Initial Data for General Relativity Containing a Marginally Outer Trapped Torus

TL;DR

This paper numerically constructs and analyzes initial data for vacuum general relativity with a marginally outer trapped torus, exploring its geometric properties and relation to black hole horizons.

Contribution

It introduces a novel numerical scheme to generate and study marginally outer trapped tori in vacuum GR initial data, linking geometric features to black hole criteria.

Findings

Tori are unstable minimal surfaces with saddle point area properties.

An apparent horizon of spherical topology surrounds the torus.

The study relates the torus properties to the hoop conjecture and isoperimetric inequality.

Abstract

Asymptotically flat, time-symmetric, axially symmetric and conformally flat initial data for vacuum general relativity are studied numerically on $R^3$ with the interior of a standard torus cut out. By the choice of boundary condition the torus is marginally outer trapped, and thus a surface of minimal area. Apart from pure scaling the standard tori are parameterized by a radius $a\in [0,1]$, where $a=0$ corresponds to the limit where the boundary torus degenerates to a circle and $a=1$ to a torus that touches the axis of symmetry. Noting that these tori are the orbits of a $U(1)\times U(1)$ conformal isometry allows for a simple scheme to solve the constraint, involving numerical solution of only ordinary differential equations.The tori are unstable minimal surfaces (i.e. only saddle points of the area functional) and thus can not be apparent horizons, but are always surrounded by an apparent horizon of spherical topology, which is analyzed in the context of the hoop conjecture and isoperimetric inequality for black holes.