# Overlapping observable universes

1. May 30, 2012

### Warp

Here's a puzzle that I have been wondered about. I know the answer, but I do not understand the reasoning for it. Could someone explain it to me in a way that a layman like could understand in an intuitive manner? (The math involved in GR equations is too complex for me to grasp.)

General Relativity does not forbid (but on the contrary predicts) the distance between two points in space increasing faster than c. This can be caused, for example, by the metric expansion of the Universe (and is, in fact, as far as we know, happening as we speak). As the Universe expands, the parts of it that are farther away from us than a certain distance are receding from us faster than c. This means that the observable universe (ie. the part of the universe that we can observe) is smaller than the entire universe. We cannot observe anything from beyond this distance. (Even though the distance between two points can increase faster than c, this still does not allow traveling at superluminal speeds. Nothing can move from one point to another faster than c.)

Now, the observable universes of, for example, two planets can overlap, even if the planets are so far apart from each other that they are receding from each other faster than c. In other words, if we have a planet A and a planet B that are so far apart that they recede from each other just slightly faster than c, A will be outside of the observable universe of B and vice-versa, and no communication whatsoever is possible between them (according to GR). However, the observable universes of A and B overlap, if their distance is not too great. This means that both A and B can, for example, observe the same star (which is in the overlapping part).

Now here's the puzzle: If both A and B can observe the same star, that means that the star can likewise observe both A and B (because they are both inside the star's observable universe). But this means that someone located at the star could, for example, take a photo of A and send it to B, all this at subluminal speeds. But this breaks the premise that no information can be sent from A to B because they are receding from each other faster than c.

This seems like a contradiction. What is the correct answer to this, and why?

2. May 30, 2012

### Mentz114

Hi Warp.

I think your logic is correct, but there's no contradiction. The situation is subtle and depends on coordinate choice. There's an explanation here http://www.astro.ucla.edu/~wright/cosmo_02.htm and in the following section.

Last edited: May 31, 2012
3. May 30, 2012

### phinds

No information is being sent from A to B. The information is being sent from a center point to B and it is in any case information that is billions of years old.

4. May 31, 2012

### Warp

I think both of those statements are incorrect (or in the case of the latter statement, it doesn't have to be so).

Firstly, of course (in this hypothetical case, assuming it were a correct interpretation) information is being sent from A to B: Photons originating from A, carrying information about A, are arriving at the star, which then relays them to B. A could be sending a message to the star, which relays it to B. This is information transferral.

Secondly, nothing in GR forces A, B and the star to be billions of light-years away from each other at the start of this experiment. They could be mere light-minutes away, and co-exist in a (hypothetical) universe that's expanding at a rate such that A and B recede from each other slightly faster than c, and the star in question lies in the middle. Besides, even if the information took billions of years to traverse the distance, it wouldn't change the apparent contradiction. The problem is not how long it takes for the information to be transferred, it's that it seems to be transferrable in the first place.

5. May 31, 2012

### Warp

Thanks for your reply. I read the article, but unfortunately I was incapable of extracting from it the information I needed to get and understand the answer to my question. Could you help me out a bit?

6. May 31, 2012

### Mentz114

I just fixed a bad typo in my first post. I meant to say

" ... but there is *no* contradiction ..."

No information passes directly between worldlines that are causally disconnected but by using a relay of observers in between them, two such worldlines can exchange signals.

Last edited: May 31, 2012
7. May 31, 2012

### Warp

Please explain in more detail, because this seems to heavily contradict the premise that no information can be transferred between two objects receding from each other faster than c (because such information transferral would obviously require faster-than-light travel).

The dilemma can be stated in another way:

If the universe expands at such a rate that two planets A and B recede from each other just slightly faster than c, this means that their observable universes overlap. This would seem to mean that A could send a probe to this overlapping zone (which is well within its observable universe) at a speed well below c, after which this probe can then proceed to travel to B (because it's now also within B's observable universe) at a speed well below c. At no point does the probe even approach speed c.

So that's the glaring contradiction: A probe just traversed from A to B, which would require a superluminal speed, without ever even approaching speed c. This should not be possible even if the probe could travel at speed c, much less at a slower speed.

The correct answer to this is not that simple.

8. May 31, 2012

### Mentz114

Sorry, I can't help any further. Look at the cosmological spacetime diagrams.

9. May 31, 2012

### Bill_K

Actually, Warp, it's even simpler.

"Information cannot be transferred faster than light because..." Why because? Because it would violate causality. Because if information is emitted at event A1 and received at event B1, and A1 and B1 are outside each other's light cone, then there must also be a way of transferring information the other way, from event B1 to A1. (Note these are events, not world lines.) Event B1 is both to the future and to the past of A1, which contradicts causality.

Your example has no such contradiction. A1 is definitely to the past of B1.

10. Jun 1, 2012

### Warp

I'm not sure I'm understanding properly, but it sounds to me like you are saying that it is, indeed, possible to send a probe at subluminal speeds from A to B even though A and B are receding from each other faster than c. So light cannot ever reach B from A (because A is outside the observable universe of B), but a probe moving slower than light can. And this is not a contradiction.

Please help me understand this, because I don't. I really want to understand the correct explanation for the answer (whichever it is).

11. Jun 1, 2012

### Vargo

It would seem that in the time that it takes for the signal to go from A to the intermediate point M, the distance between M and B will have grown far enough apart so that they would no longer lie in each other's light cones.

Simply put, for any two points A and B, if their distance increases at a rate that increases with their distance, then eventually they will lie outside each other's light cones.

But I am not a physicist and do not understand GR, so please correct me if I am wrong.

12. Jun 1, 2012

### A.T.

I agree with Vargo.

Just because the intermediate point M still receives signals from A & B at some time point, does not mean that a signal from M sent at that time point will still reach A or B.

13. Jun 1, 2012

### Warp

Please explain the answer in a more concrete way.

If we assume that the universe were to expand at such a rate that the distance between A and B increases at a constant speed (that's slightly faster than c), then at which point exactly does the star in the middle stop being able to send signals to either planet?

14. Jun 1, 2012

### A.T.

The distance doesn’t increase at a constant rate. The rate of distance increase increases with distance.

15. Jun 1, 2012

### Warp

I didn't say "let's assume the universe expands at a constant rate". I said "let's assume that the universe expands at such a rate that the distance between A and B increases at a constant speed".

16. Jun 1, 2012

### Vargo

Let A,B be two points, and let x(t) measure the distance between them which is increasing due to the metric expansion of the universe. According to the hypothesis of your question, there exists a minimum distance x_1 such that

$x'(t) > c \,\, if \,\, x > x_1$

I propose the stronger hypothesis that x(t) satisfies a differential equation of the form $$x'(t)=F(x),$$ where F is a positive function that is smooth away from 0, and is monotonically increasing ($F'(x)>0$ ) with $F(x_1)=c$. I would also assume that F(x) tends to zero as x tends to zero but that does not seem relevant to this question. Not knowing much about GR I cannot say anything about the validity of such an assumption.

This implies $$x''(t) = \frac{d}{dt} F(x(t)) = F'(x(t))x'(t)=F'(x(t))F(x(t)) >0.$$ In other words, the distance between any pair of points is growing and accelerating. Moreover, the velocity does not asymptotically increase to a speed less than or equal to c because of our assumption that F(x_1)=c. This model implies that given any two points, their distance apart will be unbounded as a function of time: x(t) goes to infinity for all initial values except x(0)=0. This model also implies that if two objects are at a distance of at least x_1, then they are outside each others light cone. So two objects within each others' light cone will not remain so for all time.

So, going back to the problem. Let M be the midpoint between A and B and assume that initially it is in the light cone of both A and B. Then A can send a signal to M. But by the time that signal reaches M, the distance between M and B will have expanded just enough so that B is now outside the light cone of M. Another interpretation of this is that A can send a signal to M and/or M can send a signal to A, but they cannot respond to each others' signals.

17. Jun 1, 2012

### Warp

I think this is a clearer explanation of what's happening. Thanks.

18. Jun 1, 2012

### Staff: Mentor

There is a nice puzzle which is very similar to this:

An immortal, point-like snail sits on one end of a rubber band of 1km length which is perfectly stretchable. Each day, the snail travels 10cm towards the other end. Each night, the rubber band is stretched by 1km. Does the snail ever reaches the other end?
Counterintuitive, it does.
Expressed as fraction of the total rubber band, the snail can travel by 10^(-4)/n on day n. The position after day n is then given by the harmonic series, which diverges.

In a similar way, "inside the observable universe" is not a static property, but depends on the time: The observable universe can grow (even in co-moving coordinates) or shrink. And while we can see stars 10 billion light years away, we can see their past only and can contact their future only - and a message from us to them would require more than 10 billion years.

19. Jun 1, 2012

### Warp

Actually, I replied too hastily. Thinking about it a bit more, I still don't get it, sorry.

If the premise is that the distance between A and B increases at a constant rate (slightly faster than c), then it follows that the distance between A and M likewise increases at a constant rate, which would be something smaller than c.

If a signal is sent from A to M, it will reach it because the distance between A and M is increasing slower than c. Now M sends a signal back. Since the premise was that the distance increases at a constant rate, it will still be slower than c, and hence the signal back will eventually reach A. (Obviously it will take a longer time because the distance has increased in the meantime, but it will eventually reach A.) I don't see why it wouldn't.

But if this is so, then the distance between M and B also increases at a constant rate that's slower than c, which would mean that M could relay the signal to B, thus breaking the premise that nothing can travel from A to B because they are receding from each other faster than c.

So... I still don't get it, sorry.

20. Jun 1, 2012

### Vargo

Great post mfb. I rethunk the model to imagine the universe as a rubber band. We can think of that like this:

For simplicity assume that space is 1-dimensional, Euclidean, and expanding in the same fashion as the rubber band. Pick two reference points P and Q. We'll say P=0 and Q = 1. Every other point is denoted by its fraction of the distance between P and Q. So x=1/2 is their midpoint. But the actual distances between the points grows according to the formula:
$$d(x,y,t) = E(t)|x-y|.$$
For the rubber band, E(t)=1+t. And as the snail shows, no matter how slow a signal is traveling, this expansion is not fast enough to prevent the signal from eventually reaching all points of space. Moreover, it does not matter when the signal is sent. All of space remains inside the observable universe for all time.

But what happens if the expansion is faster? Suppose E(t)=1+t^2. Let x(t) denote the location of a signal traveling at constant speed c. It can be shown that,
$$x(t) - x(t_0) = c( \arctan(t) - \arctan(t_0)).$$
This implies that at time 0, your observable universe consists of things of distance less than $c\arctan(\infty)=c\pi / 2$.

So lets say point A is 0, point B is at point $c2\pi / 3$, and point M is the midpoint at $c\pi / 3$. At time 0, M lies inside the light cone of both A and B. So A sends a signal to point M. The time it takes to get there is $t_0 = \sqrt{3}$.

At this time, the observable universe for M consists of those points whose x coordinate satisfies: $|x-c\pi/3|< c(\arctan(\infty)-\arctan(\sqrt{3}))=c\pi/6$. In other words B is no longer inside the observable universe of M (and neither is A). M was able to observe the signal from A, but it cannot send one back because they are now too far apart. Or if you prefer, the snail can't make it back because the rubber band is just stretching too fast for the poor little guy.

Interestingly, if you measure the size of the observable universe at that time, you get $(1+\sqrt{3}^2) c\pi /6=2cpi/3$. So you see farther, but you see less!

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