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If we take a snapshot of the train at the moment the lightening strikes hit (i.e. the moment of simultaneity to the embankment observer) why does the velocity of the train have anything to do with this moment? In my mind I'm stopping everything at this exact moment in time (even though there isn't any one "exact" moment) and since the speed of light is constant, wouldn't the lightening bolts reach the midpoint of the train simultaneously regardless of the velocity of the train?

Writing it out now I feel like my problem is my attempt to "snapshot" the train. In reality time is still moving forward, and thus velocity does become a factor. Is this where my confusion lies?

Again, I understand the Lorentz transforms and how velocity effects these, and I must admit I feel safer in the realm of math than the thought experiment. As stated in this post "there is absolutely no ambiguity in the math, and it is quite clear what Einstein is saying." and I agree.

Thanks!