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Volume 14: Pages 280-285, 2001

A General Relativstic Law of Addition for Two Velocities and Successive Lorentz Transformations along Perpendicular Axes

Jair Lucinda

Departamento de Física, Universidade Federal do Paraná, C.P. 19044, 81531‐990, Curitiba, Paraná, Brazil

The aim is to derive the composition of two boosts along perpendicular Cartesian axes as a single‐boost form. Contrary to the composition of two successive boosts along the same direction exemplified in the current literature, the composition of two successive Lorentz boosts in more than one space dimension cannot be developed with a relative velocity given by the law of addition for two velocities along the same direction. In order to attain a composition for two boosts along perpendicular axes it is necessary to derive a general transformation of velocities for a relative motion along an arbitrary direction. The result is a single equivalent boost with a nonvanishing determinant, which means that two boosts along perpendicular axes can be undone by a third. Though it is written in a single‐boost form, it has matrix elements asymmetrically distributed with respect to its principal diagonal, which is a reminder of the noncommutative property of two successive Lorentz boosts along perpendicular axes.

Keywords: addition of velocities, composition of boosts, inverse transformation, relativity

Received: August 25, 2000; Published online: December 15, 2008