The Michelson and Morley experiment
REFERENCE: None
Recall the addition of velocities in Newtonian physics:
(S) to (S') (S') to (S)
Vx' = Vx - v Vx = Vx' + v
Vy' = Vy Vy = Vy'
Vz' = Vz Vz = Vz'
Consider the following scenario:
A stationary sound source is projecting a sound toward an observer at Vsound, the speed of sound in air (about 373 m/s), and the observer is traveling toward the sound source at vobs. Let (S') be the IRF attached to the observer. Then from the perspective of the source (in (S)), v = - vobs so that from Eq. 1, Vx' = Vx - v tailors into
V'sound = Vsound + vobs.
It has been shown that a wave's speed v is related to its wavelength l and frequency f by
v = lf
so that Eq. 2 becomes, with the proper subscripts,
l'soundf'sound = lsoundfsound + vobs.
It has also been shown that the wavelength is a property of the medium through which the wave travels, and is thus the same for all observers. Thus l'sound = lsound º l so that Eq. 4 becomes
f'sound = fsound + vobs/l.
From Eq. 3 vsound = lsoundfsound = lfsound so that l can be substituted out of Eq. 5:
f'sound = fsound(1 + vobs/vsound).
Eq. 6 is the familiar Doppler effect and predicts that the frequency of the sound heard by the moving observer will increase with the speed of the observer. It also predicts that the frequency will decrease if vobs is reversed.
Two important properties of waves are revealed in the analysis of the Doppler effect.
(1) There is a medium through which the wave must travel. In the case of sound, the medium is air, and the wave energy travels via pressure-oscillations of the air molecules.
(2) There is a favored IRF where the wave has its "natural" speed. In this case, it is in (S), the IRF in which the source is stationary.
Around 1860, James Maxwell derived the equations describing electromagnetic phenomena (called Maxwell's equations), which predicted that light was a wave and that it had a particular speed c = 3´108 m/s associated with it. At the time of Maxwell, physicists knew that all waves required a medium through which to propagate, and they thus postulated a medium for light waves and called it the luminiferous ether. Just as sound has a favored reference frame (the air's rest frame), so does light (the ether's rest frame). Quite a bit of effort was put into the discovery of the ether rest frame (through which the earth must be traveling).
It was not until 1886 that an experiment of sufficient precision was designed by Michelson and Morley to detect the ether frame, using the Doppler effect as applied to light. Here is how it worked:
From the Doppler effect we know that when we approach a source, the frequency increases. Consider a light source and a stationary observer (Fig. 2). The observer then measures a frequency f' = f(1 + 0/c) = f. This means that the frequency measured by the observer is the same as the frequency of the light source (say starlight from a distant star). If, on the other hand, we approach the light source (Fig. 3), we will measure a frequency given by f' = f(1 + vobs/c). Note that in this case, f' > f by the small correction vobs/c. The main reason no experiment had detected this frequency change to the time of Michelson and Morley was precisely because the ratio vobs/c was so small. After all, c = 186,000 miles per second.
The properties of wave addition and interference illustrated in Fig. 4 were used in the design and interpretation of the Michelson-Morley experiment. The heavy black wave is the sum of the other two.
A heavy marble platform floating in a vat of mercury held the apparatus, absorbing street vibrations and allowing the experimenters to rotate the apparatus through any desired angle. A light source sent a beam of light to a beam-splitter (Fig. 5), which then sent two in-phase beams down perpendicular arms of length L, to be reflected at mirrors A and B, collected at the detector, and observed for interference. The platform was rotated so that one arm was parallel to the direction of motion of the earth through the ether. It was expected that the frequency would be higher in arm A than in arm B (perpendicular to the motion VE through the ether). Interference fringes could be tallied as the apparatus was rotated slowly through 90°, and frequency differences could be calculated. Then f' = f(1 + vobs/c) could be solved for vobs, and we would thus know what our relative speed through the ether frame was.
The results were null. No interference fringes were detected.