Abstract
Two Cartesian forms and a vectorial one for the Einstein law of composition of three non-collinear velocities are obtained for the first time, and their dependence on the Thomas rotation angle is studied. In particular, one of those Cartesian forms has the interesting property to be explicitly independent of the Thomas rotation angle and similar to the known Cartesian form for the Einstein law of composition of two non-collinear velocities. In this case, the Thomas rotation angle is simply present as the angle between the corresponding axes of the coordinate systems which are related by successive Lorentz transformations. Those facts make the treatment of kinematic problems with respect to two coordinate systems related by successive Lorentz transformations as simple as that of kinematic problems with respect to two coordinate systems related by a pure Lorentz transformation. As an illustrative application, the light aberration between two coordinate systems related by successive Lorentz transformations is considered. It is found that the direction of a light source in the moving frame depends on the Thomas rotation angle. Original vectorial diagrams are given for the first time.
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