Apparent Horizon Formation and Hoop Conjecture in Non-axisymmetric Spaces

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 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: C ∼ 4πM . PACS numbers: 04.20.Dw; 04.25.Dm; 04.70.Bw; 97.60.Lf Typeset using REVTEX ∗Current address: Department of Physics, University of Tokyo, Tokyo 113-0033, Japan