A synthetic Gannon-Lee incompleteness theorem

TL;DR

This paper extends the Gannon-Lee incompleteness theorem to weighted and low-regularity spacetimes using synthetic energy conditions, broadening its applicability in mathematical relativity.

Contribution

It generalizes the classical Gannon-Lee theorem to weighted spacetimes under synthetic null energy and asymptotic regularity conditions.

Findings

Proves the theorem for globally hyperbolic spacetimes.

Introduces synthetic null energy and asymptotic regularity conditions.

Suggests potential extensions to low regularity scenarios.

Abstract

We prove the Gannon-Lee incompleteness theorem for globally hyperbolic spacetimes. We assume the synthetic null energy condition of Ketterer and a trappedness condition we call "synthetically asymptotically regular". Our result generalizes this classical result to the weighted case. It also motivates and indicates extensions to low regularity, which are deferred to future work.