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I have currently been introduced to relativity of simultaneity with the following thought experiment:

Consider a train in which there are two presidents, one sitting at the front and the other at the back of the train. The two presidents must sign an agreement simultaneously for everyone to be happy. The way that they are going to carry this out is by having a laser shoot a light beam simultaneously towards each of the presidents and when the light beam gets to each president they will sign the agreement. Now consider this train moving from left to right and we will analyse this situation from the two inertial reference frames "train frame" and "platform frame". It then follows that from the "platform frame" the two presidents have not signed the agreement simultaneously while regarding this situation from the "train frame" the presidents did sign the agreement simultaneity. Until this point I understand everything

Q: Now consider that there is a bomb in the middle of the train and that bomb would blow up if the two presidents don't sign the agreement simultaneously and won't blow up if they do (this can be achieved via sensors added at the back and front of the train). Now this experiment is carried and from " platform frame" train blew up while in "train frame" train didn't blow up. Now this poses a paradox as the train can't blow up and not blow up in the same time even if its a matter of reference frames, since reality is connected. How can this paradox be resolved?

Moreover from this paradox I get the feeling that, even though observers in different inertial reference frames don't agree on simultaneity, they must agree on something, maybe cause of their action on nature around them? I don't know how to call this agreement. Because if they wouldn't agree on this (idk how to put it into words) then we would get paradoxes as the one above. Thank you for any help

ps: The though experiment with the presidents was presented by Brian Greene in his course on special relativity

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  • $\begingroup$ The clocks between the track and the train weren't synchronized. $\endgroup$ – Cinaed Simson Dec 8 '19 at 2:15
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The real problem is that the trigger on the bomb is ill specified. It's not as complete as you might think. The entire point of these simultaneity exercises is to demonstrate that simultaneity is not preserved between frames of reference. It's an odd artifact of relativity, and that's why it needs thought experiments like these.

Thus, a physical bomb cannot blow up "if the presidents do not sign simultaneously," because simultaneity is specific to the reference frame one is talking about. The most natural answer is "in the reference frame within which it is at rest." That would be the only meaningful natural assumption.

You could have the bomb blow up if the presidents do not sign simultaneously in the platform frame. This could be done by signalling to the bomb from an observe in the platform frame, or by the bomb doing relativistic calculations to adjust its readings to put them into the platform frame.

However, you cannot do both. The whole point of these experiments is that you can't do both because simultaneity is not preserved between reference frames. This is the same paradox as a bus with a bomb that will blow up if the bus dips below 55mph, but the bus is already at 0 mph in its own frame of reference. In the bus case, it is because the magnitude of absolute velocity is not preserved between Newtonian reference frames, and in the presidents on the train case it is because simultaneity is not preserved between relativistic reference frames.

Both the train bomb which blows up if the presidents do not sign simultaneously and the bus bomb that blows up if the bus dips below 55mph are incompletely defined scenarios because the triggers do not specify the reference frame. In the case of the bus bomb, we will intuitively translate this as 55mph "in the ground frame," and that intuition fills the gap.

In the train scenario, the typical intuition is simultaneous "in the bomb's reference frame (aka the train reference frame)." If everyone agrees to this intuition, we have no paradox.

A coin can land heads up. A coin can land tails up. A coin cannot land heads up and tails up at the same time.

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Here's an even more alarming paradox: I'm about to flip a coin. Next to me there is a bomb with two properties:

1) The bomb blows up if and only if the coin comes up heads.

AND

2) The bomb blows up if and only if the coin comes up tails.

Whoa! The bomb either has to blow up or it doesn't, and either way we have a paradox, right?

What is the solution to this paradox? It is that there is no such bomb.

What you seem to have imagined is a bomb with these two properties:

1) The bomb is defused if and only if the signings are simultaneous in the train frame.

AND

2) The bomb is defused if and only if the signing are simultaneous in the station frame.

There can be, of course, no such bomb because signings that are simultaneous in one frame cannot also be simultaneous in the other, just as a coin cannot be both heads-up and tails-up.

You can posit a bomb with property 1) or a bomb with property 2). But you have to pick one or the other. You can't have both. And once you pick, the paradox goes away.

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    $\begingroup$ I think you missed the part about if AND ONLY IF. $\endgroup$ – WillO Dec 7 '19 at 0:49
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    $\begingroup$ @SKDash Then the question is no longer a physics question. What you are proposing is an actual contradiction as to how we know physics to work. So if you decide to ask this "what if", then there is no longer an actual answer that can be grounded in physics. The question becomes "what if special relativity was no longer true", which is a type of question that is closed on this site. Therefore, the correct resolution of the OP's question is that there is no paradox because if there were then we aren't talking about physics as we know it anymore. $\endgroup$ – Aaron Stevens Dec 7 '19 at 14:07
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    $\begingroup$ @skdash: yes, replace the bomb with the snipers and my example is still correct and relevant. $\endgroup$ – WillO Dec 7 '19 at 14:32
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    $\begingroup$ @LucaIon: You're going to carry this out in real life by building a bomb that can't be built? $\endgroup$ – WillO Dec 7 '19 at 20:03
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    $\begingroup$ @LucaIon : If you haven't digested the answer, then I doubt very much that it will help for me to type it in again. $\endgroup$ – WillO Dec 7 '19 at 22:34
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If the bomb, the laser, the sensors, and the presidents are all in the same frame of reference, it will not matter what other frames of references see the laser do.

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There is no paradox. The question you pose is exactly analogous to this:

Two presidents must sign a treaty at 6pm or a bomb will go off. They sign it at 6PM London time, so the bomb doesn't go off. But in New York it appears that they have signed it at 1PM, so the bomb does go off. How can this be?

An event cannot be at the same time in two different time zones, so you have to specify the timezone for the presidents' 6PM deadline. Likewise an event cannot be simultaneous in two reference frames moving relative to each other, so you must specify the frame of reference in which the presidents must sign simultaneously. You must either require them to sign simultaneously when measured on the train, or to sign simultaneously when measured from the platform- they cannot do both.

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The answer lies in the question itself. The bomb does not explode because the simultaneity is considered only from the reference of the train. This is similar to the limo in the garage question in Leonard Susskind's Theoretical Minimum:Special Relativity. What I mean is that the bomb explodes only if the two light beams don't reach to the detector at the same time, the notion of time in special relativity is not the same as classical mechanics, what time is measured by someone might not be the same as measured by other, so in this case we must consider the proper time, which here is in the reference frame of the train, so the bomb doesn't explode from both reference frames even though the flashlight don't reach simultaneously to the ends with respect to the ground frame. In case you don't understand this explanation, you can refer to the above stated book.

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  • $\begingroup$ Can someone please tell what's wrong in the explanation before giving a downvote $\endgroup$ – SK Dash Dec 7 '19 at 4:05
  • $\begingroup$ I didn't downvote, but I'm pretty sure I know why people did. Let's imagine Bomb T, which explodes if and only if the signings are not simultaneous in the train frame and Bomb P, which explodes if and only if the signings are not simultaneous in the platform frame. The OP does not tell us which bomb he's imagining, then goes on to talk as if his bomb is both Bomb T and Bomb P at the same time, which is impossible (and this is the source of his confusion). Your answer assumes that he's talking about Bomb T, but we don't know that because he's never told us which bomb he means. (CONTINUED) $\endgroup$ – WillO Dec 7 '19 at 14:36
  • $\begingroup$ (CONTINUED) And it appears that he's never told us which Bomb he means because he has forgotten that "simultaneous in the train frame" and "simultaneous in the platform frame" are as different as a coin landing heads and the same coin landing tails. What he needs to understand is that these conditions can't both hold at the same time, therefore there can be no bomb that depends both on one and on the other. $\endgroup$ – WillO Dec 7 '19 at 14:39
  • $\begingroup$ @WillO if you read the question "Now consider that there is a bomb in the middle of the train and that bomb would blow up if the two presidents don't sign the agreement simultaneously and won't blow up if they do (this can be achieved via sensors added at the back and front of the train)." $\endgroup$ – SK Dash Dec 7 '19 at 14:44
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    $\begingroup$ Those sensors can be set to test whether the signings are simultaneous in the train frame or they can be set to test whether the signings are simultaneous in the platform frame. They can't be set to test both at once. The OP hasn't told us how they're set. You start off by assuming you know how they're set. The OP needs to be reminded that it's his job to tell us how they're set, and that once he makes up his mind, there will no longer be a paradox. $\endgroup$ – WillO Dec 7 '19 at 15:29
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It's not enough to say that the bomb will blow up if the two presidents don't sign the agreement simultaneously. You have to define how the bomb will decide whether or not the signing is reckoned to be simultaneous or not. How is it wired? It is wired to compensate for time delays? If so, are the supposed time delays calculated by assuming that the speed of light is fixed for the railway carriage or for the platform? Or for some other arbitrary state of motion?

You can imagine the bomb having a switch with two settings, which either put it into a mode where it compensates for assumed simultaneity by assuming fixed c in the train, or on the platform. That physical switch setting then decides the outcome.

So there's no paradox. The outcome is deterministic, but it depends on an arbitrary choice by a human being as to how to physically wire the bomb, or which switch setting to use. The bomb does not have the magical ability to compensate simultaneously for an infinite continuous range of possible beliefs about which frame lightspeed is "really" fixed in. Once you make a decision about which (arbitrary) belief to use, and set the trigger system accordingly, you have your outcome.

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