The speed of light in vacuum (but which speed?)

Question

Since there are several different definitions of speed describing light propagation like Group velocity, Front velocity and Phase velocity.

Which speed is meant when the phrase "The speed of light in vacuum" is used?

Like for example the second postulate of special relativity by Einstein that "the speed of light in free space has the same value c in all inertial frames of reference" - which speed was he referring to?

As I currently understand it - it is the maximal speed at which an information can be sent from one place to another. But if it is so then isn't the addition of "in the vacuum" (or "in free space") redundant?

Answer 1

Calling the constant $c$ the "speed of light" is something of a misnomer. In special relativity, time and space are taken on equal footing. To achieve this in a physical sense, one needs a "conversion factor" between time and space; It turns out that this factor is $c$. Of course, this conversion factor must be the same in all reference frames, otherwise, it would be very hard (if not impossible) to build a consistent theory.

Furthermore it can be shown that a massless particle in the absence of external influences (i.e. the vacuum) will always propagate with speed $c$. As is known, photons are massless particles, and since electromagnetic radiation was the first phenomenon discovered to propagate with speed $c$, the constant was named "speed of light". Of course, the speed of electromagnetic waves can be altered by introducing external influences, so the somewhat cumbersome "in the vacuum" had to be added to the name of the constant. But this does not mean that there is any significance of light per se to this speed. In the vacuum, a gravitational wave also travels with $c$, for example.

You are right about $c$ being the highest speed at which information can propagate. I think from this it should be easy to figure out which part of a light wave (first) propagates with $c$.

Answer 2

In a vacuum an electromagnetic wave has the same phase and group velocities, so the speed of light in a vacuum is unambiguous.

If the light is propagating through some medium this is generally not true. The electric field of the light interacts with the electrons in the medium and in effect the photons and electrons get mixed together. See Can speed of light be $c$ in air or other medium? for more on this. The group velocity of the resulting mixture is less than $c$ and the phase velocity is greater than $c$. So the speed of light is only $c$ in a vacuum.

Having said this, in relativity the $c$ isn't special because it's the speed light travels at, or indeed that any massless particle travels at. $c$ is a constant associated with the geometry of flat spacetime. For more on this see What is so special about speed of light in vacuum?

Answer 3

Einstein used the constant $c$ that occurs in Maxwell's equations for electricity and magnetism, the idea being that Maxwell's equations should remain true under any change of inertial coordinate systems. This idea fails when Newtonian coordinate change laws are used; but it succeeds when Lorentzian coordinate change laws are used, as long as that same constant $c$ is the one that is built into the Lorentz coordinate change formulas.

It was already known, before Einstein, that the constant $c$ in Maxwell's equations was the speed of the "electromagnetic waves" that are predicted by those equations. And it was also already known that light waves are one special type of electromagnetic wave. The reason that "vacuum" is an issue is that the crap that pollutes a vacuum (e.g. the odd hydrogen ion here, or lump of dielectric there) can interrupt the transmission of those waves; in other words, Maxwell's equations only predict electromagnetic waves of speed $c$ if there is no other "electromagnetic crap" for those waves to interact with.

So that's why the "speed of light in a vacuum" is the speed that Einstein was referring to in his formulation of special relativity.

Answer 4

For light in a vacuum, those speeds are experimentally as close to the same as can be measured. Based on this, and other experimental results, the theory assumes they are the same.