# What is expansion ? A thought experiment

1. Oct 18, 2011

### matthias31415

what is "expansion"? A thought experiment

Dear Cosmologists,

I am eager to understand the concept of "expansion". I've heard and read a lot about red shift, background radiation and the fact that distances between galaxies are increasing. I think I understand the basics of special relativity while general relativity is still a mystery to me ;-)

But I still have the feeling that I don't know how "expansion" is defined, quantitatively speaking. In other words: what is the difference between a movement within space and a movement that follows space?

To better understand the underlying concept/definitions, I invented a small thought experiment.

(a) Let us assume a static space. An observer and a light source (currently switched off) move away from each other with constant speed v. At t=0 they had met. At t=t1>0, the light source sends a light pulse. Let t2a be the timepoint, when the light arrives at the observer (time is measured by a clock next to the observer)

(b) Let us assume the same setting as in (a) above with the same speed v, except for the fact that while all this happens, space is expanding. Let t2b be the timepoint when the light arrives at the observer.

What is the relation between t2a and t2b? Are they the same?

matthias314159

2. Oct 19, 2011

### Chalnoth

Re: what is "expansion"? A thought experiment

Well, that all depends upon the contents of the universe.

Basically, if you have an empty universe, then it doesn't matter whether you say it is expanding or not. In an empty universe, you can certainly write down equations that look like an expanding universe (this is called the Milne universe). But if you actually work things through, you find that everything behaves exactly as you would expect in special relativity: it's just a weird way of looking at a flat space-time.

To seriously talk about expansion, you have to have a universe with stuff in it. And specifically we deal with a universe that is nearly homogeneous and isotropic. What this means in detail is that you can take a slice of the universe where observers at every point in the slice see that things look approximately the same in every direction, and every observer sees approximately the same thing no matter where in the slice they are. Since you have a bunch of stuff in the universe, however, this stuff bends the geometry of space-time. So you can't just reduce it simply to "the universe is expanding," you actually have to take into account the gravitational properties of the stuff that fills the universe.

That said, your thought experiment doesn't make sense in General Relativity. The problem is that there is no non-arbitrary way to synchronize clocks separated by some distance in General Relativity. There's also the problem that we don't know the relationship between the motion of these two observers and the expansion.

3. Oct 19, 2011

### matthias31415

Re: what is "expansion"? A thought experiment

thanks, chalnoth!

my question (what is the difference between motion IN space as opposed to motion WITHIN space) of course doesn't make sense in an empty universe, since, in an empty universe, there is nothing that can move. I also understand that, in an non-empty universe, as you say, things are more complicated and space isn't just expanding, but also "curved". But isn't it possible, for the didactic prupose, to temporarily assume that space is not empty (so that there is at least something that moves), but still approximately euklidic (and isotropic). In that case, do I understand you correctly that "expansion" becomes a meaningless term (which is what you say about an empty universe)?

A friend of mine also told me that, in general relativity, my thought experiment would not be as simple as it appears. But I am still not buying it! You say that there is no non-arbitrary way to synchronize clocks. I understand that. Indeed, this may be a problem when I claim that t refers to the observer's clock while I also claim that t=t1 refers to the timepoint where a distant light source emits something. But I am not looking for a precise quantitative answer (for that, of course, more details are needed), I am only interested in a qualitative comparison between t2a and t2b (smaller, equal oder greater). To this end, one can make things even more simple by assuming that the expansion in scenario b only take place after the light pulse has been emitted.

-matthias31415

4. Oct 19, 2011

### Chalnoth

Re: what is "expansion"? A thought experiment

Expansion only becomes meaningless if the universe is empty. If you have test particles in that system (which don't affect the gravitational curvature, but can still be affected by it), then you can sensibly talk about their motion. And their motion will always be exactly as predicted by special relativity. Of course, test particles don't exist in reality, but can be useful for thought experiments.

Yeah, you can't even do that. Because once you are talking about times at different places, you lose the ability to make much of any comparison in that way.

You can bring the comparison back to reality if you have only one observer sending, and have the other observer just holding a mirror that bounces the light back to the first. Then, the first observer can look at their own clock and measure the time it takes for the light beam to travel to the receding observer and back.

But then, unfortunately, it all depends upon the details of the expansion. You would have to define things much more explicitly to get an answer. The problem here is defining the motion of the other observer. That's easy to do in the flat case, but not so easy in the curved case.

5. Oct 19, 2011

### matthias31415

Re: what is "expansion"? A thought experiment

thanks again, chalnoth!

what you said seems to confirm my intuition that expansion is actually something quite difficult to grasp, contrary to the notion that "expanding" simply means becoming larger...

Two more questions come to my mind:

I saw expansion, or a(t), to be more precise, occur in the Friedman metric, where spatial distance, but not temporal distance, is multiplied by a(t) before spatial and temporal distance are combined to a 4-dimensional distance. So I could look at a(t) as a sort of weighting factor when space and time are combined, right? What would happen if, instead of space being expanded by a factor of 2, time would be shortened by a factor of two? That would leave the relative weight between space and time the same. Would it also result in similar physics or is this a stupid idea?

Second, expansion somehow seems to include the notion that particles (unless they are bound by some sort of forces) naturally follow the movement of space. Is that really so? What if, in a temporarily static corner of our universe, two objects would have a constant distance, but suddenly space starts expanding. Would this mean that these two objects experience some acceleration?

6. Oct 19, 2011

### phinds

Re: what is "expansion"? A thought experiment

Relative to what?

The creation of space doesn't exert any force on the objects, it just makes them farther apart (very NON-Newtonian, I know).

7. Oct 19, 2011

### Chalnoth

Re: what is "expansion"? A thought experiment

You mean like instead of having the FRW metric:

$$ds^2 = dt^2 - a^2(t)(dx^2 + dy^2 + dz^2)$$

...you have this:

$$ds^2 = b^2(t) dt^2 - dx^2 - dy^2 - dz^2$$

?

Well, in this case, the two metrics are really the exact same thing but with a coordinate transformation. If instead you meant this:

$$ds^2 = a^2(t)(dt^2 - dx^2 - dy^2 - dz^2)$$

...then that is just flat space-time in different coordinates.

Basically, FRW completely describes any homogeneous, isotropic space-time. You can do the same stuff in different coordinates, and many things will look very different in the math. But you can't introduce new physics without breaking the assumptions of homogeneity or isotropy. For example, we might have a universe that expands differently in different directions:

$$ds^2 = dt^2 - a_x^2(t) dx^2 - a_y^2(t) dy^2 - a_z^2(t) dz^2$$

Because this breaks the assumption of isotropy, it is genuinely different from FRW. Or, alternatively we might have an expansion rate that depends upon position:

$$ds^2 = dt^2 - a(x, y, z, t)(dx^2 + dy^2 + dz^2)$$

This would also be genuinely different.

P.S. Note that these metrics ignore spatial curvature, which you also have to take into account to fully-define a homogeneous, isotropic space-time. This was done for simplicity, as it doesn't change the overall conclusion.

Well, you can't separate the expansion of space from the motion of the objects within it. The two are just different descriptions of the same phenomenon. Space won't "suddenly" start expanding, for example: expansion only occurs if the matter content of the universe causes it to occur. And the matter itself only follows motions that are given by gravity (and the other forces).

Instead what happens is the gravitational attraction/repulsion of the matter/energy content of the universe changes the rate of expansion over time.

8. Oct 20, 2011

### matthias31415

Re: what is "expansion"? A thought experiment

"The creation of space doesn't exert any force on the objects, it just makes them farther apart"

Thanks, phinds, this makes things intuitively understandable.

Chalnoth, instead of having the FRW metric:

$$ds^2=dt^2−a^2(t)(dx^2+dy^2+dz^2)$$

I mean this:

$$ds^2=\frac{1}{a^2(t)}dt^2−dx^2−dy^2−dz^2$$

of course, it isn't algebraically equivalent, but it preserves the relative weight between space and time before they are combined.

Would this metric belong to a different universe? If not, is it OK, instead of talking of an expansion of space, to talk of a contraction of time?

you said I couldn't separate expansion of space from the motion of objects within it. I am not sure what you are saying to me here. In my question, I referred to these "test particles" that do not affect the geometry of spacetime. My question was what happens to the test particles if space suddenly expands (due to other objects that do exist but which I do not onsider at the moment)? If I understand phinds correctly, the test particles will not be "accelerated", but they do become farther apart since new space is created between them. But, conversely, that would mean that test particles that preserve their distance (e.g. because they are bound by gravitation) will experience the sudden expansion as a form of acceleration towards each other, right?

9. Oct 20, 2011

### Chalnoth

Re: what is "expansion"? A thought experiment

Yes, this is identical to what I wrote above with $b(t) = 1/a(t)$. It's the same thing as FRW but in different coordinates.

You could if you wanted, but then you'd find that atoms shrink in these coordinates. So while you absolutely can describe the universe in this way, it often isn't useful, and furthermore the time coordinate would no longer be directly-related to the time we measure on our clocks.

Well, space doesn't "suddenly" expand, as I mentioned. It expands due to the gravitational action of the matter of the universe. And these test particles would be effected by gravity in the same way as everything else. So if, for example, both particles were stationary with respect to the expansion, then they would remain stationary with respect to the expansion as the expansion continues.

10. Oct 20, 2011

### matthias31415

Re: what is "expansion"? A thought experiment

Thanks, chalnot, for your comments on the metric, it helped me understand things better.

If I understand you correctly, the notion that "our universe is expanding and that galaxies are following this expansion" is just the same as saying that "there is gravitation that pulls galaxies (and everything else that remains stationary) apart". However, in my naive understanding of gravity this would require that there is mass outside of our universe. How can this be?

11. Oct 20, 2011

### Chalnoth

Re: what is "expansion"? A thought experiment

Ah, well, if all you have is galaxies made of normal and dark matter, then the expansion can do nothing but slow down.

If, on the other hand, you have some exotic matter with sufficient negative pressure, then the gravitational effect of that is to speed the expansion up. This is the case with dark energy now, which is causing the expansion to accelerate.

12. Oct 21, 2011

### matthias31415

Re: what is "expansion"? A thought experiment

it's getting more and more clear to me. Still, I feel a bit unsure about what "stationary with respect to space time" precisely means. As far as I understood, a stationary observer...

...doesn't "feel" the changes in distance to other stationary observers in terms of gravitation/acceleration

...strictly speaking, doesn't experience any gravity/acceleration if "stationary" also accounts for local disturbances of spacetime (I'm unsure about this one!)

...shares the time coordinate with all other stationary observers if "stationary" means "stationary since singularity".

Is this correct? If so, is it really correct for all types of universes?

13. Oct 21, 2011

### Chalnoth

Re: what is "expansion"? A thought experiment

It basically means that you aren't moving with respect to the local matter. In our universe, you can directly measure your velocity with respect to the expansion by observing the cosmic microwave background: one side of the CMB is redshifted, while the other is blueshifted, if you are moving with respect to the expansion. And it turns out we are. We are moving at 627km/s in the direction of the constellation of Virgo.

Right.

Well, that's true for any free-falling observer.

Not so much shares the time coordinate, but rather you can define a time coordinate for all of them that is also the proper time for each, and it turns out that when their clocks read the same value, the universe looks the same to each one.

It's correct only for homogeneous, isotropic universes.

14. Oct 21, 2011

### phinds

Re: what is "expansion"? A thought experiment

Let me say up front that I'm on somewhat shaky ground here but I THINK I have this right. "Stationary relative to spacetime" (since the singularity), if I understand it correctly, can ONLY mean stationary relative to the CMB (which in this sense stands in for the singularity). That is, if you measure the CMB and it has the same red shift in all directions, then you are "stationary" in that sense. Our galaxy, for example, is NOT stationary in that sense because there are very slight variations, the greatest delta of which are "forward" and "backward" --- directions arbitrarily assigned in that reference. The difference is pretty slight and by the way, just be clear, I am NOT talking here about the hot/cold variations in the CMB that you see in those nifty graphics --- those graphics are created by taking into account the "speed" I'm talking about.

HOWEVER, if there is another galaxy, say 10billion light years away from us, that exhitits the exact same motion relative to the CMB, we and they are still moving apart a ferocious speed due to the expansion of the universe, and that would also be the case were we both stationary relative to the CMB.

EDIT: OOPS ... When I started typing the above, Chalnoth's post wasn't there and I see that I've duplicated some of it. I dislike it when it appears that folks are totally ignoring other people's posts, so don't like to seem as though I've done just that.

Last edited: Oct 21, 2011
15. Oct 21, 2011

### matthias31415

Re: what is "expansion"? A thought experiment

thanks for all your patience ;-)

I am moving forward, but I still don't get the difference. I understand that, if galaxies are drawn apart by some miraculous form of gravitation, they are considered stationary with respect to an expanding spacetime. But to me this doesn't look much different from a free fall. E.g. if I am falling down from a large tower, why can this not be described analogously, that is, me being stationary with respect to local disturbances of spacetime?

I am a bit disappointed that all what I've learned should be true only for the very special case of a homogeneous, isotropic universe. Isn't there a definition of what "stationary with respect to spacetime" means, that can be directly derived from general relativity, so that it would be true for all kinds of universes whether they are isotropic or not, whether there is background radiation or not?

16. Oct 21, 2011

### phinds

Re: what is "expansion"? A thought experiment

No, not really. First the "miraculous form of graviation" that you refer to is ANTI-gravitational in its action and it is not, I believe, discussed as gravitational in nature. It is SOMETHING (which we call "dark energy") that creates more space.

They would NOT necessarily be stationary with respect to anything else. Our galaxy, and I believe most if not all galaxies have some motion with respect to the CMB. I think you're getting confused on this point because expanding spacetime does not ADD to whatever motion they have relative to spacetime (although it seems to from the point of view of a distant observer).

In other words, galaxies (ours at least) have some relatively small velocity relative to the CMB and they have a HUGE velocity relative to far-away galaxies but this is because of the expansion. If you could disregard the expansion, the huge velocity would not be there, but that would NOT make them stationary.

17. Oct 21, 2011

### Chalnoth

Re: what is "expansion"? A thought experiment

It's not "miraculous". This is how gravity works. Even Newtonian gravity works this way in a homogeneous, isotropic universe.

No. This sort of idea is only relevant in a space-time where the dynamics pick out a particular reference frame. In the case of a space-time like our own, that is a reference frame where the universe looks homogeneous and isotropic (a different observer will, in general, see neither homogeneity nor isotropy in our universe). The fact that some particular observers see these properties makes our universe special. A general universe won't have observers that see these properties, such that while some alternative ideas may also have their own special reference frames, I don't believe there is any requirement that they have to.

18. Oct 25, 2011

### matthias31415

Re: what is "expansion"? A thought experiment

thanks, phinds and chalnot, I think, I am getting close to it.

As to "miraculous", well, chalnot, you said it would be more exotic than dark matter and phinds called it "antigravitational", so I thought "miraculous" would be suitable. But I don't insist on this term ;-)

what I still feel unsure about is the following: I now understand that the term "expansion" refers only to particular reference frames in our universe. But the notion of "space being created" suggested otherwise to me. "Create" means that there will be something that wasn't there before. Therefore, intuitively, when something is "created", I thought, it would appear in all reference frames (maybe distorted, but still). Probably I was wrong and this "creation" is only a metaphor that must be used with caution...

19. Oct 25, 2011

### phinds

Re: what is "expansion"? A thought experiment

I'm not sure I understand what you're saying here, but the creation of space is not a metaphor, it is physical reality.

20. Oct 25, 2011

### matthias31415

Re: what is "expansion"? A thought experiment

"I'm not sure I understand what you're saying here, but the creation of space is not a metaphor, it is physical reality."

so, phinds, can you define what "creation of space" means in terms of general relativity, strictly avoiding any references to the specific geometry of our universe (isotropy, background radiation...)?