Can anyone show me some maths to explain the train problem with relative simultaneity

solarflare

if we set off at the same time and was moving at the same speed -would we both be 1 mile away from that same point at the same time?

Doc Al

If we started from the same point, sure. (Assuming we went straight, of course.)

solarflare

when you watch the video at 28 seconds the train is paused to show the motion of the light from the two strikes. clearly the light is coming from the two ends of the train to his eyes. the video is agreeing with what im saying. by saying that the light that hit the train at the same time gets to the platform observer at different times means that you will disagreeing with the video.

DaleSpam

The strikes cannot happen at the same time in both frames. See:
http://en.wikipedia.org/wiki/Relativity_of_simultaneity
[tex]\Delta t'=\gamma (\Delta t - v \Delta x / c^2)[/tex]
Since both v and Δx are nonzero then it is impossible for Δt and Δt' to both be zero.

Doc Al

Uh, no. The video has the light hit the train at the same time according to the platform frame and thus the flashes reach the platform observer at the same time. (The flashes reach the passenger at different times.)

solarflare

yes and because he is equidistand from each end then the flashes must have come from the train at the same time

solarflare

you are saying that if he sees the flashes at the same time then the flashes occur at different times - but if he is equidistant from the falshes then that would mean light travelled at different speeds to reach his eyes at the same time

Doc Al

If the lightning strikes occur at the same time according to platform clocks (as in the video), and at the moment of the strikes the platform observer is equidistant from the ends of the train, then the flashes will reach him at the same time.

The flashes will reach the middle of the (moving) train at different times.
No, the flashes will reach the woman (train passenger) at different times, who therefore concludes that according to her train clocks the lightning strikes were not simultaneous. It is a basic principle of relativity that the light travels at the same speed with respect to all observers.

Chestermiller

If the two lightning strikes hit the ends of the train at the same time, as reckoned from the platform observer's frame of reference, then the two flashes will not arrive at the same time in the train rider's frame of reference, even though the distances between the rider and each of the two flashes as measured on the train were the same. If there were a train rider positioned at the front of the train and another train rider positioned at the rear of the train, then according to their synchronized clocks, the train rider at the front of the train would measure the flash at his location to occur at an earlier time than the train rider at the rear of the train. So the rider at the center of the train would have to reckon that the flash from the front of the train occurred first, followed by the flash from the rear of the train (if the speed of light is the same in all directions in his frame of reference).

Now, lets consider the opposite situation.

If the two lightning strikes hit the ends of the train at the same time, as reckoned from the train rider's frame of reference, then the two flashes will not arrive at the same time in the platform guy's frame of reference, even though the distances between the platform guy and each of the two flashes as measured on the ground were the same. If there were a platform guy positioned exactly at the location of the front lightning strike and another platform guy positioned at exactly the location of the rear lightning strike, then according to their synchronized clocks, the platform guy at the location of the front lightning strike would measure the flash to occur at a later time than the platform guy at the location of the rear lightning strike. So the guy at the center of the platform would have to reckon that the flash at the rear of the train occurred first, followed by the flash at the front of the train (if the speed of light is the same in all directions in his frame of reference).

Chet

solarflare

so what your saying is that they do strike at the same time but because she has forward momentum she sees the front stike first and the back strike second?

solarflare

here you say that if they strike in the trains frame the flashes will reach him at different times

Doc Al

No, they strike at the same time in the platform frame. You've got to remember that saying 'at the same time' is meaningless unless you specify according to what frame.

Right! If the strikes occur simultaneously in the train frame, they occur at different times in the platform frame.

solarflare

but how can light from the trains frame that travels the same distance from each end not reach the platform frame at the same time.

you have to apply the same maths to the platform frame as you do to the passengers frame.

solarflare

here is what i think happens - the two observers see the same thing - they both see the two bolts hit simultaneously - but they will disagree on the time that the two bolts hit. the passenger might say they hit at 3:00 and the platform observer might say they hit at 3:01

they agree on what happened but they disagree on when it happened

Chestermiller

No. If the two strikes occur at the same time according to the set of synchronized clocks in the platform frame of reference, then they will be observed to occur at different times according to the set of synchronized clocks in the train frame of reference. If there are actually observers from each of the two frames of reference present at the locations of the lightning strikes when they hit, and if the 4 observers at these locations (2 on the platform and 2 on the train) write down on pieces of paper the times on their clocks that the two strikes hit, the times written down on the platform observers' pieces of paper will be identical to one another; the times written down on the train observers' pieces of paper will not be identical to one another. According to the team of observers on the train, the clocks on the platform are out of synchronization, and according to the team of observers on the platform, the clocks on the train are out of synchronization.

Doc Al

For the same reason that light from each end of the train (simultaneously emitted in the platform frame) can arrive at the middle of the train at different times.
Exactly! You must apply the same math and the same rules for all frames. But what you cannot do is just ASSUME that the lightning strikes are simultaneous in both frames. (If you do, you'll contradict the basic assumption that the speed of light is the same for everyone.)

So don't keep flipping back and forth between two physically different scenarios. Pick one scenario, such as the lightning strikes being simultaneous in the platform frame (as in the video) and analyze it properly.

Doc Al

They do agree on what happened, but not in the way that you think.

Since the lightning bolts hit simultaneously in the platform frame, we can mathematically deduce that the flashes must arrive at the middle of the train at different times.

The train observers agree of course. But the train observers also say that the lightning strikes were not simultaneous according to their clocks. Furthermore, they claim that the clocks on the platform are not synchronized.

Simultaneity is frame dependent, just like length and clock rates.