Two black hole initial data

In [1] Misner proposed two-body wormhole initial data and in [2] he used the method of images to describe the time-symmetric initial data for an arbitrary number N of particles (see also [3] for a review of time-symmetric initial data). Such data may be viewed as a collection of Einstein-Rosen bridges connecting two isometric sheets. If we restrict ourselves to just one of them, we can view such initial data as a three-dimensional metric on a manifold with internal boundary. The boundary consists of N minimal surfaces representing surfaces of the “black holes” corresponding to the particles. In this paper I am concerned with the N = 2 case only. If the masses of black holes are equal (i.e. when the horizons are isometric) then there is a well-known formula for the three-metric: