Junction equations for two spherically symmetric spacetimes and the distributional method

TL;DR

This paper uses the distributional formalism to derive junction equations for matching two spherically symmetric spacetimes across a thin shell, clarifying sign conventions in the Darmois-Israel method.

Contribution

It introduces a systematic way to incorporate sign functions into junction equations using the distributional approach, enhancing the understanding of thin shell dynamics in general relativity.

Findings

Derived junction equations using distributional formalism.

Clarified the role of sign functions in the Darmois-Israel method.

Provided a consistent framework for thin shell analysis.

Abstract

Applying the distributional formalism to study the dynamics of thin shells in general relativity, we regain the junction equations for matching of two spherically symmetric spacetimes separated by a singular hypersurface. In particular, we have shown how to define and insert the relevant sign functions in the junction equations corresponding to the signs of the extrinsic curvature tensor occurred in the Darmois-Israel method.