A dynamical system approach to inhomogeneous dust solutions

Abstract

We examine numerically and qualitatively the Lema\^ıtre--Tolman--Bondi (LTB) inhomogeneous dust solutions as a 3--dimensional dynamical system characterized by six critical points. One of the coordinates of the phase space is an average density parameter, $<\Omega >$, which behaves as the ordinary $\Omega$ in Friedman-Lema\^ıtre--Robertson--Walker (FLRW) dust spacetimes. The other two coordinates, a shear parameter and a density contrast function, convey the effects of inhomogeneity. As long as shell crossing singularities are absent, this phase space is bounded or it can be trivially compactified. This space contains several invariant subspaces which define relevant particular cases, such as: ``parabolic'' evolution, FLRW dust and the Schwarzschild--Kruskal vacuum limit. We examine in detail the phase space evolution of several dust configurations: a low density void formation scenario, high density re--collapsing universes with open, closed and wormhole topologies, a structure formation scenario with a black hole surrounded by an expanding background, and the Schwarzschild--Kruskal vacuum case. Solution curves start expanding from a past attractor (source) in the plane $<\Omega >=1$, associated with self similar regime at an initial singularity. Depending on the initial conditions and specific configurations, the curves approach several saddle points as they evolve between this past attractor and other two possible future attractors: perpetually expanding curves terminate at a line of sinks at $<\Omega >=0$, while collapsing curves reach maximal expansion as $<\Omega >$ diverges and end up in sink that coincides with the past attractor and is also associated with self similar behavior.

Enhanced version accepted for publication in Classical and Quantum Gravity. Compiled with PdfLatex. RevTex style, 18 pdf figures

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