Lost in velocity space: Problems with the single-boost Lorentz transformation

摘要

A Lorentz velocity boost preserves spatial coordinate values perpendicular to the boost direction. Composing a Lorentz x-boost with a y-boost does not preserve any spatial coordinates. Hence, the composition of an x-boost with a y-boost cannot be represented as a single boost. In (2 + 1) space, a corrective spatial rotation renders the double boost into a single-boost form. This rotation can be expressed in terms oft he total relative velocity between initial and final reference frames. However, the new formula still does not restore the x-axis to the correct direction, and also does not admit an inverse transformation in the same set. These problems urgently call for a rethinking of the Lorentz transformations in more than one spatial dimension. [References: 4]