One might wonder where Bethell went astray in his prediction, since the Michelson-Morley experiment is performed on the surface of the rotating Earth, and we know the speed of light actually is not invariant in terms of rotating coordinates, as demonstrated in the Michelson-Gale experiment. In view of this, wouldn’t we expect to see a difference in travel times for pulses of light moving in different directions in an apparatus on the Earth’s surface? For example, according to special relativity, a light pulse will take more time for the westward trip from New York to San Francisco than for the eastward trip from San Francisco to New York – in terms of clocks at the two cities synchronized in the Earth Centered Inertial (ECI) system. Why then shouldn’t we expect a (small) non-null result for a Michelson-Morley experiment? The answer is two-fold. First, the pulses of light in a Michelson-Morley apparatus make round trips, back and forth on each arm, so all effects proportional to v/c cancel out. (The Sagnac effect demonstrated in the Michelson-Gale experiment is proportional to v/c.) This is why the Michelson-Morley experiment was designed specifically to measure much smaller anticipated effects proportional to (v/c)². Second, as Michelson discovered to his surprise, even these much smaller effects are absent, at least those corresponding to the Earth’s orbital velocity around the Sun, and none have appeared in subsequent much more precise experiments (using modern laser optics), capable of measuring effects corresponding to the tangential velocity of the Earth’s rotating surface. In retrospect it’s easy to see why: The relevant synchronization of the various parts of the apparatus is not an artificial one imposed by the Bureau of Standards (as is the ECI synchronization of clocks in cities around the world), it is the actual physical synchronization corresponding to the Lorentz-invariant equilibrium configuration of the apparatus in its own rest frame. The equilibrium configurations of physical entities, such as the arms of the interferometer, are established by forces that vary with velocity exactly as do the forces of electromagnetism (this is the meaning of Lorentz invariance), from which it follows that optical and mechanical synchronizations are identical, and hence no fringe shifts will occur.

Note that, during the time it takes for a pulse of light to traverse the arm of a Michelson interferometer, the acceleration of the apparatus is nil, so the entire apparatus can be accurately regarded as an equilibrium configuration in a single inertial reference system. In contrast, the various cities around the globe are never all at rest in terms of any single system of inertial coordinates (on the time scale required for light to travel from one city to another), so it’s more convenient to use the ECI synchronization. The absence of fringe shifts in Michelson-Morley experiments (falsifying Bethell’s prediction) confirms that optical and mechanical synchronizations in terms of any single system of inertial coordinates are identical, in perfect accord with special relativity.

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