Relativity Simultaneity Problem

[hy23]

Homework Statement

There's a train traveling at 0.5c when a light at the center of the train flashes. When the light reaches the front, it rings a bell. Light reaching the back rings a siren.

Question is: are the bell and siren simultaneous events for a train passenger?

are the bell and siren simultaneous events for an observer outside of the train (i.e. standing beside the track)?

If any of the above are not simultaneous, which event occurs first and by how much time?

Homework Equations

t=t'y

The Attempt at a Solution

I think for the passenger, it should be simultaneous events because in that frame, the light should take the same amount of time to travel to both ends of the train.
For the outside observer, it should appear that the light reaches the back first. But does the train passenger measure proper time or the outside observer?

 

[kuruman]

Your thinking about the passenger is correct. To answer the second question, note that the outside observer is moving relative to the passenger. So if two events are simultaneous in the train frame, can they also be simultaneous in a frame that is moving relative to the train frame?

 

[hy23]

ok i did the calculations and i said that the the explosion happens at t=0 in the train passenger's frame, using Lorentz transformations I found that from the outside reference frame, the light from explosion reaches the back of the train at a t'<0, and reaches the front at t'>0...(you probably don't need to know the actual values I got, just that one is negative and one is positive), but then this violates causality because the outside observer would see the light reaching the back before the explosion happens?

 

[kuruman]

Obviously causality cannot be violated. Assuming that you calculated correctly that the light reaches the back of the train at t' < 0, when (in the outside observer's frame) does the light from the explosion reach the outside observer? Have you done that calculation? If it's before the light reaches the back of the train, you're OK.

I should add that I don't really like the way this problem is formulated, but that's not your fault. Instead of a bell and a siren, there should be flashes of, say, red and blue light at each end. This train is traveling at about Mach 400,000 (absurdly supersonic) so what the outside observer would hear is questionable.