Paper detail

New variables for the Lema\^ıtre-Tolman-Bondi dust solutions

We re-examine the Lemâitre-Tolman-Bondi (LTB) solutions with a dust source admitting symmetry centers. We consider as free parameters of the solutions the initial value functions: $Y_i$, $ρ_i$ and $\Ri$, obtained by restricting the curvature radius, $Y\equiv \sqrt{g_{θθ}}$, the rest mass density, $ρ$, and the 3-dimensional Ricci scalar of the rest frames, $\R$, to an arbitrary regular Cauchy hypersurface, $\Ti$, marked by constant cosmic time ($t=t_i$). Using $Y_i$ to fix the radial coordinate and the topology (homeomorphic class) of $\Ti$, and scaling the time evolution in terms of an adimensional scale factor $y=Y/Y_i$, we show that the dynamics, regularity conditions and geometric features of the models are determined by $ρ_i$, $\Ri$ and by suitably constructed volume averages and contrast functions expressible in terms of invariant scalars defined in $\Ti$. These quantities lead to a straightforward characterization of initial conditions in terms of the nature of the inhomogeneity of $\Ti$, as density and/or curvature overdensities (``lumps'') and underdensities (''voids'') around a symmetry center. In general, only models with initial density and curvature lumps evolve without shell crossing singularities, though special classes of initial conditions, associated with a simmultaneous big bang, allow for a regular evolution for initial density and curvature voids. Specific restrictions are found so that a regular evolution for $t>t_i$ is possible for initial voids. A step-by-step guideline is provided for using the new variables in the construction of LTB models and for plotting all relevant quantities.