Global symmetries: matter from geometry, hoop conjecture, and cosmic censorship

TL;DR

This paper proves that certain four-dimensional spacetimes with translational symmetry cannot contain apparent horizons, supporting the hoop conjecture and raising questions about cosmic censorship.

Contribution

It demonstrates a novel link between global symmetries, dimensional reduction, and the absence of apparent horizons in vacuum and matter-filled spacetimes.

Findings

Spacetimes with a global spacelike translational symmetry lack apparent horizons.

Dimensional reduction to (2+1) spacetime simplifies horizon analysis.

Supports the hoop conjecture and suggests potential violations of cosmic censorship.

Abstract

We show that four-dimensional Lorentzian metrics admitting a global spacelike Lie group of isometries, $G_{1}={\mathbb R}$, which obey the Einstein equations for vacuum and certain types of matter, cannot contain apparent horizons. The assumed global isometry allows for the dimensional reduction of the (3+1) system to a (2+1) picture, wherein the four-dimensional metric fields act formally as matter fields. A theorem by Ida allows one to check for the absence of apparent horizons in the dimensionally reduced spacetime, with the four-dimensional results following from the topological product nature of the corresponding manifold. We argue that the absence of apparent horizons in spacetimes with translational symmetry constitutes strong evidence for the validity of the hoop conjecture, and also hints at possible (albeit arguably unlikely) generic violations of strong cosmic censorship.