Towards a theory of thin self-gravitating crossing shells

TL;DR

This paper derives conservation laws and velocity relations for crossing thin dust shells in general relativity, extending existing formulas to new configurations.

Contribution

It introduces a new simple formula relating shell velocities before and after crossing for mixed isotropic and time-like shells.

Findings

Conservation law for crossing dust shells derived.

't Hooft-Dray formula confirmed for isotropic shells.

New velocity relation for mixed shell crossings established.

Abstract

In the case of crossing thin dust shells the momentum conservation law is found. For two crossing isotropic shells it coincides with the 't Hooft-Dray formula. The system of one isotropic and one time-like shell is considered. In this case we found a very simple formula which relate velocities of dust shell before and after crossing.