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

TL;DR

This paper analyzes the asymptotic behavior of self-similar spherically symmetric perfect fluid solutions with a linear equation of state, classifying their limits and associated geometries for different parameter ranges.

Contribution

It provides a comprehensive classification of asymptotic forms of these solutions, including power-law, quasi-static, Minkowski, and Kasner types, with detailed parameter dependencies.

Findings

Solutions are asymptotically power-law at large and small z.

Different asymptotic geometries occur depending on alpha, including Friedmann, Kantowski-Sachs, static, Minkowski, and Kasner.

Some solutions depend on logarithmic powers of z, indicating complex asymptotic behaviors.

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.