A new approach to initial value variables for the Lemaître–Tolman–Bondi dust solutions

Abstract

We re-examine the Lemaître–Tolman–Bondi (LTB) dust solutions by considering as free parameters the initial value functions: Yi, ρi, (3)ℛi, obtained by restricting the curvature radius, Y ≡ , the rest mass density, ρ, and the three-dimensional Ricci scalar of the rest frames, (3)ℛ, to an arbitrary regular Cauchy hypersurface, \U0001d4afi, marked by constant cosmic time (t = ti). Assuming the existence of symmetry centres, we use Yi to fix the radial coordinate and the topology (homeomorphic class) of \U0001d4afi, while the time evolution is described in terms of an adimensional scale factor y = Y/Yi. We show that the dynamics, regularity conditions and geometric features of the models are determined by ρi, (3)ℛi and by suitably constructed volume averages and contrast functions expressible in terms of invariant scalars defined in \U0001d4afi. These quantities lead to a straightforward characterization of initial conditions in terms of the nature of the inhomogeneity of \U0001d4afi, as density and/or curvature overdensities (‘lumps’) and underdensities (‘voids’) around a symmetry centre. In general, only models with initial density and curvature lumps evolve without shell-crossing singularities, though special classes of initial conditions, associated with a simultaneous big bang, allow for a regular evolution for initial density and curvature voids. Specific restrictions are found so that a regular evolution for t ≥ ti is possible for initial voids. A brief guideline is provided for using the new variables in the construction of LTB models and for plotting all relevant quantities.