What's so special about the speed of light?

What's so special about the speed of light?

Why do many equations in physics include the speed of light in vacuum $c$?

Why do so many thing depend upon it?

Why can't it be the speed of sound?

Or as a matter of fact any other variable?

For example, in mass-energy equivalence $E=mc^2$, why the speed of light $c$?

Answer

I'd like to add to Chris White's excellent answer and summarise things thus:

$c$ is a number that parameterises the family of all possible linear transformations that follows from Galileo's relativity with the assumption of absolute time relaxed.

If you do Galileo's relativity with the assumption of an absolute time (that two relatively moving inertial observers will measure the same time interval between two given events), you can derive the intuitive vector addition of velocities rule to transform between relatively moving inertial frames.

Indeed, with an assumption of absolute time, this intuitive addition rule is the unique linear transformation rule in keeping with Galileo's notion of relativity, most wonderfully, poetically (and scientifically rock-solid-accurately) described in his allegory of Salviati's Ship.

However, if we relax the assumption of absolute time, and this was Einstein's bold and uniquely Einstein step, then derive all the possible linear transformations consistent with Galileo's postulates with the relaxed time assumption, you find that there are now a whole family of possible linear transformations, each characterised by a parameter $c$ with dimensions of speed. Galilean relativity does indeed belong to this family. It is the family member with $c=\infty$ (if we think of the parameter $c$ living on a compactified real line).

The deeply characteristic thing about the value $c$ for any member of the family of transformation laws is that if anything travels at this speed, then it will be observed to travel at this speed $c$ in all inertial frames.

It then follows that for any "universe", i.e. set of observers which could in theory possibly compare experimental results, then the particular value of $c$ characterising Galileo-consistent transformation laws must be unqiue for that "universe".

So now, it becomes a wholly experimental question:

What value of $c$ characterises transformations between inertial frames in our universe? Is it even finite?

So now, experimentally, we must look for something - anything- whose speed is the measured to be the same for all inertial observers. If we find anything like this, then we know that a finite $c$ characterises transformations between inertial observers in our universe, and its speed is the special parameter $c$.

You know of course that we did indeed find something with a speed that transforms between inertial observers in this very special way: it was the famous Michelson-Morley experiment, and light was that thing. But it turns out that any massless (i.e. with rest mass of nought) thing must be always observed to travel with this speed relative to any inertial observer. So it is also the speed of, for example, the massless graviton, if indeed this particle turns out to be real. For many years neutrinos were thought to be massless and indeed we observe them to travel at $c$ to within most experimental error bounds. We know that they must have mass indirectly through the observation of Flavour Oscillation. But we don't yet have a definite measurement for their nonzero rest mass.

This value of $c$ also appears, as discussed in Chris White's answer in the generalised $m^2 c^4 = E^2-p^2 c^2$, thus answering your last question. An intuitive way of getting to the $E=m\,c^2$ special case is to use the method that Einstein used in his 1905 paper (the one AFTER the big one):

A. Einstein, "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?", Ann. der Phys. 18:639,1905 English translation "Does the Inertia of a Body Depend upon its Energy-Content?" is here

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The following might look a bit daunting at first but it really is intuitive and we're not talking about at first anything that gainsays everyday Galilean relativity, so I'd urge you to think about applying these ideas to the simple problem where we have three frames: $F_1$, the street, $F_2$ a bus driving along the street and $F_3$ a person walking down the aisle of the moving bus. In the following, let us call the shift from one frame to another, uniformly relatively moving frame a boost:

1. (Linearity) If I transform from frame $F_1$ to a frame $F_2$ moving at a constant speed $v_{1,2}$ in some direction then my distance and time co-ordinates $(x, t)$ are transformed by some $2\times2$ matrix $T(v_{1,2})$, i.e. $X=\left(\begin{array}{c}x\\t\end{array}\right)\mapsto T(v_{1,2}) X$;



2. (Transitivity and Associativity): If I then transform to a third frame $F_3$, one moving at velocity $v_{2,3}$ in the same (original) direction relative to the transformed frame $F_2$ (using the matrix $T(v_{2,3})$, this has to be equivalent to a single transformation $T(v_{1,3}) = T(v_{2,3}) \circ T(v_{1,2})$ from the first to the third frame with some relative velocity $v_{1,3}$. Or, with our "boost" word: a boost combined with another boost in the same direction is still the same as a boost with some relative speed: transformations in the same direction do not change their character by dent of their being composed of boosts or indeed how (our of an infinite number of ways) they might be composed of boosts. If I walk at some speed along a bus itself moving along the road, then my motion should be describable as my moving along the road at some relative speed, forgetting about the bus;


3. (Symmetry of Description) In particular, if frame $F_3$ is moving relative to frame $F_2$ at velocity $-v$, then frames $F_1$ and $F_3$ have to be the same and $T(v) T(-v) = I$ (here $I$ = identity transformation - my running away from you at velocity $v$ should seem the same as your running away from me at the same speed in the opposite direction). This symmetry arises from a basic "homogeneity" (space and time are the "same" in some sense everywhere) and the Copernican notion that there is no special frame. Think carefully about these and you will see that the Galillean transformation fulfills all these intuitive symmetries.

Now for the killer question:

Do the conditions 1 through 3 fully define a Galilean transformation? Or, more mundanely, What is the most general form of the matrix $T(v)$ that fulfils conditions 1 through 3?

It turns out that, not only does the Galilean law $v_{1,2}+v_{2,3} = v_{1,3}$ fulfill all the above axioms, but there are a whole family of possible transformations, each parameterised by a parameter $c$, with the Galilean law being the transformation law we get as $c\to\infty$. Such laws are the Lorentz transformations. See the section "From group postulates" in the "Derivations of the Lorentz transformations" Wikipedia page. Notice how one has NOT assumed that $v_{1,2}+v_{2,3} = v_{1,3}$, aside from in the special case of when $v_{1,2} = -v_{2,3}$. It seems likely that Ignatowsky (see Wikipedia page) was one of the first to understand that one could derive relativity from these assumptions alone in 1911, although Einstein actually mentions the group structure of the Lorentz transformations in his BIG 1905 paper "Zur Elektrodynamik bewegter Körper", Ann. der Phys. 18:891,1905 (english translation "On the Electrodynamics of Moving Bodies" is here).

Things get a bit more complicated with three-dimensional velocities: there is no Lie group of three dimensional boosts. This is a mathematical fact, not a physical one, so it's not simply a question that no-one has yet demonstrated such a group. It can be proven to be impossible. So when we make our bus problem three dimensional, we get a group that includes both boosts and rotations. The composition of two three dimensional boosts does not in general give a boost, but a boost composed with a rotation. This rotation begets the phenomenon of Thomas Precession (see wiki page of same name). This can also be thought of in terms of the "polar decomposition" that allows any square matrix to be written as a product of an orthogonal and upper triangular matrix (itself a special case of the Iwasawa decomposition of Lie groups).

The smallest Lie group containing three dimensional boosts is the famous Lorentz group, which I discuss in Example 1.10 "General Classical Groups and The Proper Lorentz Group $SO^+(1,3)$" on the page "Some Examples of Connected Lie Groups" on my website. There are also the Wikipedia references on the Lorentz group Chris White gave you; also see the Wikipedia page "Derivations Of The Lorentz Transformations" to see whether there is an explanation there that grabs you.