Contents

16.1 Static solutions 249suitably. Moreover, if (by some other method) a solution (ζ,w) is known,possibly new solutions (ˆζ,ŵ) can be generated byˆζ = ζ,[ ∫ŵ = w + C(ζ +2xζ ,x ) −2 exp 4]ζ ,x (ζ +2xζ ,x ) −1 dx(16.11)(Heintzmann 1969), or byŵ = w,ˆζ = Cζ∫ζ −2 (2 − 4wx) −1/2 dx. (16.12)Once a solution (ζ,w) of (16.10) is known, λ, µ and p can be computedfrom (16.9).Introducing a new function α by e 2λ = (1 + rν ′ ) 2 /α, Fodor (2000)transformed the condition of isotropy (16.4) into a linear equation for α,r(1 + rν ′ )α ′ + 2[(1 − rν ′ ) 2 − 2]α + 2(1 + rν ′ ) 2 = 0 (16.13)(see also Burlankov (1993)), which for given ν can be solved by quadratures(or, for prescribed α, gives a quadratic equation for rν ′ ).Sometimes isotropic coordinatesds 2 =e 2λ (r 2 dΩ 2 +dr 2 ) − e 2ν dt 2 (16.14)prove useful. In these coordinates, the condition of isotropy of pressurereadsλ ′′ + ν ′′ + ν ′2 − λ ′2 − 2λ ′ ν ′ − (λ ′ + ν ′ )/r =0, (16.15)which is a Riccati equation in either λ ′ or ν ′ . It can also be written asLG, xx =2GL, xx , L ≡ e −λ , G ≡ Le ν , x ≡ r 2 (16.16)(Kustaanheimo and Qvist 1948), cp. §15.6.2. This equation is linear inboth L and G, and solutions can be easily found by prescribing one ofthese two functions appropriately.The condition of isotropy (16.15) is invariant under the substitutionˆν = −ν, ˆλ = λ +2ν (16.17)(Buchdahl (1956), cp. §10.11). This substitution can be used to generatestatic perfect fluid solutions from known ones.Also the general form (15.9) of the line element (Buchdahl 1967, Simon1994, Roy and Rao 1972) or coordinates with ν(r) =r (Roy and Rao1972) may make the analytic expression for the solution simple and/orlead to physically interesting solutions.