Apparent horizons of an N-black-hole system in a space-time with a cosmological constant

We present the analytic solution of N Einstein-Rosen bridges (N black holes) in the space-time with a cosmological constant and analyze it for one- and two-bridge systems. We discuss the three kinds of apparent horizons: i.e., the black-hole, white-hole, and cosmological apparent horizons. In the case of two Einstein-Rosen bridges, when the total mass is larger than a critical value, the black-hole apparent horizon surrounding two Einstein-Rosen bridges is not formed even if the distance is very short. Furthermore, in this case, the cosmological apparent horizon enclosing both bridges does not appear. Hence, it seems that when the total mass is very large, the Einstein-Rosen bridges cannot collapse into one black hole unless the gravitational mass is released in some way.