Applications of Lobachevsky Geometry to the Relativistic Two-Body Problem

In this talk we consider the geometrical basis for the reduction of the relativistic 2-body problem, much like the non-relativistic one, to describing the motion of an effective particle in an external field. It is shown that this possibility is deeply related with the Lobachevsky geometry. The concept of relativistic reduced mass and effective relativistic particle is discussed using this geometry. Different recent examples for application of relativistic effective particle are described in short.