Einstein's relativity of simultaneity train/embankment thought experiment

Question

Einstein's thought experiment I'm referring to is this one:
http://www.bartleby.com/173/9.html

briefly: train/embankment experiment is where lightning strikes at either ends of the running train (points A and B) and there's an observer inside the train (call him INSD), and outside the train (call him OTSD). if the observer OTSD sees the lightning strikes to be simultaneous, the observer INSD will see that one happens after the other. detailed explanation can be found in the link above.

anyways, so i get the reasoning. the speed of light is always constant, and from the observer INSD's frame of reference, the lightning strikes happened equi-distance from him, and since lightning on the front (B) reached him before the lightning on the back (A), then the one on the front (B) must have happened first... so what's simultaneous to OTSD is not simultaneous to INSD; i get it.

now, what i'm actually confused is;

assuming lightning stroked not the train, but the embankment outside, also at the same points A and B; and left a mark where it hit.

the observer INSD will still see the lightning strike from B first before A; but this time, observer sees where B actually hit the embankment; so can see the distance it actually traveled; which should be shorter..

that's the part i'm confused about; if the lightning striked on the embankment and left a mark there, can you still make the argument that both traveled equi-distance in INSD's frame of reference?

i know i'm missing something, but i struggled alone on this problem for a full day; and i don't have anyone personally to ask this question (not in school anymore); so please be nice and teach it to me like a little kid :)

Engr. Ravi · Answer

This paper discusses the train and embankment though experiment (TETE) in detail and answers your question:

https://arxiv.org/abs/physics/0606135

the correct relativistic analysis of Einstein’s original TETE shows
  that the ‘lightning strokes’ will be judged to be simultaneous by both
  the train and the embankment observers, in contradiction to Einstein’s
  conclusion.

fishinear · Answer

If we assume that the lightning strikes leave marks on both the train car, and the platform, then both observers can measure the distance independently. And yes, the OTSD observe will judge the marks to be closer together than the INSD observer.
But the key to the lightning-and-train experiment is not whether those distances are the same, but whether the distance from the front to the middle point is the same as the distance from the rear to the middle point. And note that each observer has their own middle point. The OTSD observer judges that the light does arrive at his middle point at the same time. The INSD observer judges that the light does not arrive at her middle point at the same time. That is, the INSD observer, sitting at the middle point of the train car, clearly sees the lightning strike the front of the train car before it strikes the back. The only conclusion that she can draw is that the lightning strikes were not simultaneous.

Einstein's relativity of simultaneity train/embankment thought experiment

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