What time do simultaneous events in rest frame occur in moving frame?

1. The problem statement, all variables and given/known data

A train is moving by a stationary observer, Bob, at 0.8c. An observer standing still in the middle of the train, Alice, measures the distance to each end of train as 9*10^11 m. Two lightning bolts hit each end of the train simultaneously, as seen by the observer at the station, Bob. Set t = 0 when the lightning bolts hit the ends of the train and t’ = 0 when Alice passes by Bob.

a) How far away from the ends of the train does Bob think Alice is?
b) How long will it take for Bob to see the lightning bolts?
c) At what times, in the Bob’s reference frame, does Alice see the lightning bolts?

2. Relevant equations

3. The attempt at a solution
I have done a) and b) but am confused on how to proceed with c).

For a) I have that Bob observes Alice to be standing 5.4 * 10^11 m away from each end of the train, and therefore it will 5.4 * 10^11 / c = 1.8 * 10^3 seconds for the lightning to be seen by Bob.

For c) I don’t really know how to find the time. I’ve drawn a spacetime diagram, and I’ve determined that Alice sees the lightning bolt from the front before she sees the lightning bolt from the back, but I don’t know how to express this quantitatively, or even how to apply which Lorentz transformation. I’m getting myself more and more confused, so if anyone could point me in the right direction that would be great.


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