is the speed of light any faster in low density space than in high density space?
What do you mean by density of space? Do you have some source for this terminology? (It is very non-standard.)
The speed of light measured in a lab (locally) will be the same everywhere, at all times, in a vacuum, according to current theories.
The speed of light in matter is slower than c, potentially much slower than c for exotic materials.
In vacuum the velocity of light is lower in a region of strong gravity. If you send an electromagnetic signal through the solar system it will travel slower close to Jupiter, the Sun etc. However, the frequency of an atomic watch will also slow down in the gravitational field by exactly the same amount so locally the velocity of light will be percieved as invariant.
The extreme case is the Schwarzschild radius of a black hole where both the rate of time and the velocity of light, according to general relativity, will become infinitely slow.
Light in a strong gravity well will appear slower to an observer further away. However, the observer in the strong gravity measuring the speed of light over any short distance will measure it as c. The definition of a semi-Riemannian manifold means that sufficiently locally light, will have speed c, at all times, and all places (there is flat Minkowski tangent plane). I don't think we are really disagreeing here.
This is different from the effect of matter on lightspeed, which occurs for local measurements.
No, we do not really disagree. The different ways in which light interacts with matter, thus affecting lightspeed is of course not a relativity issue. The velocity of light in a spherically symmetric gravitational field varies, if I am not misinformed?, with the factor:
but time slows down with the same factor, so using a local clock the velocity is always c. I am not exactly sure how one would correctly add up the slowdown of the light due to the Earth, the Sun and so forth, perhaps someone could inform me?
If you send an electromagnetic signal to some far off place away from the sun and even the milky way, such as the andromeda galaxy, then the velocity of light in the space between the galaxies where there are no significant masses nearby, even though always equal to c locally, should be higher than in our solar system.
Does anyone have any estimate on how high the velocity of light would be under such circumstances, compared to here?
We agree on the observation that a local measurement of lightspeed in a vaccuum is always c. You attach signficance to a coordinate speed of light that I don't feel has any fundamental significance. At any point, if I set set up Fermi-Normal coordinates, coordinate speed is c. Why should your choice of coordinates have more fundamental meaning than mine?
It is true that if you measure 'ruler distance' divided by 1/2 round trip time over significant distances in curved space time you can end up with speeds either greater than c or less than c. This is a real physical prediction, however I think it has never been measured. (Non light based distance measures of sufficient length in regions of sufficient gravity have not been achieved, I think).
When one talks about light speed variation over great distances, in practice, one is talking about choice coordinate conventions which could just as easily be made differently.
As has been pointed out, the locally-measured speed of light always has the same value, c. There is no unique answer for the "remotely measured" speed of light; it depends what coordinate system (or convention) you choose to use, and in general relativity all coordinate systems have equal status.
Here's an analogy. When you draw a map of part of the world on a flat sheet of paper, the map has a scale, e.g. 1:25000 (or equivalently 4 cm to 1 km), or 1 inch to the mile, or whatever. For maps that cover a relatively small area, say, less than 100 × 100 miles, the scale is near-enough constant across the whole map. For larger maps, the curvature of the Earth's surface cannot be ignored, and it's impossible to create a map with exactly the same scale everywhere. In this analogy, the speed of light is equivalent to the ratio of the vertical map scale to the horizontal map scale. On small maps, or (usually) in the centre of a large map, this ratio is 1:1, but the ratio may be different in other parts of the same map. But you have a choice of lots of different map projections and different projections may give different ratios at the same place.
If, for instance, the sun had been a really compact object like a black hole but with the same mass as now then, at some small radial distance (A) from the sun, the velocity of light would be only half of the velocity of light at the radial distance from the Sun to the Earth, due to the gravity of the Sun. Local measurement would measure the same c in both cases, but the velocity of light, if Sun is used as a reference point, will still be only half as high at A. I am not saying that the velocity of light at a certain gravitational potential has a more fundamental meaning than that at another potential, but still If you want to calculate how long time it takes to send a signal through the solar system, and do it correctly, you should consider that light travels sligthly faster at the radial distance of Pluto than at that of Mercury... At least that is what I think... Let us use whatever coordinates Nasa uses to calculate the ephemerides, the position and velocity of the bodies of the solar system. A cartesian coordinate system based on the barycentre of the solar system?
I think it has been measured, originally buy a guy named Shapiro, thus it is sometimes called the "Shapiro effect". When sending light from the Earth to Venus and Venus is on the other side of the Sun but the three is almost on a line, it takes longer to send signals than if the Sun had not been so close to the path of light.
Well I assume that the variation of the speed of light within spherically symmetric (probaly also other shaped) gravitational potentials is a real physical effect and not something one could make go away with a clever choise of coordinate system...
Just be sure you are measuring the same photon.
Density of space can mean different things. If that density is comprised of some things, those things may absorb the photon and "a little later" emit another one.
So even if each photon goes c, the intermediate rest stops may add up to show a measure of assuming "in" photon and "out" photon as the same photon to be slower than c.
Thanks for the note about Shapiro time delay - I knew about this, but never thought that it is equivalent to the theoretical measurement I described. On further thought, I see that it is equivalent, in particular the part about needing a distance determination independent of light (or radar).
On the rest, I continue to disagree that what you describe is anything more that a particular, useful coordinate choice, not fundamental physics required by GR. The way I look at is:
- every small piece of spacetime is like every other; really, not just apparently.
- how these are fit together produces the geometry that make predictions for sets of non-local measurements that cannot all be made consistent with flat Minkowski space.
- How you interpret this discrepancy: localizing variations to particular regions of spacetime is mostly coordinate dependent and not fundamental.
I agree if you use spherically symmetric coordinates in which planetary orbits take a particular simple shape, you see radial variation in light speed. However I can use other coordinates that distribute deviation from flatness completely differently.
1) I pick a 'stationary' point in earth's orbit and define all distances as c times 1/2 round trip radar time (this will also define my definition of surfaces of simultaneity - which will be different from more common coordinates, leading to different proper distances between objects - as this is dependent on chosen simultaneity surface). Now, for example, the Shapiro time delay is not an observed phenomenon. Instead I have the 'Shapiro orbital bump' - a small deviation from ellipticity as other bodies pass behind the sun from me.
2) I can choose a preferred radial line out from the sun, and measure distances and times from this line such the speed of light near this preferred radius is constant, for arbitrary precision, for some distance from this preferred radius for some time from a starting time. (This follows from the fact that a line has no curvature, and that I can make the metric essentially flat for a thin 4-tube around a chosen spacelike path - the initial radius at some 'synchronization time'). Now I will conclude that the speed of light does not vary radially. Instead it varies over time and also with distance from this initially chosen radial line (in a rather complicated way). Admittedly, this would be a strange coordinate system, but according to GR, equally valid to the standard one. And its interpretation of variation of light speed over spacetime is just as valid.
I am sure you can use cartesian coordinates if you like. The fact that planets are roughly spherically symmetric do not depend on the choice of coordinate system. I do not think that the shape of the planetary orbit matters much either but maybe, in a Shapiro-like setup, the one way speed of light will vary depending on wheter the "planet in the middle" is moving towards the sender or the observer so fast it can not be viewed upon as stationary...?
I do not quite get what you are saying but ignoring the gravitational effects from the planets, which will distort things, saying that the speed of light does not vary radially with respect to the sun to me is a very strange thing to say.
If I am right in my speculations in another thread (on flat sub-manifolds), then I believe you can validly interpret the Schwarzschild geometry in a way they says light speed along some chosen, preferred radial line is constant, period (not just for a short time period). Then light speed varies in a complex way as you move away from that preferred radial line (in any direction).
[Edit: In fact I think you can define distance using light such that all radial measures of light speed are constant. Light speed measured in other directions would then vary. This would be an alternative to the normal interpretation of radial variation of light speed. The latter is certainly more natural as it fits with space time symmetry - but it is not a necessary interpretation.]
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