Passive Lorentz transformations with spacetime algebra

Abstract

In special relativity, spacetime algebra (STA) provides a powerful and insightful approach to an invariant formulation of physics. However, in this geometric algebra of spacetime, relativistic physics is usually considered a misnomer: STA provides us with an invariant (coordinate-free) formulation and hence it seems that (passive) Lorentz transformations are misplaced. However, in order to interpret experimental results, it is important to relate invariant physical quantities to some reference frame. In this paper, passive Lorentz transformations are derived using STA but without the second postulate on the speed of light used by Einstein. This derivation sheds light on the physical background of STA and, at the same time, clarifies Lorentz transformations in the proper setting of (Minkowski) four-dimensional spacetime.