I understand (supposedly) the mathematics concerning the relativity of simultaneity in Special Relativity, but I have a nagging question regarding the original example given by Einstein supporting it (I'm only disagreeing with this specific example, not the concept).
It is normally given as a person on an embankment and a person on a train. There is a relative speed between them (usually presented as the train passing the embankment). Now, when both people are at the same x-position (x=0), there is a flash of light at x = +dx and x = -dx. The argument as I keep seeing it is that the person on the embankment will say that both flashes reach him at the same time, whereas the person in the train will say that the flash in front of him reaches him before the other because he was moving toward it, and thus the observers will disagree on the simultaneity of the flashes.
But given that the flashes occurred at the same distance from each of them, the speed of light is constant in both frames, and either one can claim to be at rest, then won't they, according to SR, necessarily see the flashes as simultaneous (both flashes have to travel the same distance in both frames since at the time of emission, the sources of both flashes were equidistant from both observers). I agree that the person on the embankment will say that the person on the train shouldn't see them as simultaneous (and vice versa) since either observer will see the other moving relative to the sources, but in each of there own frames, they must see the flashes as being simultaneous shouldn't they? Am I just misunderstanding the example?
Thanks.
Answer
I agree that the person on the embankment will say that the person on the train shouldn't see them as simultaneous
Well, then the person on the train shouldn't see them as simultaneous. Some things change between reference frames, but conclusions of the form "in frame $S$, an observer will see..." do not change, since the statement itself specifies which frame you have to be in to understand what it is saying.
The observer on the embankment could easily see the train observer intercept the forward flash before the rear flash. (Of course, the embankment observer couldn't do this in real time; one has to wait until after one's hypothetical grid of rulers and clocks reports back what happened when and where.) One nice thing about SR is that time-ordering is invariant. That is, two events $A$ and $B$ can have one of three relations to one another: $A$ is in $B$'s past light cone (and $B$ is in $A$'s future), the reverse of that statement, or $A$ and $B$ are spacelike separated. Whichever one of these holds will hold for all observers.
So we know, just from the embankment analysis, that "in the frame of the train, the forward flash reaches the observer's eyes first," and this statement is always true for anyone who speaks it in its entirety.
What about the train observer? Indeed, as you say,
the flashes occurred at the same distance from each of them, the speed of light is constant in both frames, and either one can claim to be at rest
Suppose two people, $C$ and $D$, stand equal distances from you and are known to pitch balls at exactly the same speed. With everyone standing at rest, $C$ and $D$ each toss you a ball. You get the ball from $C$ before the one from $D$. This is not a logical inconsistency. It simply means $C$ threw a ball before $D$ in your reference frame. That is, the person on the train, operating under the SR assumption of "the speed of light is constant," and using the data (retroactively obtained from a ruler-clock grid, or maybe obtained in real time based on brightnesses) that the flashes were equidistant, must conclude that the forward flash went off first.
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