# Relativity of Simultaneity - Why doesn't the train/lightning example contradict the absolute speed of light?

I know similar questions have been asked here before, but none of them seem quite to address my particular confusion.

I'm not afraid of math (I did well in calc III last semester, for example) but my familiarity with physics is still primarily conceptual. My basic, high-level understanding of special relativity is that space/time must be malleable in order to preserve the constant speed of light regardless of reference frame.

Working from that understanding, the train & platform example where a flash of light originates from the center of the train makes sense to me. I think these two pictures (taken from Wikipedia) make it pretty clear that the two observers will disagree about whether the light reaches both ends of the train simultaneously:

However, I don't understand this example where lightning strikes both ends of the train simultaneously from the perspective of the stationary observer on the platform (also pictured on the previously linked Wikipedia page). It makes sense that the observer on the platform will see the front flash reach the passenger before the rear flash -- since the passenger is moving toward the front flash from the perspective of the observer on the platform -- but shouldn't the the passenger see the two flashes reach her location in the middle of the train simultaneously as well? Otherwise, how can we say that the speed of light is unaltered by the motion of the passenger's reference frame?

In other words: if moving toward a flash of light allows you to reach it sooner, wouldn't that contradict the fundamental premise that the speed of light is unaffected by relative motion?

I know I'm misunderstanding; I'm just not sure how to improve my understanding... Maybe someone can help me with the math, or at least explain what I'm missing conceptually?

This is one of those situations whereby it is much clearer what the fundamental issue is, when you draw out the Minkowski diagram.

The person on the ground observed that the lightning hits both ends of the train at the same time. But that is a "same time" that is horizontal on the Minkowski diagram. The person on the train will simply disagree that it is the "same time", because "same time" is the slanted line to people on the train.

Learning about relativity is a lot about learning to be careful in stating what happens where and who thinks what. All the "same time" and "same place" have to be very carefully stated clearly, and then many conceptual issues will clear up too.

You are completely correct that if the two lightning strikes occur simultaneously in the passenger's frame, then the passenger must see the light from both of them simultaneously. This is exactly the point: since the passenger sees one flash before the other, we must conclude that the strikes cannot have occurred simultaneously in the passenger's frame.

You say 'if moving toward a flash of light allows you to reach it sooner, wouldn't that contradict the fundamental premise that the speed of light is unaffected by relative motion?'. Isn't that exactly what happens in the diagrams you have included in your question? In the frame of the platform, the rear of the train meets the flash of light before the front of the train, because the rear is moving toward it while the front is moving away.

It is impossible for the light from both flashes to reach the person at the centre of the train at the same time while also reaching the person on the platform at the same time, since the light has to meet at one place or the other. So, if the person on the train sees the light from the front of the train before they see the light from the rear, they must conclude that the forward strike occurred sooner than the rear strike.

Suppose you are walking past me just as a person a light-second away from us flashes a light. The light will take a second to reach me. Suppose that during that second you move a metre towards the light. You have to see the light before I do, because the light has to pass you to reach me. Given that the flash was a light second away from both of us, and yet you see it sooner than I do because of walking towards it, we must disagree on the timing of the flash. In your frame it must have happened slightly before it did in my frame.