Spherical universes with anisotropic pressure

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, τ, experienced by the dust particles. The general solution contains four arbitrary functions of R—M(R), L(R), E(R) and τ0(R). The solution is described by two quadratures which are, in general, elliptic integrals. It provides a generalization of the Lemaǐtre–Tolman–Bondi solution. We present a discussion of the types of solution, and some examples. The relationship to the Einstein clusters and the significance for gravitational collapse are also discussed.