Does the Hubble expansion of space have a preferred frame?

[Strilanc]

I have a paradox that I don't know the answer to. Special relativity has no preferred frame, but it seems like adding an exponential expansion to space introduces such a frame.

The Setup

Suppose we're in a universe with a much faster expansion rate, where the space between any two objects doubles every minute. Place two balls a meter apart, wait one minute, and they'll be two meters apart.

Additionally, say we have some inertial frame F and two ships. One ship is at rest with respect to F, while the other is moving at 0.99c. The ships are equipped with arms, for placing balls that then drift away due to the expansion of space.

As the moving ship crosses above the stationary ship (with respect to F), they both release a ball one meter away from their side (in a direction perpendicular to the relative velocity between them).

Before release:

                      ship
Stationary Ship:  ------◇------○ ball
                         arm (1m)             ^                      ^
Moving Ship: ^    ------◆------●    ^
             ^                      ^
            0.99c                  0.99c

During release (retract the arms), when the ships are on top of each other:

retracting arms...
   ---◈---   ◉

(*ships on top of each other; length contraction not to scale at all)

One minute later:

      ◆            ●

... far away ...

      ◇            ○
       -----2m-----

Two minutes later:

      ◆                        ●

... far far away ...

      ◇                        ○
       -----------4m-----------

Etc.

Same Factor, Different Durations

The stationary ship can check that the ball's distance does in fact double each minute by bouncing photons off of the ball and seeing that it takes twice as long for the photons to return.

But now consider things from the perspective of the moving ship. Things are twice as far apart one minute later in the rest frame, but in the moving frame clocks run slower by a factor of ~7. In order for things to stay consistent, it seems like the ball's distance must double every 8.4 seconds (with distance measured by bouncing photons off of the ball, and time measured by a clock on the moving ship) in order for there to be a consistent picture of spacetime. Meaning the ships will disagree about the expansion rate.

So it seems like moving increases the apparent expansion rate of space, and there should be some special frame where the expansion rate is minimal.

Question

The obvious question is: where's the stupid oversight? Probably something to do with the relativity of simultaneity like it always is...

 

[PeterDonis]

Special relativity has no preferred frame, but it seems like adding an exponential expansion to space introduces such a frame.

There is a particular frame in which the expansion is homogeneous and isotropic; in other frames it isn't. But that isn't a "preferred frame" in the sense in which there aren't supposed to be preferred frames in relativity. The "preferred frame" isn't built into the laws of physics; it just happens to be a feature of a particular solution to those laws, because of the special symmetry of that solution. Any solution that has a symmetry will have a "preferred frame" in which that symmetry is manifest.

 

[PeterDonis]

it seems like moving increases the apparent expansion rate of space

No, it makes it not isotropic. Try working out how it looks in a direction perpendicular to the relative motion of the two ships. (Or, for that matter, in a direction opposite to the relative motion.)

 

[Strilanc]

There is a particular frame in which the expansion is homogeneous and isotropic; in other frames it isn't. But that isn't a "preferred frame" in the sense in which there aren't supposed to be preferred frames in relativity. The "preferred frame" isn't built into the laws of physics; it just happens to be a feature of a particular solution to those laws, because of the special symmetry of that solution. Any solution that has a symmetry will have a "preferred frame" in which that symmetry is manifest.

But it is a frame that everyone agrees on, right? Something you could talk about and use for coordination of velocities without referring to some particular physical arrangement of matter known to both parties, similar to how you can use CP violations to explain left-handed vs right-handed?

(Is the expansion "rest" frame the same as the CMB "rest" frame?)

No, it makes it not isotropic. Try working out how it looks in a direction perpendicular to the relative motion of the two ships. (Or, for that matter, in a direction opposite to the relative motion.)

Do you happen to have a link to an article or post that works this out? That would be helpful because I find it really easy to make mistakes when Lorentz transforms are involved.

 

[PeterDonis]

it is a frame that everyone agrees on, right?

I'm not sure what you mean by this. If you mean, everyone agrees on which observers (in which state of motion) are at rest in this frame, yes, that's true. (These observers are usually called "comoving" observers in the literature, and which ones they are is an invariant, that everyone agrees on.)

(Is the expansion "rest" frame the same as the CMB "rest" frame?)

Yes.

Do you happen to have a link to an article or post that works this out?

No. But it should be straightforward.

 

[Strilanc]

Yes.

Interesting! Is there a mathematical reason they're the same, or is it a coincidence? Does the expansion somehow dilute the CMB's motion with respect to the expansion "rest" frame so that they converge over time?

No. But it should be straightforward.

Alright, I'll give it a try.

 

[PeterDonis]

Is there a mathematical reason they're the same, or is it a coincidence?

It's not a coincidence; they have to be the same by definition. The "comoving" frame is the frame in which the universe, including the matter and energy in it, appears homogeneous and isotropic. The CMB rest frame is the frame in which the CMB appears homogeneous and isotropic. The CMB is part of the matter and energy in the universe, so the latter frame has to be the same as the former.

In fact, the CMB, since it's composed entirely of radiation, is the best measure of the universe's homogeneity and isotropy that we have, because, unlike the matter in the universe, which has clumped into gravitationally bound systems, the CMB really does fill all of the universe pretty much evenly, at a constant density.

 

[bcrowell]

It's not a coincidence; they have to be the same by definition. The "comoving" frame is the frame in which the universe, including the matter and energy in it, appears homogeneous and isotropic. The CMB rest frame is the frame in which the CMB appears homogeneous and isotropic. The CMB is part of the matter and energy in the universe, so the latter frame has to be the same as the former.

I would put this in a slightly different way. A perfect fluid can be defined as a matter field or fields such that a frame exists in which the stress-energy tensor is isotropic. We refer to this frame as the fluid's rest frame. So I would say there are two nontrivial things going on that are not just matters of definition. (1) The matter fields that we have are ones that are capable of behaving as perfect fluids. (2) The Einstein field equations connect the isotropy of the matter fields to the isotropy of spacetime.

 

[PeterDonis]

I would say there are two nontrivial things going on that are not just matters of definition. (1) The matter fields that we have are ones that are capable of behaving as perfect fluids. (2) The Einstein field equations connect the isotropy of the matter fields to the isotropy of spacetime.

I agree that these are not matters of definition. But the question is whether the two frames--"comoving" and "CMB rest"--being the same is a matter of definition. I would say it depends on whether we define the "comoving" frame as the frame in which spacetime (or, more precisely, a spacelike slice of constant time) is homogeneous and isotropic, or the frame in which the matter/energy is homogeneous and isotropic. The latter must be the same as the CMB rest frame by definition. The former being the same is, as you say, a matter of physics, since the EFE has to be invoked to connect the two.

 

[loislane]

I would put this in a slightly different way. A perfect fluid can be defined as a matter field or fields such that a frame exists in which the stress-energy tensor is isotropic. We refer to this frame as the fluid's rest frame. So I would say there are two nontrivial things going on that are not just matters of definition. (1) The matter fields that we have are ones that are capable of behaving as perfect fluids. (2) The Einstein field equations connect the isotropy of the matter fields to the isotropy of spacetime.

Nicely worded. I would add to (2) that this connection necessarily follows an improper conservation theorem as described by Emmy Noether in 1918 in her "Invariant variational problems", section 6. This was expressed in more accessible way by Penrose in his science popularization tome "Road to reality"(page 456):
"The problem can be phrased as the fact that the extra index b in Tab prevents it from being the dual of a 3-form, and we cannot write a coordinate-independent formulation of a ‘conservation equation’"

 

[Strilanc]

Is the expansion rest frame determined by the distribution of matter in the universe, or an independent thing? I was thinking of it as an independent thing, and if that were false it would explain some of my confusion.

 

[PeterDonis]

Is the expansion rest frame determined by the distribution of matter in the universe, or an independent thing?

It's determined by the distribution of matter. The "expansion rest frame" is the frame in which, on average, the matter looks homogeneous and isotropic.

 

[Strilanc]

That explains it then; the thought experiment with just the two ships can't have the expansion simply defined to be at rest w.r.t. one of the ships because it's already defined by their relative positions and velocity.

I also assume "interesting" things happen if the matter distribution isn't isotropic in any frame, as would be the case if the universe was just two ships and two balls.

 

[PAllen]

It is also worth pointing out that 'frames' in General Relativity are local. Thus, associated with an observer who detects isotropy in CMB (for example) is a frame, and local observers moving in this frame do not detect isotropy.

Globally, there are coordinates which manifest isotropy everywhere, and in which comoving observers have constant position over time.

However, these coordinates are different from the most natural extension of a comoving observer's frame. In the extension of comoving observer's frame using, e.g. the procedures used to establish Minkowski coordinates (specifically, for example, setting up Fermi-Normal coordinates from a given comoving observer), you find that other comoving bodies are in relative motion to the chosen one [but never superluminal relative motion]. These frame expanding coordinates are completely different from standard cosmological coordinates.