A covariant four-dimensional expression for Lorentz transformations

Abstract

References on special relativity, despite denoting Lorentz transformations abstractly in covariant four-dimensional notation, express the explicit formulas for these transformations only in the three-vector form which treats space and time separately. Rewriting the explicit formulas covariantly yields an expression more natural for the four-dimensional context where space and time are unified. A slight generalization valid in any inertial frame transforms any proper velocity four-vector into any other. The direct decomposition of the product of two pure Lorentz transformations into the product of a pure spatial rotation and a pure Lorentz transformation illustrates how the covariant expression can simplify difficult problems.