A complete classification of spherically symmetric perfect fluid similarity solutions

TL;DR

This paper provides a comprehensive classification of all spherically symmetric perfect fluid solutions to Einstein's equations that are self-similar, detailing their asymptotic behaviors, exact solutions, and parameter dependencies for different equations of state.

Contribution

It offers the first complete classification of self-similar spherically symmetric perfect fluid solutions, including exact power-law solutions and asymptotic behaviors based on the equation of state parameter a.

Findings

Identified three exact power-law solutions: Friedmann, static, and Kantowski-Sachs.

Described two families of asymptotically Minkowski solutions for a>1/5.

Provided explicit analysis of dust solutions and extended discussion to pressure solutions.

Abstract

We classify all spherically symmetric perfect fluid solutions of Einstein's equations with equation of state p/mu=a which are self-similar in the sense that all dimensionless variables depend only upon z=r/t. For a given value of a, such solutions are described by two parameters and they can be classified in terms of their behaviour 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 that all similarity solutions must be asymptotic to solutions which depend on either powers of z or powers of lnz. We show that there are only three similarity solutions which have an exact power-law dependence on z: the flat Friedmann solution, a static solution and a Kantowski-Sachs solution (although the latter is probably only physical for a<0). For a>1/5, there are also two families of solutions which are asymptotically (but not exactly) Minkowski: the first is asymptotically Minkowski as z tends to infinity and is described by one parameter; the second is asymptotically Minkowski at a finite value of z and is described by two parameters. A complete analysis of the dust solutions is given, since these can be written down explicitly and elucidate the link between the z>0 and z<0 solutions. Solutions with pressure are then discussed in detail; these share many of the characteristics of the dust solutions but they also exhibit new features.