New variables for the Lema\^{\i}tre-Tolman-Bondi dust solutions

TL;DR

This paper introduces new variables for Lemaître-Tolman-Bondi dust solutions, enabling a clearer initial condition characterization and regular evolution criteria, especially for inhomogeneous models with density and curvature lumps or voids.

Contribution

It proposes a novel set of initial value functions for LTB models, simplifying the analysis of their dynamics, regularity, and geometric features based on invariant scalars and initial inhomogeneity types.

Findings

Models with initial density and curvature lumps evolve without shell crossing.

Special initial conditions allow regular evolution for voids.

Guidelines for constructing and analyzing LTB models with new variables.

Abstract

We re-examine the Lem\^aitre-Tolman-Bondi (LTB) solutions with a dust source admitting symmetry centers. We consider as free parameters of the solutions the initial value functions: $Y_i$, $\rho_i$ and $\Ri$, obtained by restricting the curvature radius, $Y\equiv \sqrt{g_{\theta\theta}}$, the rest mass density, $\rho$, 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 $\rho_i$, $\Ri$ 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.