Meno,
“In less than one second? That would mean light is travelling faster than the speed of light in train man’s reference frame. That’s impossible.”
No, on his watch it’ll be a second. On the platform man’s watch, it’ll be less than a second.
“The discrepancy is not in when the light hits each observer’s eyes. Train man will see two lightning strikes simultaneously and platform man will see two lightning strikes simultaneously.
The discrepancy is in when each observer sees the light hitting train man’s eyes. From train man’s reference point, the light beams from both strikes hit his eyes simultaneously (that’s why he observes the strikes as occurring simultaneously). From platform man’s reference point, the light beams from both strikes do not hit train man’s eyes simultaneously. In other words, platform man sees the strikes occur simultaneously, but he does not see them reaching train man’s eyes simultaneously.”
The platform man observes the following: Two flashes that happened simultaneously at the moment the middle of the train (and the train man) passed him, by calculating that the light from each flash reached him exactly 1 second after the actual strike.
The train man observes the following: Two flashes, with the flash at the front of the train happening just before he passed the platform man, and the flash at the back of the train happening just after he passed the platform man, by calculating that the light from each flash reached him exactly 1 second after the actual strike.
The platform man observes the following about the train man: he was speeding towards one of the flashes at the same time the light from it was heading towards him, so the light reached him in under a second. He was moving away from the back flash, so that light reached him after slightly longer than a second.
The train man observes the following about the platform man: he was speeding towards the back flash, which happened later than the front flash, so the light from the later back flash reached him at the same time as the light from the earlier front flash, which he was moving away from.