Complete classification of spherically symmetric self-similar perfect fluid solutions - art. no. 044023

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 t he sense that all dimensionless variables depend only upon z=r/t. This exte nds 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 pf ttl but (due to a coordinate anomaly) it may also correspond to finite z. We base o ur analysis on the demonstration (given elsewhere) that all self-similar so lutions must be asymptotic to solutions which depend on either powers of z at large and small /z/ or powers of In\z\ at finite z. We show that there a re only three self-similar solutions which have an exact power-law dependen ce on z: the hat Friedmann solution, a static solution and a Kantowski-Sach s solution (although this is probably only physical for alpha<-1/3). At lar ge values of \z\ we show that there is a 1-parameter family of asymptotical ly Friedmann solutions, a 1-parameter family of asymptotically Kantowski-Sa chs solutions and 2-parameter family which we describe as asymptotically "q uasi-static." For alpha>1/5, there are also two families of asymptotically Minkowski solutions at large distant es from the origin, although these do not contain the Minkowski solution itself: the first is asymptotical to the Minkowski solution as \z\-->infinity and is described by one parameter; th e second is asymptotical to the Minkowski solution at a finite value of z a nd is described by two parameters. The possible behaviors at small distance s from the origin depend upon whether or not the solutions pass through a s onic point. If the solutions remain supersonic everywhere, the origin corre sponds 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. As z-->0, there is again a 1-parame ter family of asymptotic Friedmann solutions: this includes a continuum of underdense solutions and discrete bands of overdense ones; the latter are a ll nearly static close to the sonic point and exhibit oscillations. Then 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 thes e solutions.