Self-similar spherical metrics with tangential pressure

Abstract

A family of spherically symmetric spacetimes is discussed, which have anisotropic pressure and possess a homothetic Killing vector. The spacetimes are composed of dust with a tangential pressure provided by angular momentum of the dust particles. The solution is given implicitly by an elliptic integral and depends on four arbitrary functions. These represent the initial configurations of angular momentum, mass, energy and position of the shells. The solution is derived by imposing self-similarity in the coordinates R, the shell label, and τ, the proper time experienced by the dust. Conditions for evolution without shell crossing and a description of singularity formation are given and types of solution discussed. General properties of the solutions are illustrated by reference to a particular case, which represents a universe that exists for an infinite time, but in which every shell expands and recollapses in a finite time.