I Transformation matrix for an expanding space

[Tendex]

Even if you don't want to consider scale factors look up the Milne universe, mentioned by another poster previously.

 

[johnconner]

Even if you don't want to consider scale factors look up the Milne universe, mentioned by another poster previously.

All right. How do I measure the spatial change for this sphere in the Milne universe?

 

[Tendex]

All right. How do I measure the spatial change for this sphere in the Milne universe?

You can compute it from the Milne metric in spherical coordinates here: https://en.wikipedia.org/wiki/Milne_model

 

[PeterDonis]

Expansion is homogeneous. it occurs everywhere.

Not in the sense you appear to be thinking of it. The Earth is not expanding. A sphere 1 m in radius made of, say, iron is not expanding. Atoms are not expanding.

In terms of the coordinates that we normally use to describe the universe as a whole, FRW coordinates, if we assume that the spatial coordinate of the center of a sphere 1 m in radius is zero (i.e., we choose this point as the spatial origin of our coordinates), the spatial coordinates of points on the surface of the sphere will decrease with time, since the scale factor $a$ in the metric in FRW coordinates is increasing with time. But this is only a statement about coordinates; it says nothing about the physics of the sphere.

 

[johnconner]

Not in the sense you appear to be thinking of it. The Earth is not expanding. A sphere 1 m in radius made of, say, iron is not expanding. Atoms are not expanding.

In terms of the coordinates that we normally use to describe the universe as a whole, FRW coordinates, if we assume that the spatial coordinate of the center of a sphere 1 m in radius is zero (i.e., we choose this point as the spatial origin of our coordinates), the spatial coordinates of points on the surface of the sphere will decrease with time, since the scale factor $a$ in the metric in FRW coordinates is increasing with time. But this is only a statement about coordinates; it says nothing about the physics of the sphere.

I said it in #22: a simple geometrical 3-D shape. a hypothetical sphere. you can consider it the Hubble flow itself but not in cosmological scale. one meter in diameter, less, even nanometer if you want. the only thing to remember is it shouldn't be affected by the scale factor so it's not in cosmological scale. what you are suggesting is not needed in this situation. I'm not looking for different cosmic times. which makes the change of scale factor in time negligible. also since the scale of the sphere is very small then scale factor would be negligible in two farthest points on the sphere, too.

I have to mention. I'm only neglecting the scale factor in order to simplify the matter. in reality it needs to be considered but that's for another time.

 

[Ibix]

You misunderstand. A solid sphere will not expand because its internal forces hold it together. An identically sized spherical ball of dust will expand because nothing holds one dust particle tight to the next one (if it is not so dense that its gravity holds it together, anyway). This has nothing to do with size - a microbe sized sphere or a galaxy sized sphere will both expand or not depending on what's holding them together (or failing to do so).

Edit: I'm assuming an actual expanding universe here, not the Milne cosmology.

 

[Ibix]

...so what do you mean by a "simple geometric shape"? A massless shell of buoys marking the boundary of a spherical region? Or a spherical lump of metal?

 

[johnconner]

...so what do you mean by a "simple geometric shape"? A massless shell of buoys marking the boundary of a spherical region? Or a spherical lump of metal?

first one.

 

[PeterDonis]

An identically sized spherical ball of dust will expand

If the individual dust grains all start out on comoving trajectories, i.e., expanding, and if the dust is far enough from all gravitating masses, and if the dust has negligible self gravity, yes. But in practice no real ball of dust will satisfy all these conditions.

 

[PeterDonis]

I said it in #22: a simple geometrical 3-D shape. a hypothetical sphere. you can consider it the Hubble flow itself but not in cosmological scale.

There is no such thing as the Hubble flow not on a cosmological scale. If you restrict to small scales the Hubble flow does not exist. The Hubble flow is not some magical background thing that makes galaxy clusters move apart. It is galaxy clusters moving apart. It's a description of the large scale motion, not a cause of it.

 

[johnconner]

If the individual dust grains all start out on comoving trajectories, i.e., expanding, and if the dust is far enough from all gravitating masses, and if the dust has negligible self gravity, yes. But in practice no real ball of dust will satisfy all these conditions.

Ok if all the problem is finding a physical system for this scenario you can consider a system of entangled atoms which are expanding on the shell of the sphere. although achieving the constant rate of expansion is a tricky thing to do (I'm not familiar with the practical experiments so it might be even impossible to do), but it's a good example of what the sphere would actually be. Each one moves at different direction on the shell of sphere covering all directions and they have the constant rate of expansion as magnitude of velocity. since atoms are entangled they are not to be seemed as individuals but as parts of a whole. I used atoms because they have mass thus they are not emitted with the speed of light. still the practical issue of velocity remains.

 

[johnconner]

There is no such thing as the Hubble flow not on a cosmological scale. If you restrict to small scales the Hubble flow does not exist. The Hubble flow is not some magical background thing that makes galaxy clusters move apart. It is galaxy clusters moving apart. It's a description of the large scale motion, not a cause of it.

it's just a hypothetical sphere. and it's expanding with the rate of expansion in small scales like meter or less.

 

[Ibix]

Right. So you want free floating buoys out in space. Given Peter's #39, the answer would seem to be: do you want it to expand or not? If you place the buoys so that they each see the CMB as isotropic (which is probably what you want - that's at rest with respect to matter near you, ideally anyway) then I think the diameter of the sphere at time $t$ is $a(t)/a(t_0)$ times the size it was at time $t_0$ (Peter will correct me if I'm wrong...)

 

[johnconner]

Right. So you want free floating buoys out in space. Given Peter's #39, the answer would seem to be: do you want it to expand or not? If you place the buoys so that they each see the CMB as isotropic (which is probably what you want - that's at rest with respect to matter near you, ideally anyway) then I think the diameter of the sphere at time $t$ is $a(t)/a(t_0)$ times the size it was at time $t_0$ (Peter will correct me if I'm wrong...)

Is that the scale factor?
If you consider it for two seconds apart would it actually be different? if that $a$ is scale factor that remains the same. for one second separating the $t$ from $t_{0}$ scale factor won't change.
And why should we change what I suggested? is it wrong to say a sphere that does not have peculiar velocity starts expanding with rate of expansion and leave it at that? I mean not that I have something against CMB but isn't it better to talk about rate of expansion instead of observing the CMB isotropic?

 

[Ibix]

Is that the scale factor?

Yes (well, $a(t)$ is the scale factor at time $t$ - I don't think the ratio of scale factors I used has a name).

If you consider it for two seconds apart would it actually be different?

Not measurably so, in the real world.

And why should we change what I suggested? is it wrong to say a sphere that does not have peculiar velocity starts expanding with rate of expansion and leave it at that?

I didn't notice you saying this. What I specified was equivalent to saying that points on the surface of the sphere have zero peculiar velocity. Whether this means "the sphere has zero peculiar velocity" is up to you - you can have a sphere whose center has zero peculiar velocity but whose surface has an outward, inward, or zero peculiar velocity.

I mean not that I have something against CMB but isn't it better to talk about rate of expansion instead of observing the CMB isotropic?

I'm not sure one way is better than the other. Viewing the CMB as isotropic means that you have zero peculiar velocity.

 

[johnconner]

Yes (well, $a(t)$ is the scale factor at time $t$ - I don't think the ratio of scale factors I used has a name).

Not measurably so, in the real world.

I didn't notice you saying this. What I specified was equivalent to saying that points on the surface of the sphere have zero peculiar velocity. Whether this means "the sphere has zero peculiar velocity" is up to you - you can have a sphere whose center has zero peculiar velocity but whose surface has an outward, inward, or zero peculiar velocity.

I'm not sure one way is better than the other. Viewing the CMB as isotropic means that you have zero peculiar velocity.

the center yes. it has zero peculiar velocity but the surface has outward velocity which in magnitude is equal to rate of expansion.

 

[Ibix]

the center yes. it has zero peculiar velocity but the surface has outward velocity which in magnitude is equal to rate of expansion.

...i.e., the points on the surface have zero peculiar velocity. So the size is as I said.

 

[johnconner]

...i.e., the points on the surface have zero peculiar velocity. So the size is as I said.

the points on the surface won't have zero peculiar velocity. they are moving away from the center. and peculiar velocity is any velocity that is not comoving. this is not comoving. this is exactly the opposite of comoving.

 

[Ibix]

the points on the surface won't have zero peculiar velocity. they are moving away from the center. and peculiar velocity is any velocity that is not comoving. this is not comoving. this is exactly the opposite of comoving.

You appear to misunderstand the term comoving. Comoving objects, in cosmology, are those with zero peculiar velocity - i.e., those moving apart at the expansion rate. Not those whose separation does not change.

The former is what you want the surface of your sphere to do, as I understand you.

 

[johnconner]

You appear to misunderstand the term comoving. Comoving objects, in cosmology, are those with zero peculiar velocity - i.e., those moving apart at the expansion rate. Not those whose separation does not change.

The former is what you want the surface of your sphere to do, as I understand you.

No I'm certain it has peculiar velocity. because the whole point was to generalize what I said in #10. the point with one directional motion in 2-D becomes a sphere in 3-D space without center. it is comoving before expansion of its own. which means that it's comoving with the Hubble flow in our universe. but then it starts expanding to cancel the local expansion of the universe and achieve the point of view that I mentioned in #10.

 

[PAllen]

If the individual dust grains all start out on comoving trajectories, i.e., expanding, and if the dust is far enough from all gravitating masses, and if the dust has negligible self gravity, yes. But in practice no real ball of dust will satisfy all these conditions.

Just to add a little more to this, if this idealized ball of dust starts with trajectories such that they are mutually at rest (i.e. some initial slice in fermi-normal coordinates based on one of them is 4-orthogonal to all of them), then:
1) if the second derivative of a(t) is positive (rate of expansion increasing), they ball will expand
2) if it is zero (the Milne cosmology), they will remain at mutual rest
3) if it is negative (decreasing rate of expansion, not decreasing scale factor), then this ball will contract.

The initial conditions of the ball matter a lot.

 

[PeterDonis]

you can consider a system of entangled atoms

What do entangled atoms have to do with anything? You have already got enough complications without introducing quantum mechanics.

since atoms are entangled they are not to be seemed as individuals but as parts of a whole

Again, this doesn't help your scenario at all, it just introduces further complications that are irrelevant to the question you are trying to answer.

 

[PeterDonis]

If you place the buoys so that they each see the CMB as isotropic (which is probably what you want - that's at rest with respect to matter near you, ideally anyway) then I think the diameter of the sphere at time $t$ is $a(t)/a(t_0)$ times the size it was at time $t_0$ (Peter will correct me if I'm wrong...)

No correction needed; this is right.

 

[PeterDonis]

Is that the scale factor?

You labeled this thread as "I" level. That means you should already know the answer to this question. But for the record, the answer is yes, $a$ is the scale factor.

the points on the surface won't have zero peculiar velocity

Yes, they will. At least, they will if you want every point on the sphere to be "expanding with the universe". The fact that you don't understand why this is true should be a huge red flag to you that you do not have the required background knowledge to investigate this topic yet.

You really, really, really, really, really need to learn some basics about the FRW models in cosmology. You are taking up several people's time with questions that an "I" level poster should already know the answers to.

 

[PeterDonis]

Since the OP does not have the required background knowledge for the topic, and plenty of information has been presented at this point to enable further research, this thread is now closed.