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I am having a hard time understanding the implications of one of the popular variations of the thought experiments associated with Relativity of simultaneity. The example goes as follows. 2 countries want to sign a peace treaty on a moving train at the exact same time, and use a light placed in the middle of a table with the 2 say leaders of each country at each end. When the light switch goes on, they sign at the same time (from the frame of reference of the people on the train). But to an observer of the train who is standing still, the frame of reference is such that the light reaches the leader at the back of the train first (since he is moving towards the the light, thereby reducing the distance the light has to travel) and this leader signs the contract first. The leader in the front of the train moves away from the light, thereby increasing the distance and the light reaches him later, and he signs later.

This makes sense to me so far, but what I don't understand is, is what does the leader who signs later do during the time that the light takes to reach him. Because it appears to me, that suddenly we have this extra time introduced, that wasn't there for the people on the train.

To make my question clearer: Let's modify the thought experiment and assume, the people on the train are clapping their hands in simultaneous fashion to the same rhythm before the light is turned on. Let's say the folks on the train count ten claps and then the light goes on and they sign. But to the outside observer, it might therefore happen, that the leader in the front of the train gets 11 claps in before the light reaches him.

Now what if we put a sound recording device right next to the person watching this train drive by. How many claps would that device record?

Tartaglia
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  • Draw a spacetime diagram of all the events. Then try to find this extra time that was introduced. You won't be able to do it. – mmesser314 Nov 23 '23 at 04:12
  • Not strictly related : Special Relativity - Regarding the Simultaneity of Events During the Train Paradox. – Frobenius Nov 23 '23 at 04:17
  • It seems like you should be going to the step before you can begin analysing the relativity of simultaneity, namely, you should be checking again how time synchronised and what defines simultaneity in any one frame. That is why mmesser said that you would not be able to find the extra time. There is no extra time. – naturallyInconsistent Nov 23 '23 at 04:25
  • I did start off with a spacetime diagram, that is where my confusion started, there is an illustration on the wiki page for time dilation (https://en.wikipedia.org/wiki/Relativity_of_simultaneity#:~:text=Einstein%27s%20thought%20experiments-,Einstein%27s%20train,to%20strike%20at%20different%20times.), To my understanding, there is extra time introduced, once the outside observer looks at it. – Tartaglia Nov 23 '23 at 04:26
  • It appears to me, that once the outside observe is introduced and the events are not synchronized any more, we take away time from the folks in the back of the train, in my example they would clap 9 times and give that extra time to the folks in the front of the train, they get 11 claps. – Tartaglia Nov 23 '23 at 04:29
  • So it is not extra time, more a shift in time from the folks in the back to the folks in the front. But doesn't that change the actual actions they can take? The back folks only clap 9 times, and the front ones clap 11 times. But that is absurd. – Tartaglia Nov 23 '23 at 04:31
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    It's not an exchange of the time from one end of the train to the other. It's a literal rotation of the "now" plane in spacetime (the line that represents "now" in the spacetime diagram is tilted). "what does the leader who signs later do during the time that the light takes to reach him" - whatever he was doing (from the on-train POV) before the light reached him. The outside observer will observe the ten claps, but they will not start at the same time at both ends of the train, and the light pulse will not be simultaneous with any of the final claps. – Filip Milovanović Nov 23 '23 at 05:15
  • Thank you for this answer, it is similar to the one above and I will add a comment below. – Tartaglia Nov 23 '23 at 17:38
  • I meant to say the answer below by Quantumplate. – Tartaglia Nov 23 '23 at 17:51

1 Answers1

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Your clapping is essentially a regular, repeating motion, e.g. a clock.

Let's replace it with a light clock. Each cycle can be representative of one clap of the hands.

You'll need to start with both clocks in the center of the train so they are in sync, then move one to the front and one to the back.

The stationary observer will note the light in the clocks taking a diagonal path. They'll also note as you move one to the back it contracts this diagonal path (e.g. the clock speeds up) and moving one to the front expands the diagonal path (e.g. the clock slows down). The two clocks are now out of sync with each other and so the clapping would also be out of sync.

Now fire the light from the center. Again the stationary observer notes it reaching the clock at the back first and it will give you the count of cycles since it started. This is the clock that is fast. The clock that is slow is still waiting for the light to catch up. When it catches up it will give its count of cycles and surprise, surprise it will be the same number as the clock at the back.

So translating to your scenario, the stationary observer will observe two people at the center of the train clapping in unison. When they move to front/back there will be a slight change in cadence/frequency of the clapping (the front one slowing down, the back one speeding up). Then when they stop at front/back the frequency will be equal again, they'll just be out of sync by some amount. Now the light is fired from the center. It reaches the back person first at exactly X claps, the front person is not quite up to X yet but in the extra time the light takes to reach them, they catch up to X claps too.

Both stationary and moving observers agree that both people clapped the same number of times. The moving observer will conclude that the claps are in sync (no matter how they try to ascertain this) and both people finished at the same time. The stationary observer will conclude that the clapping got out of sync during their move to front/back, then the back person finished first and the front person kept going until they reached the same number.

So in conclusion, the leader who signed later didn't experience any extra time, it was just that they experienced more time dilation so the same number of claps took longer for them than the leader who signed first (at least according to the stationary observer).

Quantumplate
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  • Thank you Quantumplate for providing this answer (I will respond here as I need a bit more space). I had similar thoughts about how time dilation might explain this, and your answer makes sense to me BUT it raises a follow up issue for me. What causes this time dilation. Has it been there all along even without the light experiment? That is, to an outside observer who looks at the train car at a 90 degree angle to the direction the train travels, is it always the case that the folks in the front of the train go through time slower than the folks in the back? But if so, why? What causes this? – Tartaglia Nov 23 '23 at 19:42
  • If you would indulge me for a moment, I thought of another modification of the experiment. Let's remove the light entirely as a trigger for the signing and let's say instead we use some fancy advanced signal transmission (super fancy 5G) via cell phone, and we are capable of sending a text message at the exact same time to both leader's cell phones, and to the folks on the train, they get the text and sign. What does the outside observer see? – Tartaglia Nov 23 '23 at 19:43
  • There is no light this time that gets there in delayed fashion, the signal is transferred from cell phone tower lines that run along the train tracks so the there is continuous coverage along the tracks. If I understand your time dilation argument correctly, it should still be the case that to the outside observer, they sign at different points, so the light doesn't have anything to do with it, right? The front of the train must be moving slower through time for the outside observer, not matter what. But why? Or am I wrong about that part? – Tartaglia Nov 23 '23 at 19:43
  • @Tartaglia time is not moving slower in the front of the train than the back. It is only the period of moving forwards (or backwards) which introduces a change in the clocks. You can consider the person walking to the front as being in their own frame of reference moving at say 5m/s relative to the train. – Quantumplate Nov 24 '23 at 00:49
  • (Visible) light and cell phone signals are both parts of the electromagnetic spectrum, just at different frequencies. They will both behave the same so this won't change your thought experiment. Since clocks represent the level of time dilation in a frame then matter will also be effected just as the light clock is. Hence any signalling you conceive of, e.g. electrical signals, physical movement, etc will be effected the same. – Quantumplate Nov 24 '23 at 00:55
  • As for what causes time dilation. I think that's a whole other question and there are probably plenty of questions/answers out there already on this. – Quantumplate Nov 24 '23 at 00:56
  • Thank you Quantumplate for taking the time to answer this. This is still not quite clear to me, and I think I need to study time dilation a bit more to get a better understanding. But in the meantime, if we get back to my original question. If we ignore the observers and any visual aspects and focus only on the audio coming from the clapping which is recorded by 2 devices, one is on the train and one is off the train. Let's say the agreement is to clap until the light goes off, and on the train there are 10 simultaneous claps. What would the device off the train record? – Tartaglia Nov 24 '23 at 02:13
  • If I understand you correctly, it would record the 10 claps being out of sync. But what if there was no light experiment and no contract signing. But rather just 2 people, one in the front one in the back clapping simultaneously, would they also be out of sync for the outside device? If so, then time dilation appears to have nothing to do with the light experiment but rather just a result of these 2 people being on different parts of a moving object and experiencing time differently as a result? – Tartaglia Nov 24 '23 at 02:15
  • Thank you Quantumplate, that works. – Tartaglia Nov 26 '23 at 19:01