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I found an explanation of time dilation using a light-pulse clock in the book 'Concepts of modern physics' by Arthur Beiser:

In this explanation, the clock is moving in a direction perpendicular to the light ray. What if the clock is moving in the direction of light ray. Will the time dilation change? How can we see the time dilated in this case?

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Michelson - Morley interferometer was a "composition" of two light clocks that had transverse and longitudinal arms.

Every arm of Michelson interferometer can be seen as a light clock having a specific rate, whose relative orientation dependendces can be tested. Therefore, such experiments are sometimes called "clock anisotropy experiments" since they compare two internal frequencies or clocks.

This experiment has shown within tight limits that in an inertial frame the two-way speed of light is isotropic and independent of the closed path considered orientation of the apparatus, so, light propagation times in all directions are the same.

In the moving frame this result can be explained by Lorentz - FitzGerald contraction of the interferometer in direction of motion.

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