The Penrose singularity theorem, MOTS stability, and horizon topology in weighted spacetimes

TL;DR

This paper extends classical theorems on singularities and horizon topology to weighted spacetimes with synthetic dimensions, under a weighted null energy condition, broadening understanding of geometric and physical properties in these generalized settings.

Contribution

It introduces weighted versions of the Penrose and Hawking theorems applicable to arbitrary dimensions, utilizing a novel weighted scalar curvature concept.

Findings

Weighted theorems hold under a weighted null energy condition

Weighted scalar curvature differs from the trace of weighted Ricci curvature

Results support the interpretation of weighted curvatures via warped product metrics

Abstract

We consider versions of the Penrose singularity theorem and the Hawking horizon topology theorem in weighted spacetimes that contain weighted versions of trapped surfaces, for arbitrary spacetime dimension and synthetic dimension. We find that suitable generalizations of the unweighted theorems hold under a weighted null energy condition. Our results also provide further evidence in favour of a weighted scalar curvature that differs from the trace of the weighted Ricci curvature. When the synthetic dimension is a positive integer, these weighted curvatures have a natural interpretation in terms of warped product metrics.