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I know that the speed of light, the universal constant of gravitation and the Planck's constant are considered to be the three fundamental constants of the universe. But, why is speed of light considered as a fundamental constant? The speed of light changes from medium to medium for satisfying Fermat's principle then, how can it be a constant?

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marked as duplicate by tpg2114, Emilio Pisanty, Qmechanic Nov 11 '13 at 1:24

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It's not constant. The speed of light in vacuum is a fundamental constant. –  David H Nov 10 '13 at 12:35
@Qmechanic these two questions didn't appear in the related questions link when I was typing it up. That's why I proceeded towards asking this one? –  Rajath Krishna R Nov 10 '13 at 12:37
@RajathKrishnaR, a fundamental constant is a physical quantity that is generally believed to be both universal in nature and constant in time. The speed of light in a vacuum is believed to be a fundamental constant because it does not vary in nature and through time. Two observers in different areas of space and time, even travelling at different velocities, will each deduce the same value for the speed of light, c. –  Mew Nov 10 '13 at 14:02
@RajathKrishnaR And by the way: a photon in a medium still travels with the vacuum speed of light. it is the scattering in the medium - or in classical terms the superposition of incoming and scattered waves that conspire to produce a phase-velocity of less than (but in rare cases even more than!) the vacuum SOL. But that is just the speed of the total wave. Each 'fundamental' wave still propagates with vacuum SOL. –  nephente Nov 10 '13 at 15:36

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But, why is speed of light considered as a fundamental constant?

c is an invariant speed; if something (not necessarily light) has speed c in one inertial reference frame (IRF), it has speed c in all IRFs.

This result follows from the Lorentz transformation which, empirically, gives the correct coordinate transformation between relatively moving IRFs.

Thus, c is a physical constant with units of speed. So it is c, the physical constant, that is constant.

Yes, light propagates with speed c in vacuum but there is nothing contradictory with light propagating with speed less than c in a medium. In other words, we do not denote the speed of light in a medium as c.

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Sir, then based on what criteria is c considered as a fundamental constant? Is it because of its frame dependent speed in vacuum? –  Rajath Krishna R Nov 10 '13 at 13:39
@RajathKrishnaR, c is considered a fundamental constant for the same reason that h and other fundamental constants are which is: the appearance in equations of fundamental importance to our understanding of the world. –  Alfred Centauri Nov 10 '13 at 14:37

$c$ is not thought of primarily as the speed of light (even in a vacuum) in modern thinking - it's a universal constant that comes to mean the speed of any massless particle. It's existance, as a unique speed which is constant for all observers, is foretold by basic symmetry and homogeneity arguments of special relativity. See my answer to "The reference frame of c ", where I discuss these symmetry arguments.

Experimentally light's speed in a vacuum is observed to transform in this very special way, so this tells us that light is a massless particle, and, because special relativity foretells that the universal speed $c$ that transforms in the special way is unique, we know we're measuring the "right" one when we measure light's speed in a vacuum and declare it to be $c$.

Interestingly, you can turn this argument around and show that, because light's speed transforms in this unique way, we know we're not immersed in any "medium" that interacts with light and which we're not aware of (e.g. "transparent" dark matter) - see here.

Note that if we were immersed in some kind of medium like this (with refractive index 1.00001, say), and Michelson and Moreley did their experiment today, they would get a non-null result. But they still wouldn't get the result foretold by Galilean relativity. So the result would not tell against special relativity, and indeed we'd be able to get good estimates on what the true value of $c$ is, even though our measured light speed would be ever so slightly lower.

Optical mediums "break" the symmetries underlying special relativity and velocities of light in them transform under boosts quite differently from $c$ . The matter of the medium throws up a "preferred" reference frame because the atoms / molecules in it "hold on to the light" for small time intervals in a process of cyclic absorption and re-emission that begets a "slowing of light" and thus an effective refractive index. The delays are nonzero and thus are dilated under boosts, leading to a dependence on frame of the refractive index and the measured speed.

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The speed of light in a vacuum is constant whilst in a physical medium it slows down. Now this would still seem to violate the principle of special relativity so it is worth noting that an individual quanta of light (the photon) at no point is traveling less than c. If we approach this from a classical perspective we see that the incident light can be thought up as having a group velocity and a phase velocity; it is the group velocity that we are primarily interested in and this will be the result of the superposition of all the waves in the medium (that is we can add the 'individual waves' to get a resultant one. Light is a type of electromagnetic radiation so its propagation can be considered as an electric field. Inside a solid for example. You have nuclei and electrons of opposite charge that act as dipoles (pairs of positive and negative charge). Our beam of light interacts electromagnetically with these dipoles in a way that causes emission of more waves. By taking the superposition of the incident waves with the ones that are a result of the interaction with the dipoles we get a wave with a group velocity that is less than the speed of light but the individual photons themselves never change speed.

From a quantum mechanical perspective we see that this principle still holds true. The quantum mechanical theory of light (quantum electrodynamics) predicts that a photon propagating through a medium will travel along every possible path avaialbe to it and the resulting photon will be the result of the superposition of every possible path. This can be demonstrated in the classical 2 slit experiment where individual photons act when sent towards an obstacle with two slits in it as if they have interfered like a wave with itself meaning that it must have simultaneously traveled through both slits.

The nuance here is appreciating the difference between collective and individual behaviour which allows for a packet of light to have slowed down whilst its individual components haven't.

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But, why is speed of light considered as a fundamental constant?

The fundamental constant relating to "speed"$\! { \, }^{\text (1)}$ is more specifically called the "signal front speed";
or, correspondingly, "speed of light in vacuum", as far as "vacuum" is understood to describe a region in which front speed and group speed of signals are equal.

The speed of light changes from medium to medium

What's "changing from medium to medium" (or what's useful for distinguishing "media" in the first place) is not the signal front speed, but rather signal group speed, or the phase speed.

Wikipedia states correspondingly that

  • It can be shown that [...] the speed of the earliest part of the pulse (the front velocity) [...] is (under certain assumptions) always equal [...]

As far as such "certain assumptions" are indeed necessary, and as far as they are considered self-evident, signal front speed is therefore considered as a fundamental constant.

($\! { \, }^{\text 1}$: Instead of "signal speed" one may of course also speak of "signal velocity" in the sense of "(average) signal velocity in the direction from sender to receiver".)

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