Apparent horizon formation and hoop conjecture in nonaxisymmetric spaces

We investigate the validity of Thorne’s hoop conjecture in nonaxisymmetric 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 nonaxisymmetric spaces. We implement, for the first time, the “hoop finder” in nonaxisymmetric 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 $\mathcal{C}$ is reduced to solving an ordinary differential equation. We find that even in nonaxisymmetric spaces the existence or nonexistence of an apparent horizon is consistent with the inequality $\mathcal{C}\lesssim 4\pi M.\mathrm{}$