Maxwell’s Vacuum and Doppler By Any Means

The classical wave theory of light in Galilean space and time leads to different predictions for the Doppler shift between emitter and receiver, depending on the states of motion of those entities in terms of the rest frame of the putative medium of propagation. In this context, the phrase “medium of propagation” refers not to any ordinary material substance, but to the presumed medium in whose rest frame the wave propagates at a characteristic speed.  It was known already by the middle of the 19^th century that the medium of propagation of light cannot be any ordinary material substance. Writing in 1864 (“A Dynamical Theory of the Electromagnetic Field”), Maxwell distinguished between ordinary matter (which he called gross matter) and whatever substance comprises the luminiferous aether. He wrote (in paragraph 4) that the region through which light travels

…may be filled with any kind of matter, or we may  endevour to render it empty of all gross matter, as in the case of Feissleh's tubes and other so-called vacua.  There is always, however, enough of matter left to receive and transmit the undulations of light and heat, and it is because the transmission of these radiations is not greatly altered when transparent bodies of measurable density are substituted for the so-called vacuum, that we are obliged to admit that the undulations are those of an aetheral substance, and not of the gross matter, the presence of which merely modifies in some way the motion of the aether.

Likewise in his Encyclopedia article on the “Ether”, Maxwell wrote

When light travels through the atmosphere it is manifest that the medium through which the light is propagated is not the air itself, for in the first place the air cannot transmit transverse vibrations, and the normal vibrations which the air does transmit travel about a million times slower than light. Solid transparent bodies, such as glass and crystals, are no doubt capable of transmitting transverse vibrations, but the velocity of transmission is still hundreds of thousand times less than that with which light is transmitted through these bodies. We are therefore obliged to suppose that the medium through which light is propagated is something distinct from the transparent medium known to us, though it interpenetrates all transparent bodies, and probably opaque bodies too.

Of course, the speed of propagation of light within a transparent substance like air or glass is affected by the index of refraction of the substance. This is why Maxwell used the word “medium” to refer to two different things, one being the gross matter that is present in a region where light is propagating, and the other being the mysterious aethereal “matter” (very unlike ordinary matter) that serves as the actual carrier of the undulations. As Maxwell noted

The velocity of light, however, is different in different transparent media, and we must therefore suppose that these media take some part in the process… but the energy of the vibrations of the gross particles must be very much smaller than that of the aether, for otherwise a much larger proportion of the incident light would be reflected when a ray passes from vacuum to glass or from glass to vacuum than we find to be the case.

Since the index of refraction of the air is extremely close to 1, Maxwell routinely equated air with vacuum, such as when he wrote indiscriminately about “the velocity of light in air or vacuum”. He also explicitly stated that at times he took the index of refraction of air to be unity. However, he recognized that this was not exactly true, since air differs slight from vacuum.  He wrote

If μ is the coefficient of magnetic induction and k is a quantity that may be called "electric elasticity" [the reciprocal of what we call permittivity]... we find V² = k/(4πμ)... shewing that the disturbance is propagated with the velocity V… According to our result, the ratio of electromagnetic to electrostatic units should be equal to the velocity of light in air or vacuum…  The unit of electricity is usually defined with reference to experiments conducted in air. We now know from the experiments of Boltzman that the diaelectric constant of air is somewhat greater than that of a vacuum, and that it varies with the density. Hence, strictly speaking, all measurements of electric quantity require to be corrected to reduce them either to air of standard pressure and temperature, or, what would be more scientific, to a vacuum, just as indices of refraction measured in air require a similar correction, the correction in both cases being so small that it is sensible only in measurements of extreme accuracy.

Mostly conclusively, Maxwell referred to the experiment in which Fizeau

…has observed that the propagation of light in a stream of water takes place with greater velocity in the direction in which the water moves than in the opposite direction, but that the change of velocity is less than that which would be due to the actual velocity of the water, and that the phenomenon does not occur when air is substituted for water.

To be precise, if v is the speed of the material medium, the inferred speed of the aether in the medium is, to the first order, just v(1 – 1/n²) where n is the index of refraction. Since the index of refraction of water is 1.33 the (putative) dragging effect of water is about 0.43v, whereas the index of refraction of atmospheric air is 1.0003, so the dragging effect of air is only about 0.0006v, which was too small to be measured by Fizeau. Now, one might wonder if the dragging of the aether within a solid object with high index of refraction is significant, perhaps there might be some convective effect for the aether in the region just outside the solid object. People sometimes speculated about this to explain why light speed seems to be isotropic in the rest frame of the earth. However, this hypothesis not only conflicts with stellar aberration, it has also been tested directly by (for example) Lodge’s experiment with a huge massive rotating disk. Lodge made extremely precise optical measurements of light skimming just outside the disk, and (to his surprise and disappointment) could detect no convective dragging whatsoever.

All these considerations had convinced physicists by around 1900 that light propagates in air or vacuum at a speed that is independent of either the speed of the source or the speed of the ordinary material substances (such as the earth) in the vicinity. But this led to some expectations for a Galilean wave theory that conflicted with experiment. One test was proposed by Maxwell himself. He noted that the time required for a pulse of light in air or vacuum to go forward and backward along a rod moving at speed v relative to the aether would be proportional to (v/c)² where v is the orbital speed of the earth, but when Michelson performed this experiment a few years later he could find no such effect (to his surprise and disappointment).

Another test involves the simple Doppler effect, and was originally discussed by Stark and Einstein, and later (in 1938) performed by Ives and Stilwell. Such tests have the advantage of not involving any distant synchronization.  Place two identically-constructed clocks, mutually at rest, some distance apart, and place a third such clock directly between them, moving toward one and away from the other at speed v. Assume this takes place in a medium with index of refraction arbitrarily close to 1. A sequence of light pulses are emitted from the middle clock at frequency ν₀, and they arrive at the “forward” clock at frequency ν₁ and at the rearward clock at frequency ν₂. This is depicted in the figure below.

How are these frequencies related? For the Galilean wave theory the answer depends on whether the emitter or receivers are at rest in the aether. If the emitter (world line 0) is moving and the receivers (world lines 1 and 2) are at rest in the aether, then it’s easy to show that

which is to say that ν₀ is the harmonic mean of ν₁ and ν₂. On the other hand, if the emitter is at rest in the aether and the receivers are in motion, we have (according to Galilean wave theory)

which is to say that ν₀ is the arithmetic mean of ν₁ and ν₂. Of course, these are just two of infinitely many possible relations, since it’s possible that neither the receivers nor the emitter are at rest in the aether. Based on which relationship actually applies, we could infer the absolute motions of the clocks relative to the aether frame (assuming the Galilean wave theory).

However, what we actually find is neither of those two relationships (nor any of the infinitely many others). What we find, instead, is that the frequencies are related by

which is to say that ν₀ is the geometric mean of ν₁ and ν₂. (Note that ν₀ is also the geometric mean of the values for the two cases of the Galilean wave theory just described.) This, again, applies to regions in which the index of refraction is sufficiently close to 1. Knowing the frequency of the source and at either of the receivers, the velocity of the middle clock is given by

On the other hand, if the index of refraction differs appreciably greater than 1, the speed c_m at which light propagates in this region in the frame of the material substance will be less than c. In that case, if the receiving clocks are at rest in the material substance, the Doppler shifts are

where we’ve denoted the index of refraction by n = c/c_m. From this it follows that, to the second order, we have

Thus for a substance with index of refraction close to 1 (such as air, which has n = 1.0003), this quantity is essentially (v/c)², whereas the Galilean wave theory predicts twice this value. The prediction of special relativity was confirmed by Ives & Stilwell.

Incidentally, solving the previous equations for v, we find that the speed of the middle clock is given by either of the two expressions

where ϕ = c_m/c. Equating those two expressions, clearing the radicals, and simplifying, we find that the product of the following factors must equal zero:

The relevant factor is the middle one, so it follows that the frequencies are related by

Naturally this reduces to the previous expression when c_m = c. (In comparison, for a Galilean wave theory with the receiving clocks at rest in the material substance, the middle clock frequency is still just the harmonic mean of the two receiving frequencies.) Substituting the above expression for ν₀² into either of the relations for v/c_m gives

The same relation applies to both of the Galilean wave theory scenarios, so this relation does not distinguish between any of the alternative theories. Also, note that as v approaches c_m the frequency ν₁ approaches infinity (all the pulses arrive at once). It’s possible for v to exceed c_m (resulting in Cherenkov radiation), corresponding to when the value of ν₁ takes a large negative value, signifying that the most recently emitted pulses arrive ahead of the pulses emitted earlier.

Combining the last two equations, we have (for any index of refraction) the relations

and equivalently

where γ = 1/√(1 – v²/c²). This is the special relativistic version of the Galilean wave theory’s harmonic mean relation, both of which are valid for any index of refraction. It is consistent with the fact that the analysis procedes as in the Galilean wave context, except for the time dilation of the source, represented by the γ factor on the left side.

To test these relations we need to isolate the second order effect, since they agree to the first order. Thus we consider the quantity

which in the case of the Galilean wave theory would have the value

whereas the relativistic value is

To the lowest non-zero order, applicable to values of v not too close to 1, these expressions equal 2v² and v² respectively, thus differing by a factor of 2. Many experiments, beginning with Ives and Stilwell in 1938 and continuing to the present day, have confirmed the relativistic prediction to high precision.

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