Topological Appearance of Event Horizon: What Is the Topology of the Event Horizon That We Can See?

Abstract

The topology of the event horizon (TOEH) is usually believed to be a sphere. Nevertheless, some numerical simulations of gravitational collapse with a toroidal event horizon or the collision of event horizons are reported. Considering the indifferentiability of the event horizon (EH), we see that such non-trivial TOEHs are caused by the set of endpoints (the crease set) of the EH. The two-dimensional (one-dimensional) crease set is related to the toroidal EH (the coalescence of the EH). Furthermore, examining the stability of the structure of the endpoints, it becomes clear that the spherical TOEH is unstable under linear perturbation. On the other hand, a discussion based on catastrophe theory reveals that the TOEH with handles is stable and generic. Also, the relation between the TOEH and the hoop conjecture is discussed. It is shown that the Kastor-Traschen solution is regarded as a good example of the hoop conjecture by the discussion of its TOEH. We further conjecture that a non-trivial TOEH can be smoothed out by rough observation in its mass scale.