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I have a twin who stays on Earth, and I will move to another planet 4 ly away. I will travel at a certain relativistic speed towards this planet. What I just described here is the first half of the Twin Paradox, isn’t it?

Now, I understand that, due to time dilation, the clock of everything that is moving relative to an observer ticks more slowly than the observer’s clock.

Since my twin on Earth watched me move away at such high speed to the planet, my clock ticks more slowly compared to his clock. So, when I arrive at the planet, I will be younger than him. But, from my point of view, my twin on Earth moves away from me at the same speed. So, his clock would tick more slowly than mine during the trip, and when I arrive my twin would be younger than me.

I know that the full version of the Twin Paradox is not really a paradox in special relativity. But just can’t make sense of the seemingly paradoxical case that I just described which is actually just the first half of the Twin Paradox.

Would anyone please help me point out the mistakes that I made in my thought process?

Qmechanic
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Mr Deevid
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    This is a duplicate of this question – Marius Ladegård Meyer Sep 25 '23 at 09:19
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    Does this answer your question? How can time dilation be symmetric? – Stéphane Rollandin Sep 25 '23 at 09:22
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    At the start of your trip, you accelerated towards the other planet. That broke the symmetry between you and your twin, who did not accelerate. Unlike velocity, acceleration is absolute so you and your twin can agree that it was you who accelerated. – Mike Scott Sep 25 '23 at 09:33
  • The issue is that asking "who is older" is meaningless unless the twins are brought back together in the end. That's why, to make the weirdness of relativity more apparent, the twin paradox story has the twins rejoin on Earth. There is no universal notion of "when I arrive." Do you mean "when the person on earth sees you arrive" or do you mean "when the person on earth sees you arrive minus four years to account for the light travel." Which one you choose determines whether the age (proper time) of the person on earth is greater than the age of the person when they arrived on the other planet. – AXensen Sep 25 '23 at 10:04
  • @MikeScott This is a common argument in response to "but the twins are in a symmetric situation! both of them think the others' time moves slower!" But the argument never really helped with calculating the difference in age-and in the end, this acceleration has no bearing on the calculation. And in this particular question it's kind of irrelevant. You could ask the same question about a spaceship flying past earth that never accelerates, but the spaceship and a ground observer synchronize clocks in the moment the spaceship passes. – AXensen Sep 25 '23 at 10:12
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    @AXensen is this going to be another "terrible twins" question that gets bogged down in comments and then "moved to chat"? Perhaps you might consider commenting on or improving my actual answer below? Of course, what would be even nicer would be for the OP to reappear and say whether any of our contributions helped! – m4r35n357 Sep 25 '23 at 10:23
  • @AXensen Thank you for your reply, it was thought provoking. I am talking about the moment when I arrive at the planet. I would thus expect that the twin on Earth would account for the light travel. In fact, if both me and him know exactly the distance to the planet and the speed that I am traveling with. We would both predict that the other would be younger at the moment when I arrive at the planet. So there would be no need to actually observe it, right? – Mr Deevid Sep 25 '23 at 11:28
  • @MrDeevid I think part of the issue here is that you're trying to make a convoluted combination of the twins understanding and accounting for relativity but not understanding enough that they still get confused by the answers the calculations spit out. Sure... we can calculate how much proper time passes on earth in the time between the twin leaving and four years prior to light hitting the earth showing the other arriving (do the calculation!). And we can calculate how much proper time passes for the twin between the earth and the other planet (do the calculation!). – AXensen Sep 25 '23 at 11:34
  • Thank you OP ;) – m4r35n357 Sep 25 '23 at 11:52

2 Answers2

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The answer is that you are younger than your twin in his frame, and he is younger than you in yours. There is no paradox here. The reason why this happens is that you and your twin have different 'nows' owing to the relativity of simultaneity. When you compare your ages, what you are effectively asking is how old is my twin 'now', where 'now' is whatever time it happens to be where you are. Because you and your twin are moving relative to each other, you have tilted planes of simultaneity, so 'now' for you on a distant planet is not the same time as 'now' for your twin back on Earth.

If you decelerate to come to rest relative to your twin, then you break the symmetry and you become younger than your twin in what is now your common rest frame. Alternatively, your twin could accelerate to reach your speed relative to the Earth and become stationary relative to you in that way, in which case it would be your twin who would be younger in the common frame.

Marco Ocram
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  • Let’s say that my twin knows exactly the distance to the planet and the speed that I travel with. He would be able to predict how much time would have passed on my clock due to time dilation. Based on that he would know how old I would get when I arrive. Same goes for me when I try to predict how old my twin would be when I arrive. So it seems that you could just ‘predict’ the outcome at the exact moment when I arrive without actually having to observe whether I arrived or not? – Mr Deevid Sep 25 '23 at 11:18
  • @MrDeevid Your answer is filled with phrases that are meaningless or have an unclear meaning - thats part of the point of this answer (and my comments on the question). "how old my twin would be when I arrive" what does it mean "when I arrive" but from the earth's perspective? Do you mean when light reaches the earth showing you arriving? Or what do you mean? What event on earth defines "you arriving" somewhere 4 light years away? There is no universal clock where we can say "how much time passed for you when the universal clock says 12pm" and that's kind of the whole point of relativity. – AXensen Sep 25 '23 at 11:28
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    This answer relies on "relativity of simultaneity", a concept which the OP is clearly not aware of. My answer is written for "poets", and gets to the root of the OPs question. – m4r35n357 Sep 25 '23 at 11:42
  • @MrDeevid: Yes, your twin can predict your age upon your arrival based on the distance you're traveling and the speed. And you can do the same for your twin. He calculates in his frame, where you are younger than he is on your arrival; you calculate in your own frame, where the opposite is true. – WillO Sep 25 '23 at 11:42
  • @WillO Thank you for confirming my thought process. So, special relativity would just say that my twin on Earth would always be younger than me from my perspective on the new planet because I moved away from Earth, and that I would be younger from the point of view of my twin on Earth just because I moved away? In other words, special relativity just tells you that, even if it seems paradoxical, it's just because different frames shows you different perspectives on the age of both of us? – Mr Deevid Sep 25 '23 at 11:47
  • @MrDeevid: What does "from my perspective on the new planet" mean? "On the new planet" seems to suggest that you are envisioning yourself stationary on the planet. In the planet frame, you are permanently younger than your twin. In the rocket frame, your twin is permanently younger than you. – WillO Sep 25 '23 at 11:50
  • @WillO Sorry that I did not explicitly clarify this. I assumed that I would stay on the new planet after arriving, so I am not moving anywhere at all anymore after my trip. Nevertheless, I thought that this would not matter since the age difference with my twin on Earth would be constant from the moment when I am on the new planet. Your last statement would be true from the point of view of the Earth, wouldn't it? From my perspective on the planet, I would be permanently older than the Earth twin, which seems to contradict your statement then? – Mr Deevid Sep 25 '23 at 11:54
  • @MrDeevid: You are permanently younger than your twin in the planet frame; permanently older in the rocket frame. – WillO Sep 25 '23 at 11:55
  • @WillO Wow, this is a new insight for me. You're saying that from my perspective, I am older than my twin while I'm traveling towards the planet. But when I arrive and stay on the planet, I suddenly am permanently younger than my twin on Earth? Is this because my frame of reference basically became the same as my Earth twin's frame when I arrive, since both are stationary? – Mr Deevid Sep 25 '23 at 12:02
  • @MrDeevid : YES! The two frames disagree about many things, including which clocks were ticking at what rate at which times, and when the clocks were set to zero. (This is a key insight, that I tried to point out in my answer.) Those disagreements add up to a disagreement about who is younger. When you stop moving, you switch frames. – WillO Sep 25 '23 at 12:05
  • Wow, and if I would turn this scenario around and say that I just keep traveling on my rocket indefinitely, and my twin decides after some time to move as well in the same direction on exactly the same speed as me for indefinite time. Would my twin then be permanently younger than me? So the take-away is that whenever you try to compare results from different inertial frames, there will always be some disagreement between different observers, but whenever two frames moves at the same velocity, the one that changed from a different velocity would now agree with the original one? – Mr Deevid Sep 25 '23 at 12:19
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"So, his clock would tick more slowly than mine during the trip," Yes, this is correct.

"and when I arrive my twin would be younger than me." This does not follow. A clock that has just spent the last hour ticking slower than yours can show either an earlier or a later time than yours does.

WillO
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  • I’m sorry that I don’t follow your last statement. How can a moving clock that has been running more slowly show a later time than your own stationary clock? – Mr Deevid Sep 25 '23 at 11:20
  • @MrDeevid ; Any clock can show any time at all, depending on how it was set in the first place. – WillO Sep 25 '23 at 11:26
  • @MrDeevid you seem to be avoiding commenting on my answer, which does tell you where you are going wrong without sarcasm or technical distraction. – m4r35n357 Sep 25 '23 at 11:36
  • @MrDeevid: It is in fact true that when you arrive, your twin is younger than you (in your frame). But that does not follow from the difference in tick rates. The fact that you seem to think it does follow from the distance in tick rates goes to the locus of your confusion, which is that there are two frames here. In the earthbound frame, you are younger than your twin; in the rocket frame you are older. – WillO Sep 25 '23 at 11:39
  • As for the answer by @m4r35n357 , he is clearly answering a question different from the one you asked, as your title specifies a one-way trip and his answer assumes a return. – WillO Sep 25 '23 at 11:40
  • @WillO you are incorrect. My answer provides the key information the OP requested: "Would anyone please help me point out the mistakes that I made in my thought process?", whereas yours is merely snobby nit-picking. – m4r35n357 Sep 25 '23 at 11:41
  • @m4r35n357 : You don't think it's a mistake in a thought process (and an important one if you're learning relativity) to think that the time shown on a clock is independent of when that clock was set to zero? I pointed to the first mistake in the OP's thought process. There are of course others. – WillO Sep 25 '23 at 11:44
  • . . . and you will gladly list them one by one without providing any guidance whatsoever. That is the letter of the law, not the spirit. – m4r35n357 Sep 25 '23 at 11:50
  • @m4r35n357 : I'm going to guess that I've taught this kind of material to a lot more students than you have. In my experience, students learn a lot better if you listen to what they're asking (so for example, if they ask about one-way trips, you don't change the subject to round trips), point out critical errors in their reasoning, and encourage them to re-think in light of those errors. – WillO Sep 25 '23 at 12:00
  • You almost certainly have! However if the OP were a student, they would not need to ask here. Perhaps @MrDeevid can let us know whether he is a "poet", or a student. – m4r35n357 Sep 25 '23 at 12:05
  • Re, "You don't think it's a mistake...to think that the time shown on a clock is independent of when that clock was set to zero?" I think you should slow down and reflect on why it's always called the twins paradox. I mean, why isn't it sometimes called the, "somebody who was selected by NASA to go on an interstellar trip and some other completely unrelated person who stays behind" paradox? – Solomon Slow Sep 25 '23 at 13:04