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19.3 The metric and the projection formalism 297The isotropic coordinates with W = x 1 = ρ are called Weyl’s canonicalcoordinates (x 1 = ρ, x 2 = z; √ 2ζ = ρ +iz). In these coordinates, thespace-time metric (19.17) is given byds 2 =e −2U [e 2k (dρ 2 +dz 2 )+ρ 2 dϕ 2 ] − e 2U (dt + Adϕ) 2 . (19.21)For stationary axisymmetric fields, one can develop a projection formalism(Geroch 1972) similar to that for stationary fields given in §18.1.We introduce the notationλ AB ≡ ξ A a ξ Ba, A,B =1, 2, (19.22)for the scalar products of the Killing vectors ξ 1 = ξ, ξ 2 = η, and identifythe projection tensorH ab ≡ g ab + W −2 λ AB ξ Aaξ Bb, W 2 ≡− 1 2 λAB λ AB, (19.23)with the metric tensor on a differentiable manifold Σ 2 . Capital Latinindices are raised and lowered with the aid of the alternating symbols(3.66), as in (3.67). Covariant derivatives (denoted by D a ) and theRiemann curvature tensor on Σ 2 are defined in analogy to (18.3) and(18.5) (h ab → H ab ).If 2-spaces V 2 orthogonal to the group orbits exist, Σ 2 can be identifiedwith these 2-spaces and the two scalars C A = ε MN ε abcd ξ Maξ Nbξ Ac;dvanish.Then the Ricci tensor R ab in V 4 can be written in terms of tensorsand metric operations on V 2 as follows,2Rab =(2W ) −2 λ AB ,aλ AB,b + W −1 D a W ,b + H c aH d b R cd , (19.24)D a (W −1 λ AB,a)= 1 2 W −3 λ ABH ab λ CD ,aλ CD,b− 2W −1 R ab ξ a A ξb B , (19.25)H c aξ d A R cd =0. (19.26)The contracted Bianchi identities (2.82) in the projection formalism aregiven by Whelan and Romano (1999) and read (for C A =0)D a (Wξ b AH ac G bc )=0, D b (WH c aH bd G cd )= 1 2 Wξc Aξ d BG cd (W −2 λ AB ) ,a ,(19.27)where G ab is the Einstein tensor.Kitamura (1978) gave an invariant characterization of the stationaryaxisymmetric metric (19.17) in terms of a tetrad and its first covariantderivatives. This characterization is independent of the field equations.