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I know that there are a lot of questions about simultaneity, but mine is based on a specific aspect of it.

In the book University physics ( with modern physics) they explain the train example and how the person inside of it sees the events happening non simultaneously.

However, they say that : "You may want to argue that in this example the lightning bolts really are simultaneous and that if Mavis at O' could communicate with the distant points

without the time delay caused by the finite speed of light, she would realize this."

and then they add "But that would be erroneous; the finite speed of information transmission is not the real issue. If 0' is midway between A' and B', then in her frame of reference

the time for a sigual to travel from A' to 0' is the same as that from B' to 0'.

Two signals arrive simultaneously at 0' only if they were emitted simultaneously at A' and B'. In this example they do not arrive simultaneously at 0', and so Mavis must conclude that the events at A' and B' were not simultaneous."

This does not really convince me, I mean I always thought that simultaneity was based on the finite speed of light (if it was infinite , the limit of the ratio of t1/t2 = 1 {t1 and t2 being the time at which the observer inside the train actually 'sees' the light after the lightning had struck}.

Thanks!