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16.2 Non-static solutions 253In this coordinate system, the field equations read (cf. (15.6))κ 0 µ = 1Y 2 − 2 (Y e−2λ Y ′′ − Y ′ λ ′ + Y )+ ′2 2 (2Y Y e−2ν Ẏ ˙λ + Ẏ 2 ), (16.21a)2Yκ 0 p = − 1Y 2 + 2 (Y e−2λ Y ′ ν ′ + Y )− ′2 2 (2Y Y e−2ν Ÿ − Ẏ ˙ν + Ẏ 2 ), (16.21b)2Y[(κ o pY =e −2λ ν ′′ + ν ′2 − ν ′ λ ′) ]Y + Y ′′ + Y ′ (ν ′ − λ ′ )[(¨λ −e −2ν + ˙λ2 − ˙λ)˙ν Y + Ÿ + Ẏ ( ˙λ]− ˙ν) ,(16.21c)0=Ẏ ′ − Ẏν′ − Y ′ ˙λ. (16.21d)Many of the known solutions have vanishing shear. In this case, (15.47)implies the relation Ẏ/Y = ˙λ, whose integral is Y =e λ g(r). Thus, by acoordinate transformation ˆr =ˆr(r), we can transform (16.20) intods 2 =e 2λ(r,t) (r 2 dΩ 2 +dr 2 ) − e 2ν(r,t) dt 2 , (16.22)i.e. we can introduce a coordinate system which is simultaneously comovingand isotropic. In (16.22), the r-coordinate is defined up to a transformation(inversion)ˆr =1/r, e 2ˆλ =e 2λ r 4 . (16.23)If the shear does not vanish, isotropic coordinates (16.22) can again beintroduced, but they cannot be comoving (and (16.21) no longer hold).16.2.2 Expandingsolutions without shearSome basic propertiesSolutions without shear and expansion are either static or can easily begenerated from static solutions, see §15.6.3.For expanding solutions without shear it was shown in §15.6.4 that onecan introduce coordinatesds 2 =e 2λ(r,t) (r 2 dΩ 2 +dr 2 ) − ˙λ 2 e −2f(t) dt 2 (16.24)and reduce the field equations to the ordinary differential equatione λ (λ ′′ − λ ′2 − λ ′ /r) =−ϕ(r), (16.25)where ϕ(r) is an arbitrary function. Introducing the variablesL ≡ e −λ , x ≡ r 2 , (16.26)