Generalized relativistic velocity addition with spacetime algebra

Using spacetime algebra, the geometric algebra of spacetime, the general problem of relativistic addition of velocities is addressed. The successive application of non-collinear Lorentz boosts is then studied in Minkowski spacetime. Even spatial vectors, such as the relative velocity of two reference frames, are analyzed in their proper setting: as vectors belonging to a four-dimensional spacetime manifold and not as vectors in ordinary three-dimensional (Euclidean) space. This is a clear and physical result that illustrates the fact that Lorentz boosts do not form a group. The entire derivation is carried through without using the second postulate on the speed of light, thereby stressing that the framework of special relativity does not depend on electromagnetism.