Classification of spherically symmetric self-similar dust models - art. no. 044022

We classify all spherically symmetric dust solutions of Einstein's equation s which are self-similar in the sense that all dimensionless variables depe nd 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 de scribed 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 t he 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 singu larity at t = 0 and are asymptotically Friedmann at large z;; models with E >0 are everywhere underdense relative to Friedmann and expand forever, whil e those with E<0 are everywhere overdense and recollapse to a black hole co ntaining 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 ju st the time reverse of the z >0 ones, having a big crunch at t = 0. The 2-p arameter solutions with D>0 again represent inhomogeneous cosmological mode ls 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 an d 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 on es with E greater than or equal to 0 expand or contract monotonically but t he latter may now contain a naked singularity. The ones with E<0 expand fro m 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-cros sing singularity and become unphysical or expand from such a state.