Spherical Universes with Anisotropic Pressure

TL;DR

This paper derives solutions to Einstein's equations for spherically symmetric universes with anisotropic pressure from angular momentum, generalizing known models and exploring implications for gravitational collapse.

Contribution

It introduces a new class of solutions with shell-dependent angular momentum, extending the Lemaitre-Tolman-Bondi model to include anisotropic pressure effects.

Findings

Provides explicit solutions involving elliptic integrals.

Shows the relationship to Einstein clusters.

Discusses implications for gravitational collapse.

Abstract

Einstein's equations are solved for spherically symmetric universes composed of dust with tangential pressure provided by angular momentum, L(R), which differs from shell to shell. The metric is given in terms of the shell label, R, and the proper time, tau, experienced by the dust particles. The general solution contains four arbitrary functions of R - M(R), L(R), E(R) and r(0,R). The solution is described by quadratures, which are in general elliptic integrals. It provides a generalization of the Lemaitre-Tolman-Bondi solution. We present a discussion of the types of solution, and some examples. The relationship to Einstein clusters and the significance for gravitational collapse is also discussed.