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

266 Infinite-dimensional symmetries in gravityThe simplest non-trivial example, that will be considered here, permits us torecover the Ferrari–Ibanez colliding plane wave metric. Let us consider for thisaim the monodromy matrix)Å(w) =( w0 −ww 0 +w00w 0 +ww 0 −w∈ SL(2, C) (14.154)and usew − w 0 =− ρ ( 1(t − t 0 ) 0)4t 0 t − t (14.155)with the special value w 0 = 1 2 and t 0 ≡ t(x; w 0 ).We use light cone coordinates, with the following notation to facilitate thecomparison with the standard literatureu ≡ x + , v ≡ x − (14.156)then the remaining conformal invariance is entirely fixed by choosing thecoordinates in such a way thatρ + (u) = 1 2 (1−2u2 ), ρ − (v) = 1 2 (1−2v2 ) ⇒ ρ(u,v) = 1−u 2 −v 2 (14.157)where ρ(u,v) > 0 because the interaction region, where the waves collide, isu 2 + v 2 < 1. Substituting (14.155) into (14.154) and defining two particularsolutions (14.146) in our gauge ast 1 (u,v) ≡ tt 1 (u,v) ≡ t(u,v; w = 1 )=2(u,v; w =− 1 )2√1 − u 2 − v√1 − u 2 + v > 0 (14.158)√1 − v=−2 + u√1 − v 2 − u < 0 (14.159)where the inequalities hold in the interaction region, we obtain in astraightforward way the desired factorization form for the monodromy matrix.Then it follows that( √ − t 2 t−t 1)t1 t−tˆv(u,v; t) =20√ . (14.160)0 − t 1 t−t 2t2 t−t 1Putting t = 0 we recover v(u,v) in the triangular gauge, and then read directlythe result for by virtue of (14.69). We get =− t 1= 1 − ξt 2 1 + ξ , B = 0 (14.161)where the oblate spherical coordinates have been introduced√ √ξ ≡ u 1 − v 2 + v 1 − u 2 (14.162)√ √η ≡ u 1 − v 2 − v 1 − v 2 . (14.163)