The stationary axisymmetric two‐body problem in general relativity

We continue our study, begun in [7], into the stationary axisymmetric solutions of the Einstein vacuum equations containing two (or more) black holes. This problem reduces to an elliptic system of nonlinear partial differential equations with prescribed blow‐up as boundary conditions. We show the existence of solutions to the reduced system for any value of the five parameters of the problem. We show some regularity at the poles of the horizons. Using this result, we simplify the formula for the angle of deficiency of the conical singularity on the component of the axis of symmetry between the two horizons. This is to be interpreted as the gravitational force between rotating black holes. We also prove that the metrics constructed from solutions of the reduced system are indeed asymptotically flat.