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So I'm reading a book called "The Elegant Universe" and here is a part of it

Imagine two countries that have been at war are sitting down to sign a treaty ending hostilities while traveling aboard a train that is moving at a constant velocity. The catch is that neither country's delegate wants to sign the treaty before the other delegate and thus, a simple system is devised to ensure that both delegates sign the peace treaty simultaneously. The solution involves setting a light bulb at the center of a table in such a way that the light bulb is exactly between the delegate from Forwardland (who is facing the direction the train is traveling) and the delegate from Backwardland (who has her back to the direction the train is traveling). When the light bulbs lights up, that is the signal for both delegates to sign the treaty.

This setup is agreeable to all parties on the train and to both security councils in the countries' respective capitals. Once the bulb lights up and the delegates have simultaneously signed the peace treaty, everyone on the train celebrates the cessation of hostilities, but they are perplexed to discover that fighting has broken out anew between the two countries. The reason given is that the delegate from Forwardland was tricked into signing the treaty before the delegate from Backwardland! How can this be? So, since the speed of light is constant how are the people on the train viewing it otherwise? How do they see light at the same time when really forward land was moving TOWARD the light and therefor saw it first. Why don't they see it like this? And how are the people off the train seeing it like it IS. (With forward land approaching the light and therefor seeing it like t is) and how does frame of reference affect this? Please don't use too many technical terms in an explanation

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  • $\begingroup$ Lets say the windows of the train car have been boarded up for safety; how can the diplomats tell that they're moving? Since the train is traveling at constant velocity they feel no acceleration, so for all they know they're sitting still. Why would one see the light before the other? $\endgroup$ – Asher May 4 '15 at 1:57
  • $\begingroup$ Yea, but if the speed of light is constant and one of them must travel less distance to reach it than the other why doesn't it seem like the light hit forward land first? $\endgroup$ – Alex Taylor May 5 '15 at 3:01
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You're right that there would be a disagreement over who signed the treaty first, but it would not be between the diplomats on the train; it would be between the people on the train and the people not on the train.

The setup initially is that the two diplomats are sitting in the dining car with the curtains drawn, for security. Let's say the light that the two diplomats are using is in the ceiling, halfway between the two. The light has to travel the same distance to reach each of them, and since light travels at the same speed in every direction in every reference frame, it should reach them at the same time. Because the curtains are drawn they can't see if they are moving, and since the train is moving at constant velocity they can't even feel if they are moving. They are in an inertial reference frame; the laws of physics work the same for them now as they would if they were sitting still. For all they know, they are sitting still. They test the method of using the light as a signal, and agree that when they flick the light on and off that they see that happen at the same time.

However, since they can't see anything when the light is off, they decide to open the curtains while they complete the signing of the treaty so that they have light to work by. Opening the curtains doesn't change the laws of physics, of course; they still agree that they see the light turn on and off at the same time as each other does. So the method is still agreeable to both parties, and they turn the light off, ready their quills, then turn the light on and sign the treaty simultaneously, witnessed by a neutral peacekeeping officer standing next to the table.

Or was it simultaneous? A farmer standing in the field next to the train sees it coming up the tracks and, having nothing better to do, watches as it passes by, thus witnessing the signing of the treaty. He sees two diplomats sitting in the train, and sees the light flick on. The light travels at the same speed toward the front of the train as it does toward the back, since the speed of light is constant; that means that the light reaches the diplomat in the rear seat first (since he is catching up to the light) and then reaches the diplomat in the forward seat second (since it has to chase him down). Thus the farmer witnesses the rearward diplomat sign the treaty slightly before the forward diplomat.

When the two witnesses are interviewed by the newspaper later, the peacekeeping officer will claim that the diplomats signed at the same time and the farmer will claim that one signed before the other. The tricky thing is, both of them are correct in their own reference frames. The solution to the apparent paradox arises from the Relativity of Simultaneity which itself arises naturally from observation of physical law: particularly the two facts that a) physics works the same in all inertial reference frames and b) light travels at the same speed through space in all reference frames, regardless of the speed of the source.

So the overall answer to your question is that the observers in the train will see things differently than the observers standing next to the tracks, because they are moving relative to each other, and neither perspective is incorrect in its own reference frame.

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[...] neither country's delegate wants to sign the treaty before the other delegate and thus, a simple system is devised to ensure that both delegates sign the peace treaty simultaneously.

The solution involves setting a light bulb at the center of a table in such a way that the light bulb is exactly between the [two] delegate[s ...] the light bulbs lights up, that is the signal for both delegates to sign the treaty.

This setup is clearly inadequate: If one delegate saw that the bulb had lit up then she has no guarantee that the other delegate had been about to see that the bulb had lit up, too (and that her view hadn't been blocked, for instance); or that the other delegate would indeed sign the treaty in good faith upon seeing that the bulb had lit up. (Given these considerations, neither delegate would be careless enough to sign the treaty for good just because of having seen that the bulb had lit up.)

The diplomatically (and physically) acceptable setup solution is rather:

  • for both parties to send trusted envoys to the bulb,

  • the two delegates (at their respective ends of the table) signing the treaty provisionally upon having seen that the bulb had lit up, and

  • having the two envoys (and the bulb) certifying that the signatures had indeed been issued simultaneously, and thus both ratifying the treaty, if (and only if) they all observed the (starts of) the two signings by the two delegates in coíncidence. (Otherwise both signatures are to be considered null and void; new prints of the treaty may be distributed and the procedure may be repeated until it succeeds.)

This setup is agreeable to all parties on the train and to both security councils in the countries' respective capitals. Once the bulb lights up and the delegates have simultaneously signed the peace treaty, everyone on the train celebrates the cessation of hostilities,

As well they should.
Note that the agreed procedure is explicitly independent of whether and how any railway sleepers are moving by the delegates and the bulb and envoys; or perhaps airplanes, etc.
(However, it is of course quite difficult to characterize this reference system, i.e. to determine whether two ends of a given table had been and remained at rest to each other; and whether a given light buld had been and remained at rest to either table end, and "at the center" of the two table ends.)

but [...] forward land was moving TOWARD the light and [...]

That's at best imprecise. There is "land" (incl. distinctive "railway sleepers") both "ahead" and "behind" the table and bulb. Correct is surely that

  • the "railway sleepers" were moving from the "Backward" end of the table towards the "Forward" end,

  • the "railway sleepers" were at rest with respect to each other, and any given pairs of the "railway sleepers" succeed in identifying who was and remaind "the center between" each other, therefore

  • applying the very same procedure (among each other), the reference system of "railway sleepers" can determine simultaneity (or dis-simultaneity, sequence) of their indications. Considering especially

  • the railway sleeper ("$J$") who was passed by the "Backward" table end just as (in coincidence) they observed that the bulb had lit up and that therefore the "Backward delegate" started putting her (provisional) signature on the treaty, and

  • the railway sleeper ("$K$") who was passed by the "Forward" table end just as (in coincidence) they observed that the bulb had lit up and that therefore the "Forward delegate" started putting her (provisional) signature on the treaty,

then, according to the given setup prescription, they find that $K$'s indication of being passed by the "Forward" table end was indeed before $J$'s indication of being passed by the "Backward" table end. (But that's hardly a reason to continue fighting despite an orderly signed peace treaty.)

Why don't they see it like this? [...]

The general point is that determining simultaneity (or dis-simultaneity) of indications of suitable participants is not (merely) a matter of plainly "seeing" and "perceiving", but of measurement;
which (by definition of the appropriate setup procedure) requires determining

  • which participants were and remained at rest to each other (and which not), and

  • for any pair of participants who were and remained at rest to each other: who was identified as "the center" (or "the middle") between the pair under consideration, and

  • finally, case by case, which signals the appropriate "center" had observed in coincidence, or in sequence.

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