Singularities and asymptotic behavior of the Tolman-Bondi model

TL;DR

This paper investigates the properties and behaviors of the Tolman-Bondi model, focusing on singularities, asymptotic limits, and the interpretation of the Bondi mass formula as a coordinate transformation.

Contribution

It provides explicit procedures for determining the arbitrary functions in the TB model and analyzes the model's properties under transformation rules.

Findings

Derived equations of motion for singularity hypersurfaces in falling TB models

Analyzed the asymptotic behavior of expanding TB models with Friedmann-like solutions

Interpreted the Bondi mass formula as a transformation rule on co-moving coordinates

Abstract

The Bondi formula for calculation of the invariant mass in the Tolman- Bondi (TB) model is interprated as a transformation rule on the set of co-moving coordinates. The general procedure by which the three arbitrary functions of the TB model are determined explicitly is presented. The properties of the TB model, produced by the transformation rule are studied. Two applications are studied: for the falling TB flat model the equation of motion of two singularities hypersurfaces are obtained; for the expanding TB flat model the dependence of size of area with friedmann-like solution on initial conditions is studied in the limit $t \to +\infty$.