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As far as I know when a person travels with some velocity relative to other person time runs slower to him relative to the other person (Because the velocity of light is constant in all frame of reference). This I have understood from twin paradox. But this becomes complex to understand when it is more than twins.

For example, here is a thought experiment,

Imagine there is a huge treadmill covering the entire circumference of the earth and there is a vehicle situated on that treadmill belt. Now there are three twins( twin A; twin B; twin C). So here twin C gets into the vehicle, twin B hold on to the treadmill belt(so that he moves along with the belt) and twin A is standing in the ground(stationary).

Now the vehicle starts to move in 99% of $c$ (speed of light) and the treadmill starts to run in the opposite direction with the same speed of the vehicle so that the vehicle appears to be stationary to twin A.

So here blindly applying the theory which was applied in twin paradox time running for twin B should be slower than the time running for twin A. And time running for twin C should be slower than that of twin B. But now if we suddenly neglected that treadmill from our experiment(Let's say it becomes invisible) now this makes no sense as the vehicle looks stationary to twin A and so the time should run in equal speed for them. So what will actually happen in this thought experiment?

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    $\begingroup$ The fact that the conveyor belt is circular makes this a bit trickier. Would you be just as happy with an answer where the conveyor belt is straight? $\endgroup$ – knzhou Nov 29 '18 at 10:20
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    $\begingroup$ Do you mean triplets? $\endgroup$ – Aaron Stevens Nov 29 '18 at 11:56
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Time dilation effects are not transitive. According to an inertial observer, any clock moving relative to that observer will be observed to be ticking slower. Therefore, A and C will observe the clock of B to be slower than their clocks. B will observe the clocks of A and C to be ticking slower than their clock. A and C are in the same inertial reference frame, so they see each of their clocks ticking at the same rate.

You have to pick a frame and then stay in that frame. You cannot compare different time dilations for different frames and hold them all to be true at once. It all depends on the reference frame.

Also, since A and C are in the same intertial reference frame, this is basically the just the original twin paradox (it's not a triplet paradox).

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  • $\begingroup$ If we consider that A and C are in the same reference frame then B will move in different speeds wrt A and wrt C, right? which means the time dilation should differ wrt A and wrt C? But this makes no sense and even if we consider the amount of time dilation to be the same wrt A&C then Speed of Light will not be in constant for all frames of reference, isn't? $\endgroup$ – multiversal Nov 29 '18 at 17:43
  • $\begingroup$ @multiversal both A and C will observe B to be moving at the same speed. $\endgroup$ – Aaron Stevens Nov 29 '18 at 18:13
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$\let\g=\gamma \let\D=\Delta$ @Aaron Stevens Each observer will view the clock moving relative to them as ticking slower. Are you saying this is not the case? Excuse me, Aaron, for my delay. I can't keep up with all threads where I would like to say something.

I'm replying to your question-comment with an answer since I'm not able to stay within limits of a comment, be it a serial one.

It is the case if properly qualified. It isn't the case if naively stated, as I see in most cases. I've been following SE for slightly more than 3 months, and remember several answers of mine where I already had to explain what should be meant when statements such that are made. I also said that I don't like the term "time dilation", just because it is prone to misunderstandings and confusions. Nor I like "clocks going slower" and worse of all "time (itself) going slower. Of course I don't expect that anybody, let alone a newcomer, may know those statements, so I'll go to repeat.

And now for the positive side. The issue of time dilation, correctly understood, refers to two events (of timelike separation), say E, F and to two (inertial) reference frames. Given E and F, there is a frame $K_0$ where they happen at the same place. In any other frame $K$ the same events happen instead at different space positions. This sets the stage.

Now for measurements. Let $\D t_0$ be the time interval between E and F, measured by a clock in $K_0$, set just in the place where both events are happening. Let $\D t$ the time measured in $K$. But here an important clarification is in order. The time measurement in $K$ can't be done with just one clock. Since E and F happen in different places, two clocks are needed, one for each place. And of course these clocks must have been previously synchronized. It is tacitly understood that synchronization is effected "à la Einstein", i.e. through a light signal (whose velocity is assumed known) between clocks.

All this being set, it can be shown that always $\D t>\D t_0$. More precisely, $\D t=\g\,\D t_0$, $\g$ being the usual Lorentz factor for the speed $v$ of $K$ relative to $K_0$.

So you see why we can't simply say "A sees B's clock running slower and viceversa". When one talks about A's clock he's implicitly using events related to two readings of A's clock, so that $K_0$ is A's frame, $K$ is B's. And when one switches to B's clock two other events are in action, related to B's clock. Now $K_0$ has become B's frame, $K$ is A's.

A final note. Twins paradox in another matter, I don't want to touch here.

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As far as I know when a person travels with some velocity relative to other person time runs slower to him relative to the other person

If you really read this somewhere, the author is guilty of having led you into a deep misunderstanding. There is no need of your gigantic treadmill to see it. A simple question is enough.

Let me call A the first person in the quotation above, B the other person. You say A travels with some velocity relative to B, then time runs slower to A relative to B's time.

I say: if A travels wrt B, then B travels wrt A. So it should be B's time to run slower. Since both things can't happen together, there is a contradiction. Two possibilities arise:

  • Your original statement is wrong.
  • Special relativity is inconsistent.

What do you believe the truth is?

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    $\begingroup$ Each observer will view the clock moving relative to them as ticking slower. Are you saying this is not the case? $\endgroup$ – Aaron Stevens Nov 29 '18 at 12:22
  • $\begingroup$ @AaronStevens: The OP says (incorrectly) that time runs slower for B than for A. Elio says (correctly and relevantly) that this is nonsense. You say (correctly but much less relevantly) that each observer views the clock moving relative to him as ticking slower --- which is an entirely different statement. What ever would make you think that Elio is denying this statement? $\endgroup$ – WillO Nov 29 '18 at 15:49
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    $\begingroup$ third possibility, A sees B's clock running slower and viceversa $\endgroup$ – Wolphram jonny Nov 29 '18 at 16:24
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    $\begingroup$ @Wolphramjonny That is what I was trying to say as well. $\endgroup$ – Aaron Stevens Nov 29 '18 at 16:25
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    $\begingroup$ @AaronStevens Yes, I agree with you and do not understand the answer, unless it is plain wrong as I suspect $\endgroup$ – Wolphram jonny Nov 29 '18 at 16:26

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