gravity - Speed of light in a vacuum

I see many references to the speed of light in a vacuum implying that it is only truly a constant measurement in a vacuum. I can live with that, but what kind of vacuum?

Are we still talking about the kind of vacuum under a bell-jar with all the air pumped out or the kind of vacuum you could find in interplanetary space, or outside the heliosphere, or between stars, or between galaxies? Each has a lower energy content and therefore a lesser effect on the light.

We know that light is noticeably bent by massive bodies, I guess it is also slightly bent by any kind of energy in its proximity, even other light (or am I wrong).

Should we be talking about the speed of light in a dark, intergalactic vacuum? If so, what would be the difference? What subtle adjustments are we failing to make by measuring this speed at the bottom of a deep gravity well here on the earth's surface?

How accurate is our measurement and could we have it sufficiently wrong to affect some other ideas we may have about the universe?

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:

We know that light is noticeably bent by massive bodies, I guess it is also slightly bent by any kind of energy in its proximity, even other light (or am I wrong).

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!