Fabry-Perot Interferometer Moving Transversely This simple animation shows the trajectory of light beams that move with a 2 mirror (Fabry-Perot) interferometer as the interferometer moves transversely relative to a fixed frame observer. The separation beween the mirrors is L so in the moving frame the time between mirror intercepts is t0=L/c. Since the speed of light is always c and the trajectory that he sees is slanted because of the motion it will take longer to complete a round trip between the mirrors than if the mirror were not moving. The velocity component along the y direction will be vy=sqrt(c*c-vx*vx) so that the mirror intercept time increase factor is t/t0=c/sqrt(c*c-vx*vx) = 1/sqrt(1-vx*vx/(c*c)) which is usually called gamma in relativity convention. The term vx/c is usually called beta. In the animation, the trace with the black dot at its leading end is the one that would be seen by an observer moving at the same velocity as the interferometer. The trace with the red dot at the leading end is what an observer watching from outside the moving interferometer would see. Since we base our time scale on the times between mirror intercepts, the outside observer time t(vx) is considerably greater than the time t(0) as expected by the longer pathlengths. For a parallel mirror interferometer like this, it is usually the time for light to go between mirror intercepts that is important. It is clear that the black dot gets ahead of the red dot so these times appear different in this animation. However we also see that the DISTANCE between mirror intercepts that is the same from both frames of reference. A question arises whether the outside observer will see the moving interferometer being resonant at a different frequency than the stationary one. Since observed time passes more slowly on the moving platform, any oscillator on it (like a laser) would be observed to have a lower frequency and its emissions would then have longer wavelength. The factor by which the wavelength is increased is the same as the factor by which the observed trajectory length is increased so the number of wavelengths between the mirrors would be the same. That means that, if the moving oscillator was resonant in the interferometer before the transverse motion started, it will remain resonant after the motion starts. This must be the case since, if it were not, then Michelson's famous interferometer experiment would have not given a null result. We might want to use this device as a clock where each mirror intercept would correspond to a single time increment or tick. Since we are trying to define elapsed time, we should not use times as starting and stopping events. A reasonable choice of starting event would be departure from the interferometer position x=A and then the ending event would be arrival at x=B. How many ticks of each clock should we expect between these two events? It is easy to show that the number of ticks in the interferometer frame is [(B-A)/vx](c/L) while the number in the laboratory frame is [(B-A)/vx](c/L)*sqrt(1-vx*vx/(c*c)) The lab frame observer would conclude that time passes more slowly (less ticks between events) when the interferometer is moving. This agrees with results for time passage (time dilation) of special relativity.
beta L Single Step