Overlapping observable universes

[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?

 

[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.

 

[phinds]

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?

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.

 

[Warp]

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.

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.

 

[Warp]

I think your logic is correct, but there's contradiction. The situation is subtle and depends on coordinate choice. There's an explanation here ... and in the following section.

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?

 

[Mentz114]

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?

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.

 

[Warp]

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.

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.

 

[Mentz114]

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

 

[Bill_K]

The correct answer to this is not that simple.

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.

 

[Warp]

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.

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).

 

[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.

 

[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.

 

[Warp]

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.

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?

 

[A.T.]

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

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

 

[Warp]

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

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".

 

[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.

 

[Warp]

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.

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

 

[mfb]

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.

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.

Spoiler

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.

 

[Warp]

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.

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.

 

[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 let's 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!

 

[A.T.]

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".

In that case the signal from A will reach B, even if that constant distance increase rate is greater than c. See the snail example.

 

[Warp]

So the final verdict is: If the metric expansion of the universe has a rate below a certain critical value (but always so that the distance between A and B grows faster than c), the probe will eventually reach B, regardless of how unintuitive that might sound at first, but if the rate of metric expansion is larger than this critical value, then the probe will never reach B.

This actually makes sense, once you get your head around the seeming unintuitive fact that a probe traveling at less than c can reach the other planet that's receding faster than c (I'm assuming that from A's perspective it will look like the probe accelerates until it surpasses c and leaves A's observable universe).

Thanks for all the answers. I think that now I'm finally happy with the explanation and can put this puzzle to rest.

 

[Warp]

Thanks for all the answers. I think that now I'm finally happy with the explanation and can put this puzzle to rest.

Actually, I spoke too soon. This puzzle is more obstinate than I thought. It doesn't want to leave my head, and only produces more questions as I think about it.

So, to recapitulate: If the metric expansion of the universe were such that two planets A and B were receding from each other slightly faster than c, light from A would still reach B because of the "snail-on-a-stretching-rubberband" effect. But only up to a maximum receding velocity (ie. if the expansion is faster than what the "stretching rubberband" makes the "snail" advance.)

But the question is: What is the limit in the receding velocity between the two planets before it becomes impossible for light to travel between them? My math fails me at this point.

My intuition tells me that if light is able to reach the mid-point M between A and B from A (because the distance between A and M is increasing slower than c), then it can reach B (because then the distance between M and B is also increasing slower than c.) Therefore if the distance between A and M is increasing faster than c, then it will never reach neither M nor B.

But: Why wouldn't it be able to reach point M in the same way as it reached B in the original scenario? We can pinpoint a mid-point between A and M, let's call it Q (as in "quarter") which may be receding at less than c from both. If light can reach point Q from A, then it can reach point M from Q by the same reasoning. And if it can reach M, it can therefore reach B by the exact same logic. Therefore, it would seem, no matter how fast the universe expands, light is able to reach any point in it.

Something doesn't feel right here, but I can't figure out what. I get an eery feeling of having a Zeno's paradox here, but I can't explain it fully. Could someone help me here?

 

[mfb]

It depends on the evolution of the expansion. With linear expansion, the signal will always reach B. With exponential expansion, there is a maximal distance - exactly at the point where the length between the two objects increases with c. With something in between, it depends on the details of the expansion.

 

[Warp]

It depends on the evolution of the expansion. With linear expansion, the signal will always reach B. With exponential expansion, there is a maximal distance - exactly at the point where the length between the two objects increases with c. With something in between, it depends on the details of the expansion.

So you are saying that if the universe were to expand at such a rate that the two planets would recede from each other at a constant velocity (which, I think, would mean an asymptotically decelerating expansion of the universe), it doesn't matter how large this recession velocity is, light from one planet will always reach the other? Is this also the mathematical result of the "snail-on-a-stretching-rubberband" problem?

(Out of curiosity, how do you solve that mathematical problem?)

 

[haael]

If two objects are gravitationally bound, then they would not escape from each other's horizons.

 

[mfb]

(Out of curiosity, how do you solve that mathematical problem?)

The easiest way is to keep the length of the band constant and modify the speed of the snail. For the universe, you can use comoving coordinates, and get the same effect.

(which, I think, would mean an asymptotically decelerating expansion of the universe)

Constant expansion (constant $\dot{a}(t)$).

 

[Samshorn]

This puzzle is more obstinate than I thought. It doesn't want to leave my head, and only produces more questions as I think about it... My intuition tells me that if light is able to reach the mid-point M between A and B from A (because the distance between A and M is increasing slower than c), then it can reach B (because then the distance between M and B is also increasing slower than c.) Therefore if the distance between A and M is increasing faster than c, then it will never reach neither M nor B.

But: Why wouldn't it be able to reach point M in the same way as it reached B in the original scenario? We can pinpoint a mid-point between A and M, let's call it Q (as in "quarter") which may be receding at less than c from both. If light can reach point Q from A, then it can reach point M from Q by the same reasoning. And if it can reach M, it can therefore reach B by the exact same logic. Therefore, it would seem, no matter how fast the universe expands, light is able to reach any point in it... Something doesn't feel right here, but I can't figure out what. I get an eery feeling of having a Zeno's paradox here, but I can't explain it fully. Could someone help me here?

I think your questions are answered on this web page, with some illustrations:

www.mathpages.com/home/kmath666/kmath666.htm

See especially the paragraph about half way down the page, beginning with the words "One benefit of presenting..."