# Help with twin paradox

1. Oct 16, 2015

### aliens123

I was hoping I could get some help on wrapping my head around the twin paradox. The problem is I have never seen the following "contradiction" addressed. I understand (at least on paper) that events simultaneous in one frame needn't be simultaneous in another. But consider the following argument:

Consider three observers, Oscar, Beth, and Jim.

Jim is tasked with retrieving a rare mineral from the planet near the star Alkaid. Alkaid is 99-light years from Earth. Jim knows that with the technology available he can travel at a speed of .98c. The trip to Alkaid will therefore take him 20 years in his frame. However, to observers on Earth, it will take 101 years.

Beth stays at home on the Earth. Jim travels to Alkaid. Oscar is in the same reference frame as Jim. Jim gets in a rocket and travels to Alkaid. The Earth and Alkaid synchronize clocks with respect to one another. Jim gets in a rocket and travels to Alkaid. When he leaves his frame, his clock, Earth’s clock, and Alkaid’s clock all read 0.

When Jim arrives at Alkaid he finds that the clock on Alkaid which was synchronized to Earth reads 101 years. Oscar however, who was in the same frame as Jim, is present whizzing over Earth at the time Jim reaches Alkaid. He looks down at the clock on Earth and it reads 3.96 years! This is because the Earth bound clock was moving relative to their frame.

Imagine Beth, when her clocks reads year 3.96, looks upward. She sees Oscar whizzing by over her, and sees that his clock reads a time of twenty years. She reasons that this means Jim has already reached Alkaid. She is able to send a message to Oscar (because Oscar is right above her) to reverse the direction of the ship. Oscar does this nearly instantaneously. Jim returns home another 3.96 years later. This means that he returned home to earth in 7.92 years (as measured by the clocks on earth), with the rare mineral intact.

But to observers on Earth this trip should have taken a total of 202 years.

2. Oct 16, 2015

### Orodruin

Staff Emeritus
The problem is that your understanding seems to stay on paper. Your assumption is that Beth will conclude from Oscar's passing by that Jim has arrived. This is not true. If Beth knew relativity, she would know that even if the event where Jim reaches the goal is simultaneous with Oscar's clock showing 20 years in Oscar's rest frame, they will not be simultaneous in her rest frame due to the spatial separation. She will therefore disagree with Oscar, instead observing that Jim at that time has hardly begun the journey at the time when Oscar passes by.

3. Oct 16, 2015

### aliens123

Thanks for the reply.

So assuming Beth orders this turnaround at 3.96 years, what would the story they all tell be? Imagine right after the turnaround Oscar "jumps out" onto Earth. Once Jim arrives wouldn't Oscar say "you were at Alkaid when I jumped out." ?

4. Oct 16, 2015

### Orodruin

Staff Emeritus
Beth orders turnaround at 3.96 years according to which frame?

If Oscar is changing inertial frame he is changing his perception of what simultaneous events are. He might conclude that Jim would have arrived at the goal at the same time he jumped out in his original inertial frame. However, if Jim stopped when the clocks in Beth's frame showed 3.98 years, he would have stopped long before that.

Another common misconception is to mix up what observers actually see with what is going onin their frames. When we talk about an observer findig two events simultaneous it does not mean that the light from those events reach him at the same time. Instead, it means that if he takes the constant speed of light and the distance to the event into account, he can compute when they occured and whether they were simultaneous or not.

Even if Jim arrives at the goal at the same time as Oscar passes the Earth (in their common rest frame), Oscar will not see the light from this happening until 20 years later (the distance between Jim and Oscar is 20 light years in their rest frame). However, when this light does reach Oscar, he can conclude that the event of Jim reaching his goal must have been simultaneous with him passing the Earth (or, more accurately in their frame, that Alkaid reaching Jim was simultaneous with Earth reaching him).

5. Oct 16, 2015

### aliens123

I guess it just seems like at 3.96 years (to Beth) Jim is both arriving and not arriving. He is not arriving because 101 years need to pass before that happens, but he has arrived because Oscar is above her.

But then we say that Oscar and Jim won't be synchronized to Beth. So when Beth looks up at Oscar - this doesn't mean that Jim has arrived. Ergo, when Beth orders the ship to turnaround, and Oscar comes back down to Earth, they will discover that Jim actually wasn't at Alkaid because they are synchronized according to Beth's frame now.

How is it though, that after Oscar comes back down to Earth, to him, Jim should have arrived? How does the frame switch stop something from ever happening which should have happened at least in Oscar's frame?

I hope I am making myself clear. When you say:

"However, if Jim stopped when the clocks in Beth's frame showed 3.98 years, he would have stopped long before that."

How is this consistent with Oscar's perspective?

6. Oct 16, 2015

### Orodruin

Staff Emeritus
No, you cannot have it both ways. In Beth's frame he definitely has not arrived. She will disagree with Oscar on this because he is in a different inertial frame. They both know this because they know simultaneity is not an absolute concept.

No he should not have arrived. He turned around long before Oscar arrived at Earth in the rest frame of Oscar and Jim.

Why do you think it is not consistent? Jim will never arrive if Beth orders him to turn around at the time (in her frame) of 3.98 years. If Jim was allowed 20 years in his travelling frame he would arrive, ergo he must have turned around earlier than when the time in his travelling frame reaches 20 years, which happens to be the time when Oscar arrives at Earth in that frame.

7. Oct 16, 2015

### aliens123

Because

"Oscar however, who was in the same frame as Jim, is present whizzing over Earth at the time Jim reaches Alkaid. He looks down at the clock on Earth and it reads 3.96 years! This is because the Earth bound clock was moving relative to their frame."

8. Oct 16, 2015

### Orodruin

Staff Emeritus
But you again fail to take the simultaneity of relativity into account! If Jim is ordered to stop when the clocks in Beth's frame shows 3.98 years, Oscar will know that this is not simultaneous (in his frame) to when he finds himself at an event where the clock in Beth's frame shows 3.98 years.

9. Oct 16, 2015

### aliens123

Okay, its starting to make sense, thank you.

10. Oct 17, 2015

### aliens123

So one more time...

Consider only the perspective of Oscar. Oscar is moving over Beth. To him this happens at the same time that Jim reaches Alkaid. A fraction of a second after he moves over Beth he decides to nearly instantaneously comes to a stop. He is now in Beth's frame at 3.98 years, meaning that Jim is no longer at Alkaid but instead has just begun his trip. Is it correct to say that something that previously was happening (to him) now hasn't happened yet (to him)?

I understand that mathematically the only reason he should conclude the event of him passing over Beth was the simultaneous with Jim passing over Alkaid is because he would receive the light signal that this occurred in his frame at a time which would correlate perfectly with the distance. But doesn't special relativity say that the events actually happen at the same time (to Oscar) so long as he is at rest with respect to Jim when he passes over Earth?

11. Oct 17, 2015

### Vitro

If and only if Jim has/will arrive(d) at Alkaid, then the events "Oscar meets Beth" and "Jim arrives at Alkaid" are simultaneous in Oscar's frame, they had both just happened. In Beth's frame Jim has only traveled a small fraction of the distance when Oscar passes by her, the event "Jim arrives at Alkaid" is still far into her future.

Of course Oscar would only know that Jim has arrived at Alkaid theoretically because he knows that's how the scenario was constructed. If something were to have happened and Jim's ship exploded half-way into the trip Oscar would have no way of knowing that, even though the explosion event would be in his past. The thing is that, regardless of when events actually happened, knowledge of them happening and any causal connection between them is limited to the speed of light.

But, let's suppose Jim was carrying a giant clock visible to both Oscar and Beth which they have been watching through telescopes, then as Oscar passes Beth they both see the exact same image of Jim's clock, and for every practical purpose they both can only know of Jim's fate up to that time and nothing further.

12. Oct 17, 2015

### Orodruin

Staff Emeritus
No it is not correct. The same thing must happen in all frames, although the description of why may differ. Jim either turns around before he reaches the destination or not. In Oscar's frame, this would be because he turns around before Oscar reaches Earth. In Beth's frame it is because he turns around at the same time Oscar reaches the Earth.

If he knew how it was constructed, he would have known Jim was going to turn around before. Of course he could say that Jim would have arrived had he continued at the same velocity - but he didn't.

You have to separate what observers actually see and what is happening in the different frames. Trying to figure out how something would actually look is a further step of complication which requires taking the light travel time into account as well.

13. Oct 17, 2015

### aliens123

Forget the whole idea of Jim turning around. Just assume Jim is going to land on Alkaid no matter what.

Oscar who is in the same frame as Jim whizzes over Earth at 3.96 years. Because he is in the same frame as Jim this happens at the same time that Jim lands on Alkaid - to Oscar. (Correct?)

Now suppose Oscar spontaneously decides to "jump out" and decelerate almost instantaneously. We just agreed that to him, Jim was on Alkaid. So after he decelerates and joins Beth in her frame, what has happened to Jim, to Oscar?

14. Oct 17, 2015

### Orodruin

Staff Emeritus
When you change your inertial frame you change your idea about what simultaneity is. In his new frame, Oscar will not consider Jim to have arrived yet.

Note that this is all irrelevant, the event of Jim arriving is not causally connected to the event of Oscar passing the Earth so there is no way these two events can influence each other.

A better (or at least less confusing) splitting of spacetime based on an event in relativity is into events which are causally connected and in the future, events which are causally connected and in the past, and events which occur "elsewhere". The elsewhere events may be in the future or past depending on the inertial frame used. Jim arriving is "elsewhere" from Oscar arriving at Earth.

15. Oct 17, 2015

### aliens123

Right, and I understand that at least theoretically I think. But according to this it seems like the following statement is true:

"Something that previously was happening (to him, in frame A) now hasn't happened yet (to him, in frame B)."

True.

What do you mean?

16. Oct 17, 2015

### Orodruin

Staff Emeritus
The concept of simultaneous in relativity is just a convention, and a confusing one at that. It depends completely on a relatively arbitrary splitting of space and time based on a particular inertial obsrver. As such, it really has no physical meaning. The concept of elsewhere is much more physical, it is the collection of events which are not causally connected to a specified event.

17. Oct 17, 2015

### Vitro

The post I replied to implied Jim actually arrives at Alkaid, I have also stated it explicitly in my reply.
That's exactly what I was doing, separating what actually happens from Beth and Oscar's knowledge of it, as in separating past light cone events (from which Beth and Oscar have received light) and elsewhere events (of which neither has any physical proof, like Jim arriving at Alkaid). When Oscar and Beth pass each other their light cones coincide, neither knows more than the other about elsewhere events. This was in anticipation of the later posts where it was hinted that when Oscar changes frames he has knowledge of events that had already happened in one frame and which haven't happened yet in the other frame.

18. Oct 17, 2015

### Staff: Mentor

You're trying to work through how Oscar can say that Jim has already landed on Alkaid, and then after he slows down to match Beth's speed say that Jim has not yet landed on Alkaid.

First we have to consider exactly what it means when Oscar says that Jim has "already landed" on Alkaid which is 3.96 light-years away. It means that the light from Jim's landing (or a radio message from Jim confirming his arrival) will reach him in less than 3.96 years from now. This is just the basic idea that when light from some event hits my eyes, I can figure out when the event happened by subtracting the light travel time from the time that the light hit my eyes. Note that because the light or radio message from Jim's arrival has not reached Oscar yet, he has not actually observed that Jim has landed; he has merely calculated from what he knows of the speeds and distances that the light from Jim's arrival 3.96 light-years away will reach his eyes in less than 3.96 years.

But when Oscar slows down to match speeds with Beth, he must redo the calculation. This time he calculates that Alkaid is 99 light-years away and that it will be more than 99 years before the light from Jim's arrival reaches his eyes. Therefore, Jim has not yet arrived. Again, this is not something that Oscar observes, it's something that he calculates.

Thus, the answer to your question is that nothing has happened to Jim, and the only thing that has happened to Oscar is that he has changed speed and position in such a way that the light from Jim's arrival will reach his eyes at a different time.

19. Oct 17, 2015

### Staff: Mentor

The problem here is with the notion that Oscar "changed frames". He didn't. A frame no more than a convention for assigning x, y, z, and t coordinates to events so Oscar is always in both frames. When he slowed down to match speed with Beth, his speed in one frame (the one in which Jim's speed is 0) changed from 0 to -.99c while his speed in another frame (the one in which Beth's speed is 0) changed from .99c to 0.

20. Oct 17, 2015

### Staff: Mentor

But where is Oscar in this frame?

"When he leaves his frame" in which frame?

This is a very common error in formulating relativity problems: forgetting that "when" has no meaning until you've specified which frame the "when" is relative to.

What I think you mean to say here is: in the Earth-Alkaid frame, the event at which Alkaid's clock reads 0 is simultaneous to the event in which Earth's clock reads 0, and Jim leaves Earth at the latter event, and Jim's clock is set to 0 when he leaves, i.e., at the latter event. But what you said is open to other interpretations.

This seems to imply that Oscar is behind Jim in Jim's frame--about 19.7 light-years behind, since in this frame, Oscar is co-located with Earth at the same instant that Jim is co-located with Alkaid, and Alkaid and Earth are 19.7 light-years apart in this frame (99 light-years length contracted by the gamma factor corresponding to 0.98c).

Yes.

Yes.

But why would she reason this way? Only 3.96 years has passed for her, so Jim can't possibly be more than 3.96 * 0.98 = about 3.9 light-years away.

What she will reason, instead, is that in her frame, Oscar's and Jim's clocks are not synchronized; Oscar's clock is running 20.1 - 3.96 = about 16.1 years ahead of Jim's.

Oscar can nearly instantly reverse the direction of his ship, but he certainly can't instantly reverse the direction of Jim's ship, which is 19.7 light-years ahead of him, in his frame (before he reverses direction). Any message he sends to Jim to reverse direction will take time to get there. When you work it out, you will see that, in the Earth-Alkaid frame, Oscar's signal cannot reach Jim until 3.96 years after Jim arrives on Alkaid; so even if Jim leaves immediately, he will not arrive back on Earth until 202 + 3.96 = about 206 years after he left, according to Earth clocks. (His own clock will read about 44 years when he returns.)

If, instead, you are imagining Oscar and Jim to occupy a single ship which is 19.7 light-years long, while that isn't impossible (just very, very improbable), that just means that the ship won't be able to turn all at once. Oscar's end can start turning as soon as he receives Beth's signal, but it will still take time for the change in direction to propagate through the ship and reach Jim's end. In other words, the ship cannot be treated as a perfectly rigid body; no object can, because the information telling each part of the ship to turn cannot reach that part faster than light.

21. Oct 17, 2015

### Vitro

By "changing frames" I meant changing his rest frame, as in going from being at rest in one frame to being at rest in another (Beth's rest) frame. I thought it's quite a common and unambiguous short-form.

22. Oct 18, 2015

### aliens123

Yessir.

I understand this, but this seems to imply that the things didn't really happen simultaneously, only that it would be calculated this way. But special relativity tells us that things really are simultaneous in a different frame.

Drawing a spacetime diagram (with simultaneity lines) helped me understand the situation more clearly.

23. Oct 18, 2015

### Orodruin

Staff Emeritus
But this is the entire point. There is no such thing as "really simultaneous" which is frame independent. It therefore makes no sense to claim that two events are simultaneous without referring to which frame they are simultaneous in.

24. Oct 18, 2015

### aliens123

A certain distance behind Earth? (Don't want to do the math)

You are correct, I don't know why I phrased it that way to begin with. I think I meant to say "when he leaves Earth.

So I've mostly dropped this line of argumentation and instead am now interested in what things look like to Oscar.

So to me this seems like dodging the question, because what I originally meant was that as soon as Oscar passes above Earth the entire really long ship reverses direction simultaneously. Of course, you could refute my original confusion by pointing out that just because the ship reverses direction simultaneously in Oscar/Jim's frame doesn't mean that this would be simultaneous in Beth's frame. So I was wrong, but I don't see why the light signal travel time matters?

Edit: Actually on rereading I realize now that you basically already said that with "the ship cannot be treated as a perfectly rigid body."

25. Oct 18, 2015

### aliens123

Didn't I specify "really simultaneous" within a certain frame?

"But special relativity tells us that things really are simultaneous in a different frame. "

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