Spherically symmetric self-similar solutions and their astrophysical and cosmological applications

We discuss spherically symmetric perfect fluid solutions of Einstein’s equations which have equation of state ($p=\alpha \mu$) and which are self-similar in the sense that all dimensionless variables depend only upon $z\equiv r\mathrm{/}t$. For each value of $\alpha$, such solutions are described by two parameters and have now been completely classified. There is a 1-parameter family of solutions asymptotic to the flat Friedmann model at large values of z. These represent either black holes or density perturbations which grow as fast as the particle horizon; the underdense solutions may be relevant to the existence of large-scale cosmic voids. There is also a 1-parameter family of solutions asymptotic to a self-similar Kantowski-Sachs model at large z. These are probably only physically realistic for $-1<\alpha <-1\mathrm{/}3$, in which case they may relate to the formation of bubbles in an inflationary universe. There is a 2-parameter family of solutions associated with a self-similar static solution at large z. This family contains solutions with naked singularities and this includes the critical'' solution discovered in recent collapse calculations for $\alpha < 0.28$. Finally, for $\alpha >1/5$, there is a family of solutions which are asymptotically Minkowski. These asymptote either to infinite z, in which case they are described by one parameter, or to a finite value of z, in which case they are described by two parameters and this includes the critical’’ solution for $\alpha >0.28$. We discuss the stability of spherically symmetric similarity solutions to more general (non-self-similar) spherically symmetric perturbations.