# B Did anyone actually see the space moving faster than light?

#### PAllen

I assume the rulers are not attached in any way. Then, if the rate of expansion is neither accelerating nor decelerating, the balls and rulers will remain exactly as they started. Because they had no relative motion to start, they will not develop any. Comoving observers move apart because they always were moving apart.

If the rate of expansion is accelerating, then the balls and rulers will separate from each other. If the rate of expansion is decelerating, the balls and rulers will develop pressure, over time.

One further thought is that a better SR analog of recession rate is to consider the growth of distance, in some inertial frame, of two oppositely moving bodies as a function of their proper time. This is really the way cosmological recession rate is defined. The result is that even in SR, there is no upper bound on this recession rate - it can easily be a trillion times c. Yet the relative velocity of the two bodies is always less than c. That is, comparing recession rate to c as a relative velocity limit is a nonsensical category error.

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#### Bandersnatch

I assume the rulers are not attached in any way. The, if the rate of expansion is neither accelerating nor decelerating, the balls and rulers will remain exactly as they started. Because they had no relative motion to start, they will not develop any. Comoving observers move apart because they always were moving apart.

If the rate of expansion is accelerating, then the balls and rulers will separate from each other. If the rate of expansion is decelerating, the balls and rulers will develop pressure, over time.
I don't want to hijack the thread, so just a quick question:
I've only seen the tethered galaxy problem (which is what this is) analysed for recession velocities <c. But, say, we separate the balls beyond the Hubble radius and expansion is steady. The distant ball should have constant proper distance w/r to the other ball (observer). Yet, the environment around the distant ball should be receding in such a way that its proper distance grows > c as seen by the observer.
Can you comment on how to approach this so that it makes intuitive sense? (or where I've go my distance definitions confused)

#### PAllen

I don't want to hijack the thread, so just a quick question:
I've only seen the tethered galaxy problem (which is what this is) analysed for recession velocities <c. But, say, we separate the balls beyond the Hubble radius and expansion is steady. The distant ball should have constant proper distance w/r to the other ball (observer). Yet, the environment around the distant ball should be receding in such a way that its proper distance grows > c as seen by the observer.
Can you comment on how to approach this so that it makes intuitive sense? (or where I've go my distance definitions confused)
I am not sure this is the same as the tethered galaxy problem. The key, in this problem, is that the geodesics followed by the balls initially have no redshift or blue shift. If one assumes one of the balls is comoving, the other has a peculiar velocity toward the first ball arbitrarily close to c relative to a a coincident comoving observer.

If you think of my SR example, the symmetrically opposite moving balls have recession rate much greater than c, while any balls starting with no spectral shift will have recession rate of zero, forever.

Consider asking your question in the Milne cosmology, which has a maximal expansion rate without cosmological constant, yet is just Minkowski spacetime in funny coordinates.

FYI, I am generally a follower of Weinberg in the sentiment that expanding space is the root of all evil, a mildly exaggerated statement reflecting the idea that it is a superfluous concept prone to many misunderstandings.

#### fbs7

Wow, most fascinating discussion! Thanks all very much!

#### PAllen

I don't want to hijack the thread, so just a quick question:
I've only seen the tethered galaxy problem (which is what this is) analysed for recession velocities <c. But, say, we separate the balls beyond the Hubble radius and expansion is steady. The distant ball should have constant proper distance w/r to the other ball (observer). Yet, the environment around the distant ball should be receding in such a way that its proper distance grows > c as seen by the observer.
Can you comment on how to approach this so that it makes intuitive sense? (or where I've go my distance definitions confused)
I am not sure this is the same as the tethered galaxy problem. The key, in this problem, is that the geodesics followed by the balls initially have no redshift or blue shift. If one assumes one of the balls is comoving, the other has a peculiar velocity toward the first ball arbitrarily close to c relative to a a coincident comoving observer.
.....
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I thought of a way of motivating my claim in the general case, without reference to SR or the Milne cosmology.

Consider that there is some galaxy that you can see at extreme red shift such that per standard cosmological coordinates its recession rate was superluminal at emission time. (Note, this argument does not apply to hypothetical galaxies beyond the cosmological horizon, which you can't see).
That you can see it means that from an event epsilon earlier than what you see, it is possible to construct a timelike path from it to you now. Given that there is a time like path from this spacetime vicinity to you, for which both standard cosmological distance and Fermi-Normal distance decrease from some large value to 0, it follows there is some other timelike path from this vicinity that maintains constant Fermi-normal distance (while the comoving galaxy, of course, increases superluminally in standard cosmological distance, and much slower - but still very fast - in Fermi-Normal distance). Posit that the far ball and rulers move along such constant Fermi-Normal distance timelike world lines.

The only requirement for this argument is that you can see the galaxy.

"Did anyone actually see the space moving faster than light?"

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