Apparent Horizon Formation and Hoop Conjecture in Non-axisymmetric Spaces

TL;DR

This paper numerically investigates the hoop conjecture in non-axisymmetric spacetimes by developing a new method to identify apparent horizons, demonstrating that the conjecture holds in these complex geometries.

Contribution

It introduces the first implementation of the hoop finder in non-axisymmetric spaces with discrete symmetry, expanding horizon detection methods beyond symmetric cases.

Findings

Apparent horizons are consistent with the inequality ${ m C} extless 4 m \pi M$ in non-axisymmetric spaces.

The method simplifies horizon calculations to solving an ordinary differential equation.

The study confirms the hoop conjecture's validity in non-axisymmetric configurations.

Abstract

We investigate the validity of Thorne's hoop conjecture in non-axisymmetric spacetimes by examining the formation of apparent horizons numerically. If spaces have a discrete symmetry about one axis, we can specify the boundary conditions to determine an apparent horizon even in non-axisymmetric spaces. We implement, for the first time, the ``hoop finder'' in non-axisymmetric spaces with a discrete symmetry. We construct asymptotically flat vacuum solutions at a moment of time symmetry. Two cases are examined: black holes distributed on a ring, and black holes on a spherical surface. It turns out that calculating ${\cal C}$ is reduced to solving an ordinary differential equation. We find that even in non-axisymmetric spaces the existence or nonexistence of an apparent horizon is consistent with the inequality: ${\cal C} \siml 4\pi M$.