Energy Conditions and Junction Conditions

TL;DR

This paper investigates the validity of Israel's junction conditions for thin walls in Einstein-Hilbert gravity, proposing that the induced metric remains continuous for all positive energy matter configurations, supported by proofs in specific cases.

Contribution

It conjectures and provides evidence that the induced metric is continuous across thin walls in positive energy spacetimes, extending the applicability of Israel's junction conditions.

Findings

Proven for static spherically symmetric spacetimes.

Validated for planar symmetric cases.

Supported in non-symmetric settings with well-behaved null geodesics.

Abstract

We consider the familiar junction conditions described by Israel for thin timelike walls in Einstein-Hilbert gravity. One such condition requires the induced metric to be continuous across the wall. Now, there are many spacetimes with sources confined to a thin wall for which this condition is violated and the Israel formalism does not apply. However, we explore the conjecture that the induced metric is in fact continuous for any thin wall which models spacetimes containing only positive energy matter. Thus, the usual junction conditions would hold for all positive energy spacetimes. This conjecture is proven in various special cases, including the case of static spacetimes with spherical or planar symmetry as well as settings without symmetry which may be sufficiently well approximated by smooth spacetimes with well-behaved null geodesic congruences.