Noncommutative nature of the addition of noncollinear velocities in special relativity and the geometric phase method (commemorating the publication centennial of A Sommerfeld’s work)
Federal Research Center Institute of Applied Physics of the Russian Academy of Sciences, ul. Ulyanova 46, Nizhny Novgorod, 603000, Russian Federation
In 1909, Arnold Sommerfeld used geometric calculations to show that the relativistic addition of two noncollinear velocities on an imaginary-radius sphere is a noncommutative operation. Sommerfeld was the first to use the geometric phase method to calculate the angle between the resulting velocities depending on the order in which they are added. For this, he related the value of this angle to the excess of the spherical triangle formed by the two original velocities and their sum. In 1931, Sommerfeld applied his method to analyze the Thomas precession.
This article was written to commemorate the publication centennial of A Sommerfeld’s paper “Über die Zusammensetzung der Geschwindigkeiten in der Relativtheorie” (On the addition of velocities in relativity theory) in Physikalische Zeitschrift.