The Lobachevsky geometry method in the relativistic kinematics of particle collisions: special frame of reference

Abstract

The use of the geometry of the Lobachevsky momentum space in the relativistic kinematics of particle collisions is demonstrated by the example of the problem of a special reference system. That system complements the geometric image of the process of elastic scattering of two particles of unequal masses. The speed of a special reference system relative to the center of mass and the angle of scattering of particles in it are determined. The conditions for the existence of such a reference system are analyzed. It is shown that in the case of a process with equal masses, the point corresponding to such a system goes into the ideal region of the extended Lobachevsky space - beyond the cone, and the lines intersecting in it become diverging lines in the sense of Lobachevsky geometry. In this case, the angle between the divergent straight lines (geodesics) of the geometric image is purely imaginary and connected to the minimum length of the segment perpendicular to the diverging straight lines (geodesics).