Ordered addition of two Lorentz boosts through spatial and space-time rotations

Abstract

The ordered addition of two Lorentz boosts is normally shown to result in a boost by utilizing concepts from group theory and non-Euclidian geometry. We present a method for achieving this addition by performing a sequence of spatial rotations and uni-dimensional Lorentz transformations. The method is first developed for two-dimensional space and it is then extended to three-dimensional space by utilizing the commutative property of the rotation of the y-z plane and a boost along the x-axis. The method employs only matrix multiplication and certain invariant quantities that are natural consequences of spatial rotations and Lorentz transformations. The combining of two boosts in different directions into a single boost cannot be expected a priori because we show that the converse of this statement is not true. That is, two rotations interspersed with a boost cannot always be reduced to a single rotation preceded and followed by boosts.