# Twin paradox and the size of the universe

• DiracPool

#### DiracPool

Ok, let's take the standard twin paradox, Alice leaves on a trip in her rocketship near the speed the light, and comes back to Earth some time later to find herself 5 years younger than her twin, Bob.

Now they go out to lunch and strike up a conversation as to how old the universe is. Alice says it's ~13.8 billion years old and is such and such a diameter across. Bob says, no no no, the universe is ~13.8 byo + 5 years and has an accordingly larger diameter. Which one is correct? How do we reconcile this? I mean, if each are living in a different-sized universe, how are they able to have lunch together? For instance, how are the physical properties of the restaurant not affected, etc.?

Stephanus and AbhijithPrakash

Indeed, you hit the nail on the head: how are they able to have lunch together?

The answer is because they have matched velocities and are in the same rest frame. Thus, they measure the universe identically.

Sure, while on their trips they made all sorts of measurements. (Hey look, that restaurant passing by my windows at .999c looks massively longitudinally contracted - as does my local universe! And everything is now moving verrrry slowly.)

But those measurements are relative. When they slow their ship and turn around, they will get different answers. They know they are changing their velocities, so they will certainly know their measurements will vary accordingly.

(Ah, now that I am in the same rest frame as the restaurant, it looks quite normal, as does my local universe.)

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Alice says it's ~13.8 billion years old and is such and such a diameter across. Bob says, no no no, the universe is ~13.8 byo + 5 years and has an accordingly larger diameter.

Bob is correct, because (by hypothesis) he has remained at rest in "comoving" coordinates--physically, he has continued to see the universe as homogeneous and isotropic. To relate how old you think the universe is to its diameter in the way you are assuming, you have to be such an observer--one who has always seen the universe as homogeneous and isotropic. Since Alice is not such an observer, she has to apply a correction when deriving the diameter of the universe from how old she thinks it is; with the correction applied, she gets the same answer Bob does.

(Note that the above applies to Alice after she has come back and met up with Bob again, so she is now at rest relative to Bob. During Alice's trip, the "diameter" she assigns to the universe in a frame in which she is at rest will be different because of her motion relative to Bob, and the relationship between this "diameter" and the age Alice assigns to the universe will also be different. I put "diameter" in quotes here because Alice's surfaces of simultaneity while she is in motion are different from Bob's, so the "diameter" she is assigning belongs to a different slicing of spacetime into space and time. Discussions of cosmology almost always assume "comoving" observers, like Bob, and the corresponding slicing; so if you deviate from that, you have to be careful not to make assumptions that are only valid for that slicing.)

(Also, I assume that the "diameter" you mean here is the diameter of the observable universe. According to our best current model, the universe itself is spatially infinite.)

A short answer - there is no absolute time in special or general relativity, so there is no true "absolute age" of the universe. We do have certain conventions, those conventions are used when "the age of the universe" is given. These conventions are that we measure the age in a co-moving frame, one in which the CMB is isotropic.

Seems simple to me, but it appears people have difficulties with giving up the idea of "absolute time".

Seems simple to me, but it appears people have difficulties with giving up the idea of "absolute time".
Well, maybe, but I don't think that's the issue here. The issue is how can the age of the universe from a single event in spacetime (the simulataneous co-existance of Bob and Alice at the same point in space) experience the age of the universe in two different ways. I had exactly the same idea as Dave but I'm still mulling over Peter's analysis.

Well, maybe, but I don't think that's the issue here. The issue is how can the age of the universe from a single event in spacetime (the simulataneous co-existance of Bob and Alice at the same point in space) experience the age of the universe in two different ways.

Yeah, that's exactly my point. My point is that, as they are having lunch together and look through the telescope the waiter brought them, they each look through the lens and see a different universe? With a different age and diameter? (visible universe, that is).

PeterDonis says Bob is correct. Does that mean that when Alice looks through the telescope, she sees the universe as Bob sees it? Is that how this is reconciled?

as they are having lunch together and look through the telescope the waiter brought them, they each look through the lens and see a different universe?

Of course not. They are both at the same point in spacetime, with the same velocity, so they see everything the same. How they got there does not affect what they see at that moment; it only affects the elapsed times on their respective clocks.

Does that mean that when Alice looks through the telescope, she sees the universe as Bob sees it?

Yes. But she calculates a different relationship between what she sees and the elapsed time on her clock, because of her different history of motion compared to Bob.

how can the age of the universe from a single event in spacetime (the simulataneous co-existance of Bob and Alice at the same point in space) experience the age of the universe in two different ways.

Because they arrived at that point in spacetime by different routes, and the "age of the universe" that you experience depends on the route, not just on the point in spacetime you are currently at. More precisely, the relationship between what you currently observe (which is the same for all observers at the same event with the same velocity), and the "age of the universe" that you experience, depends on the route.

Because they arrived at that point in spacetime by different routes, and the "age of the universe" that you experience depends on the route, not just on the point in spacetime you are currently at. More precisely, the relationship between what you currently observe (which is the same for all observers at the same event with the same velocity), and the "age of the universe" that you experience, depends on the route.
Makes sense. Thanks for continuing to clarify that for us. Now that I think about it, a thought experiment using the decay of a radioactive element shows clearly that the age would be different for Bob and Alice. I just hadn't thought it through.

Makes sense. Thanks for continuing to clarify that for us. Now that I think about it, a thought experiment using the decay of a radioactive element shows clearly that the age would be different for Bob and Alice. I just hadn't thought it through.

That doesn't really make complete sense to me... If the age (of the visible universe I'm assuming) would be different for Bob and Alice, then they would not see the same thing looking through the telescope, would they?

the "age of the universe" that you experience depends on the route

Ok, sounds like there may be differences, but...

They are both at the same point in spacetime, with the same velocity, so they see everything the same.

Ok, that sounds like there are no differences...

But she calculates a different relationship between what she sees and the elapsed time on her clock, because of her different history of motion compared to Bob.

Could you elaborate on that a little bit? What kind of different relationship? I'm imagining her looking through the telescope and looking at her watch and going, ahhh, I get it. But I don't get it.

TheDemx27
That doesn't really make complete sense to me... If the age (of the visible universe I'm assuming) would be different for Bob and Alice, then they would not see the same thing looking through the telescope, would they?
Yes, as Peter explained, they WOULD see the same thing, but that is NOT relevant to the age that they see the universe as being. Just using the differences in their ages should be enough to make that clear but somehow, for me, it didn't so I thought about a radioactive element. Use magic to consider that this radioactive element has been around since the beginning of the universe and is exactly 13,000,0000,000 years old (round number for ease of demonstration). They break it in half and one half goes on the trip and one half stays home. When the traveling half gets back it's, say, 13,000,0000,005 years old, BUT ... the half that stayed home is 13,000,0000,020 years old. They have taken different paths through spacetime and thus have different ages. Similarly, Bob and Alice had taken different paths through spacetime and so see the age both of each other and of the universe as being different.

Ok, let's take the standard twin paradox, Alice leaves on a trip in her rocketship near the speed the light, and comes back to Earth some time later to find herself 5 years younger than her twin, Bob.

Now they go out to lunch and strike up a conversation as to how old the universe is. Alice says it's ~13.8 billion years old and is such and such a diameter across. Bob says, no no no, the universe is ~13.8 byo + 5 years and has an accordingly larger diameter. Which one is correct? How do we reconcile this? I mean, if each are living in a different-sized universe, how are they able to have lunch together? For instance, how are the physical properties of the restaurant not affected, etc.?

What is special about universe, so that universe is not just another aging object?

Alice: "During my 1 year trip universe and Bob aged so quickly that they aged 6 years."

Bob: "During Alice's 6 year trip universe and me aged 6 years. But Alice aged so slowly that she aged 1 year."

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What is special about universe, so that universe is not just another aging object?
Nothing. The universe IS just another aging object, but like Bob and Alice, and everything else, how you measure the age of the universe depends on your path through spacetime.

Alice: "During my 1 year trip universe and Bob aged so quickly that they aged 6 years."
Yes, so?
Bob: "During Alice's 6 year trip universe and me aged 6 years. But Alice aged so slowly that she aged 1 year."
Yes, so?

What you seem to not be getting is that the age of an object ALWAYS happens at one second per second for that object, BUT ... when compared to the age of a different object, then what matters is the path through spacetime that each have taken.

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If the age (of the visible universe I'm assuming) would be different for Bob and Alice, then they would not see the same thing looking through the telescope, would they?

The age of the universe is not something you directly observe; it's something you calculate, and the calculation depends on your own past history, not just on what you're currently observing. Bob and Alice each have the same current observation, but they have a different past history, so they calculate a different age for the universe.

What this really means is that the term "age of the universe" is really something of a misnomer for what we are all talking about. See below.

What kind of different relationship? I'm imagining her looking through the telescope and looking at her watch and going, ahhh, I get it.

You're imagining it way too simply. Alice's watch registers the proper time elapsed along her worldline, and Bob's registers the proper time elapsed along his worldline. If we suppose (which of course can't be the case in a real scenario) that both watches were set to time zero at the Big Bang, then the reason the two watches read differently is simply that Alice and Bob followed different worldlines to get to the same current event in spacetime. So the readings on their watches aren't really telling you "how old" the universe is, because the readings don't depend on the universe itself--both worldlines are in the same universe. They're just different worldlines, and the watch reading depends on the worldline. So Alice should not expect a simple relationship between her watch reading and what she sees through the telescope, and she certainly should not expect to see the same relationship as Bob does.

When cosmologists use the term "age of the universe", what they really mean is the proper time elapsed since the Big Bang along a "comoving" observer's worldline, like Bob's. The special property that picks out "comoving" worldlines from all other worldlines (like Alice's) is that "comoving" observers always see the universe as homogeneous and isotropic; observers following any other worldline will, for at least some portion of their history, see the universe as non-isotropic or non-homogeneous (like Alice does during her trip). So the "age of the universe" in cosmology is really "the proper time elapsed since the Big Bang along a comoving worldline". A non-comoving observer, like Alice, can still calculate this number from their watch reading and what they see through a telescope, but the calculation won't be as simple as it is for a comoving observer, who can just read it directly off his watch (again, assuming an idealized case where the watch is set to time zero at the Big Bang).

When cosmologists use the term "age of the universe", what they really mean is the proper time elapsed since the Big Bang along a "comoving" observer's worldline, like Bob's.

Ah yes, the "co-moving" term I've been coming across most frequently recently and have been putting off investigating in full. I guess that's going to be my project for tomorrow. I'll give you the report

I'm sorry if this answer isn't in the spirit of the thread...
Alice and Bob, both knowing she would be making this trip at a significant and measurable fraction of the speed of light, would adjust Alice's clock just like we do for the clocks on GPS and other satellites. They would agree on the amount of time passed and the corresponding expansion of the universe.

I'm sorry if this answer isn't in the spirit of the thread...
Alice and Bob, both knowing she would be making this trip at a significant and measurable fraction of the speed of light, would adjust Alice's clock just like we do for the clocks on GPS and other satellites. They would agree on the amount of time passed and the corresponding expansion of the universe.
What do you mean "the amount of time passed" ? This whole thread has been about explaining that there is no "THE" amount of time passed, just the differing amounts of time passed relative to each other. They are not the same amounts of time for Bob and Alice. Do you dispute that?

Ok, let's take the standard twin paradox, Alice leaves on a trip in her rocketship near the speed the light, and comes back to Earth some time later to find herself 5 years younger than her twin, Bob.

Now they go out to lunch and strike up a conversation as to how old the universe is. Alice says it's ~13.8 billion years old and is such and such a diameter across. Bob says, no no no, the universe is ~13.8 byo + 5 years and has an accordingly larger diameter. Which one is correct? How do we reconcile this? I mean, if each are living in a different-sized universe, how are they able to have lunch together? For instance, how are the physical properties of the restaurant not affected, etc.?
The universe is a composite object, with no detectable center. All its components are moving at varying velocities, with varying degrees of time dilation. On that basis the universe has a range of ages. Alice has made an excursion relative to Earth. On reuniting, her accumulated time is 5 yr less than that for Bob, and anything else still existing there. This difference only applies to the Alice-Earth system. When they observe a distant component, it is only a relative doppler effect, i.e. a comparison of frequencies (clocks).
I agree with pervect, as to the CMB being the only thing serving as a "fixed" reference, since events don't move.
Considering the distances involved, any age comparisons of the universe with any component would be vague and uncertain.

In relativity, there is no preferred frame of reference, but in cosmology, there certainly is. (The most obvious frame is the one where the cosmic microwave background is closest to isotropic. We call this the co-moving frame.)

The special property that picks out "comoving" worldlines from all other worldlines (like Alice's) is that "comoving" observers always see the universe as homogeneous and isotropic; observers following any other worldline will, for at least some portion of their history, see the universe as non-isotropic or non-homogeneous (like Alice does during her trip).

In relativity, there is no preferred frame of reference, but in cosmology, there certainly is. (The most obvious frame is the one where the cosmic microwave background is closest to isotropic. We call this the co-moving frame.)

These comments makes sense to me. I like using the standard of the observer always seeing the universe as homogeneous and isotropic as the "comoving" frame. A few questions:

1) So in the scenario above, are we assuming that Bob is "comoving" or in the same comoving frame as the CMB, that this qualifies as Bob and the CMB being in the same inertial rest frame (IRF), and the "test" of such is that the universe appears to be homogeneous and isotropic, to Bob?

2) There is a galaxy with an earth-like planet far far away receding from the Earth at 0.9c, Does an inhabitant on that planet also see the universe as homogeneous and isotropic as well as being in the same rest frame or comoving with the CMB? We can say that Alice during her trip breaks symmetry with Bob and travels a different "distance" through spacetime due to her departing from Bob's (and the CMB's) IRF, and thus shows a different age when she shows up back on Earth. But what about the guy in the receding galaxy? Is he also traveling a different distance through spacetime than Bob seeing as he's traveling at 0.9c relative to Bob? Or is there some special feature about this scenario whereby the guy in the distant galaxy also sees the universe as homogeneous and isotropic? In other words, does a symmetry apply in this case where it doesn't in Alice's case.

So in the scenario above, are we assuming that Bob is "comoving" or in the same comoving frame as the CMB, that this qualifies as Bob and the CMB being in the same inertial rest frame (IRF), and the "test" of such is that the universe appears to be homogeneous and isotropic, to Bob?

Yes.

here is a galaxy with an earth-like planet far far away receding from the Earth at 0.9c, Does an inhabitant on that planet also see the universe as homogeneous and isotropic as well as being in the same rest frame or comoving with the CMB?

If the galaxy is "comoving" yes. As you have stated it, it's impossible to tell; the recession velocity of the galaxy alone isn't enough. You would also need to know the proper distance from Earth to the galaxy, so you could compare the recession velocity of the galaxy with what it would be for a comoving object at the same distance.

(Note that we are assuming here that the Earth is comoving, which it actually isn't; the CMB as we see it from Earth has a significant dipole anisotropy, corresponding to a velocity for Earth relative to a "comoving" observer of about 600 km/s. This dipole anisotropy is almost always removed before displaying charts of CMB observations, since the purpose of those charts is to look for patterns as they would be seen by a "comoving" observer.)

We can say that Alice during her trip breaks symmetry with Bob and travels a different "distance" through spacetime due to her departing from Bob's (and the CMB's) IRF, and thus shows a different age when she shows up back on Earth.

This is OK except for the term "IRF". There is no such thing as an IRF covering the universe, because the spacetime of the universe is not flat. If Alice doesn't go that far away from Bob, and returns in a short enough time (anything we humans are going to achieve in the foreseeable future will count as "not very far away" and a "short enough time"), then we can analyze the entire scenario in a single local inertial frame covering Bob's and Alice's worldlines during the experiment. But that won't work for objects farther away; see below.

what about the guy in the receding galaxy? Is he also traveling a different distance through spacetime than Bob seeing as he's traveling at 0.9c relative to Bob?

"Distance through spacetime" is ambiguous; "distance" between what events? Alice and Bob share a pair of events (the start and end of the experiment), so that question has an easy answer. But for the distant galaxy, you have to adopt a simultaneity convention in order to pick out "corresponding" events on the two worldlines (Bob's and the galaxy's). If both of them are "comoving" (which we can test by seeing if they both see the universe as isotropic), then the obvious simultaneity convention is the one for "comoving" observers; each spacelike hypersurface of constant time in the standard FRW coordinate chart is a surface of simultaneity for comoving observers. Then each such observer will have the same elapsed proper time ("distance" in spacetime) between a given pair of such surfaces (i.e., between two given values of FRW coordinate time--in fact, FRW coordinate time is the same as proper time for "comoving" observers).

The fact that the above is true even though Bob and the galaxy are in relative motion should make clear why you can't construct an inertial frame that covers the universe. (Bob and the galaxy are also in free fall, i.e., they are inertial observers, which makes this point even clearer.)

This question is not as complicated as many are making it.
Let's say that before Alice leaves on the trip, they both device a mechanism for measuring the age of the universe precisely and set two clocks to indicate that age. So as the clock passes 18,000,000,000 year, Alice departs.
When Alice returns, she notes that when her clock reported 18,000,000,000.1, Bob's reports 18,000,000,005.1 - a five year difference.
At this point, Alice can repeat the measurement made to set the clocks to begin with and discover that Bob's clock is the one that accurately represents that age of the universe from their Earth-bound frame of reference.

They are both in the same universe - with the same universe age.

They are both in the same universe

Yes.

with the same universe age

Only if you define "universe age" in a way that makes that true. The standard definition of "universe age" is "proper time elapsed since the Big Bang for a comoving observer". By that definition, yes, all observers see the same "universe age". But that definition is a convention, an agreement about how to use words. There's nothing in the physics that requires us to define "universe age" that way.

The way I'd normally put this is that Alice and bob both see the same Universe as each other when they meet up, look through the waiter's telescope, etc. but disagree about how it got to be like that. Just like Alice and Bob agree about which of them is younger (remember why the Twin's Paradox was supposed to be a paradox?) but they disagree about how that came about.

... Alice and Bob agree about which of them is younger ... but they disagree about how that came about.
Simon, I don't get why they would disagree about how that came about since they know they took different paths through spacetime. If they have two different opinions on how it came about, what would those two opinions be?

2) There is a galaxy with an earth-like planet far far away receding from the Earth at 0.9c, Does an inhabitant on that planet also see the universe as homogeneous and isotropic as well as being in the same rest frame or comoving with the CMB?

At cosmological scales, when someone says "inertial rest frame" they mean "local inertial rest frame". There are no cosmos-wide inertial frames, so both Alice and Bob can be at rest relative to the cosmic microwave background yet not one another.

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Simon, I don't get why they would disagree about how that came about since they know they took different paths through spacetime. If they have two different opinions on how it came about, what would those two opinions be?
Same reason they may disagree about how their ages got to be different.
They have different personal experiences and observations. Remember why the twins paradox was supposed to be a paradox?

Same reason they may disagree about how their ages got to be different.
They have different personal experiences and observations. Remember why the twins paradox was supposed to be a paradox?
I don't know if this is just a matter of semantics and definitions or what, but I just don't get it. I understand that they had different experiences, but are they to be assumed to be so stupid that they don't recognize that? Why would they not both see that their difference in experiences IS what caused the age difference?

It is semantics.
Each describes their own experiences and notice that their experiences differ - I'm calling that a "disagreement".
But they know about relativity so they can figure it out.

The solution to the original question is to note that each observers personal experiences lead up to the same result when looking through the telescope - just like it does when they compare their ages. (One of them just says "of course you are older than me, for a while there your clock totally went way faster than mine!", while the other says, "sure you're younger than me, your cock was running slow the whole journey." Or something. Different experiences, same result.)

You are thinking as someone used to thinking in terms of relativity, so of course you find this a bit of a puzzling way to put things.
People still learning need to take smaller steps - it's still so mysterious to them.

OK, got it. Thanks.

I am probably over-simplifying this, but it seems to me that they would agree on the size of the observable universe, and they would disagree on the rate of expansion of the universe. I am assuming that they infer the age of the universe by mathematically running the trajectory of observable matter backwards until we see it converges to a point, and calculating how long that takes. That might be way off base.

Alice would say that the cosmological constant (I am dropping a phrase hear that I am not qualified to drop, please help me if its being used in a nonsensical context) had a different value during her trip than before and after her trip.

Am I missing something in taking that perspective?

it seems to me that they would agree on the size of the observable universe

Since Alice and Bob are at rest relative to each other when they make the comparison, yes, they would. But while Alice is moving relative to Bob, she would not assign the same "size" to the observable universe, at least not if she defined "size" as "size in the frame in which Alice is at rest". (There are plenty of technicalities involved here, but I don't think we need to go into them.)

and they would disagree on the rate of expansion of the universe.

If they calculate "rate of expansion" as "amount of proper time it took by my clock to get to its current size", then yes, they would. However, this is not necessarily the correct way to calculate "rate of expansion". See below.

I am assuming that they infer the age of the universe by mathematically running the trajectory of observable matter backwards until we see it converges to a point, and calculating how long that takes.

That's basically what is done in cosmology, yes (although there are plenty of complications which we don't need to go into here). However, as was noted in earlier posts in this thread, "how long" is defined as "how long according to a comoving observer's clock". The reason for that is that comoving observers have a special property: they are the only ones who see the universe as homogeneous and isotropic, all the time. Alice is not a comoving observer (because during the time when she moves relative to Bob, she does not see the universe as homogeneous and isotropic), so if she runs the same calculation but defines "how long" as "how long according to Alice's clock", she will get a different answer.

However, there is another way to define the "rate of expansion" of the universe which does not have this problem. There is a quantity called the "expansion scalar" which is frame invariant; it's the same for all observers. This quantity has the right units to be a "rate of expansion" (it basically is the fractional rate of increase of the scale factor), so it can be used to give an invariant meaning to that term. If Alice measures the expansion scalar while she is moving relative to Bob, she will get the same answer Bob does; so in this sense, they will agree on the rate of expansion of the universe.

Alice would say that the cosmological constant (I am dropping a phrase hear that I am not qualified to drop, please help me if its being used in a nonsensical context) had a different value during her trip than before and after her trip.

No, she wouldn't. The cosmological constant is frame invariant, like the expansion scalar above; it has the same value in all frames and for all observers.

Thanks for the very digestible reply. You have given me some interesting hooks for further reading - much appreciated.