Can We Pair Composition of Velocities with an Expanding Universe?

[Tibriel]

Hi All,

I get that one object can't travel faster than light with respect to another object because of SR and length and time contraction. What I don't get is how this meshes up with expanding universe.

If the space between 2 objects is expanding then wouldn't two objects be accelerating away from each other at a pace which would eventually exceed the speed of light? Yet my understanding of SR says this isn't possible. Does this mean there is a limit to how fast the universe can expand with respect to any observer?

Please help, thinking about this is making my brain short out.
Oh, and I realize the above questions are yes/no questions so if you could also explain that would be helpful.

Thanks!

 

[PAllen]

Velocity addition formula of SR only applies locally in GR, which is required to discuss expanding universe. In GR, there is no unique definition of relative velocity at a distance, so no strong statement can be made.

Instead, different velocities are used for different purposes. Recession velocity (which is based on growth of proper distance between galaxies) can greatly exceed c. On the other hand, if one parallel transports the 4-velocity of one galaxy along its light path to another galaxy, and compare the 4-vectors, you always get a sub-luminal relative speed.

Upshot: your intuition that expanding universe creates a problem for the SR statement is correct.

 

[Khashishi]

The distance between two objects in expanding space can increase faster than c, but velocity (relative to the local background radiation) cannot exceed c. That's because distance is a global quantity, and velocity is measured locally. Unfortunately, you can't define velocity as change in global distance over time. It is the change in infinitesmal distance over infinitesmal time, measured between two objects at the same place and time. Kinda makes it hard to actually use for anything.