Light can only travel at one speed (in a vacuum), approximately 300,000 km/s. It doesn't matter what frame of reference it is created in, it never goes faster or slower than this speed, and it doesn't matter what frame of reference you are measuring it from, you will always measure it to be the same speed.
This is given by Maxwell's equations, Einstein's Theory of Relativity and backed up by all experiments we have done to test this. I don't think anything contradicts this including Julian Barbour's theory.
So in this sense the speed of light is special. In another sense the speed of light has to be something so whatever figure it is is not remarkable in itself, it is simply another constant of physics.
From what I can gauge from that article, Julian Barbour's theories are essentially the same as Einsteins theory of relativity in that they predict the same things. Where they differ seems to be in the separation of time from space-time and in the definition of a theory of gravity which is not based on space-time. In Julian Barbour's theory time is emergent (not tided to space) but otherwise roughly the same (e.g. time dilation still occurs). This had some repercussions for gravity at larger distances (due to the different interpretation of time) that may effect our understanding of dark matter and dark energy. But nothing conclusive so far.
Another "absolute" in Einstein's relativity, which the article did not go over too much, is light. Einstein thought light is the "speed limit" of the universe. The way I understand it, any 2 people, no matter how far apart they are or how fast they are going, see an event at a 3rd location as happening at the same time, relative to their locations.
Your understanding is incorrect. You cannot compare clocks in different frames of reference, nor agree on an event in a third frame of reference as having occurred at the same time in those two frames of reference. You are correct that the speed of light seems to be the 'speed limit' of the universe. All experiments we have done bear this out (nothing has ever been measured going faster than the speed of light) and it seems likely that nothing can be. I couldn't see anywhere that Barbour's theory differed from or contradicted this.
Therefore, whenever you move, you travel through not just space, but also time (called time dilation), because others must see you at each different place at the same time as each other.
We are always travelling through time (time always travels forwards) and essentially Earth is moving through space so we are always moving. I don't think this the right way to look at things. Its better to think of time dilation as occurring as speeds approach the speed of light. Time dilation simply means that time slows down for that frame of reference (relative to a stationary one). Both frames are still there and can measure things, its just that their clocks may differ in the times they record for events outside their frame of reference.
Therefore, we can not travel at the speed of light, because then others would see us at all places at once.
No this is wrong. We can't travel at the speed of light as it requires increasingly more energy to accelerate us to that speed (infinitely so) so we can never reach it. But extrapolating (thought experiment only) then if we see someone in a spaceship travelling past us at the speed of light, we see time frozen for them (e.g. no movement whatsoever in the space ship, including no aging of the pilot, no electrical signals, no movement of air particles, etc). The spaceship is entirely frozen relative to itself, but relative to us it is flying by at the speed of light. It's not everywhere, its still a single entity moving past.
I believe that if we go faster than the speed of light, we travel backwards in time because others see us going backwards. If any of this is wrong or confusing (I'm sure it is), please feel free to ask for clarification or just edit if you have enough rep.
People have postulated this but currently it is just extrapolating the laws of physics past their known limits. As far as we know you can't go faster than the speed of light so this can't occur.
Is the speed of light special in Julian Barbour's theories of relativity like it is in Einstein's?
I'm just going by what I read in the link to the article you provided but from what I understand Barbour's theory doesn't treat the speed of light any differently from Einstein's theory of relativity. E.g. its still the speed limit for all matter and nothing can go past it. His theory does separate time from space, but I don't think this leads to any new behaviors of time (e.g. travelling backwards in time) or for time dilation at relativistic speeds (e.g. it would still behave like Einsteins models predict).
Science is full of ideals in its wordings. This is one of them.
SI has fixed the speed of light in a vacuum to be 299,792,458 m/s. If there was indeed light propagating through a perfect vacuum, that would be its speed ... because we define it to be.
For practical purposes, however, we need to be able to design experiments with which to measure distances using this definition. We have done these sorts of experiment regularly in high vacuum, on par with or more extreme than the vacuum of interstellar space. When we look at the effect matter has in slowing the speed of light, we find that the difference between its speed in a perfect vacuum and an achievable vacuum is smaller than the measurement error on our experimental devices. Before we had fixed the speed of light to be a constant, we had measured it to within 1 m/s.
How much of an effect does it have? I'm having trouble finding sources to give a definitive answer, but based on the refractive index of hydrogen as a function of pressure, I would expect interstellar levels of hydrogen to slow light by a factor on the order of µm/s. It's very difficult to measure physical things to 8 or 9 digits, and µm/s is 15 digits away from the speed of light, so our measurements in a high vacuum are as usable as if they were in a perfect vacuum.
If, at some point in the future, we discover that this approach is flawed, we will amend it, as has been done several times before—the most recent amendment being fixing the kilogram as a function of several fundamental constants.
Best Answer
I think there are two quite separate points to make in response to your question.
The first is that the speed of light is only locally constant. This means if you measure the speed of light at your position you will find it's always a bit under $3 \times 10^8$ m/sec. However if you measure the speed of light at some distance away from you the speed you measure may be different. The classic example of this is a black hole. If a light ray passes you on it's way towards a black hole you'll measure the velocity as it passes you to be $c$. However as the light approaches the black hole you'll see (I'm using the word see loosely here!) the light slow down as it approaches the event horizon. If you waited an infinite time you would see the light actually come to a stop at the event horizon.
Effects like this arise whenever spacetime is curved. The speed of light is only guaranteed to be $c$ when spacetime is flat. The reason a local measurement of the speed always returns the result $c$ is because spacetime in your vicinity always looks flat if you look at a small enough area around you. The usual analogy for this is that the surface of the Earth looks flat around you if you only look a few metres, but look further and you'll know it's curved because you can see the horizon.
Incidentally, this is a bit of a diversion, but you ask:
You aren't wrong. We normally think of gravity, i.e. spacetime curvature, being caused by matter, but actually it's caused by an object called the stress-energy tensor. Matter does contribute to this, but so does energy and even surprising things like pressure. So light is bent by energy, but because energy and mass are related by Einstein's famous equation $E = mc^2$ it takes a lot of energy to have the same effect as a small amount of matter.
But back to the speed of light and the second point.
Light is an electromagnetic field and it interacts with any charged particles it encounters. Mainly it interacts with electrons because electrons are relatively light; it does interact with atomic nuclei as well but the interaction is inversely proportional to the mass of the charged particle, and nuclei are so heavy that (for visible light) it's only the electrons that interact significantly.
When light encounters an electron it makes the electron oscillate and transfers energy to it, but the electron re-emits the energy and the light travels on unchanged. Be a bit careful trying to make a mental image of this. The light doesn't get absorbed, wait a bit, then get re-emitted - life is more complicated than this. The electromagnetic wave and the electron form a composite system and the resulting mixture has a velocity of less than $c$ i.e. in the presence of electrons light travels more slowly. Sadly I don't know of a simple analogy for this process.
Anyhow, the reason that the refractive index of say glass is greater than one is because it contains lots of electrons for the light to interact with. This interaction slows the light and increases the refractive index. The point of all this is that whenever there are electrons about the speed of light will be less than $c$.
So to summarise: the speed of light is only $c$ when it's travelling in a (locally) flat spacetime and there are no electrons (or other charged particles) about. This is pretty close to what you have in your bell jar, so yes it is the kind of vacuum you get in your bell jar. True, spacetime is a bit curved in the bell jar because it's in the Earth's gravitational field, but the bell jar is small enough that the spacetime it encloses is almost flat. It's also true that your bell jar contains more stray gas molecules than say intergalactic space, but with a decent vacuum the density of gas molecules (and the electrons they contain) is so low it makes little difference to the speed of the light.
I get the impression you were hoping a vacuum (at least as far as the speed of light is concerned) would be something more special than just pumping out a bell jar, but it isn't. I hope I haven't disappointed you!