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I understand einstein's train paradox. Where one man on a platform is passed by a man in a traincar, at the moment they meet a flash of light is given off in the middle of the train car. To the man on the platform the light hits the ends of the car at opposite moments in time. To the man in the train car they hit at the same time.

The man on the train sees the ends of the car at a fixed distance from each other, thus the light hits each end at the same time (same distance to travel). The man on the platform however sees the rear of the car catching up to the light while the front is moving away from it (light going to the front of the train has longer distance to go).

So, the only reason the light ever reaches the front of the wall is because it is not moving at the speed of light. Light catches up to it.

How would this experiment be different if the train is moving at the speed of light?

To the man on the train nothing would change I would have to say, because he is in his same reference frame of the train still. The man on the platform though, what would he see?

Would he see light hit the back of the train at the exact same time the flash happens, and never see the flash hit the front of the train but always [length of train car]/2 meters away from the front of the train car?

Edit: Looks like I forgot to say this is a hypothetical question. Unless one of you all have a speed of light train, this question is purely a thought experiment. If that does not satisfy you, then yes I have a train that does 671 million mph and I am just trying to understand my results of the first test run.

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Train couldn't move at the speed of light. Only some separate particles, like photon, neutrino or gluon, can. Bodies can't, and observers and frames of reference can't. Division by zero. –  firtree May 13 '13 at 16:36
Ok say (just for funs) we do have a train capable of this speed. If you would re-read the question with this train in mind and try to answer it? –  BumSkeeter May 13 '13 at 16:57
Hi BumSkeeter. This post asks hypothetical questions outside accepted standard physics theories, such as, e.g., asking about a massive object traveling at the speed of light. Fictional questions are off-topic, cf. faq. If you can modify your question, so it stays within the realm of standard physics, I would be happy to reopen it. –  Qmechanic May 13 '13 at 19:24
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closed as off topic by Qmechanic May 13 '13 at 19:23

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2 Answers

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The observer on the train is the least well defined part of this thought experiment. The thing is, Lorentz transformations and such are only valid for relative velocities of strictly less than the speed of light. All sorts of things go to $0$ and/or $\infty$ if you start boosting at $c$, and so you cannot boost into and out of a photon's frame.

We can still ask about the person on the ground. Let's recast the problem as there being three equally-spaced photons moving along the tracks in the same direction. $$ \underbrace{\odot\!\!\rightarrow \qquad \odot\!\!\rightarrow \qquad \odot\!\!\rightarrow}_L $$ At some point, the middle one splits into two photons, one moving in the same direction, the other moving backward. $$ \odot\!\!\rightarrow \qquad \leftarrow\!\!\odot\!\!\rightarrow \qquad \odot\!\!\rightarrow $$ Then clearly the backward-propagating photon would meet the forward-propagating photon at the rear in time $L/(2c)$. On the other hand, the new forward-going photon would forever remain a distance $L/2$ behind the original front photon. Thus your intuition for this frame (the only legitimate one of the two) is correct.

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You're incorrectly assuming that there is a frame of reference (the train) that moves with speed c relative to another frame of reference (the station).

But, there isn't such a frame of reference.

Mathematically, there is no spacetime coordinate transformation from the station's frame of reference to a "frame of reference" with relative speed c.

say (just for funs) we do have a train capable of this speed.

But that's like asking "say, imagine something that's not possible in Special Relativity is true, what happens according to Special Relativity?"

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Well what's wrong with asking, "say, imagine something that's not possible in Special Relativity is true, what happens according to Special Relativity?" To me its no more different than saying "imagine a pink unicorn in your house, could you ride it"? Yes both are impossible, but to me there is value in the answer and asking the question. I would argue if you think it odd I ask this, then I figure you never watch fictional movies because impossible events take place. –  BumSkeeter May 13 '13 at 17:14
It is wrong in asking because that would violate a law in the first place which is making the theory logically and experimentally correct to follow everywhere . Suppose in newton's laws you say suppose for a body $F=m \vec v$ then obviously in some particular problem , this is going to create a paradox because that violates the theory in the first place and it was these little in-violations that made the theory to get accepted in the first place –  nonagon May 13 '13 at 17:38
There's no value in an answer to "imagine a pink unicorn in your house, could you ride it"? because no one knows if an imaginary creature is rideable (maybe they use their horn to reflexively skewer anyone that tries to ride), just as there's no value in giving a physics answer to a question that's outside the realm of current physics since any answer will just be imaginary. –  Johnny May 13 '13 at 18:13
There's no value in an answer to [ride unicorn] I guess now that I think about it, its more that I value the idea of thinking like that; outside the realm of what is possible. So I wonder about it more. Yes now I realize that I shouldn't ask questions like that on here per the FAQ. Sorry. –  BumSkeeter May 13 '13 at 21:12
@BumSkeeter To me its no more different than saying "imagine a pink unicorn in your house, could you ride it"? - No, the mathematical impossibility is something stronger, it means that something doesn't make sense. For your example, it would be closer to "imagine a ghost-dragon-unicorn-mermaid in your house, could you ride it?" Then you would open your eyes wide, and ask "But what is a ghost-dragon-unicorn-mermaid?". That's what happens with contradictory suggestions like yours. –  firtree May 14 '13 at 8:51
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