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16Spherically-symmetric perfectfluid solutionsContrary to what may be the common belief, only a minority of spherically-symmetricsolutions is known. Most of the known solutions are staticor shearfree, and only very few of them satisfy fundamental physical demandssuch as a plausible equation of state or the absence of singularities.16.1 Static solutions16.1.1 Field equations and first integralsStatic spherically symmetric perfect fluid solutions are hypersurfacehomogeneousspace-times, cp. Chapter 13. They have been widely discussedas models of stars in mechanical and thermodynamical equilibrium.One often takes Schwarzschild (or canonical) coordinates defined byds 2 = r 2 dΩ 2 +e 2λ(r) dr 2 − e 2ν(r) dt 2 , dΩ 2 ≡ dϑ 2 +sin 2 ϑ dϕ 2 , (16.1)and the field equations then readκ 0 µr 2 = −G 4 4r 2 =[r(1 − e −2λ )] ′ ,κ 0 pr 2 = G 3 3r 2 = −1+e −2λ (1+2rν ′ ),κ 0 p = G 1 1 = G 2 2 =e −2λ [ν ′′ + ν ′2 − ν ′ λ ′ +(ν ′ − λ ′ )/r].(16.2a)(16.2b)(16.2c)These field equations should be supplemented by an equation of statef(µ, p) =0. (16.3)From the four equations (16.2)–(16.3), the four unknown functions µ, p, λand ν can be determined. Physically, and to get a realistic stellar model,one should start with a reasonable equation of state and impose some247