Symmetries of the Stephani Universes and Related Solutions

Some years ago Stephani derived the general solution of Einstein's eld equations for a conformally at perfect uid. In these metrics the spacelike hypersurfaces orthogonal to the uid ow have constant curvature. The metrics with non-zero expansion involve ve arbitrary functions a, b, c 1 , c 2 and c 3 of time t. It is shown that any Killing vectors must be tangent to the constant curvature hypersurfaces and that the number of Killing vectors depends on the rank of the ve functions of t. The expansion-free conformally at solutions also depend on ve arbitrary functions of t, but the isometry structure is more complicated. Killing vectors which arètilted' out of the constant curvature hypersurfaces may exist if four of these functions satisfy a certain system of linear diierential equations. Also the number of independent Killing vectors orthogonal to the uid ow is shown to depend on the rank of the same four functions. The isometry structure of Stephani's type D rotating`dust' solutions is also investigated and found to have many features in common with that of the non-expanding conformally at solutions. These solutions admit a family of timelike hypersurfaces of constant curvature and involve eight arbitrary functions of a spacelike coordinate z. It is found that in general they admit no Killing vectors, but that`tilted' Killing vectors exist if these functions satisfy a linear system of diierential equations. Killing vectors tangent to the constant curvature hypersurfaces may also exist if these functions satisfy certain linear algebraic equations. We consider the symmetries of Stephani's 1 conformally at perfect uids and Stephani's 2 rotating`dust' solutions. Two classes of conformally at uid spacetimes exist depending on whether the uid expansion vanishes or not. When the expansion is non-zero the solution is known as the Stephani universe and is given by 3 ds 2 = (a(t) + b(t)r 2 + 2r c(t)) ?2 (dx 2 + dy 2 + dz 2) ? V 2 dt 2 (1) with V = 3(_ a(t) + _ b(t)r 2 + 2r _ c(t)) (t)(a(t) + b(t)r 2 + 2r c(t)) (2) where a(t), b(t), c(t) and the uid expansion (t) are arbitrary functions of time. The energy density of the uid is given by = 12(ab ? c 2) + 2 =3.