Complete classification of spherically symmetric self-similar perfect fluid solutions

We classify all spherically symmetric perfect fluid solutions of Einstein’s equations with an equation of state $p=\alpha \mu$ which are self-similar in the sense that all dimensionless variables depend only upon $z\equiv r/t.\mathrm{}$ This extends a previous analysis of dust $(\alpha =0)$ solutions. Our classification is “complete” subject to the restrictions that $\alpha$ lies in the range $0$ to $1$ and that the solutions are everywhere physical and shock-free. For a given value of $\alpha ,$ such solutions are described by two parameters and they can be classified in terms of their behavior at large and small distances from the origin; this usually corresponds to large and small values of $|z|$ but (due to a coordinate anomaly) it may also correspond to finite z. We base our analysis on the demonstration (given elsewhere) that all self-similar solutions must be asymptotic to solutions which depend on either powers of z at large and small $|z|$ or powers of $\ln |z|$ at finite z. We show that there are only three self-similar solutions which have an exact power-law dependence on z: the flat Friedmann solution, a static solution and a Kantowski-Sachs solution (although this is probably only physical for $\alpha <-1/3).$ At large values of $|z|,$ we show that there is a 1-parameter family of asymptotically Friedmann solutions, a 1-parameter family of asymptotically Kantowski-Sachs solutions and a 2-parameter family which we describe as asymptotically “quasi-static.” For $\alpha >1/5,$ there are also two families of asymptotically Minkowski solutions at large distances from the origin, although these do not contain the Minkowski solution itself: the first is asymptotical to the Minkowski solution as $|z\overrightarrow{|}\infty$ and is described by one parameter; the second is asymptotical to the Minkowski solution at a finite value of z and is described by two parameters. The possible behaviors at small distances from the origin depend upon whether or not the solutions pass through a sonic point. If the solutions remain supersonic everywhere, the origin corresponds to either a black hole singularity or a naked singularity at finite z. However, if the solutions pass into the subsonic region, their form is restricted by the requirement that they be “regular” at the sonic point and any physical solutions must reach $z=0.\mathrm{}$ As $\overrightarrow{z}0,$ there is again a 1-parameter family of asymptotic Friedmann solutions: this includes a continuum of underdense solutions and discrete bands of overdense ones; the latter are all nearly static close to the sonic point and exhibit oscillations. There is also a 1-parameter family of asymptotically Kantowski-Sachs solutions but no asymptotically static solutions besides the exact static solution itself. The full family of solutions can be found by combining the possible large and small distance behaviors. We discuss the physical significance of these solutions.