Análise da rotação de Wigner pela propriedade não comutativa da composição relativística de velocidades não paralelas

Abstract

Lorentz transformations should be used to relate spacetime coordinates of two frames when the relative velocity between them is comparable to the speed of light. In this field, the laws of Classical Mechanics do not foresee satisfactory results in accordance with the experiments. When two successive boosts of Lorentz are applied to obtain the velocity of an object in a reference of interest, we can substitute them by a single boost followed by a rotation in the coordinate space. The angle of rotation is called Wigner Angle. In this work, we evaluate the relativistic combination of velocities using Lorentz boosts. Note that when two boosts are applied in different orders, the velocities obtained do not commute, because even though they have the same modules, they point in different directions. We will show that the angle between the velocity directions is just the Wigner Angle. Finally we calculate the Wigner angle by two different methods, one based on the hyperbolic space of Lobachevsky and the other by a generalization of the Law of Addition of Einstein Speeds in the Complex Plane.