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

The Geroch group 259a nonlinear way. We have (with the suffix 0 for Matzner–Misner)e 0 () = 0, e 0 (B 2 ) =−1 (14.103)h 0 () = 2, h 0 (B 2 ) =−2B 2 (14.104)( ρ 2f 0 () =−2B 2 , f 0 (B 2 ) = B2 ) 2 − . (14.105)Again, the generator f 0 acts nonlinearly.14.4 The Geroch groupThe aim of this section is to combine the two groups, SL(2, R) E with fields(, B) and SL(2, R) MM , with (, B 2 ), into a unified group, the infinitedimensionalGeroch group. The associated Lie algebra is an affine Kac–Moodyalgebra.We return first to duality relationρ −1 2 ∂ µ B 2 = ɛ µν ∂ ν B (14.106)which is invariant under the Kramer–Neugebauer transformation. We need thisequation because we now have to evaluate the action of SL(2) E on B 2 and ofSL(2) MM on B.14.4.1 Action of SL(2, R) E on ˜λ, B 2Keeping in mind that δρ = 0, we haveB → B + δB ⇒ ɛ µν ∂ ν (δB) = δ( 2 ρ −1 ∂ µ B 2 ) (14.107)after the functional differentiation and the usage of duality∂ µ (δB 2 ) = ρɛ µν (∂ ν δB − 2 −3 δ). (14.108)Consequently, from the change of B calculated before, we have the variation ofB 2 due to the SL(2) E generators.e 1 :0= ∂ µ (δB 2 ) ⇒ e 1 (B 2 ) = c 1 (= constant) (14.109)h 1 : ∂ µ (δB 2 ) = 2ρ −2 ɛ µν ∂ ν B ⇒ h 1 (B 2 ) = 2B 2 (14.110)f 1 : ɛ µν ∂ ν (δB 2 ) = 2ρ( −2 B∂ ν B + −1 ∂ ν ) ⇒ f 1 (B 2 ) = 2φ 1 . (14.111)Here a dual potential φ 1 has been introduced, which is defined such thatρ −1 ɛ µν ∂ ν φ 1 = −2 (B∂ µ B + ∂ µ ). (14.112)Careful inspection of these relations now shows the following. The contributionsdue to e 1 and h 1 are linear in the fields and local; the difference is in the