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?
Special Relativity – Understanding the Reference Frame of c in Inertial Frames and Observers
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By compression of light you mean the Doppler shift?
Then yes you can measure your speed relative to the light source by comparing the Doppler shift in different directions.
It's been used for a number of different radio positioning systems - but it only gives you a motion relative to the light sources
It sounds like your confusion is coming from taking paraphrasing such as "everything is relative" too literally. Furthermore, this isn't really accurate. So let me try presenting this a different way:
Nature doesn't care how we label points in space-time. Coordinates do not automatically have some real "physical" meaning. Let's instead focus on what doesn't depend on coordinate systems: these are geometric facts or invariants. For instance, our space-time is 4 dimensional. There are also things we can calculate, like the invariant length of a path in space-time, or angles between vectors. It turns out our spacetime has a Lorentzian signature: roughly meaning that one of the dimensions acts differently than the others when calculating the geometric distance. So there is not complete freedom to make "everything" relative. Some relations are a property of the geometry itself, and are independent of coordinate systems. I can't find the quote now, but I remember seeing once a quote where Einstein wished in reflection that instead of relativity it was the "theory of invariants" because those are what matter.
Now, it turns out that the Lorentzian signature imposes a structure on spacetime. In nice Cartesian inertial coordinates with natural units, the geometric length of a straight path between two points is:
$ds^2 = - dt^2 + dx^2 + dy^2 + dz^2$
Unlike space with a Euclidean signature, this separates pairs of points into three different groups:
$> 0$, space like separated
$< 0$, time like separated
$= 0$, "null" separation, or "light like"
No matter what coordinate system you choose, you cannot change these. They are not "relative". They are fixed by the geometry of spacetime. This separation (light cones if viewed as a comparison against a single reference point), is the causal structure of space time. It's what allows us to talk about event A causing B causing C, independently of a coordinate system.
Now, back to your original question, let me note that speed itself is a coordinate system dependent concept. If you had a bunch of identical rulers and clocks, you could even make a giant grid of rulers and put clocks at every intersection, to try to build up a "physical" version of a coordinate system with spatial differences being directly read off of rulers, and time differences being read from clocks. Even in this idealized situation we cannot yet measure the speed of light. Why? Because we still need to specify one more piece: how remote clocks are synchronized. It turns out the Einstein convention is to synchronize them using the speed of light as a constant. So in this sense, 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.
The modern statement of special relativity is usually something like: the laws of physics have Poincare symmetry (Lorentz symmetry + translations + rotations).
It is because of the symmetry of spacetime that we can make an infinite number of inertial coordinate systems that all agree on the speed of light. It is the structure of spacetime, its symmetry, that makes special relativity. Einstein discovered this the other way around, postulating that such a set of inertial frames were possible, and derived Lorentz transformations from them to deduce the symmetry of space-time.
So in conclusion:
"If all motion is relative, how does light have a finite speed?"
Not everything is relative in SR, and speed being a coordinate system dependent quantity can have any value you want with appropriate choice of coordinate system. If we design our coordinate system to describe space isotropically and homogenously and describe time uniformly to get our nice inertial reference frames, the causal structure of spacetime requires the speed of light to be isotropic and finite and the same constant in all of the inertial coordinate systems.
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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:
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:
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.