A classification of spherically symmetric self-similar dust models

Abstract

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 $z\equiv r/t$. We show that the equations can be reduced to a special case of the general perfect fluid models with 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 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, the ones with $E \ge 0$ 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.