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420 27 The line element for κ = σ =0, Θ+iω ̸= 027.1.3 Admissible tetrad and coordinate transformationsThe vectors l and k are fixed up to a Lorentz transformation (27.5), whichwill preserve (27.18) and (27.20) only in combination with a coordinatetransformation. The metric (27.27) is, therefore, invariant underk ′ = kF ,u , l = l ′ F ,u , u ′ = F (u, ζ, ζ), r = r ′ F ,u , F ,u > 0. (27.28)This transformation will induce a transformation of the metric functions,ρ ′ = ρF ,u , P = P ′ F ,u . (27.29)The vectors m (and m) are fixed up to a rotation (27.4), which keeps themetric invariant and P real if combined with a coordinate transformationζ ′ = ζ ′ (ζ) (ζ ′ analytic in ζ):m ′ =e iC m, ζ ′ = ζ ′ (ζ), e iC =( ′ ) 1/2dζ /dζdζ ′ , P ′ = P/dζ∣dζ ′dζ∣ . (27.30)The possible transformations involving only the coordinates and notthe tetrad are (i) changes of the origin of the affine parameter rr ′ = r + f(u, ζ, ζ), (27.31)f being constant along the rays, and (ii) the transformationsu ′ = u + g(ζ,ζ). (27.32)These two types of transformations are just the degrees of freedom inherentin the respective definitions (27.18) and (27.20) of the coordinates rand u.27.2 The line element in the case with non-twisting rays(ω =0)If ω vanishes,ρ = ρ = −Θ ⇔ ω =0, (27.33)then the line element (27.27) can be further simplified. For ω = 0, thevector field k is normal, i.e. proportional to a gradient, and thus a transformationk ′ = Ak will lead toω 3 = −k i dx i =du. (27.34)(The same result, L = 0, could be achieved by starting from (27.22).Because of (27.26) and (27.33), ω 3 ∧ dω 3 is zero, and a transformation(27.28) gives L = 0.)