Successive Lorentz transformations for energy and momentum. Application: relativistic elastic scattering of two particles having non-collinear velocities and its dependence on Thomas rotation

Abstract Successive Lorentz transformations (SLT) for energy and momentum are derived, and the involved Thomas rotation angle (TRA) is highlighted. In particular, if the axes of the coordinate systems related by SLT are suitably chosen, the SLT for energy and momentum do not depend on the TRA explicitly and take a form analogous to that of the Lorentz transformation for energy and momentum. Then the study of dynamical problems involving SLT becomes as easy as that of dynamical problems involving a Lorentz transformation. Those results are applied to the study of a relativistic elastic scattering of two particles having non-collinear velocities as a dynamical problem involving SLT. The expressions for the energies and momentums of the scattered particles in the laboratory system are obtained in terms of the TRA. If the considered particles are protons with given non-collinear velocities, the TRA, the energies and directions of the scattered protons are determined. Original diagrams for momentum are obtained.