I was thinking of galaxies that are only moving away from each other in one dimension, like another galaxy at the other end of the universe moving oppositely away. The apparent velocity is simply a logical conclusion: because of acceleration, at a certain point, the distance at which the the galaxy moves over a period of times is greater than light would travel normally. Since the two galaxies accelerate at the same rate, you can calculate every instantaneous position from a central reference point, and thus you know the two galaxies relative speed, which looks v>c. Its the same way you would calculate regular speed: distance/time. However, i said apparent velocity, because if things were actually gaining kinetic energy, it would violate special relativity.Usually you want to calculate velocities locally. Calculating relative velocity of something which is half way across universe will not work like you'd expect, because velocity is a vector, which lives on a curved surface. If you want to compare velocities on two different points, you have to first move them to the same point (parallel transport, if you're familiar with the jargon), and if you do this in the case of expanding universe, you find that on average, the galaxies are not moving.
Of course, nothing stops you from talking about some sort of apparent velocity, where you forget about the curvature of the universe and just calculate what you see. You just cant' expect the apparent velocity to satisfy the usual rules, like v<c.
So let's take a concrete example. Suppose you're an observer on the north pole of the earth, and a car is driving at a constant velocity southwards. Now, if the surface of the earth were a manifold, the car would always move on a straight line in it's own reference frame. But viewed from north pole, it seems that it's motion is accelerating because it's moving on a curved surface. The apparent velocity of the car changes, while physical (local) velocity remains constant.The apparent velocity is simply a logical conclusion: because of acceleration, at a certain point, the distance at which the the galaxy moves over a period of times is greater than light would travel normally. Since the two galaxies accelerate at the same rate, you can calculate every instantaneous position from a central reference point, and thus you know the two galaxies relative speed, which looks v>c.
This makes sense, but doesn't this assume that the universe is in fact curved, and not infinite etc?So let's take a concrete example. Suppose you're an observer on the north pole of the earth, and a car is driving at a constant velocity southwards. Now, if the surface of the earth were a manifold, the car would always move on a straight line in it's own reference frame. But viewed from north pole, it seems that it's motion is accelerating because it's moving on a curved surface. The apparent velocity of the car changes, while physical (local) velocity remains constant.
It's just an analogy, not a model of the universe :) If you want, you can imagine the car to be driving on whatever surface you can think of and visualize. I just wanted to keep it as simple as possible. Also, the Einstein equations imply that spacetime has to be curved if there is stuff in it, so that assumption is in the end not very constraining. If the spacetime is flat, then there's no distinction with apparent and real velocities, no gravity and everything works everywhere by the rules of special relativity.This makes sense, but doesn't this assume that the universe is in fact curved, and not infinite etc?
Google "Metric Expansion" for a full discussion.Since the most distant galaxies appear to be faster than the speed of light, due to the insertion of new space, does this mean that galaxies haven't actually gained any velocity as the universe accelerates?