Confusion regarding acceleration in SR

[Orodruin]

So the resolution of the paradox would seem to be based on which of the two twins has undergone some acceleration during the experiment. One of them has 'stepped outside' Special Relativity conditions and the amount of the 'time disparity' would have to depend on the amount of Energy that has been put into the exercise in reversing the velocity of the moving twin.

No, acceleration is not the key issue and SR is perfectly capable of handling it. The key issue in how the paradox arises in the first place is a misapplication of what statements made in the theory actually mean.

Geometrically, the accelerating twin having a shorter proper time is equivalent to a curved path being longer. Is it longer because the pen had to change or is it longer because you had to move the pen a longer distance to draw it?

 

[Nugatory]

GR says in gravity generated by acceleration the higher, the faster time goes. higher is farther here.

That's not a prediction of GR. It's a prediction of SR, which is all that's needed to handle acceleration and the inertial forces that appear in accelerating frames. As long as there is no gravitating mass present, GR is never needed.

 

[sweet springs]

I didn't say it necessary .. I would like to say
GR also works.

 

[PeterDonis]

I didn't say it necessary .. I would like to say
GR also works.

A better way of stating the point @Nugatory was making is that in flat spacetime, "GR" is the same thing as "SR". They are not two different theories; they are the same theory.

 

[sophiecentaur]

The key issue in how the paradox arises in the first place is a misapplication of what statements made in the theory actually mean.

OK it's still within SR - sorted. But the essence of who ages slower (the asymmetry) is to do with the Energy / Work expended on him/her. The 'missing' bit in the diagram, during the instant acceleration explains the process well.

 

[Orodruin]

OK it's still within SR - sorted. But the essence of who ages slower (the asymmetry) is to do with the Energy / Work expended on him/her.

I disagree. You can construct "twin paradoxes" without any acceleration by using three observers.

 

[stevendaryl]

OK it's still within SR - sorted. But the essence of who ages slower (the asymmetry) is to do with the Energy / Work expended on him/her. The 'missing' bit in the diagram, during the instant acceleration explains the process well.

It's true in flat spacetime that you can always tell which of two twins has aged the least by which one accelerated, but I wouldn't call that the "essence", since in a simple generalization of flat spacetime, you can have a twin paradox with no acceleration. If you generalize SR to a "cylindrical" universe (by saying that the point with coordinates $x=0, t=0$ in some frame is identified with the point $x=L, t=0$), then you can have two twins, one of who sits at $x=0$ and the other who travels at constant velocity in the $x$ direction. When they get back together, the "traveling" twin will be younger. But neither accelerates.

 

[jbriggs444]

But the essence of who ages slower (the asymmetry) is to do with the Energy / Work expended on him/her.

But an acceleration that takes an outward velocity and produces an inward velocity of equal magnitude requires zero work.

 

[SiennaTheGr8]

It's true in flat spacetime that you can always tell which of two twins has aged the least by which one accelerated, but I wouldn't call that the "essence", since in a simple generalization of flat spacetime, you can have a twin paradox with no acceleration.

Or one where both twins accelerate!

 

[sophiecentaur]

If you generalize SR to a "cylindrical" universe

But I suggest that the twins paradox would not be a paradox when the curvature was involved. For a start, all distances could be Modulo(The circumference of the Universe)

 

[stevendaryl]

But I suggest that the twins paradox would not be a paradox when the curvature was involved. For a start, all distances could be Modulo(The circumference of the Universe)

I'm not sure how that helps. But it seems to me to be equally paradoxical. You have one twin, Alice, at rest. You have a second twin, Bob, moving away at 99% of the speed of light. From Alice's point of view, Bob is aging slower, and from Bob's point of view, Alice is aging slower. It's just as paradoxical as the flat space version. But eventually, Bob will meet up with Alice from the other direction, and one of them will be older than the other. I don't see how viewing distances modulo the circumference of the universe helps.

 

[A.T.]

he can claim that the simultaneous events on Earth changed so much due to the acceleration..

Since it's just a convention, he can claim whatever he wants. He is not actually physically affected by those distant events at the time he claims they happen.

 

[sophiecentaur]

I'm not sure how that helps. But it seems to me to be equally paradoxical. You have one twin, Alice, at rest. You have a second twin, Bob, moving away at 99% of the speed of light. From Alice's point of view, Bob is aging slower, and from Bob's point of view, Alice is aging slower. It's just as paradoxical as the flat space version. But eventually, Bob will meet up with Alice from the other direction, and one of them will be older than the other. I don't see how viewing distances modulo the circumference of the universe helps.

If you change the Cylindrical Universe to ants on the surface of the Earth then there are loads of 'surprises' about relative lengths of journeys but I wouldn't say that they are paradoxes. But perhaps the ants would think of them in that way.
Perhaps we have done this to death. There is no limit to how many scenarios we could think up. With Bob moving away fast around the cylinder (and Alice moving equally fast away from him, they could both meet on the other side but they would each see the other one as younger (in years) rather than just observing their clocks being slow as they go past each other. Isn't that an even more paradoxical paradox if they could be standing next to each other and both be older and younger than their twin? I hope I've missed something obvious about this because it's quite uncomfortable to contemplate.

 

[A.T.]

I hope I've missed something obvious about this because it's quite uncomfortable to contemplate.

In the cylindrical universe there would be a preferred frame globally.

 

[stevendaryl]

If you change the Cylindrical Universe to ants on the surface of the Earth then there are loads of 'surprises' about relative lengths of journeys but I wouldn't say that they are paradoxes. But perhaps the ants would think of them in that way.
Perhaps we have done this to death. There is no limit to how many scenarios we could think up. With Bob moving away fast around the cylinder (and Alice moving equally fast away from him, they could both meet on the other side but they would each see the other one as younger (in years) rather than just observing their clocks being slow as they go past each other. Isn't that an even more paradoxical paradox if they could be standing next to each other and both be older and younger than their twin? I hope I've missed something obvious about this because it's quite uncomfortable to contemplate.

Well, the point of introducing the cylindrical universe was just to show that acceleration isn't really necessary for the paradox; differential aging can happen even when nobody accelerates.

The cylindrical universe is interesting because it's actually equivalent in every way to an infinite flat universe with periodic boundary conditions. Alice sitting on Earth can look through a powerful telescope and see, far away, another copy of the Earth, and beyond that, another copy and another copy, ad infinitum. After adjusting for the delay for light to travel such a distance, she would compute that all the Earths are the same age. Similarly, there is an infinite line of Bobs all the same age, traveling at the same velocity. Each Bob travels from one Earth to the next, and when he arrives, he is younger than the Alice of that Earth.

From the perspective of Bob, though, things are weirder. He also sees a line of infinitely many Earths, and infinitely many Alices and infinitely many Bobs. But the difference for Bob is that looking in one direction, each Alice is older than the last. From his point of view, his Alice leaves and a second, older Alice starts moving toward him. The second Alice is aging slower than Bob is, but she starts out older, so when she gets to Bob, she's still older than he is.

 

[Kontilera]

Okey, I feel like I need to explain some missunderstandings about the whole thread now. :)First of all, I've taken SR and GR courses on university level.
And I feel like I do understand the SR and at least some GR.

The twin paradox itself is not a problem for me. I understand that the acceleration boosts Bobs inertial frame of reference. That his notion of what direction time has changes and that he will realize that he his the younger one when he travels back to earth.
My fault is that I put my confusion into a twin-paradox-context and now it feels like many people want to explain the twin paradox for me.
An effort I, of course, am very, very thankful for!
My confusion is about what you do see when you do change your inertial frame of reference by accelerating, if we play with the thought that our telescopes could give us a good resolution on very large distances.

Take the scenario described by Penrose in 'The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics' as an example. In this case he discusses a potential invation of Earth by the civilization in the andromeda galaxy.

"Two people pass each other on the street; and according to one of the two people, an Andromedean space fleet has already set off on its journey, while to the other, the decision as to whether or not the journey will actually take place has not yet been made. How can there still be some uncertainty as to the outcome of that decision? If to either person the decision has already been made, then surely there cannot be any uncertainty. The launching of the space fleet is an inevitability. In fact neither of the people can yet know of the launching of the space fleet. They can know only later, when telescopic observations from Earth reveal that the fleet is indeed on its way. Then they can hark back to that chance encounter, and come to the conclusion that at that time, according to one of them, the decision lay in the uncertain future, while to the other, it lay in the certain past. Was there then any uncertainty about that future? Or was the future of both people already "fixed"?" - R. Penrose

Wikipedia solves this "paradox" by stating:

"The "paradox" consists of two observers who are, from their conscious perspective, in the same place and at the same instant having different sets of events in their "present moment". Notice that neither observer can actually "see" what is happening in Andromeda, because light from Andromeda (and the hypothetical alien fleet) will take 2.5 million years to reach Earth. The argument is not about what can be "seen"; it is purely about what events different observers consider to occur in the present moment."

My question is: If they see the same thing in the andromeda galaxy. Then how can the person approaching the Andromeda galaxy even justify that his present is simultaneous to the fleet already moving? I mean, when calculating your simultaneous universe I assume that you can use the formula:

What the time is now in Andromeda = The time I observe right now in Andromeda + The time that the andromeda people experience that it took for light to reach me.

You might say that the second term here changes due to time dilation and length contraction... but that doesn't seem to solve the problem, because those effects are independent of the sign of the velocity. So if one person is walking with the velocity +v and the other one is walking with the velocity -v, they will both experience those effects but still disagree of what is simultaneous in the Andromeda galaxy.

 

[sophiecentaur]

according to one of the two people, an Andromedean space fleet has already set off on its journey,

Do you mean he just thought it was a possibility and happened to voice that exactly the 2.5 million years before Earth observers happened to spot the fleet? If that was just a random idea then why is it of any consequence? Plenty of people are convinced they dreamed about the winner's name and the actually manage to back the winner. How does that fit in with this stuff? Clearly, Penrose had a serious message there.

 

[jartsa]

What the time is now in Andromeda = The time I observe right now in Andromeda + The time that the andromeda people experience that it took for light to reach me.

You might say that the second term here changes due to time dilation and length contraction... but that doesn't seem to solve the problem, because those effects are independent of the sign of the velocity. So if one person is walking with the velocity +v and the other one is walking with the velocity -v, they will both experience those effects but still disagree of what is simultaneous in the Andromeda galaxy.

Let's use the same Olympics sprint example as before, and the same Bob that acceletes.

Bob before acceleration: "At this moment on Earth people should be saying the Olympic event is happening right now. And those people should be saying that Bob is one light day away"

Bob after acceleration: "At this moment on Earth people should be saying the Olympic event was two weeks ago. And those people should still be saying that Bob is one light day away. And then they should say that two weeks ago Bob was 10 light days away. Because at the time of the Olympic event Bob started moving towards the Earth quite fast".

 

[alejandromeira]

My confusion is about what you do see when you do change your inertial frame of reference by accelerating, if we play with the thought that our telescopes could give us a good resolution on very large distances.

Unfortunately light don't have infinite speed, and then all these questions have to be solved with the minkowsky diagram for radio signals, and the equations of the contraction of length and dilation of time.
We can perfectly know the time measured in andromeda when our radio signal comes out and when it arrives, (we can know ablolutely the time meausured in both reference frames).

In a real world, the acceleration is not instantaneous, and the trajectory of the traveling twin, in the Minkowsky diagram, will be rounded in the area of acceleration. In this area we can draw their axes, if we do it and we extend the traveling X axis until we cut the T axis at rest, we will see how the time of the resting twin begins to increase. Specifically, at the point of V = 0 for the traveler, the time at rest is half of what the trip takes. Later it is symmetric about the outward trip.

The question of when we 'perceive' it by the telescope, then we would have to go back to the other Minkowsky diagram of the radio signals.

Regarding Penrose's paradox, in that case I have to say clearly that I do not understand anything about that.

 

[Kontilera]

Do you mean he just thought it was a possibility and happened to voice that exactly the 2.5 million years before Earth observers happened to spot the fleet? If that was just a random idea then why is it of any consequence? Plenty of people are convinced they dreamed about the winner's name and the actually manage to back the winner. How does that fit in with this stuff? Clearly, Penrose had a serious message there.

No, I´m with you. Of course he cannot observe whether the fleet leaves the Andromeda galaxy or not.
But if you watch the minutephysics video that seems to be the explanation.

As for me I´m trying to ask: In what sense is the future of the Andromeda galaxy part of his 'present-universe' if he can't observe it and none of the terms in my pseudoequation changes?

 

[Janus]

I feel that I understand the twin paradox perfectly when looking at the signaling diagram and the plane-of-simultanity diagram separate. But I can't figure out how they can be compatible with each other.My question is: If they see the same thing in the andromeda galaxy. Then how can the person approaching the Andromeda galaxy even justify that his present is simultaneous to the fleet already moving? I mean, when calculating your simultaneous universe I assume that you can use the formula:

What the time is now in Andromeda = The time I observe right now in Andromeda + The time that the andromeda people experience that it took for light to reach me.

No, the formula is, T now in Andromeda = T I observe + time it took the light to reach me as measured by me from the distance Andromeda was when the light left it x the time dilation factor for Andromeda's speed.

If you and and Andromeda are approaching each other, then Andromeda was further from you when the light left then it is when you see the light. If you are receding from each other, then it was closer to you when the light left it than it is when you see the light. Thus while two people passing each other at the same speed relative to Andromeda, but in different directions will agree that Andromeda is the same distance away from them at the moment they pass, they will not agree as to how far Andromeda was from them when the light left Andromeda, and thus how long it took the light to make the trip.
The time dilation factor for Andromeda will be the same for both, So if they take the time they see, How long the light has traveled according to them, and factor in time dilation, they get what time it is now at Andromeda according to them, and come up with different answers as to what time it is at Andromeda when they pass each other.

To illustrate, here are the space-time diagrams for our approaching and receding observers.

Approaching:

Red line is Andromeda, Green is approaching observer, Yellow line is the light traveling from Andromeda and reaching observer when he passes receding observer (blue line). The black arrow is the distance between observer and Andromeda when the light left. The white line marks off what time it is "now" at Andromeda according to our observer when he sees the light.

Receding:

Same meaning for the Lines. We see that Andromeda was closer to our Observer when the light left, and thus took less time to reach the observer. Thus the "Now" line for the observer crosses Andromeda's line at a different point then it did for the approaching observer ( between ticks 1 and 2 instead of between ticks 2 and 3.)

 

[sophiecentaur]

In the cylindrical universe there would be a preferred frame globally.

Could you translate that please? Which would be the 'preferred frame"?

 

[pervect]

My question is: If they see the same thing in the andromeda galaxy. Then how can the person approaching the Andromeda galaxy even justify that his present is simultaneous to the fleet already moving? I mean, when calculating your simultaneous universe I assume that you can use the formula:

What the time is now in Andromeda = The time I observe right now in Andromeda + The time that the andromeda people experience that it took for light to reach me.

This doesn't look right to me. Possibly I'm misunderstanding your question.

Let's suppose that "I" use coordinates in an unprimed frame, (t,x). We'll forget about y and z for now, they aren't relevant to the problem.

Let's suppose that the Andromedean's use the coordinates in a primed frame, (t', x').

Then if I'm understanding your words correctly, the time now in Andromeda = t'
And the time I observer right now in Andromada = t

A big concern I have is what you mean by the notion of "now". You haven't clarified which frame the notion of "now" you are using applies to. This is usually indicative of not understanding the relativity of simultaneity. If it is an accidental omission on your part, it's still not at all clear what you think you mean when you say "now", and that issue seems to be close to the heart of the problem you are struggling with.

If we assume that I have translated your words correctly into mathematics, then we can apply the Lorentz transform and write:

$$t' = \gamma(t - v\,x/c^2)$$

This gives us the "time now in the Andromeda frame", t', as a function of the time now in my frame, t, and the distance now in my frame, x. We could do some math to try to get rid of the x (distance in my frame) and replace it with x' (distance in the Andromedean frame). I actually did so, but I don't want to get into the details unless it appears to be useful, which at this point it doesn't.

Your words are suggesting

$$t' = t + x'/c$$

And this just isn't right. And more importantly I don't know why you think it would be.

 

[sophiecentaur]

Well, the point of introducing the cylindrical universe was just to show that acceleration isn't really necessary for the paradox; differential aging can happen even when nobody accelerates.

Except that the whacky geometry of space seems to change everything - not the least because, as you say, there will be multiple images of bob out there. But the possibility of both twins going in opposite directions and meeting on the other side has to introduce the possibility of a real paradox which cannot happen in flat space. Is my reasoning about that flawed? In flat space, the only 'odd 'thing' is that each twin will observe the other as aging slower. They will never stand together unless one of them does some accelerating to bring them together and there will definitely be an older and a younger twin. In your cylindrical universe they both start their engines and accelerate apart. The visual ageing rates will be different. When they pass each other on the other side they will actually see each other as younger. Now what happens if they both apply retro rockets and land back on Earth? Will their body clocks be back in sync with both of them being younger than the staff on Earth?

 

[Orodruin]

But the possibility of both twins going in opposite directions and meeting on the other side has to introduce the possibility of a real paradox which cannot happen in flat space

It doesn’t. Does the fact that you can draw different lines between two points on a torus mean that there is an ambiguity in the length of those lines? Also, the cylindrical universe is flat.

 

[alejandromeira]

To illustrate, here are the space-time diagrams for our approaching and receding observers.

Approaching:

Red line is Andromeda, Green is approaching observer, Yellow line is the light traveling from Andromeda and reaching observer when he passes receding observer (blue line). The black arrow is the distance between observer and Andromeda when the light left. The white line marks off what time it is "now" at Andromeda according to our observer when he sees the light.

Receding:

Same meaning for the Lines. We see that Andromeda was closer to our Observer when the light left, and thus took less time to reach the observer. Thus the "Now" line for the observer crosses Andromeda's line at a different point then it did for the approaching observer ( between ticks 1 and two instead of between ticks 2 and 3.)

Thanks a lot.
I will to work whit these diagrams for understand what they mean.

 

[jbriggs444]

If they see the same thing in the andromeda galaxy. Then how can the person approaching the Andromeda galaxy even justify that his present is simultaneous to the fleet already moving?

It's just a coordinate choice. It has no measurable consequences. There is nothing to justify. The launching of the Andromedan battle fleet is neither in our past nor our future light cone.

The boundaries of our past and future light cones are coordinate independent. Those are physical and observable. The event of the Andromedan battle fleet's launch is in neither cone. The hyper-plane of simultaneity is an artifact of coordinate choice, not physical at all. It is only a matter of accounting.

When does the bill for the launch cost for the Andromedan battle fleet come due? It is insured at launch? That's is a question for accountants and the insurance men, not a question about physical reality.

 

[Kontilera]

Thank you!


I just read your posts and I think its clear now.
I will contemplate on this and seee if I have any futher questions, but I think I understand my missunderstanding now.

So, again: Thank you!

 

[A.T.]

Could you translate that please? Which would be the 'preferred frame"?

See links here:
https://www.physicsforums.com/threads/the-cosmological-twin-paradox.51197/#post-361621

 

[sophiecentaur]

See links here:
https://www.physicsforums.com/threads/the-cosmological-twin-paradox.51197/#post-361621

Perhaps you could translate those links instead and help with identifying and explaining the 'preferred frame'. Those links introduce so much other stuff that my paths is getting more divergent than the convergent I was hoping for.