Receding movement of a stopped object in an accelerating universe?

[Suekdccia]

TL;DR Summary

Receding movement of a stopped object in an acceleraring universe?

Imagine we attach an imaginary cosmological scale rope to an object that is very far away from us. Before attaching the string, the object would be receding from us due to spacetime expansion. After attaching it, tension would form in the string and we would eventually stop the object. After doing that, we let the object be free by cutting the rope. What would happen next? :

If the universe has a decelerating expansion, the object would be completely stopped and then it would begin to be attracted towards us by the force of gravity

If the universe has an accelerating expansion, the object would be completely stopped but then it would begin to move away from us again (provided the object is sufficiently far away that the speed of the expansion of the universe overcomes the force of gravity). So that if we reattach another string to the object and stop the receding movement again, after we cut the new rope,it would start moving away again and so on...

Are these scenarios right?

 

[Jorrie]

Check out the Tethered galaxy problem. It will probably answer your questions and more.

 

[Bandersnatch]

But in short, yes that's correct. Accelerated expansion = tension in the string. Deceleration = slackening of the string.

 

[PeterDonis]

in short, yes that's correct

I'm not so sure. The paper @Jorrie linked to says that the object's peculiar velocity, i.e., its velocity relative to the Hubble flow in its vicinity, asymptotes to zero. That would indicate that, whether the universe's expansion is accelerating or decelerating, the object's peculiar velocity will decrease, meaning it will not move towards us, it will start moving away from us, since that is what the Hubble flow in its vicinity is doing. Accelerating vs. decelerating just changes the details of how its movement away from us changes over time.

 

[PeterDonis]

If the universe has a decelerating expansion, the object would be completely stopped and then it would begin to be attracted towards us by the force of gravity

No, it wouldn't, because our galaxy is not the only gravitating object in the universe. The object is surrounded on all sides by gravitating matter, which is spherically symmetric around it, so to a first approximation it feels no "force of gravity" at all. When you work out the details, as the paper @Jorrie linked to does, you find that the object gets "dragged along" with the Hubble flow in its vicinity.

 

[Bandersnatch]

I'm not so sure.

That's literally the conclusion stated in section II.B.

 

[Jorrie]

When you work out the details, as the paper @Jorrie linked to does, you find that the object gets "dragged along" with the Hubble flow in its vicinity.

As I understand it, that hypothetical object would fall right through our galaxy and join the Hubble flow on the opposite side of us.

 

[Bandersnatch]

That would indicate that, whether the universe's expansion is accelerating or decelerating, the object's peculiar velocity will decrease, meaning it will not move towards us, it will start moving away from us, since that is what the Hubble flow in its vicinity is doing.

This is not how this works. The joining with the Hubble flow is not due to expansion exerting drag, as the paper stresses. The decrease of peculiar velocity comes from something else.

Consider the edge case universe with q=0, i.e. steady expansion. Neither accelerating nor decelerating, so these effects don't play any role.
You start with a tethered galaxy, so with non-zero peculiar velocity w/r to the local Hubble flow, at constant proper distance from the observer.
As time passes, the Hubble parameter goes down towards zero. This makes the Hubble flow at the constant proper distance slow down. In this way, as time goes on, the galaxy asymptotically joins the local Hubble flow. Even as it stays at constant proper distance, zero recession velocity w/r to the observer, and zero tension in the tether.
I.e. this is due to how the local Hubble flow 'dilutes' in strength as the universe expands, not due to the expansion carrying the galaxy with it.

 

[PeterDonis]

As I understand it, that hypothetical object would fall right through our galaxy and join the Hubble flow on the opposite side of us.

Yes, on reading through the paper I see that. However, I'm having some trouble understanding the paper's analysis.

The key issue is that the "distances" involved do not lie along spacelike geodesics of the spacetime. This is seen most simply in the "empty universe" case, described in the paper as (0, 0) (both $\Omega_M$ and $\Omega_\Lambda$ are zero), which is just Minkowski spacetime (or more precisely the "wedge" of it corresponding to the future light cone of the origin) in funny coordinates. The "comoving" spacelike surfaces (i.e., the surfaces of "constant time" in comoving coordinates) are then hyperbolas, whereas the spacelike geodesics are straight lines.

If we look at things in standard Minkowski coordinates, the event of us on Earth "now" can be given coordinates $(t, x) = (1, 0)$. The "tethered" galaxy, at the instant it is released, is then at some "distance" $D$ from us. But this "distance" is measured along the hyperbola $t^2 - x^2 = 1$ (the hyperbola passing through $(1, 0)$), not along the spacelike geodesic $t = 1$ (the horizontal line passing through $(1, 0)$). So the event in question is not $(1, 0.5)$, as one would "naively" expect; it is actually $(\cosh D, \sinh D)$.

Since the tethered galaxy is at rest relative to Earth, its worldline once it is released is thus the vertical line $x = \sinh D$. The comoving coordinate of this worldline will decrease with time since the comoving coordinate is just $x / t$, and $x$ is constant while $t$ is increasing. So the part about the peculiar velocity approaching zero asymptotically makes sense.

However, light emitted from this worldline has zero redshift relative to the worldline of Earth, which is just the vertical line $x = 0$. So I don't understand how the paper is concluding that the redshift of the tethered galaxy is nonzero in the empty universe case.

More generally, the mismatch between the comoving "now" surfaces and the spacelike geodesic surfaces makes me wonder about any analysis in comoving coordinates that involves peculiar velocities.

 

[PeterDonis]

this is due to how the local Hubble flow 'dilutes' in strength as the universe expands, not due to the expansion carrying the galaxy with it.

Yes, per the fourth paragraph in my post #9 just now, that part makes sense to me.

 

[PeterDonis]

More generally, the mismatch between the comoving "now" surfaces and the spacelike geodesic surfaces makes me wonder about any analysis in comoving coordinates that involves peculiar velocities.

To give an example: in Fig. 2 of the paper, for the empty universe (0, 0) case, proper distance is shown as constant. But that proper distance assumes inertial coordinates, not comoving coordinates. It's not clear to me that the "proper distance" measured along the comoving "now" hyperbolas is constant.

 

[Jorrie]

The key issue is that the "distances" involved do not lie along spacelike geodesics of the spacetime. This is seen most simply in the "empty universe" case, described in the paper as (0, 0) (both and are zero), which is just Minkowski spacetime (or more precisely the "wedge" of it corresponding to the future light cone of the origin) in funny coordinates. The "comoving" spacelike surfaces (i.e., the surfaces of "constant time" in comoving coordinates) are then hyperbolas, whereas the spacelike geodesics are straight lines.

Is their (0,0) scenario really Minkowski spacetime, or is it uniformly expanding empty spacetime (Milne)? Or are they the same?

 

[Suekdccia]

@Jorrie @PeterDonis @Bandersnatch Thank you for your answers. So, as far as I understand, two objects at relative rest may later gravitate towards each other, or if Λ>0, move away, right?

However, I was having a discussion with another physicist about this, and, while he agreed that with Λ>0 things at relative rest would move away from each other, he told me that in this case of tethering galaxies, "what doesn't exist is a frictional/Aristotelian effect that makes things want to go at a certain velocity. If you tether a galaxy, it doesn't "remember" its old speed and return to it, nor is it dragged back to it by galaxies you didn't tether, or by expanding-space aether"

What does this mean? Would the galaxies separated by spacetime expansion (after put into rest by the rope) move away at less speed than when it was initially separating? So each time we'd attach and re-attach the string to the galaxy, the tension would be weaker and weaker, as the initial speed of the galaxy could never be recovered?

 

[Bandersnatch]

What does this mean?

I think all it says is what was already said in this thread - that it's the acceleration (whatever the sign) of the expansion, not the recession that pulls the tethered galaxy from rest.
If you tether a galaxy at some distance d, the galaxies in its vicinity recede from the observer with some recession velocity given by Hubble's law. If you untether it, it does what the acceleration/deceleration dictates - it does not try to catch up to the same velocity as its neighbourhood. I.e. it's not that those receding galaxies nearby drag it along.
In Newtonian mechanics we'd say that they move inertially with some large initial velocities, while your tethered galaxy has zero initial velocity. If you'd want to change the state of motion of the tethered galaxy, you'd have to apply an accelerating/decelerating force. The initial velocities of anything don't do anything to change the state of motion.
When you think about expansion of space as a balloon, you may end up taking this picture too far, and start thinking of space as something tangible ('fabric' of space is suggestive too) that can drag things along as it expands. That comment seems to pre-emptively address this misconception.

Whether the tension in the string changes each time you reattach the galaxy depends only on whether the acceleration/deceleration changes. If it's constant, there's always the same tension. In the LCDM model for our universe, the acceleration increases - but only asymptotically. Eventually, you'll get the same tension each time you reattach a galaxy at the same distance.

or if Λ>0, move away

Not when lambda>0, but when the nett effect of lambda and the universe's self-gravitation results in acceleration. I.e. when the deceleration parameter q is <0. Lambda has been greater than 0 since forever, but it took approx. half of the universe's age to expand enough for the matter density to dilute to the point where acceleration took over from deceleration.

 

[Jorrie]

For what it's worth, I did this set of graphs a decade ago in an attempt to show, on an engineering forum, what things "look like" (taking Davis, Lineweaver, Webb at face value) for our standard model.

The object is 'untethered' at T=0.2 Gyr, at D_proper = -0.2 Glyr, with the distance scaled down to make it plottable on the scale of the velocities.
:

 

[PAllen]

Is their (0,0) scenario really Minkowski spacetime, or is it uniformly expanding empty spacetime (Milne)? Or are they the same?

They are the same. The Milne cosmology is a portion of Minkowski spacetime foliated by hyperbolic spatial slices. The foliation has nothing to do with intrinsic geometry or physics.

 

[PeterDonis]

Is their (0,0) scenario really Minkowski spacetime, or is it uniformly expanding empty spacetime (Milne)? Or are they the same?

They're the same.

 

[Suekdccia]

I think all it says is what was already said in this thread - that it's the acceleration (whatever the sign) of the expansion, not the recession that pulls the tethered galaxy from rest.
If you tether a galaxy at some distance d, the galaxies in its vicinity recede from the observer with some recession velocity given by Hubble's law. If you untether it, it does what the acceleration/deceleration dictates - it does not try to catch up to the same velocity as its neighbourhood. I.e. it's not that those receding galaxies nearby drag it along.
In Newtonian mechanics we'd say that they move inertially with some large initial velocities, while your tethered galaxy has zero initial velocity. If you'd want to change the state of motion of the tethered galaxy, you'd have to apply an accelerating/decelerating force. The initial velocities of anything don't do anything to change the state of motion.
When you think about expansion of space as a balloon, you may end up taking this picture too far, and start thinking of space as something tangible ('fabric' of space is suggestive too) that can drag things along as it expands. That comment seems to pre-emptively address this misconception.

Whether the tension in the string changes each time you reattach the galaxy depends only on whether the acceleration/deceleration changes. If it's constant, there's always the same tension. In the LCDM model for our universe, the acceleration increases - but only asymptotically. Eventually, you'll get the same tension each time you reattach a galaxy at the same distance.Not when lambda>0, but when the nett effect of lambda and the universe's self-gravitation results in acceleration. I.e. when the deceleration parameter q is <0. Lambda has been greater than 0 since forever, but it took approx. half of the universe's age to expand enough for the matter density to dilute to the point where acceleration took over from deceleration.

So if one would always get tension in the string in an accelerating universe, one could mine energy from it? (as this article indicates: https://adsabs.harvard.edu/full/1995ApJ...446...63H)

By the way, here it's the discussion I was having if you want to take a look (it's mainly in the comments of the answer): https://physics.stackexchange.com/q...asing-internal-energy-as-the-universe-expands

 

[Bandersnatch]

So if one would always get tension in the string in an accelerating universe, one could mine energy from it?

I want to say: 'I guess?' But I get the feeling this might be somewhat more nuanced than a naive view would suggest, and I don't feel knowledgeable enough to tackle it.

 

[Suekdccia]

I want to say: 'I guess?' But I get the feeling this might be somewhat more nuanced than a naive view would suggest, and I don't feel knowledgeable enough to tackle it.

Thank you

@PeterDonis @Jorrie do you have any corrections on this?

 

[PeterDonis]

Thank you

@PeterDonis @Jorrie do you have any corrections on this?

Looks like you have opened a separate thread on the "can we extract energy from accelerated expansion using a string" question:

https://www.physicsforums.com/threads/mining-the-universes-expansion.1045027/

 

[Suekdccia]

Looks like you have opened a separate thread on the "can we extract energy from accelerated expansion using a string" question:

https://www.physicsforums.com/threads/mining-the-universes-expansion.1045027/

Yes, perhaps this can clarify the problems of conservation of energy in cosmology

 

[kimbyd]

I'm not so sure. The paper @Jorrie linked to says that the object's peculiar velocity, i.e., its velocity relative to the Hubble flow in its vicinity, asymptotes to zero. That would indicate that, whether the universe's expansion is accelerating or decelerating, the object's peculiar velocity will decrease, meaning it will not move towards us, it will start moving away from us, since that is what the Hubble flow in its vicinity is doing. Accelerating vs. decelerating just changes the details of how its movement away from us changes over time.

The statement that the object will always move away from us after release does not follow from the statement that its peculiar velocity will decrease with time. The statement that an object's peculiar velocity will decrease with time is true for any object of constant velocity moving through an expanding universe.

For the case of a spatially-flat universe with no cosmological constant, I believe that once released, the far-away object would simply stay at that distance. Its peculiar velocity would decrease with time because the recession velocity at that distance will decrease with time.

I'm pretty sure this very specific situation can be examined in a purely Newtonian way using the spherical Newtonian model.

 

[PeterDonis]

The statement that the object will always move away from us after release does not follow from the statement that its peculiar velocity will decrease with time.

Yes, agreed. It depends on the specifics of the expansion profile. I am posting more detailed math for this in the other thread on this topic (linked to in post #21).

I'm pretty sure this very specific situation can be examined in a purely Newtonian way using the spherical Newtonian model.

Not for the case of nonzero $\Lambda$, since there is no such thing in Newtonian gravity. And not for cases where the geometry of slices of constant comoving coordinate time is not Euclidean (which is the case for at least the empty universe (0, 0) case given in the article), since Newtonian gravity assumes Euclidean spatial geometry.

You might be able to do this for the matter-dominated critical density case (the (1, 0) case in the article), since for that case neither of the above objections applies, but any solution you obtain won't validly generalize to the other cases. (And even in this case, there are still objections, one of which I posted in post #9 of this thread, the other of which I posted in post #3 of the other thread--the second objection in that post is the one that I'm referring to here.)