A classification of spherically symmetric self-similar dust models

We classify all spherically symmetric dust solutions of Einstein’s equations which are self-similar in the sense that all dimensionless variables depend only upon z ≡ r/t . We show that the equations can be reduced to a special case of the general perfect fluid models with equation of state p = αµ . The most general dust solution can be written down explicitly and is described by two parameters. The first one (E) corresponds to the asymptotic energy at large | z | , while the second one (D) specifies the value of z at the singularity which characterizes such models. The E = D = 0 solution is just the flat Friedmann model. The 1-parameter family of solutions with z > 0 and D = 0 are inhomogeneous cosmological models which expand from a Big Bang singularity at t = 0 and are asymptotically Friedmann at large z ; models with E > 0 are everywhere underdense relative to Friedmann and expand forever, while those with E < 0 are everywhere overdense and recollapse to a black hole containing another singularity. The black hole always has an apparent horizon but need not have an event horizon. The D = 0 solutions with z < 0 are just the time reverse of the z > 0 ones. The 2-parameter solutions with D > 0 again represent inhomogeneous cosmological models but the Big Bang singularity is at z = − 1 /D , the Big Crunch singularity is at z = +1 /D , and any particular solution necessarily spans both z < 0 and z > 0. While there is no static model in the dust case, all these solutions are asymptotically “quasi-static” at large | z | . As in the D = 0 case, expand or contract monotonically but the latter may now contain a naked singularity. The ones with E < 0 expand from or recollapse to a second singularity, the latter containing a black hole. The D < 0 models either collapse to a shell-crossing singularity and become unphysical or expand from such a state.