I Coordinate dependence of recession velocities

[timmdeeg]

Superluminal recession velocities of far away galaxies are due to the choice of FRW-coordinates. As @Ibix said here #71 "The key point, in this context, is that you will never see these galaxies overtake a light pulse."

But is there any other choice? Riemann normal coordinates don't seem to be the right answer because - if I understand it correctly - they describe the neighborhood of the origin and are thus not applicable globally.

Is the choice of FRW-coordinates compelling because curved spacetime can not be transformed away?

 

[PeterDonis]

is there any other choice?

It depends on what you mean by "other choice".

In a curved spacetime, any global choice of coordinates will have counterintuitive properties like coordinate speeds not always obeying the "nothing goes faster than the speed of light" rule. That is just a manifestation of the fact that that rule is not valid for curved spacetimes. You have to use the more general form of the rule, which is "everything moves within the local light cones"--or, as @Ibix said, nothing can overtake a light pulse. Coordinate speeds have no physical meaning anyway, so fixating on them is a mistake.

 

[PeterDonis]

Is the choice of FRW-coordinates compelling because curved spacetime can not be transformed away?

No. FRW coordinates are a compelling choice because they match the particular symmetries of this particular family of curved spacetimes. Different curved spacetimes with different symmetries make different coordinate choices compelling--for example, Schwarzschild coordinates in Schwarzschild spacetime.

 

[George Jones]

Superluminal recession velocities of far away galaxies are due to the choice of FRW-coordinates.

What can be said about "velocities" in FRW (like) coordinates in Minkowski spacetime (the Milne Universe)?

 

[timmdeeg]

What can be said about "velocities" in FRW (like) coordinates in Minkowski spacetime (the Milne Universe)?

You mean the "empty universe". In flat spacetime (the Milne universe) special relativity holds globally. So recession velocities aren't superluminal.

@PeterDonis thanks for your answers, I will come back to that tomorrow.

 

[Orodruin]

You mean the "empty universe". In flat spacetime (the Milne universe) special relativity holds globally. So recession velocities aren't superluminal.

This was not the question. The question was about coordinate velocities in Milne coordinates.

 

[PeterDonis]

In flat spacetime (the Milne universe) special relativity holds globally. So recession velocities aren't superluminal.

This is not correct. As @Orodruin said, you should look at the coordinate velocities in FRW coordinates for this case (what @Orodruin called "Milne coordinates"). You will find that those coordinate velocities are superluminal for comoving objects sufficiently far apart.

 

[timmdeeg]

This is not correct.

I am not sure why this is not correct. I've been talking about the Milne universe and thus about Minkowski spacetime with non-superluminal velocities, see The Milne universe occurs ... in flat Minkowski spacetime.

This example is very illustrative as it reveals coordinate dependent velocities in the case of the empty universe. The OP refers to curved spacetime though.

 

[Orodruin]

I am not sure why this is not correct. I've been talking about the Milne universe and thus about Minkowski spacetime with non-superluminal velocities, see The Milne universe occurs ... in flat Minkowski spacetime.

The issue was the superluminality of the coordinate speed of the separation in FRW coordinates. This is also the case for Milne coordinates. Just as in Minkowski space, parallel transport of a local (subluminal) 4-velocity between any events in the general RW spacetime will result in a (subluminal) 4-velocity at the new point.

 

[timmdeeg]

No. FRW coordinates are a compelling choice because they match the particular symmetries of this particular family of curved spacetimes. Different curved spacetimes with different symmetries make different coordinate choices compelling--for example, Schwarzschild coordinates in Schwarzschild spacetime.

But isn't the difference that for Schwarzschild geometry one can use Schwarzschild coordinates, Kruskal-Szekeres coordinates and others, whereas the geometry of our universe permits the choice of coordinates only the empty case?

 

[PeterDonis]

I am not sure why this is not correct.

That is because you are guessing instead of doing the math. Do the math.

I've been talking about the Milne universe

Yes, we know that. We know what the Milne universe is.

and thus about Minkowski spacetime with non-superluminal velocities

Minkowsi spacetime has non-superluminal coordinate velocities in standard Minkowski inertial coordinates. It does not have all non-superluminal coordinate velocities in FRW coordinates. Do the math and see.

This example is very illustrative as it reveals coordinate dependent velocities in the case of the empty universe.

Exactly: coordinate dependent velocities. Do you understand what that means?

The OP refers to curved spacetime though.

You think it does, but you are wrong, since the Milne universe is precisely a non-curved spacetime example of the phenomenon you describe. That is what we are trying to tell you.

 

[PeterDonis]

the geometry of our universe permits the choice of coordinates only the empty case?

No. There are an infinite number of possible coordinate charts you can choose on any spacetime.

 

[timmdeeg]

@PeterDonis thanks for you very instructive answers and encouraging remarks.