Fermat's Principle and Lorentz Transformations for a Moving Mirror

 In (1) it is stated that Fermat’s principle of least time applied to light is equivalent to a statement that light travels along a geodesic i.e. is a fundamental principle. In fact, this principle seems to be used frequently (i.e. light travels in straight lines ignoring curved space) without being explicitly called Fermat’s principle. In (2) it is shown that one may solve for the reflected angle in terms of the incident angle for a light ray reflecting from a mirror moving at constant velocity v. It is stated that this allows one to find such a relationship without using Lorentz transformations. 

    In this note we argue that Lorentz transformations follow from using Fermat’s principle in two frames. The first frame is that of the lab and is used in (2) to directly find the reflected angle in terms of the incident angle. The second frame is the moving frame such that one has a stationary mirror. Again Fermat’s principle applies. One may deduce the Lorentz transformation by assuming that c is the same in all frames (Einstein’s assumption) and that x in the direction perpendicular to motion is unchanged in different frames. This has already been done by Einstein in 1905 (3). 

  In this note we argue that conservation of momentum perpendicular to frame motion suggests that not only x, but perpendicular momentum is unchanged by frame motion. Consider a frame moving in the x direction. In this frame light is sent in the y direction and hits a wall and bounces back and forth. This light is then energy trapped at a fixed x point in the moving frame, in other words localized or rest energy. From the point of view of the lab frame, the light moves in a triangle, but the momentum also follows this triangle. If both momentum and length perpendicular to the motion of the frame remain unchanged then (x,t) and (p,E) transform in the same manner for light because E=pc. Furthermore the Lorentz transformation obtained seems to hold to particles with rest mass because the light in the moving frame bouncing in the y direction is rest energy i.e. rest mass cc. In fact, the result for a moving mirror (i.e. reflected angle in terms of incident) given in (3) can be obtained from a transformation of p,E without considering x or t. Thus Fermat’s principle together with the assumption that light has the same speed in all frames (an assumption used in (2)) and the idea that momentum and length components perpendicular to the direction of motion do not change in different frames, allow one to obtain the Lorentz transformation which applies to both (x,t) and (p,E) not only for light, but also for particles with rest mass. The final result is consistent with Fermat’s principle applied directly to the lab frame because following (1) light follows a geodesic which means least time.