Speed of light in an expanding universe

[Warp]

From answers given to me here and elsewhere, I finally got a (what I think should be the right) answer to the problem:

Suppose that the universe were expanding in such a way that two planets were receding from each other at a constant speed of 1.1c (or, in other words, their distance would grow at a constant speed of 1.1c). Assume that one of the planets sends a light ray to the other. Will the light ray ever reach the other planet?

Quite surprisingly and rather unintuitively, the answer seems to be yes: The light ray does reach the other planet eventually, even though that planet is receding faster than c. The mechanism behind this lies in how the geometry of the universe is changing, and is easiest understood with an analogy: If there's an infinitely stretchable rubber band with a snail on one end, and the snail advances 1 meter per hour and the rubber band is stretched 1.1 meters per hour, will the snail ever reach the other end? The answer is yes.

This got me thinking: The snail reaches the other end because it's effectively traveling faster than 1 m/hour (because the stretching rubber band is "pulling" the snail).

Doesn't it likewise mean that the ray of light is effectively traveling faster than c, or else it would never reach the other planet?

Does this mean that the speed that light effectively travels is larger in an expanding universe than in a (hypothetical) steady-state universe? (And conversely, light would travel slower in a shrinking universe.)

If the answer is yes, then what consequences does this have to the definition of c?

 

[clamtrox]

This got me thinking: The snail reaches the other end because it's effectively traveling faster than 1 m/hour (because the stretching rubber band is "pulling" the snail).

I think an easier way to visualize this is that after the snail has moved 1/10 of the rubber band's length, then the stretching of the rubber band ahead of the snail is smaller than 1m/hour. The snail moves with respect to the rubber band always at 1m/s. Likewise, photon move with respect to space always at the speed c.

 

[Warp]

I think an easier way to visualize this is that after the snail has moved 1/10 of the rubber band's length, then the stretching of the rubber band ahead of the snail is smaller than 1m/hour. The snail moves with respect to the rubber band always at 1m/s. Likewise, photon move with respect to space always at the speed c.

But that's why I used the word "effectively". The distance between the two planets grows faster than c, and thus in order for light to be able to catch up with that receding planet, it has to effectively move even faster.

In other words, light is effectively moving faster than c, which kind of sounds like a contradiction (especially since the speed of light in vacuum is supposed to be constant).

 

[jtbell]

In GR, the local speed of light in vacuum is c. This is the appropriate generalization of the statement in SR that "the speed of light in vacuum is c." Any "contradictions" come from trying to use an inappropriate generalization.

If you could measure the speed of a light pulse as it whizzes right past you, using measurements confined to a region where spacetime curvature is insignificant, you would always get c. Distance/time for the same light pulse, over a distance large enough for spacetime curvature to be significant, is another matter.

 

[clamtrox]

But that's why I used the word "effectively". The distance between the two planets grows faster than c, and thus in order for light to be able to catch up with that receding planet, it has to effectively move even faster.

In other words, light is effectively moving faster than c, which kind of sounds like a contradiction (especially since the speed of light in vacuum is supposed to be constant).

Suppose I have point A and point B 1 meter apart. Then I fire a bullet from A to B. While it's moving, I move point A 1 meter backwards. Did I just double the bullet's "effective speed"?

If we return to the snail analogy, I can take a guess of what you are thinking. You are thinking that the snail is on top of the rubber band, and the stretching is making it move faster with respect to the ground. But now when we switch from rubber bands to space, there is nothing physical acting as the "ground" in this analogy. You can of course embed an expanding universe into a static one, and in that case things work exactly like in the snail analogy. But you have no reason to talk about quantities in the static background, because that's just a mathematical trick you've done. It corresponds to nothing physical. Velocities are only meaningful if you define them with respect to the rubber band.

 

[Bill_K]

Suppose that the universe were expanding in such a way that two planets were receding from each other at a constant speed of 1.1c (or, in other words, their distance would grow at a constant speed of 1.1c). Assume that one of the planets sends a light ray to the other.

How does one calculate the redshift in a case like this?

EDIT: I'll answer my own question. The redshift is λ_now/λ_then = a_now/a_then, where a(t) is the "radius", a length characteristic of the size of the expanding universe. The redshift we observe depends only on how much the radius changes from emission to reception, not how fast it took. So it's independent of da/dt, and having da/dt > c would not be a contradiction.