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In his popular book on relativity, in chapter IX, "The relativity of simultaneity", Einstein describes an experiment in which a flash happens simultaneously on A and B, as defined by the fact that an observer at the middle point M can see the light coming from A and B at the same moment:

              --train-->
---embankment---A-------M-------B----

Then, he proceeds to say that an observer on the train "is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A".

But I find the opposite result, from the train point of view:

  • the embankment moves to the left (in the diagram) at a speed of v.
  • so, after dt, A moved by v*dt to the left, the light coming from A moved by c*dt to the right, and the distance between A and the light increased by (c+v)*dt.
  • similarly, the distance between B and the light coming from B increased by (c-v)*dt.
  • so, the light coming from A is going to reach the middle point M before the light coming from B, so the observer is going to see the light from A first (if he happens to be at M at the correct moment), which is the opposite of what is said in the book.

Another way to state about the same thing is: the light from both points go towards M, but M is moving to the left, so the light from A is going to reach M before the light from B does.

Where is the error in my reasoning?

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In space-time (i.e. special relativity theory), there's no absolute reference frame. You are making the observer on the train the reference frame to analyze what happens at M. This isn't valid. –  Diego Nov 7 '13 at 21:17
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2 Answers

up vote 1 down vote accepted

Your result is correct and, of course, there is no problem with relativity. When you're on the train and you see how events unfold for an observer on the platform at point M, then you conclude that he will see the light from A first. When you're on the platform and you see how events unfold for an observer on the train, then you conclude that he sees the light from B first. There is no problem with that—that's actually the essence of the theory of relativity: things look different from different reference frames, and every observer is right in their respective reference frame. When the two observers compare their conclusions, they will be able to attribute the difference to the fact that the speed of light is constant.

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It looks to me that you've left out other important parts of the passage that are needed to understand what's going on here.

The motion of A and B from the point of view of an observer on the train is irrelevant to determining which flash reaches the midpoint of the train first. What matters is that both flashes travel at speed c, and the times at which they were emitted from A and B. The main problem in your reasonsing is your assumption that they are both emitted from A and B simultaneously for an observer on the train.

If you read the passage more carefully, I think you'll find that the event of the two flashes occuring at the midpoint of the train doesn't exist for an observer on the embankment, and so doesn't exist either for a person on the train. Since the light flashes travel at the same speed c on the train, the only conclusion left is that the flashes coudn't have occured at the same time. And this highlights the relativity of simultaneity in a simple way as Einstein was trying to do here: Events which are simultaneous in one frame are not so in any other frame.

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