Half period theorem of binary black holes

Merging event horizons of the binary black holes will be investigated. Though recent development of the numerical study of the binary black hole coalescence has shown that their apparent horizons can orbit for many periods in the case of appropriate initial conditions, the apparent horizons only suggests the existence of an event horizon surrounding them. In the present article, we discuss how many periods their event horizons orbit before their coalescence. One finds that they soon merge into one and the black holes cannot orbit for a half period, though the apparent horizons can. We will realize that, after the merging, the event horizon rotates including the apparent horizons inside.