On The Vaidya Limit of the Tolman Model

TL;DR

This paper investigates the conditions under which Tolman models can approach the Vaidya metric limit, revealing that only hollow dust distributions, like the self-similar case, permit such a limit, and clarifying assumptions in previous demonstrations.

Contribution

It identifies the specific Tolman models that can approach the Vaidya limit and corrects a hidden assumption in prior work on this limit.

Findings

Only hollow dust Tolman models have a Vaidya limit.

Dense Tolman models with shell-focusing singularities lack a Vaidya equivalent.

A different coordinate transformation is needed to properly demonstrate the Vaidya limit.

Abstract

We show that the only Tolman models which permit a Vaidya limit are those having a dust distribution that is hollow - such as the self-similar case. Thus the naked shell-focussing singularities found in Tolman models that are dense through the origin have no Vaidya equivalent. This also casts light on the nature of the Vaidya metric. We point out a hidden assumption in Lemos' demonstration that the Vaidya metric is a null limit of the Tolman metric, and in generalising his result, we find that a different transformation of coordinates is required.