A simple electromagnetic model for the light clock of special relativity

Abstract

Thought experiments involving a light clock are common in introductory treatments of special relativity, because they provide a simple way of demonstrating the non-intuitive phenomenon of time dilation. The properties of the ray or pulse of light that is continuously reflected between the parallel mirrors of the clock are often stated vaguely and sometimes involve implicitly other relativistic effects, such as aberration. While this approach is adequate for an introduction, it should be supplemented by a more accurate analysis of the light clock once the formulae for the Lorentz transformation and the transformation of the electromagnetic field have been developed. A simple yet accurate electromagnetic model for the light clock is presented for this purpose. In this model, the ray of light in the qualitative treatment is replaced by a guided wave in a parallel-plate waveguide. Expressions for the electromagnetic field and energy density within the waveguide are determined in the inertial frame in which the clock is at rest and the laboratory frame in which the clock is moving with constant velocity. The analytical expressions and graphical results obtained clearly demonstrate the operation of the clock and time dilation, as well as other interesting relativistic effects.