On a weak solution of Einstein equations for expanding dust

TL;DR

This paper proposes a method to construct weak solutions for expanding dust in general relativity with shell-crossing singularities by cutting and matching spacetime regions, extending the Lemaitre-Tolman-Bondi solution.

Contribution

It introduces a novel approach to handle shell-crossing singularities in dust solutions by using thin shell junctions within the Einstein equations.

Findings

Method effectively models shell-crossing scenarios.

Solution matches known LTB solutions before singularity.

Provides a toy model for underdense universe regions.

Abstract

An expanding spherically symmetric dust cloud is considered in a framework of general relativity. Initial conditions leading to a shell-crossing singularity are chosen. The way to construct a weak solution for such a case is proposed. Suggested method consists in cutting off the region containing the shell-crossing and matching the remaining parts of space-time at a thin shell. Junction conditions determine the motion of that thin shell. The singular part of dust stress-energy tensor is nontrivial only after the shell-crossing occurs. Before that the solution coincides with Lemaitre - Tolman - Bondi one. A toy model representing an underdensed region in Universe is discussed.