I'd like to add a little to GreenBeans wonderful answer that not everything is relative. He makes a few points at the end of his/her already long answer hurriedly (not meant as a criticism):
...it is a choice[, ]a choice of coordinate system[?] There are many coordinate systems in which the speed of light is not constant, or even depends on the direction. So, is that it? It's a definition? That is not a very satisfying answer, and not a complete one. What makes relativity work is the amazing fact that this choice is even possible.
and further on ...
... If we design our coordinate system to describe space isotropically and homogenously and describe time uniformly ...
We sometime forget in relativity and in differential geometry that there is still an objective reality in co-ordinates, however bizarrely and human-centrically they may be defined and even though we think of them as human constructs. At least in physics, for co-ordinates to be useful, there must be an objective, physical procedure for finding the physical point in spacetime labelled by given co-ordinates. Let's look at this objective, nonrelative physics.
To use GreenBeans' words again, the choice that makes an invariant $c$ possible and gives it physical meaning is the choice of affine co-ordinates. Roughly, these are co-ordinates defined by the of rational multiples of displacements along linearly independent directions in space and time of uniform intervals marked out by unit measuring rods and clock ticks in each of the inertial frames. Physics enters our geometry insofar that we make the physical postulate that the Euclidean geometrical notion of "straightedge" (more generally, geodesic segment) and the idealized constructions, defined by Euclid's postulates, of marking out a rational number times a unit length along a straight line are a good mathematical model of what we do when we take a ruler and do the same. This is an experimental, objectively testable result. Likewise, the time co-ordinate enters an analogous description by marking out rational multiples of unit "ticks", where the ticks are defined either by Einstein's procedures with light, or, one can use the definition in Chapter 1 of [1] that "uniform" ticks are ones that the make the motion of a body uninfluenced by forces look uniform from an inertial frame.
There is another piece of objective, nonrelative physics essential to the invariant speed concept and that is Galileo's principle: the notion that an observer that there is no measurement that an observer in an inertial frame can do from within their own frame that can detect the observer's motion relative to any other frame. This is most poetically described in Galileo's own 1632 Allegory of Salviati's Ship within his famous "Dialogue Concerning the Two Chief World Systems" (the one that got him into heaps of trouble with Pope Urban II when the latter, having a bad hair day, got a bit bolshie at the implied slight on Papal Infallibility).
Once you accept Galileo's relativity postulate as a piece of objective, reproducible, nonrelative physics, this means that co-ordinate transformations between inertial frames must form a group (there's a little bit more to this assertion, as I show on my website [2] and also in a hopefully (subject to review) a forthcoming EJP article). So a general transformation on co-ordinates $X$ is of the form $X\mapsto f(T,\,X)$ where the transformation $T$ belongs to a group and $f$ is the group's action on the co-ordinates. Then, once you accept that affine geometry models real systems of surveying procedures and time measurement, then the Copernican notion that Nature doesn't care where we put our origin translates to
$$f(T,\,X_1+Y) - f(T,\,X_2+Y) =f(T,\,X_1) - f(T,\,X_2)$$
i.e. the affine components of vectors linking two spacetime points with affine co-ordinate $X_1$ and $X_2$ are unaffected by an arbitrary shift $Y$ of origin. It then follows that the group action fulfils the equation:
$$h(X+Y) = h(X)+h(Y),\text{ where }h(X)\stackrel{def}{=}f(T,X)-f(T,0)$$
which is Cauchy's famous functional equation. There is one and only one continuous solution to this equation and that is $h(X) = \Lambda\, X$, where $\Lambda$ is a matrix. So if we make the further physical postulate that co-ordinate transformations are continuous, then:
$$\text{Galileo's Postulate } + \text{Copernican Spatial Homogeneity Postualte }+\text{Continuity of Transformation Postulate} = \text{Co-ordinate Transformations Between Inertial Frames Form a Matrix Group }\\\text{Acting Linearly on Affine Co-ordinates}$$
The physical postulate of continuous transformation encodes the everyday experimental result that, as we ride on a bus, we see trees and walkers in the street as we pass even though the bus is moving: we don't see their images shattered into disconnected chaotic sets!
Another active user on this site, Benjamin Crowell, has a wonderful description about how affine geometry with metric structure leads to the Lorentz transformation and the invariant speed concept in chapter 2 of his general relativity book[3]. The following is my own take on it.
If we further postulate that there are collinear motions that are descibed by a matrix group parameterized by a real parameter such that group composition is a continuous function of this parameter, then the only transformation group in keeping with this physical postulate as well as Galileo's, Copernicus's and Transformation Continuity is of the form:
$$\mathfrak{L} = \{\exp(\eta\,K)|\,\eta\in\mathbb{R}\}$$
so any co-ordinate transformation $\Lambda$ transforming the spacetime co-ordinates between inertial frames belongs to a group of $4\times4$ matrices this form, where $K$ is a constant matrix defining the direction of motion and $\eta$ is a generalized swiftness parameter, called the rapitity. Just think of it as a speedometer reading transformed in a nonlinear way that we'll discover below. Although this postulate sounds a bit technical, here is the physical idea: when we ride in a bus, as we accelerate from the busstop to cruising speed, and as we look out the window, we see the motions of trees and walkers relative to us change continuously and not jerkily.
So now we need to find the matrix $K$. Four more objective, experimentally testable, non relative pieces of physics now enter:
- Spatial isotropy: No direction in space has any preference over any other;
- The experimental notion of spatially "Orthogonal" corresponds in the Physical World to the idealized construction, following Euclid's axioms, of the perpendicular bisector of the line joining two points;
- "Movie Reversal": if we reverse the time co-ordinate, we invert co-ordinate transformations. A movie of an object undergoing a change of inertial motion state played backwards shows the same transformation that the object would undergo if the time co-ordinate were reversed (note that this does not generally apply to quantum states of particles, which follow the more general CPT Symmetry, but it does apply, as far as we know, to co-ordinate transformations);
- Our universe is Causal: i.e. the time co-ordinate of a cause is always less than that of its effect: "all causes come before their effects".
Given the spatial isotropy postulate, we can align our co-ordinate system so that the $x$ axis points along the direction relative motion. We then use the orthogonal postulate together with isotropy to conclude that a co-ordinate transformation is unchanged if we rotate the co-ordinate system through any angle about the direction of motion. We actually need the metric notion of orthogonal to define the rotation, and we assume that the Euclidean geometrical notion of rotation, expressed by a rotation matrix conserving the Euclidean inner product, corresponds to the physical notion of rotation. So, if we rotate our co-ordinates about the $x$-axis thus through angle $\phi$, we transform our co-ordinates so that $R_x(\phi)\,\Lambda\,R_x(\phi)^{-1} = \Lambda$ and $R_x(\phi)\,K\,R_x(\phi)^{-1} = K$, thus $K$ must commute with $R_x(\phi)$ and so the invariant subspaces of $R_x(\phi)$ and $K$ must be the same. The eigenvectors of $R_x(\phi)$ are $(0,\,0,\,1,\,\pm i)$ together with any pair of linearly independent superpositions of $\hat{T}=(1,\,0,\,0,\,0)$ and $\hat{X}=(0,\,1,\,0,\,0)$. This assertion together with the understanding that $K$ must be real implies that the most general $K$ matrix must have the form:
$$K=\left(\begin{array}{cccc}\kappa_{t\,t}&\kappa_{t\,x}&0&0\\\kappa_{x\,t}&\kappa_{x\,x}&0&0\\0&0&\kappa_{y\,y}&-\kappa_{y\,z}\\0&0&\kappa_{y\,z}&\kappa_{z\,z}\end{array}\right)$$
Here we assume our co-ordinates are column-vectors of the form $(t,\,x,\,y,\,z)^T$. Now the "Movie Reversal" postulate shows that $K$ must become $-K$ when we reverse the time co-ordinate; thus $K$ anti-commutes with $M=\mathrm{diag}(-1,1,1,1)$. Imposing this anitcommutation, we find:
$$K=\left(\begin{array}{cccc}0&\kappa_{t\,x}&0&0\\\kappa_{x\,t}&0&0&0\\0&0&0&0\\0&0&0&0\end{array}\right)$$
and only the $x$ and $t$ co-ordinates mix by the following $2\times 2$ matrix:
$$\Lambda(\eta) = \exp\left(\eta^\prime\left(\begin{array}{cc}0&\kappa_{t\,x}\\\kappa_{x\,t}&0\end{array}\right)\right) = \left(
\begin{array}{cc}
\cosh\left(\sqrt{\zeta }\, \eta \right) & \frac{\sqrt{\zeta } }{c} \sinh\left(\sqrt{\zeta} \,\eta \right)\\
\frac{c}{\sqrt{\zeta }} \sinh\left(\sqrt{\zeta }\, \eta \right)& \cosh\left(\sqrt{\zeta}\, \eta \right) \\
\end{array}
\right)$$
where $\zeta=\pm1$ is the sign of $\kappa_{t\,x}\,\kappa_{x\,t}$ and the constant with dimensions of velocity defined by:
\begin{equation}
\kappa_{t\,x} = \frac{\zeta}{c^2}\,\kappa_{x\,t}
\end{equation}
Since we can absorb any real constant we like into the rapidity parameter and still get an additive rapidity parameter (i.e. $\Lambda(\eta_1)\Lambda(\eta_2) = \Lambda(\eta_1+\eta_2)$), we have replaced $\eta\,\kappa_{x\,t}/c_I$ by $\eta$ in the above.
Lastly, we look at the signature $\zeta$. If $\zeta=-1$, then the matrix above becomes the rotation matrix:
$$\left(\begin{array}{cc}\cos\eta &-\frac{1}{c} \sin\eta\\c \sin\eta& \cos\eta \\\end{array}\right)$$
which means that for every vector between the spacetime co-ordinates of a cause and its effect, we can find an inertial frame, defined by $\eta=\pi$, where this vector is reversed in direction. This clearly violates the causality postulate, so we conclude $\zeta=+1$. When $\zeta=+1$, the transformation is
$$\left(\begin{array}{cc}\cosh\eta &\frac{1}{c} \sinh\eta\\c \sinh\eta& \cosh\eta \\\end{array}\right)$$
whose eigenvectors are $(t\,x) \propto (1,\,\pm c)$. This means that $c$ is invariant: it is the same constant for all inertial observers. Furthermore, with this choice, we recover our intuitive notion of causality: the everyday observation that causes come before effects but only if we further postulate that the invariant speed $c$ is the speed limit for the propagation of cause-effect, as in my other answer. Causality is not the only physics that would be radically changed if $\zeta=-1$: we know $\zeta=+1$ experimentally without even getting up from our seat: other physics and relationships that would arise with $\zeta=-1$ are explored by science fiction author Greg Egan in his trilogy Orthogonal[4]. A wonderful and correct summary of some of these weird changes in a non Lorentzian universe are given as a primer for his trilogy on Egan's website[5] and include a variable lightspeed depending on wavelength, thus a spectral spread of colors in the night sky, the {\it decrease} of a body's total energy as its speed increases and the {\it emission} of light by plants to allow them to gain energy by photosynthesis.
Now, if we set $c\to\infty$ we recover Galileo's Relativity and it is thus seen to be the unique relativity in keeping with our postulates that has absolute time i.e. all inertial observers measure the same time interval between two events. Thus we see that Special Relativity is simply Galileo's relativity with the assumption of absolute time relaxed. When we relax this assumption, Galileo's relativity foretells a whole family of relativites, each parameterized by a different value of $c$.
So in summary, $c$ doesn't need to be thought of as a speed, but rather simply a universal, nonrelative constant that chooses which of Galileo's relativities is followed by our Universe, and it has the experimental meaning of the maximum speed of propagation of a cause-effect link as I describe in my other answer. To arrive at this conclusion, we have used the nonrelative, objective, experimentally reproducible physical postulates discussed in this answer.
It should be stated that the first person to think along the lines of a relativity not predicated on light was Vladimir Ignatowski in 1910[6]. Other references describing and building on his approach are given in the bibliography of my paper [2], of which a preprint can be seen on my website.
References
[1]: Charles Misner, Kip Thorne & John Wheeler, Gravitation - the famous "big black book", quite a tough project to read, but ultimately very clear and a definitely worthwhile acquisition for anyone interested in either special or general relativity. A hardcover version is also essential (get it second hand: $202 for a paperback is outrageous), as this beast is so large its own weight utterly destroys even the best quality paperback bindings after a few months of fireside reading. You could also wait for a Kindle edition, or, as I did, buy a paperback and have it scanned.
[2]: Rod Vance, "Of Galileo, Groups and What's So Special about the Speed of Light", under review by the European Journal of Physics
[3]: Benjamin Crowell, General Relativity See Chapter 2 for an excellent discussion of how the form of the Lorentz transformation follows from affine and metric geometry. Incidentally, he also has a special relativity book
[3b] Benjamin Crowell, Special Relativity but, for fundamentals like we are thinking about here, I actually find the relevant sections of his GR text clearer. He also has a fun read:
[3c] Benjamin Crowell, Relativity for Poets wherein he gives a great presentation of the underlying ideas, philosophies and history of relativity. Although it's meant to be "Relativity Lite" for nonspecialists, nonetheless it does give some clear insights not present in more mathematical treatments, and so is a good read for physicists as well.
[4]: Greg Egan, The Orthogonal Trilogy: book 1 The Clockwork Rocket, (2011) book 2 The Eternal Flame, (2012) both published Nightshade Books; book three The Arrows of Time, (2013) Orion Publishing Group
[5]: Greg Egan, "Plus, Minus: A Gentle Introduction to the Physics of Orthogonal"
[6]: Vladimir Ignatowski, "Einige allgemeine Bemerkungen über das Relativitätsprinzip", Physikalische Zeitschrift 11 pp972–976, 1910 English Translation, "Some General Remarks on the Relativity Principle" is here
