Universe Expansion: Understanding the Role of Geometry in the Expansion of Space

[marcus]

... continuing to move apart following the initial kick given by inflation. ...

I see, you picture it as "continuing to move".
In which direction did this initial kick point?

 

[Wallace]

I see, you picture it as "continuing to move".
In which direction did this initial kick point?

Marcus, you surprise me. I thought you were beyond this! It is elementary physics to realize that the spherical symmetry of the FRW model requires that the forces are also spherically symmetric as well. Let me explain more fully though;

Imagine a very large (possibly infinite) blob of dust of uniform density. Let it start completely from rest. What will the blob do? Because of gravity, all bits of the dust will be attracted to every other bit and the blob will start to contract, in the sense that everything gets closer to everything else (thus even if it is infinite we can still say that is contracts). Now, if you do the sums you will see that the material in the blob will be following Hubble law contraction (same as Hubble flow, but contracting). We don't need any funky relativity here, this thought experiment is entirely Newtonian. If you want, you could describe this situation as 'contracting space' as long as you knew that you were just inventing a metaphor, not describing physics. You could derive co-moving co-ordinates and all the rest.

Now, imagine that we turn gravity OFF after a small amount of time. What happens? Does everything stop moving? No, of course not, the dust blob continues to follow a Hubble law contraction and space continues to contract in the invented metaphor. The second derivative of the contraction rate would go to zero, but the contraction would continue.

So, in this situation, in which direction was the kick that was applied before we turned gravity off? Easy, every particle can say they every other particle received a kick towards it. All kicks are inwards to every point. This is an inevitable consequence of Gauss's law and the homogeneity of the material. It is Aristotelean to suggest that motion requires something to perpetuate it, so let's get with the Renaissance and realize that mass has momentum and we are done.

Now, how do we go from this thought experiment to our Universe? Easy. Inflation was (we think) driven by some inflaton field that had negative pressure. This means that it acts as a kind of 'anti-gravity'. For some reason the inflaton field turned off after a small period of time, and ever since* the expansion has been simply due to momentum originally given by that kick. You just need to go back to the original though experiment and apply anti-gravity in the beginning. You still get the same answer, all 'kicks' can from each point be viewed as going outwards in all directions.

* Note in fact that the expansion has slowed of course due to gravity, and is now once again accelerating due to dark energy, which like the inflaton has negative pressure. The point is that forces operate on the second derivate of motion, not the first and the symmetry of the Universe means that all forces and radial and isotropic from every Hubble flow observers viewpoint.

 

[v2kkim]

Marcus, you surprise me. I thought you were beyond this!

* Note in fact that the expansion has slowed of course due to gravity, and is now once again accelerating due to dark energy, which like the inflaton has negative pressure. The point is that forces operate on the second derivate of motion, not the first.

Some people may say I am wrong but I like to post my opinion of the expansion rate. Hopefully it helps us to have a better perspective of our universe.

The expansion rate does not accelerating but decreasing in a sense that the expansion rate means the rate of length change for a unit length. Its unit could be %.
But when we monitor actual objects like very far away galaxies their recession speed is accelerating, which is not surprising considering that the recession speed goes up with distance.
In my option, we 'd better use the expansion rate (%) when talking of acceleration universe.

 

[Wallace]

If expansion was simply units changing then we would not see it, since everything we measure with would also change and we would simply see the Universe as static.

We do not observer any individual objects to be accelerating. To do this requires measuring 'red-shift drift' which would take a 42 metre telescope 10 years of continual observations to measure (our biggest today are 10 metres). Instead we infer acceleration by measuring, effectively, the distance vs redshift of a range of objects at a single instant (in cosmological terms data we take over human lifetimes is an instant).

 

[marcus]

... continuing to move apart following the initial kick given by inflation. ...

I see, you picture it as "continuing to move".
In which direction did this initial kick point?

...
Imagine a very large (possibly infinite) blob of dust of uniform density. Let it start completely from rest. What will the blob do? Because of gravity, all bits of the dust will be attracted to every other bit and the blob will start to contract, in the sense that everything gets closer to everything else (thus even if it is infinite we can still say that is contracts). Now, if you do the sums you will see that the material in the blob will be following Hubble law contraction (same as Hubble flow, but contracting). We don't need any funky relativity here, this thought experiment is entirely Newtonian. If you want, you could describe this situation as 'contracting space' as long as you knew that you were just inventing a metaphor, not describing physics. You could derive co-moving co-ordinates and all the rest.

Now, imagine that we turn gravity OFF after a small amount of time. What happens? Does everything stop moving? No, of course not, the dust blob continues to follow a Hubble law contraction and space continues to contract in the invented metaphor. The second derivative of the contraction rate would go to zero, but the contraction would continue.

So, in this situation, in which direction was the kick that was applied before we turned gravity off? Easy, every particle can say they every other particle received a kick towards it. All kicks are inwards to every point. This is an inevitable consequence of Gauss's law and the homogeneity of the material. It is Aristotelean to suggest that motion requires something to perpetuate it, so let's get with the Renaissance and realize that mass has momentum and we are done.

Now, how do we go from this thought experiment to our Universe? Easy. Inflation was (we think) driven by some inflaton field that had negative pressure. This means that it acts as a kind of 'anti-gravity'. For some reason the inflaton field turned off after a small period of time, and ever since* the expansion has been simply due to momentum originally given by that kick. You just need to go back to the original though experiment and apply anti-gravity in the beginning. You still get the same answer, all 'kicks' can from each point be viewed as going outwards in all directions.
...

We are potentially talking to newbies (newcomers). There are pedagogical issues.
You speak of a kick. I picture a kick as having a direction and resulting in ordinary physical momentum, with a direction.
In which direction is the kick?

If it has no direction then perhaps a different bunch of words?

I think we are trying to describe the Friedmann-Lemaître model, the classic 1920s model that cosmologists have always used.
"Inflation" is an unproven mix of scenarios that were proposed starting when? Around 1980? It's typically thought of as involving superluminal rates of distance-increase. Correct me if I'm wrong.
It sounds like you are invoking inflation as your "kick".

So you visualize inflation as giving ordinary physical momentum to matter. Familiar-type motion thru space. And that is how you imagine starting off the Friedmann universe?
Maybe you would repeat and spell it out in a little more detail.

I want to be sure you are talking about ordinary real momentum and not some curious kind of metaphorical momentum. I'm trying to grasp this a newcomer to PF might.

 

[Wallace]

We are potentially talking to newbies (newcomers). There are pedagogical issues.
You speak of a kick. I picture a kick as having a direction and resulting in ordinary physical momentum, with a direction.
In which direction is the kick?

Please read my post. As I explained, the kick is radially outwards from every point. Yes, I am talking about ordinary physical momentum. You don't ask why a ball keeps moving after you give it a kick do you? Why do so for galaxies then?

If it has no direction then perhaps a different bunch of words?

Please read my post. As explained, the kick is radially outwards from every point. That is a direction.

I think we are trying to describe the Friedmann-Lemaître model, the classic 1920s model that cosmologists have always used.

Right, and the most important thing about easy explanations is that they should reduce the familiar when possible. Relativity also goes to Newtonian in various limits, and you can understand the expansion of the Universe perfectly well without any relativity. The 'expansion of space' appears when you do an equivalent Newtonian treatment. The problem is that the way it is explained people think it is some strange effect of relativity, it is not, it is a metaphor deriving from one non unique co-ordinate system.

Physically it is far easier to think of the expansion of the Universe in familiar Newtonian terms. If you do the sums Newton and Einstein disagree for very distant galaxies, but the qualitative behaviour is the same.

"Inflation" is an unproven mix of scenarios that were proposed starting when? Around 1980? It's typically thought of as involving superluminal rates of distance-increase. Correct me if I'm wrong.
It sounds like you are invoking inflation as your "kick".

You are wrong. We know SOMETHING got the expansion going. At some point the Universe began expanding. In the standard model this happened in a particular way which we call inflation, so that is what I used in my explanation. However, ever since the 1920's when the FRW model was formulated it was known that something needed to provide an initial impetus to expand, that's why we have the name 'the Big Bang', in any version the theory something goes bang, push, shove or inflate in the beginning. It doesn't necessarily have to happen in the e-folding kind of way from inflation, but every bit of motion has to start somewhere.

So you visualize inflation as giving ordinary physical momentum to matter.

Yes

Familiar-type motion thru space.

There is another kind of motion??

And that is how you imagine starting off the Friedmann universe?

I didn't imagine it, Friedmann did.

Maybe you would repeat and spell it out in a little more detail.

Please read my post. There is sufficient detail there. My may have to unlearn some of the peculiar notions you've picked up, but if you come at it with a fresh mind (just think of the blob not as a universe, forgot any ideas about expanding space, just think Newtonially).

I want to be sure you are talking about ordinary real momentum and not some curious kind of metaphorical momentum. I'm trying to grasp this a newcomer to PF might.[/QUOTE

Yes I'm talking about real momentum. What other type of momentum is there?

I really think from reading some of the recent discussion around here that this whole issue has been made unnecessarily complicated.

 

[marcus]

So every dust grain has been given a kick in a direction which is away from all its neighbors.

As you said in earlier post, it can be a finite dust cloud. And the assumption in cosmology is uniformly distributed matter---isotropic homogeneous. So the dustcloud fills space.
And it's finite volume, so let's say space is a 3-sphere.

In what direction, in that 3D space, is the ordinary motion and the ordinary momentum?
Pick a dustgrain and let's say.

To take a simpler 2D analog as example, I'm having trouble understanding you because on the 2D surface of a sphere I find it impossible to imagine a point moving (in a direction along the surface, which here represents space) which is away from all the other points on the sphere.

I don't wish to argue with you personally, Wallace, but your finite dustcloud universe with each grain given a kick seems likely to lead to a pedagogical trainwreck. It is too full of contradictions which newcomers will stumble upon. Also it is atypical---runs counter to the usual balloon model that one gets, for example, at Ned Wright's cosmology tutorial.

 

[Wallace]

Okay, this is going to sound harsh, but I think you need to be shocked out of some mental rut you've gotten into.

Your problem is not understanding cosmology or Einsteinian relativity, it is simple Newtonian physics and Galilean relativity that you do not understand the basics of.

Forgot about your fancy analogies and try some real physics, some very simple intuative, Newtonian physics. The problem with what you advocate is that not only is it wrong, but it is more complicated that the right answer!

So, let's try and understand a self gravitating cloud of stuff. Forget Ned Wright and all the rest, if you can't understand the physics behind this you will not be able to really know what is going on with cosmology. How you choose to relate cosmology to 'newbies' is up to you, but until you yourself understand the physics, you are going to continue making a tangled mess even worse. I'm not suggesting that the explanation below (or above) is the ideal way to teach newbies, that is something that still eludes me, but it has to be better than teaching something that is wrong.

Right, so first thing to remember is that the only force operating in our cloud is gravity, which is a non-inertial force. You don't feel gravity pulling you down if you fall of a building, you only feel the force of the ground pushing up when you reach it. Actually techincally you don't really feel that either, what you feel is each bit of your body pushing the next, but I digress. So, gravity being a non-inertial force each dust grain will not feel the tug of the others the way you feel a car accelerating. Therefore imagine you are sitting on such a grain, you can be getting all kinds of acceleration but because it is all gravitational you feel nothing.

We can choose to define any individual grain as the centre of some co-ordinate system. In this system, we can use Gauss's law to infer that a dust grain at distance R from us will be accelerated towards us by a rate given by Newtons law of gravity, placing all the mass in a sphere of radius R at the origin. You should derive the expression for the second time derivative of R. What you will see in a few lines is that you have just derived the exact second order Friedmann equation (with no relativity required!).

Anyway the derivation is not so important, but it should convince you that cosmology really is pretty much just Newtonian physics, no mystical expansion of space required. Now, you still seem to be mystified by the 'direction' of the 'kick' each particle gets. If you use simple Galilean relativity, we can instead move our origin by a distance R to a new dust grain. What we find now is that our original origin is now accelerated towards the new one instead of the other way around. This is no more complicated than asking which of two astronauts on a space-walk is moving if they are drifting closer to each other. Clearly you can define either or both as moving, depending on where you place the origin. Take our two astronauts. They are moving towards each other, but 'in which direction' is the momentum? If you define one as moving and the other stationary and then vice versa you will apparenetly change the direction of the momentum in the system! This will only trouble you if you don't realize that momentume is frame dependant (Galileans frames, just simple linear shift in the origin, no relativity).

The same applies for our dust cloud. If we let gravity get everything moving then turn it off, clearly at any origin you choose you can define the momentum of every other particle as being directed towards that origin. Turning gravity off means that the second derivative of displacement goes to zero, but the first derivative (that gets you momentum) clearly does not go to zero just because the 'kick' has been turned off. Momentum is conserved and the cloud drifts together. The 'direction' of the momentum of any individual dust cloud depends on your reference frame because velocity is always relative. You seem to be demanding that if momentum exists then it must have an absolute direction, when clearly from high school physics we know that all motion is realtive.

The simple symmetries of the cloud however permit the simple result that the momentum on any particle is always radial with respect to any chosen origin.

Your comment "I'm having trouble understanding you because on the 2D surface of a sphere I find it impossible to imagine a point moving (in a direction along the surface, which here represents space) which is away from all the other points on the sphere." saddens me greatly. You seem to not be grasping to most basic message of the analogy that you hold so dear. The balloon analogy is nothing but a demonstration of the fact that the motion of all points is always radial with respect to every other point. Draw any two dots on the balloon and as it inflates or deflates the motion is always radial, the points don't change their angular orientaion, regardless of which points you choose.

The point is that the balloon analogy gives the bogus idea that it is the rubber that is doing something. In fact we can see that expansion is simply due to momentum, everything is moving away from everything else because it was so in the past. Let me put it more simply, the Universe expands because of Newtons first law of motion, inertia. Why oh why do people keep wanting to invoke some kind of mechanism, like expansion of space, to explain why expansion occurs when the truth is as simple as why soaps slides across a wet floor.

If you don't like inflation then just thing of the Big Bang as an explosion. This seems like an anathema because we are always told NOT to think of this. But it is only wrong if you think of a finite stick of dynamite exploding inside a larger space. However, if you imagine an infinitely large stick of dynamite, or at least one that is big enough you have no hope of seeing the edge from where you are, then this perfectly describes the Big Bang. The pressure generated by the dynamite going off produces a force that from each point in the dynamite is directed radially towards every other point. After the explosion, clearly the momentum imparted by this kick does not dissappear, instead everything moves away from everything else, only being slowed by the gravitational pull of everything towards everything else. In any chosen reference frame there will be no net force at any point. The initial kick pushes everything away from you and the subsuquent gravitational effects slow them down.

The spherical symmetry of the universe means that you can always just think about how two point behave. If you understand that then you will understand how the whole expansion goes.

The whole expansion of space analogy mixes up first and second derivates in a very clumsy way, and this is precisely why it causes trouble when people ask the obvious question 'why don't galaxies expand with space?'. They would not ask the question if they realized that they can just use everyday intuituion about momentum to see that because galaxies aren't in the process of expanding then they simply have no reason to start doing so (Newtons first law! Really it's that simple. You don't ask why your toaster doesn't spontanteously jump upwards when no force is applied do you?). Instead we hear that 'galaxies are so small that the expansion of space on that scale is negligable' or 'the gravity of a galaxy overcomes the expansion of space'. Both of these statements explicately equate first and second derivates as if they were the same. This cannot be the basis of a satisfactory understanding, no matter how seductive it may seem.

 

[marcus]

That's enough of this, I think.

 

[v2kkim]

I see two different point of view on far away galaxy expansion:
(1) Space expansion point of view : space is expanding everywhere.
(2) Continuing expansion motion of objects : a kind of simple Newton inertia.

One common result is Galaxies will expand continuously each other if there is no disturbing factors like gravitation. And the 2 theories will result in the same mathematical formula -- continuous expansion. But they show quite different picture in dealing with within galaxy motion.
Here, one problem of the 2nd theory might be the redshift explanation, because when we use the word motion then people tend to associate relativity. However remote galaxy redshift calculation does not use special relativity but use only space scale factor.

 

[marcus]

I see two different point of view on far away galaxy expansion:
(1) Space expansion point of view : space is expanding everywhere.
(2) Continuing expansion motion of objects : a kind of simple Newton inertia.

Here, one problem of the 2nd theory might be the redshift explanation, because when we use the word motion then people tend to associate relativity. However remote galaxy redshift calculation does not use special relativity but use only space scale factor.

You can calculate the redshift using special relativity and the SR doppler formula, but it is complicated and clumsy. You imagine a chain of observers and the whole expansion history is involved.

The simple direct way is just what you said: use the scale factor.
1+z = scale(now)/scale(then)
or however you want to write it.

=================

Vakkim, if you want my perspective, what I like to do is stick close to conventional cosmo language practice and coordinates, so there is as little confusion as possible.
Cosmologists use Hubble Law and they use Friedman Model.

In practice that means using coordinates (like so-called comoving ones) where stationary objects are at rest relative to
1. the CMB (no doppler hotspot ahead of them)
2. the expansion process itself
3. the ancient matter of the universe, if you want to think of it that way.

It is a natural idea of rest and coordinates like that are natural to use in cosmo.
Other special purpose coordinates are normally defined in terms of them.
And the Hubble Law is expressed in terms of the corresponding distance.

So as far as I can see there is no issue! There is no good alternative way for a beginner to get a solid introduction. Hubble Law and Friedman model have to come first. Scalefactor. The idea of comoving---being at rest relative the expansion process or the CMB. That gives the universe timescale that works in the Friedman model. It all hangs together.

=======================

The one thing you have to remember when approaching things the standard cosmology way is that space is not a substance. When we talk about expansion we are not actually talking about a material space expanding, we are talking about geometry---distances are expanding.
I think you are sophisticated enough about this that even if you might sometimes talk about space expanding you have something more abstract in mind.
I feel confident that you aren't thinking of a piece of rubber!
In the balloon model, people should concentrate on the geometry of the pattern of white dots painted on the balloon. They should not concentrate on the rubber. It is the geometric pattern that obeys Hubble Law. The balloon is only there to "carry" the geometry. If we could de-materialize her and still keep the white dots we would