Exactly how is the constant measured velocity of light deduced from Maxwell's equation?

Question

For electromagnetic radiation the velocity of propagation is $c = 1/\sqrt{\mu_0 \epsilon_0}$. Since both $\mu_0$ and $\epsilon_0$ do not vary in any inertial frame, then $c$ must be constant in any inertial frame.

Now apply the argument to sound. The velocity of sound in a rod clearly doesn't change regardless of the velocity of the rod, but it's obvious that if the rod moves past me at the velocity of a sound wave in that rod, the sound wave will be standing still as I view it.

In other words, there appears to be something wrong with the argument in the first paragraph since the same argument fails for sound, which is also a wave with a propagation velocity.

So how exactly does light differ from sound here? I'm looking for an explanation that would be understandable to a budding scientist who is a non-physicist. (I'm trying to prepare to explain special relativity to my daughter (a geologist who's taking a course in cosmology), and want to be able to explain special relativity from first principles, in such a way that special relativity becomes obvious and simple).

Answer 1

The key is, I think, that when you establish the equations for the rod, you implicitly assume a frame of reference, namely at rest w.r.t. the rod. However, Maxwell's equations are more fundamental and based on laws that shouldn't depend on the reference frame, at least they didn't seem to.

So, when you get an equation pointing to a fixed speed, you naturally ask the question about w.r.t. what reference frame. But there was no implicit assumption of a reference frame in the case of the Maxwell equations. So that was puzzling. So the first reaction was to postulate there must be one. An aether or something like that. But this turned out to be a failure.

Answer 2

Are you really asking why the speed of light is invariant regardless of the observer's relative motion? I will try to answer that, although I'm really rusty on Maxwell.

It was commonly thought that there was an "aether" through which light propogated, just like sound waves in matter. If so, since the earth is moving through space at pretty high velocity, if you measure the speed of light along tubes oriented at right angles, you should see a difference. That was the Michaelson-Morley experiment, and oddly enough, there was no difference. It appears as long as you're moving steadily in a straight line, no matter where or how fast, if you measure the speed of light it comes out the same.

In order to explain this puzzle, a new theory was born, Special Relativity. It says our clocks and our measuring sticks (the things we measure speed with) start acting funny when we go fast with respect to someone else. Here's how it works.

Suppose you build a clock by having two parallel mirrors in a vacuum one half meter apart, and you bounce a small pulse of light vertically between the mirrors. You count the bounces. 299792458 round trips takes one second, because that's how fast light goes.

Now you mount this clock on a railroad car and look at it as it travels past you. Since the light is now traveling in a slanted direction, because the whole clock is moving, the light pulse has to travel further between bounces, which will take longer from your perspective. So from your perspective the clock on the train is running slower, but from the perspective of the person on the train, it appears to be going the same speed because that's the only clock he's got.

In fact, if you the supposedly stationary person also had a clock, the person on the train would say that your's appears to be running slower.

So, if the Michaelson-Morley experiment is correct, this is what you and the person on the train should observe. Well, experiment bears it out. That's what actually is seen. When very precise clocks are compared, and one of them is moving in a straight line at a constant high speed relative to the other, it does run slow. The faster it's moving, the slower it runs. In theory, if it could go at the speed of light, it would stop.

Also, measuring rods aligned in the direction of motion are shortened, for similar reasons.

So if you're trying to explain why the speed of light is constant regardless of the observer's motion, the answer is, we don't really know, but it appears to be, and the implications are that clocks and measuring rods should do funny things. And in fact, they do. This has deep implications, for better and worse, including nuclear energy and warfare.

Answer 3

Modified @Raskolnikov’s answer (hence community wiki because insufficient part of Incnis Mrsi to claim authorship):

It is not “deduced from Maxwell's equation” alone, as a purely mathematical consequence, but also relies on surrounding physics. Maxwell's equation and wave equation for sound (even if we ignore P/S dispersion in solids for simplicity, as well as shock waves) are similar in admitting O(1, 3) symmetry. BTW, not an unexpected fact for a hyperbolic PDE to have this symmetry. But is physics actually O(1, 3)-symmetric in each of cases? Physics is not only field equations – in case of electrodynamics we have (relativistic) dynamics of particles together with Maxwell's equation.

In a continuous medium there is no O(1, 3) symmetry, even locally (in a small piece), although if the medium is isotropic then it admits O(3) symmetry. We have no physically sensible analog of “Lorentz boosts” because any small piece of medium has a preferred reference frame. Why is it so? There are several arguments:

• Direct observation: a continuous medium “appears” to have definite velocity in any its piece. Motion of distinct pieces (such as small inhomogeneities) can be traced visually in many cases.
• Microscopic: there is a reference frame where momenta of molecules (constituent particles) sum to zero, whereas in other frames they do not.
• Friction: forces on particles interacting with the media exhibit a “preference” for the reference frame of the media.
• Optics: light propagating through a transparent medium does not show any respect for the “sound cone”, but has its own O(1, 3) thing, much more faster.

But if we look on electrodynamics in vacuum, we see that none of this obstacles take place. Dust particles in Solar System do not move in the same fashion as in air (or water) because there is no friction◗, there are no constituent particles of vacuum itself with their momenta, and all known and envisaged waves propagating in vacuum (such as hypothetical gravitational waves) obey the same light cone of electromagnetism. Is is not just an equation – it’s relativity.

Conclusion: continuum mechanics and electrodynamics in vacuum are only superficially similar.

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◗ Interstellar dust sometimes does, but this is actually the case of very sparse continuous medium, not vacuum.

Answer 4

Actually, you can only measure the relative motion of sound so easily by using something that travels faster than sound. In this case, the rigid bodies (rod, for example) are composed of quantum electromagnetic bonds which can transmit information at up to the speed of light.

In an interesting 2008 paper which you can find here http://arxiv.org/abs/0705.4652 Barcelo and others report on objects composed of phonons in condensed matter, and a Lorentz system based on the speed of sound rather than light. "Observers" composed of phonons using an interfermoter composed of phonons would find the speed of sound to be a limiting velocity.

If we had access to something that traveled faster than light, we'd likely be able to measure the relative motion of light, but we don't. Basically, we are composed of light. According to E=mc^2 light converts freely back and forth to the material particles of which we are composed. From a quantum mechanics perspective, it appears the light becomes trapped somehow into a standing wave, but I know of no physicist who claims to understand it. If you find one, I'd like to hear the explanation myself.