Photon under repeated transverse Lorentz boosts: An apparent paradox

Abstract

We investigate the effects of the repeated application of Lorentz boosts to the four momentums of a photon in the transverse direction and observe that this can take us to a reference frame in which the direction of the photon's momentum is apparently reversed. This raises an apparent paradox: how could two observers see the same photon traveling in opposite directions? We further extend this process to an infinite succession of infinitesimal transverse Lorentz boosts and show it amounts to a rotation of the photon's momentum, but as part of a Lorentz transformation and not as a simple rotation. These possibly surprising results can be understood in light of the Wigner rotation: the combination of Lorentz boosts along different directions amounts to a Lorentz boost combined with a rotation. The presented exercises offer illustrations of the counterintuitive behavior of Lorentz boost combinations.