Thin shell model revisited

TL;DR

This paper revisits the fundamental aspects of the thin shell model in general relativity, clarifying issues with coordinate definitions and conservation laws, and proposing a refined construction for shell collisions.

Contribution

It provides a rigorous analysis of the coordinate system and manifold structure in thin shell models, introducing conditions for consistent shell collisions and conservation laws.

Findings

The 'cut and paste' method does not ensure a well-defined manifold.

Continuity of the metric helps specify the manifold and tangent space.

Under certain conditions, the areal radius r can remain a good coordinate during shell collisions.

Abstract

We reconsider some fundamental problems of the thin shell model. First, we point out that the "cut and paste" construction does not guarantee a well-defined manifold because there is no overlap of coordinates across the shell. When one requires that the spacetime metric across the thin shell is continuous, it also provides a way to specify the tangent space and the manifold. Other authors have shown that this specification leads to the conservation laws when shells collide. On the other hand, the well-known areal radius $r$ seems to be a perfect coordinate covering all regions of a spherically symmetric spacetime. However, we show by simple but rigorous arguments that $r$ fails to be a coordinate covering a neighborhood of the thin shell if the metric across the shell is continuous. When two spherical shells collide and merge into one, we show that it is possible that $r$ remains to be a good coordinate and the conservation laws hold. To make this happen, different spacetime regions divided by the shells must be glued in a specific way such that some constraints are satisfied. We compare our new construction with the old one by solving constraints numerically.