Contents

21.2 Perfect fluid solutions 337but no new solutions of the types in this book have been obtained. Forthose types the only cases we know of where such a trick applies are thosecovered by Talbot (1969) and Quevedo (1992).21.2.4 Perfect fluid solutions with differential rotationFor differential rotation (Ω ,a ̸= 0), the field equations for the metric(21.41), with √ 2ζ = ρ +i z, take the formW ,ζζ= κ 0 pe 2k−2U W,(21.62a)U ,ζζ+ 1 2 W −1 [U ,ζ W ,ζ+ U ,ζW ,ζ ]+ 1 2 W −2 e 4U A ζ A ,ζ= 1 2 e2k−2U [ p − 1 2 (µ + p)H−1 e 2U { (1+ΩA) 2 + W 2 Ω 2 e −4U }] ,(21.62b)A ,ζζ− 1 2 W −1 (A ,ζ W ,ζ+ A ,ζW ,ζ )+2(U ,ζ A ,ζ+ U ,ζA ,ζ )= κ 0 W 2 e 2k−4U (µ + p)Ω(1 + AΩ)H −1 ,v ,ζζ+ U ,ζ U ,ζ+ 1 4 W −2 e −4U A ,ζ A ,ζ= −κ 0 H −1 e 2k[ µΩW e −4U + p(1 + ΩA) 2] ,2W ,ζ k ζ = W ,ζζ +2W (U ζ ) 2 − 1 2 W −1 e 4U (A ,ζ ) 2 ,(21.62c)(21.62d)(21.62e)with v defined bye −2v =2e 2k W ,ζζ. (21.63)Equation (21.62e) determines k via a line integral from U, A and W.An immediate consequence of the integrability condition T ab ;a = 0 and˙u n = H ,n2H − 1 ∂H2H ∂Ω Ω ,n, H ≡ W 2 Ω 2 e −2U − (1 + AΩ) 2 e 2U (21.64)is the equationdp = −(µ + p)u n du n = − µ + p2H(dH − ∂H∂Ω dΩ ). (21.65)It shows that the pressure p (which trivially is a function of only twovariables) is a function of H and Ω even if these two functions are functionallydependent or constant. If this function p(H, Ω) is known, one cancompute the mass density fromµ + p = −2H∂p/∂H. (21.66)Equation (21.65) also shows that if we think of H as a function ofU, W, A, Ω, then p does not depend on Ω, and if µ + p is non-zero, then