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 $z\equiv r/t.\mathrm{}$ We show that the equations can be reduced to a special case of the general perfect fluid models with an equation of state $p=\alpha \mu .$ 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 $E=D=0\mathrm{}$ solution is just the flat Friedmann model. The 1-parameter family of solutions with $z>\mathrm{}0\mathrm{}$ and $D=0\mathrm{}$ are inhomogeneous cosmological models which expand from a big bang singularity at $t=0\mathrm{}$ and are asymptotically Friedmann at large z; models with $E>\mathrm{}0\mathrm{}$ are everywhere underdense relative to Friedmann and expand forever, while those with $E<\mathrm{}0\mathrm{}$ 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 $D=0\mathrm{}$ solutions with $z<\mathrm{}0\mathrm{}$ are just the time reverse of the $z>\mathrm{}0\mathrm{}$ ones, having a big crunch at $t=0.\mathrm{}$ The 2-parameter solutions with $D>\mathrm{}0\mathrm{}$ 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 $z<\mathrm{}0\mathrm{}$ and $z>\mathrm{}0.\mathrm{}$ While there is no static model in the dust case, all these solutions are asymptotically “quasi-static” at large $|z|.$ As in the $D=0\mathrm{}$ case, the ones with $E>~0$ expand or contract monotonically but the latter may now contain a naked singularity. The ones with $E<\mathrm{}0\mathrm{}$ expand from or recollapse to a second singularity, the latter containing a black hole. The 2-parameter solutions with $D<\mathrm{}0\mathrm{}$ models either collapse to a shell-crossing singularity and become unphysical or expand from such a state.