An asymptotic analysis of spherically symmetric perfect fluid self-similar solutions

Abstract

The asymptotic properties of self-similar spherically symmetric perfect fluid solutions with equation of state p=alpha mu (-1<alpha<1) are described. We prove that for large and small values of the similarity variable, z=r/t, all such solutions must have an asymptotic power-law form. Some of them are associated with an exact power-law solution, in which case they are asymptotically Friedmann or asymptotically Kantowski-Sachs for 1>alpha >-1 or asymptotically static for 1>alpha >0. Others are associated with an approximate power-law solution, in which case they are asymptotically quasi-static for 1>alpha >0 or asymptotically Minkowski for 1>alpha >1/5. We also show that there are solutions whose asymptotic behaviour is associated with finite values of z and which depend upon powers of ln z. These correspond either to a second family of asymptotically Minkowski solutions for 1>alpha>1/5 or to solutions that are asymptotically Kasner for 1>alpha>-1/3. There are some other asymptotic power-law solutions associated with negative alpha, but the physical significance of these is unclear. The asymptotic form of the solutions is given in all cases, together with the number of associated parameters.