Questions about the accelerating universe

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.

 

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.

 

Google "Metric Expansion" for a full discussion.

 

We have a FAQ about this: https://www.physicsforums.com/showthread.php?t=508610 [Broken]