Time-varying media, relativity, and the arrow of time

Research on time-varying media has recently enjoyed renewed interest, especially in photonics.Despite the large amount of research done in this field in the past few years, the attention has been focused on electromagnetic waves solely, while a comprehensive framework describing how wave phenomena in general are influenced by time-varying media has not been fully developed yet. To this aim, we study the implications of time-varying wave mechanics, and show how the standard wave equation is modified if the speed of a wave is not constant in time. In particular, waves that experience longitudinal acceleration are shownto have clear relativistic properties when a constant reference speed exists.Moreover, the accelerating wave equation admits only solutions propagating forward in time, which are continuous across material interfaces. We then consider the special case of electromagnetic waves, finding that the Abraham-Minkowski controversy is caused by relativistic effects, and the momentum of light is in fact conserved between different media. Furthermore,we showthat the accelerating waves conserve energy when the wave ismoving along a geodesic and demonstrate two example solutions.We conclude with some remarks on the role of the accelerating wave equation in the context of the arrowof time.