜λ −1 ∂ µ ˜λ∂ µ ρ + 1 8 ρ−2 ∂ µ ρ(ρ −1 ∂ µ ρ − 2 −1 ∂ µ ) =− 1 2 λ−1 ∂ µ λ∂ µ ρ (14.81)namely if˜λ ≡ λρ 1/4 −1/2 . (14.82)Therefore, the two-dimensional reduced gravity in conformal gauge is given by apart of pure two-dimensional gravity, characterized by the conformal factor λ andthe dilaton ρ, and a matter part given by the bosonic fields and B, or ˜B: thisone has the structure of a nonlinear G/H sigma model.Following the first section of this paper, the complete Lagrangian reduced totwo dimensions in conformal gauge, for any G/H σ -model isÄ =− 1 2 λ−1 ∂ µ λ∂ µ ρ + 1 4 ρ Tr(P µ P µ ) (14.83)and we can recover, as before, the field equation for the conformal factor λ, thistime with the general σ -model matter part. It is given by the traceless part ofλ −1 ∂ µ λ∂ ν ρ = 1 2 Tr(P µ P ν ) + 1 2 ∂ µ∂ ν ρ. (14.84)This will be useful in the foregoing sections when recovering the colliding planewave solutions of Einstein’s theory.The Kramer–Neugebauer transformationNote now that the two models, that of Ehlers and that of Matzner–Misner, arerelated by the Kramer–Neugebauer transformation, defined by ↔ ρ , B ↔ B 2It is worth remembering that the fields B and B 2 are related by duality too, namelyɛ µν ∂ ν B = 2ρ ∂ µ B 2 . (14.85)To sum up: in this section we have seen that the dimensional reduction of Einsteintheory from D = 4toD = 2 can be done in two ways, leading to two differentSL(2, R)/SO(2) σ-models.We discover two different isometry groups, that of Ehlers and that of Matzer–MisnerSL(2, R) E , SL(2, R) MM . (14.86)Combining these two groups, one gets the (infinite-dimensional) Geroch group.