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404 25 Collision of plane wavesAssuming a diagonal form of the line element,ds 2 = −2e M du dv + W (e V dx 2 +e −V dy 2 ), (25.68)the metric function V satisfies the same linear equation as σ,2V uv + W −1 (W u V v + W v V u )=0. (25.69)Bičák and Griffiths (1994) fixed W = f(u)+g(v) =1+u+v and calculatedσ for a Friedman–Robertson–Walker (FRW) open (ε = −1) model (§14.2)with a stiff equation of state (p = µ),√ (√ √ )3 c +1+v − c − uσ =2 ln √ √ , (25.70)c +1+v + c − uidentified plane surfaces in this background (the group G 3 of Bianchitype VII contains an Abelian subgroup G 2 ) and determined the metricfunction V from (25.69) and appropriate boundary conditions at the wavefronts u = 0 and v = 0. The solutions V to this characteristic initial valueproblem can be expressed in terms of hypergeometric functions. In theinteraction region of the colliding plane gravitational waves in the FRWbackground, the function V is the sum of two terms, V = V 1 + V 2 , whereV 1 = c n1 u n 1(v +1) −1/2 F ( 1 2 , 1 2 + n 1;1+n 1 ; −u/[v + 1]), n 1 ≥ 1 (25.71)(Griffiths 1993a) and V 2 is obtained from V 1 by replacing u ←→ v andn 1 → n 2 ,c n1 → c n2 . In the interaction region the space-time is algebraicallygeneral (Petrov type I); the gravitational waves are scatteredin the background. The remaining metric function M can be determinedfrom a line integral,M = M 0 +Ω, Ω u = 1 2 (1 + u + v)V u 2 , Ω v = 1 2 (1 + u + v)V v 2 , (25.72)where M 0 refers to the background metric and is given bye M 0=(1+u + v)[(c − u)(c +1+v)] −3/2 + const. (25.73)The gravitational waves in this expanding background slow down the rateof expansion, but, contrary to the case with Minkowski background, futurespacelike singularities do not arise after the collision; only the big-bangsingularity (u + v + 1 = 0) occurs. The same is true in a spatially flat(ε = 0) FRW background (Griffiths 1993a). However, in a closed (ε =1)FRW background with p = µ the collision of gravitational waves leadsto future spacelike curvature singularities (Feinstein and Griffiths 1994).Gravitational waves propagating into FRW universes were investigatedalso in Bičák and Griffiths (1996) and Alekseev and Griffiths (1995).