Successive Lorentz boosts: Equivalent boost and rotation using geometric algebra

Abstract

In this paper, we derive known expressions for the equivalent Lorentz boost and rotation of two successive Lorentz boosts. We use a Clifford algebra G₄ formalism of special relativity. The salient features of this formulation are: it uses only dot and cross products; and the Minkowski metric has a symmetric form. Our results include an exact expression for the Thomas rotation angle. The derivation is computationally efficient, pictorially-guided, and straightforward, making it suitable for a demonstration in an undergraduate course in special relativity.