The Tolman-Bondi--Vaidya Spacetime: matching timelike dust to null dust

TL;DR

This paper demonstrates how to match the Tolman-Bondi dust solution with the Vaidya null fluid solution in a single spacetime, ensuring smoothness and continuity across a null surface.

Contribution

It introduces the Tolman-Bondi--Vaidya spacetime, a new solution that combines dust and null fluid regions with a smooth null surface.

Findings

The combined spacetime can be at least $C^1$ across the null surface.

The stress-energy tensor remains continuous across the matching surface.

A specific solution illustrating the matching is constructed.

Abstract

The Tolman-Bondi and Vaidya solutions are two solutions to Einstein equations which describe dust particles and null fluid, respectively. We show that it is possible to match the two solutions in one single spacetime, the Tolman-Bondi--Vaidya spacetime. The new spacetime is divided by a null surface with Tolman-Bondi dust on one side and Vaidya fluid on the other side. The differentiability of the spacetime is discussed. By constructing a specific solution, we show that the metric across the null surface can be at least $C^1$ and the stress-energy tensor is continuous.