special relativity - The reference frame of $c$

I don't have a lot of knowledge of special relativity and associated topics; some of the few things I know are that "all motion is relative" (that is, there is no 'stationary reference frame'), and the speed of light in vacuum ($c \simeq 3 \cdot 10^8~\mathrm{m~s^{-1}}$) is the absolute asymptotic speed limit (asymptotic meaning that you can never equal it, only get arbitrarily close). What escapes me is how those concepts work together - to my naive understanding, an object would never move in its own reference frame (and so would never reach $c$). What reference frame is $c$ measured against? (Is it measured against a reference frame?) Or am I looking at this the wrong way?

Answer

Another way of thinking that might be helpful to you is to take heed that $c$ is not primarily the speed of light. It comes indirectly to mean the observed speed by any observer of any massless particle, and because, as far as we know, light is massless, it comes indirectly to mean the speed of light. But, in its most fundamental form, $c$ is only a parameter that happens to have the dimensions of speed. It doesn't primarily refer to a speed: here's how we go about defining it.

Think about the intuitive Galilean addition of velocities. The combination law is linear. So, assuming a linear combination law, there are some basic symmetries and characterisics of this everyday law you might like to think about. 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 $T(v_{1,3})$ 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 famous 1905 paper "On the Electrodynamics of Moving Bodies".

So imagine we had carefully reviewed Galilean relativity as above but we didn't know anything about special relativity. This might well have been how science might have progressed in the late nineteenth century were it not for the Michelson-Morley experiment. We would now understand that our everyday Galilean looking laws might actually arise from a universe wherein we have this weird $c$ parameter that is not infinite but simply very big: this would still be consistent with our everyday addition of velocity laws with a big enough $c$. At this point, we'd only know the form of the Lorentz transformation and that there were a $c$ parameter (maybe infinite) with dimensions of velocity, so we'd like to come up with some experiment to measure whether our universe had a finite $c$ value. It would not be apparent straight away that this velocity parameter were the velocity of anything in particular or indeed whether it even could be the velocity of anything. But, now we say to ourselves, what if something were going at this velocity relative to us? A simple study of the Lorentz transformation would show us that:

1. The speed of this body $c$ would be measured to be the same in all inertial reference frames. Moreover, so as to enforce this invariance of $c$, there would be a peculiar addition rule for velocities not quite the same as the parallelogram rule;


2. No material object can go faster than $c$ and indeed something can travel at speed $c$ only if it has a rest mass of nought.

So now the Michelson Moreley experiment can be thought of not so much as validating relativity, but rather of showing that light, if made of particles, must be made of massless particles. The Michelson Morely experiment found something whose speed transforms precisely as foreseen by the general Lorentz transformation with a finite $c$, so it would then be a strong hunch (not a proof) that our universe indeed has a finite $c$ and that light is something that travels at this speed. In this context, a positive result of the Michelson Morley experiment (i.e. one showing a dependence of lightspeed on frame) could be thought of either as (i) detecting an aether (medium for light) but equally well (ii) it could be thought of saying that there is no aether but that the light particle has a small mass. Neither result would gainsay our newly found relativity laws.

Of course, many other experiments have since confirmed everything that a relativity grounded on a finite $c$ with $c$ set to the speed of light would foretell, so its quite reasonable to speak of $c$ as the speed of light in relativity. But I hope I have shown that this is not its primary meaning.

Footnote: Unfortunately these ideas don't quite work in more than one dimension. In one dimension, two boosts indeed compose to a boost, but a sequence of boosts in different directions in general compose to one boost together with a rotation. This rotation is called Thompson Precession. So we speak of the Lorentz group as the smallest group of all transformations that can be gotten from a sequence of rotations and boosts, but there is no multidimensional group of boosts, only the "one parameter" one dimensional group of boosts.