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 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 IE) 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 Bat 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, having a big crunch at t = 0. 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, the ones with E greater than or equal to 0 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 2-parameter solutions with D<0 models either collapse to a shell-crossing singularity and become unphysical or expand from such a state.