Equivalence of Darmois-Israel and Distributional-Methods for Thin Shells in General Relativity

TL;DR

This paper demonstrates that a distributional approach to Einstein's equations for thin shells aligns with the traditional Darmois-Israel formalism, providing a rigorous foundation for their equivalence in general relativity.

Contribution

It establishes the formal equivalence between the distributional method and Darmois-Israel formalism for thin shells in general relativity, clarifying their relationship.

Findings

Distributional method reproduces Darmois-Israel jump conditions

Bianchi identities hold as distributions in thin shell scenarios

Formal proof of equivalence between methods

Abstract

A distributional method to solve the Einstein's field equations for thin shells is formulated. The familiar field equations and jump conditions of Darmois-Israel formalism are derived. A carefull analysis of the Bianchi identities shows that, for cases under consideration, they make sense as distributions and lead to jump conditions of Darmois-Israel formalism.