Lorentz transformations in a Real Geometric Algebra formalism

Abstract

This paper presented a concise description of the Lorentz transformation without the use of matrices. The Real Geometric Algebra formalism used neatly unifies the Gibbs, Quaternion, Complex, Pauli, Dirac, and Hestenes algebras. The hyperbolic geometry of Minkowski space is easily accounted for by the coupling of the spatial vectors to the temporal vector in the spacetime point. The Lorentz boost is compactly represented by an exponential of a vector, while that of rotation is through a vector times the trivector i which behaves like an imaginary number. The Lorentz transformation equations for boost may be extracted from the Lorentz transformation equation by using the quaternionic expansion of the dyad product of spatial vectors, and separating the scalar, vector, and bivector parts. The Einstein addition law for parallel speeds may be derived by taking two parallel boosts.