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

Acknowledgments 267From (14.161) we have P± 2 = Q ± = 0, with P± 1 = 1 2 −1 ∂ ± . Puttingequation (14.161) into this second relation, we gain12 −1 ∂ ± = 1 2 t−1 1 ∂ ±t 1 − 1 2 t−1 2 ∂ ±t 2 (14.164)which using the formulaet −1 ∂ + t = ρ −1 ∂ + ρ 1 − t1 + t , t−1 ∂ − t = ρ −1 ∂ − ρ 1 + t1 − t(14.165)becomesP± 1 = 1 ( 1 ∓ t12 ρ−1 ∂ ± ρ − 1 ∓ t )2.1 ± t 1 1 ± t 2(14.166)Now we use the expression given here for P+ 1 to integrate the equation for theconformal factor. Some further calculations show thatλ 2 (1 − t 1 t 2 ) 2= 8uv(1 − t1 2)(1 − t2 2 ) (14.167)where the undetermined overall factor has been chosen for convenience. Then,this result yields the four-dimensional metric( )dξds 2 = (1+ξ) 2 21 − ξ 2 − dη21 − η 2 −ρ 2 1 − ξ1 + ξ (dx 2 ) 2 − 1 + ξ1 − ξ (dx 3 ) 2 . (14.168)This is (a special case of) the so-called Ferrari–Ibanez colliding plane wavesolution.AcknowledgmentsA Nagar wishes to thank Professor Pietro Fré and Professor Vittorio Gorini fortheir encouragement, and two students of Como University, Valentina Riva andMassimo Bosetti, for their hospitality and the help given to him in arranging theselectures.Further readingKramer D et al 1980 Exact Solutions of Einstein Field Equations (Cambridge: CambridgeUniversity Press)Hoensealers C and Dietz W (ed) 1984 Solutions of Einstein’s Equations: Techniques andResults (Berlin: Springer)Maison D 1978 Phys. Rev. Lett. 41 521Belinskii V A and Zakharov V E 1978 Zh. Eksp. Teor. Fiz. 48 985Breitenlohner P and Maison D 1987 Ann. Inst. Poincaré 46 215Julia B and Nicolai H 1996 Nucl. Phys. B 482 431