The Tolman-Bondi Model in the Ruban-Chernin Coordinates. 1. Equations and Solutions

TL;DR

This paper analyzes the Tolman-Bondi model using Ruban-Chernin coordinates, focusing on transformation rules, classification of solutions, and explicit dependence of solutions on initial conditions and coordinates.

Contribution

It introduces a classification of Tolman-Bondi solutions based on invariant mass transformations and clarifies the role of Ruban-Chernin coordinates in these solutions.

Findings

Ruban-Chernin coordinates correspond to an identical transformation.

Solutions can be classified into nonintersecting classes based on functions $f$ and $F$.

Explicit dependence of flat solutions on the coordinate $M$ is established.

Abstract

The Tolman-Bondi (TB) model is defined up to some transformation of a co-moving coordinate but the transformation is not fixed. The use of an arbitrary co-moving system of coordinates leads to the solution dependent on three functions $f, F, {\bf F}$ which are chosen independently in applications. The article studies the transformation rule which is given by the definition of an invariant mass. It is shown that the addition of the TB model by the definition of the transformation rule leads to the separation of the couples of functions ($f, F$) into nonintersecting classes. It is shown that every class is characterized only by the dependence of $F$ on $f$ and connected with unique system of co-moving coordinates. It is shown that the Ruban-Chernin system of coordinates corresponds to identical transformation. The dependence of Bonnor's solution on the Ruban-Chernin coordinate $M$ by means of initial density and energy distribution is studied. It is shown that the simplest flat solution is reduced to an explicit dependence on the coordinate $M$. Several examples of initial conditions and transformation rules are studied.