Junction Conditions and local Spacetimes in General Relativity

TL;DR

This paper introduces a geometric framework for joining different spacetime regions in General Relativity, enabling the characterization of local spacetimes and addressing limitations of standard methods.

Contribution

It presents a novel approach to match Lorentzian manifolds via local metric deformations, extending the applicability of junction conditions in spacetime modeling.

Findings

Successfully matches spacetimes using local deformations

Characterizes local spacetimes beyond standard techniques

Addresses problems previously unsolvable with standard gluing methods

Abstract

In the present work, a theoretical framework focussing on local geometric deformations is introduced in order to cope with the problem of how to join spacetimes with different geometries and physical properties. Using this framework, it is shown that two Lorentzian manifolds can be matched in agreement with the well-known Darmois-Israel junction conditions by locally deforming the associated spacetime metrics in relation to each other. Based on the insight that metrics can be suitably matched in this way, it is shown that the underlying geometric approach allows the characterization of local spacetimes in General Relativity. In addition, it is shown that this approach allows the treatment of problems that cannot be treated by using standard gluing techniques.