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assumes the form34.9 The case of two space-like Killingvectors 551ds 2 =e −2U [e 2k (dz 2 − dt 2 )+W 2 dx 2 ]+e 2U (dy + Adx) 2 , (34.128)cp. (17.4) and (22.40), and the Killing vectors are ξ = ∂ x and η = ∂ y . Thefield equations are to be derived from the Lagrangian given in (34.31) withthe difference that here ∇f ∇g = ∂ z f∂ z g − ∂ t f∂ t g, i.e. the metric formanipulating the ∇ operator is diag(1, −1). Equation (34.27) translatesinto∇ =(∂ z ,∂ t ), ˜∇ =(∂ t ,∂ z ), ∂ = ∂ z + ∂ t , ∂ ∗ = ∂ z − ∂ t (34.129)and the transformed Lagrangian analogous to (34.33) becomesL ′ =2∇k∇W − 1 2 W [4(∇U) 2 − e −4U (∇ψ) 2] . (34.130)N.b. (34.32) remains unchanged. Now two real Ernst potentials are introducedbyE =e 2U + ψ, E ∗ =e 2U − ψ, (34.131)and the Lagrangian becomesL ′ =2∇k∇W − 2W ∇E ∇E ∗ (E + E ∗ ) −2 . (34.132)This is the same function as (34.36) and consequently every solutionof the corresponding field equations gives rise to two different spacetimes.The rule for translating formulae from the stationary axisymmetricsituation into the one with two spacelike Killing vectors is that in everyoccurrence of the Ernst potential E or the derivative operator ∂ an ‘i’should be replaced by a ‘±’, the conjugation operation being ± → ∓.By this replacement the complex Ernst potential E splits into two realpotentials, i.e. E =e 2U +iψ → e 2U ± ψ, cp. (34.131). Note that (34.39)can be used without change after a reinterpretation of the symbols M kand ∂.The field equation for the metric function W becomes∇ 2 W = W ,zz − W ,tt =0, (34.133)and in contrast to the stationary case there are the possibilities of ∇Wbeing spacelike, timelike, null or even zero. The first case is commonlyreferred to as ‘cylindrical waves’ (§22.3), the second one as ‘cosmologicalsolutions’ or ‘colliding waves’ (Chapter 25) and the third case gives riseto plane waves (§22.3). The fourth case implies that space-time is flat for