Twin paradox in a closed universe

[JesseM]

This is true until you mention the CMB. To work out this problem one needs to consider relative motion between the travellers frames and the Earth's frame. Neither the CMB nor any other frame is necessary. I wish you would stop talking absolutist nonsense.

It seems unclear whether heldervelez was talking about circumnavigating a closed universe or just making an ordinary round trip of the type that could be taken in an infinite SR spacetime. In the latter case of course you're right that there'd be no need to consider the CMB frame, but in the case of circumnavigating a closed universe you do have to keep track of which frame is the preferred one where copies of identical events happen simultaneously.

 

[heldervelez]

We can detect the absolute rotation using a laser-ring interferometer.
IMO in a closed universe, and in a universe with a finite set of components, a preferred frame can be defined and choosen at will provided it encloses all the universe. The symmetry of this problem apeeals centering a preferred observer at the center. In any rotational motion the center point/line is a preferred reference.
we can think of the travelers at the rim of a rotating circle (or two) rotating clockwise and anticlockwise.
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we can think on a synchronization based on the center, emitting rays to the rim and use mirrors to reflect the light back to the center.
whith this configuration the center will have a 'master-clock' and the whole rim segments (travelers) became synchronous with the central allowing the central observer to know what he needs to monitor the evolution of a traveller allong the trip.
this kind of synchronization is no good to comunicate events between distinct travelers, but it is not the question in this problem.
Ahah, it occurs to me now that each traveller can know is position in space and time using the time,common to all, and use a ring-laser interferometer or a Foucault pendullum to know the 'angle' and add it to the common time, like a gps. (does it works ?)
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I do not know, at this time, how to continue the exploration of this reasoning.
I think I did no errors until now and I will try to continue later.
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[JesseM]

We can detect the absolute rotation using a laser-ring interferometer.

True, but in an infinite non-closed universe that doesn't imply a preferred inertial frame--the rotation is absolute because all inertial frames agree whether something is accelerating, and rotation is a form of acceleration. Hopefully you agree that the ordinary Sagnac experiment doesn't pick out a preferred inertial frame?

 

[Mentz114]

It seems unclear whether heldervelez was talking about circumnavigating a closed universe or just making an ordinary round trip of the type that could be taken in an infinite SR spacetime. In the latter case of course you're right that there'd be no need to consider the CMB frame, but in the case of circumnavigating a closed universe you do have to keep track of which frame is the preferred one where copies of identical events happen simultaneously.

Thanks for the clarification, I guess I wasn't paying attention.

The necessity for a preferred frame other than, say the earth, isn't clear to me. If we assume the Earth isn't going to accelerate, that is.

I withdraw my previous remarks, in any case.

 

[heldervelez]

...Hopefully you agree that the ordinary Sagnac experiment doesn't pick out a preferred inertial frame?

Indeed the Sagnac effect is not sufficient.
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[Daniel42]

The problem with this calculation is that you're not taking into account what I and DaleSpam mentioned earlier, that there are multiple copies of everything: multiple copies of twin 1, multiple copies of twin 2, and multiple copies of each of those 1000 clocks, with the different copies of a given clock not synchronized in twin 1's frame because twin 1 is not in the preferred frame. [...]Try redoing your calculations taking all this into account and see what happens.

I can`t do this. So please let me participate on your knowledge.
It`s your argumentation and so it`s your turn.

 

[JesseM]

I can`t do this. So please let me participate on your knowledge.
It`s your argumentation and so it`s your turn.

OK, let me get you started and see if you can carry on from there. In the twin1 frame you wanted the size of the universe--which is the same as the distance between copies of twin 1--to be 1 light year. If two copies of twin 1 are 1 light year apart in their rest frame, then in the Earth frame where they are moving at 0.5c, the distance between them must be sqrt(1 - 0.5^2)*1 light year = 0.8660254 light years due to length contraction. Since we want to measure time on the clocks in seconds it's more convenient to have distances in light-seconds (that way c=1 light second/second), so since 1 year = 31536000 seconds, 0.8660254 light years = 27310977 light-seconds.

Since the Earth frame is the preferred frame, this is the only frame where identical events happening to different copies are simultaneous. So, take the event of twin 1 departing from twin 2 when the clock twin1 carries (clock1000) reads 0 years. In the Earth frame this happens simultaneously with the event of twin 1B departing from twin 2B when clock1000B reads 0 seconds. If the second event happening to twin 1B is assumed to happen at coordinates x=0 light-seconds, t=0 seconds in the Earth frame, then the first event happening to twin 1 must happen at x=27310977 light-seconds, t=0 seconds in the Earth frame (x=27310977 light-seconds because that's the distance between copies in the Earth frame as mentioned in the first paragraph, t=0 seconds because these identical events must be simultaneous in the Earth frame).

Now we know the coordinates of these identical events (event #1 is twin 1 and twin 2 departing from one another with clock 1000 reading 0 seconds, event #2 is twin 1B and twin 2B departing from one another with clock 1000B reading 0 seconds) in the Earth frame. Can you use the Lorentz transformation to find the coordinates of these same events in twin 1's own rest frame, given that twin 1 is traveling at v=0.5c relative to the Earth? Remember, the equations of the Lorentz transformation are:

x' = gamma*(x - vt)
t' = gamma*(t - vx/c^2)
with gamma = 1/sqrt(1 - v^2/c^2)

It's not actually necessary to find the x' coordinates of these events in the twin 1 frame (if you do you'll find that they take place 1 light year = 31536000 light-seconds apart), but what you want is the t' coordinates of both these events in twin 1's frame. That way you'll know how much earlier event #2 happened than event #1 in twin 1's frame. And since you know clock1000 read 0 seconds at event #1 and clock1000B read 0 seconds at event #2, that'll tell you how out-of-sync the two clocks are. Can you try this calculation, and see if you get an answer or if you run into problems? Then if you get an answer but don't know where to go from there I can give you some more tips (hint: in twin 1's frame, clock1 - clock999 should all be synchronized with clock1000, so if clock1000B is out-of-sync with clock1000 by a certain amount, then clock1000B should be equally out-of-sync with clock1 which is immediately to its right).

 

[Daniel42]

OK, let me get you started and see if you can carry on from there.

I now give it up. It seems that you really want to help me.
But there is one thing you have to know:
It is of no real help to get an question answered with an question (although there may be the best motives) but only frustrating.

Thanks.

 

[JesseM]

I now give it up. It seems that you really want to help me.
But there is one thing you have to know:
It is of no real help to get an question answered with an question (although there may be the best motives) but only frustrating.

Thanks.

Did you at least try plugging in the coordinates of the events I mentioned into the Lorentz transformation to see what the time difference would be in twin 1's frame? This is nothing more than algebra, and if you have trouble with it, you can show your work so I can see where you got stuck and help you out. The advantage of answering questions with hints as to how to figure it out yourself is that you'll probably learn better and gain confidence with using and understanding the equations if you follow the hints, whereas if I just give you the answer you probably won't have much understanding of the logic behind that answer, you'll basically just be taking whatever numbers I give you on faith and having your eyes glaze over at the explanation I give for where the numbers came from.

Also, like I said, if you can figure out the time difference in twin 1's frame but don't know where to go from there, I'm happy to give some additional hints.

 

[LukeD]

So I haven't done the calculations on paper, but what I've done in my head suggests to me that in a closed dimension, velocities are no longer anti-symmetric.

This means that if I see you moving at 0.9c, you might not see me moving at 0.9c in the opposite direction.

I wonder if my intuition is correct...

This should occur because the size of the dimension changes with the velocity.

 

[JesseM]

So I haven't done the calculations on paper, but what I've done in my head suggests to me that in a closed dimension, velocities are no longer anti-symmetric.

This means that if I see you moving at 0.9c, you might not see me moving at 0.9c in the opposite direction.

I wonder if my intuition is correct...

This should occur because the size of the dimension changes with the velocity.

In a closed but flat spacetime velocities would still work the same way as they do in SR. As I've said, a closed but flat spacetime should be exactly equivalent to an infinite flat spacetime where patterns of matter and energy just happen to repeat in a periodic way, like a hall of mirrors but with each image equally real.