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Background

The twin paradox is the popular thought experiment involving twins, one of whom makes a journey to a nearby star in a high-speed rocket (travelling at a velocity close to the speed of light) and when he reaches the star, changes direction and returns home at the same speed. When he gets home, he finds that time has slowed down during his journey and that his twin who has remained on Earth has aged more. (The paradox being that because both twins have travelled at the same speed with respect to each other, why has one aged faster than the other?)

My understanding is that the solution is as follows. When he arrives at the star and changes direction, the travelling twin switches from a reference frame travelling away from Earth at a velocity near the speed of light to another reference frame travelling towards Earth with a velocity of the same magnitude but in the opposite direction. In contrast the non-travelling twin stays in the same reference frame. It is often pointed out that the travelling twin experiences an acceleration when he switches from one reference frame to another, whereas the non-travelling twin does not experience an acceleration. (This acceleration could be a Dirac delta function i.e. an instantaneous switch of direction)

Questions

My first question is what is the travelling twin’s acceleration with respect to? Since in special relativity there are no absolute reference frames. I assume it is the Earth??

Secondly, and this is the question which is intriguing me the most, suppose hypothetically there were only two objects in the Universe namely the two twins. If, in this hypothetical universe, one twin travelled on a rocket for a distance of say ten light years and then returned back to meet the original twin. Would the travelling twin’s clock still run slow with respect to the other twin? since there would be no distant background stars/galaxies in this hypothetical universe to measure the acceleration against? Is it just as valid to say that the “stationary twin” is the one in motion? Additional comment

In a various replies to this question, it has been mentioned that the moving twin feels an acceleration and the stationary one does not, but in this hypothetical Universe containing only the two twins I cannot see how this can be the case

John Davies
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    The twin paradox occurs when measured in any inertial frame of reference. There is nothing special about the one attached to the Earth. – ProfRob Feb 01 '21 at 22:50
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    Acceleration is not relative. It is absolute, unlike velocity. – D. Halsey Feb 01 '21 at 22:58
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    However, the twin paradox can be explained without any mention of acceleration, using only times, speeds, and distances. – D. Halsey Feb 01 '21 at 22:59
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    The key to the twin paradox is that the travelling twin switches inertial frames, but the other twin stays in one frame. The acceleration is merely the means by which the traveller switches frames. – PM 2Ring Feb 02 '21 at 00:21
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    @D.Halsey Agree with your comments. However to me the intriguing question is the second one. Would the twin paradox still apply if the the two twins were the only objects in the universe?? – John Davies Feb 02 '21 at 20:07
  • @ProfRob agree with your comments. Obviously there is nothing special about the (approximately) inertial frame attached to the Earth – John Davies Feb 02 '21 at 20:12
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    @D.Halsey Although it could be argued that when the travelling twin switches frames, then have an instantaneous change in velocity (so have Dirac Delta function acceleration) – John Davies Feb 03 '21 at 14:22
  • "in this hypothetical Universe containing only the two twins I cannot see how this can be the case" Why do you think that you wouldn't feel acceleration in an otherwise empty universe? Ok, Einstein was partly inspired by ideas of Mach, and Mach postulated that inertia was somehow caused by the distant stars, but that conjecture is really not relevant to relativity. GR says that an isolated observer can't distinguish between uniform acceleration and a uniform gravitational field. It does not say that you can't detect acceleration, or that you need to measure it relative to something else. – PM 2Ring Mar 09 '21 at 22:26

2 Answers2

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Velocity is relative, but acceleration is not: you can feel it (and it feels just like gravity). But as I said in the question comments, acceleration is a red herring.

The real issue is that the travelling twin occupies two (or more) distinct reference frames. That means their worldline (their trajectory through spacetime) is not a straight line, and any inertial observer measuring that worldline will agree on that, although different observers will disagree on the lengths of the segments of that trajectory, depending on their speed relative to the traveller.

So the important factor is that the worldline has a bend in it, and so it cannot be transformed into an unbent line. It doesn't make much difference whether it's a perfectly sharp bend (your Dirac delta function), or a smooth gradual curve.

There's a variant of the twin paradox that requires no acceleration, but it has 3 participants, A, B, and C. A is based on Earth, C is based on Alpha Centauri (which we can assume is at rest relative to Earth).

B's rocket takes off. It's moving at constant speed $v$ (relative to Earth) as B passes A, and at that moment they use radio to synchronise their clocks to zero. B then travels towards Alpha Centauri.

One year later (according to B's shipboard clock), C sees B approaching. C starts their rocket and heads towards Earth. C's rocket has a speed of $v$ relative to Alpha Centauri. (According to the relativistic law of composition of velocities, B & C have velocity relative to each other of $2v/(1+v^2)$, using natural units, where the speed of light $c=1$). Just as C passes B, C synchronises their clock to B's clock. That is, C "clones" the value of "1 year" from B's clock the instant they pass one another.

One year later, according to that clock, C arrives at Earth, and as C passes A, they compare clocks. C's clock says "2 years". A's clock says "9 years".

PM 2Ring
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    A bend or change of direction in a world-line IS an acceleration. – ProfRob Feb 03 '21 at 19:01
  • @ProfRob Sure, but as I said earlier, "The acceleration is merely the means by which the traveller switches frames". Clearly, the traveller can't change frames without some kind of acceleration. OTOH, by doing the kind of relay operation that I described in the second part of my answer, we still get a definite difference in clocks even though no acceleration was involved. – PM 2Ring Feb 04 '21 at 15:22
  • Right, but then that isn't the twin paradox. I agree that it is the change of inertial frames that is important. – ProfRob Feb 04 '21 at 16:08
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As Richard Feynman said about a similar situation, we can't do the experiment of removing all of the matter from the universe to see what would happen, so we don't really know.

But from what we do know, it seems like there would be spacetime even if there was hardly any matter in it. The geometry of spacetime defines the meaning of acceleration and elapsed time. The twins move "relative to spacetime" and that determines who's really accelerating and what their ages are at the end. There's nothing suggesting it's possible to have some sort of pre-spacetime that the twins could move in, but that would leave it ambiguous which one was accelerating.

benrg
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    Thank you for your interesting reply. What really intrigues me is that according to (my understanding of) special relativity there is no absolutely spacetime. Therefore, with only two objects in the Universe it would not be possible to say who was accelerating even if one was firing her/his rocket and the other wasn’t. So arguable there could no twin paradox in such a universe… an interesting thought – John Davies Feb 03 '21 at 14:18
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    A twin can still have an accelerometer with him/her. But I agree that this implies that is really acceleration that solves the problem, while often it is said that is the mere switching of reference frame. @JohnDavies – Alchimista Feb 03 '21 at 16:23
  • @Alchimista I guess the thing that is still puzzling me. Would there be any acceleration felt by either twin in an otherwise empty universe? – John Davies Mar 09 '21 at 16:29
  • @JohnDavies if I think physically we should. In the sense that our internal machinery at least has a reference frame... "accelerating" too much should still be like a knock out. These things are really hard to grasp – Alchimista Mar 09 '21 at 16:45
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    @JohnDavies read the very last part of Wikipedia article Twin paradox. There are two quotes from prominent physicists that seem to me related. Also I tend to solve the twin paradox using somehow the comoving frame... But when I have tried to ask on physics SE either they didn't understand me or it is me that don't fully understand quite some things ;) – Alchimista Mar 10 '21 at 12:35
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    Also I have found this. http://adsabs.harvard.edu/full/1958AuJPh..11..279B – Alchimista Mar 12 '21 at 14:01
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    @Alchimista thank you. To me it doesn't make sense for either twin to feels an acceleration if they are the only objects in the Universe. I like the quotation from AP French at the bottom of the Wikipedia article "......Would such effects as the twin paradox exist if the framework of fixed stars and distant galaxies were not there? Most physicists would say no. Our ultimate definition of an inertial frame may indeed be that it is a frame having zero acceleration with respect to the matter of the universe at large......" – John Davies Mar 14 '21 at 17:16
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    @JohnDavies is what seems to give some sense to me humbling about comoving coordinates as a kind of absolute (in the sense that it can be choose, or most natural...) ref frame. However, I admit that in spite of thinkibg about this and other questions time to time, and reading around, I am really confused. I've yet to solve the twin paradox if not phenomenologically...... – Alchimista Mar 15 '21 at 08:59