I am missing something in the train paradox

[merlinisproof]

The train paradox used to demonstrate relativity of simulatneity says that ( I will assume most people are familiar with it and therefore be brief) the observer on the train moves into the light from the front of the train and away from the light coming from the back of the train, therefore seeing the lightning strike at the front first. But if the speed of light is measured at C in all reference frames, then surely the observer would measure the speed of light aproaching him/her from the back of the train as C, and the fornt also. Since they are in the middle of the train, does this not mean that the light form each should reach him/her at the same time?

I know I must be going wrong somewhere

 

[Fredrik]

Yes, if the flashes had been emitted at the same time in the train's rest frame, the light from the front and the light from the back would have reached the middle of the train car at the same time in the train's rest frame. That's why we can conclude that the two flashes weren't emitted simultaneously in the train's rest frame.

 

[Janus]

The train paradox used to demonstrate relativity of simulatneity says that ( I will assume most people are familiar with it and therefore be brief) the observer on the train moves into the light from the front of the train and away from the light coming from the back of the train, therefore seeing the lightning strike at the front first. But if the speed of light is measured at C in all reference frames, then surely the observer would measure the speed of light aproaching him/her from the back of the train as C, and the fornt also. Since they are in the middle of the train, does this not mean that the light form each should reach him/her at the same time?

I know I must be going wrong somewhere

Remember, in the embankment frame, the light from the two flashes do not reach the train observer at the same time. This is one thing that the two frames must agree on. Otherwise, you will arrive at a contradiction.

For example, imagine that the train observer is carrying a bomb that will explode if and only if the light from the two flashes reach it simultaneously. We have already established that the lights don't reach the bomb simultaneously according to the embankment, and thus according the the embankment, the bomb does not explode. We can't have the lights strike the bomb at the same time according to the train because then the bomb would explode according to the train and we would be left with both an destroyed and not destroyed train.

Thus we are left with both observers agreeing that the train observer saw the flashes at different times, and as Fredrik has already pointed out, this means that the train observer must conclude that the lightning strikes did not occur simultaneously by his clock.

 

[ghwellsjr]

...But if the speed of light is measured at C in all reference frames, then surely the observer would measure the speed of light aproaching him/her from the back of the train as C, and the fornt also. Since they are in the middle of the train, does this not mean that the light form each should reach him/her at the same time?

It is correct to say that "the speed of light is measured at C in all reference frames" if by that you mean the "average" round trip speed of light.

For example, if the observer in the middle of the train wants to measure the speed of the light coming from the lightning at the front of the train, he needs to place a mirror at the back of the train, start his stopwatch as soon as he sees the lightning flash from the front and stop it when he sees the reflected flash coming from the mirror at the back of the train. Then, after measuring the distance between himself and the mirror, he can calculate the speed as twice the distance divided by the measured time.

In a similar manner, he can place a mirror at the front of the train, start another stopwatch when he sees the lightning flash from the back of the train and stop it when he sees the reflected flash from the front of the train and repeat the same calculation.

It will always turn out that no matter how fast the train is going in any direction, or if it is "stopped", he will always get the same answer for the measured round-trip speed of light coming from the two lightning flashes.

But it is not possible for the observer to "measure the [one-way] speed of light approaching him/her from the back of the train" (or from the front) because how would he know when to start his stopwatch? He cannot see the flash of lightning until it gets to him and if you say he can stop it when it gets to the back of the train, then how can he know when that happens?

Please note that these measurements have nothing to do with Special Relativity. Scientists were doing that long before Einstein came along. However, they expected that the measured round-trip speed of light would change as they were traveling at different speeds because they thought there would be only one valid reference frame and it came as quite a shock to them when the measured results were always the same no matter how fast they were traveling or in which direction. So they "explained" the strange result as their own train shrinking in the direction of travel through the supposed valid reference frame and their own stopwatches running slower in just the right amount to create the "illusion" that the measured round-trip speed of light was a constant. They believed that the two one-way trips were not the same when they were traveling and that the reflections occurred at different times, even when the lightning flashes appeared to have occurred at the same time. This was a valid, consistent, and legitimate way to "explain" the measurements.

However, Einstein came along and said, if you just assume that the two one-way trips are equal, no matter how fast or in which direction you are traveling, then you will end up with another valid, consistent, and legitimate way to "explain" the measurements and you won't have to worry about finding the one and only valid reference frame--they're all valid. Then you will conclude that everyone else who is traveling at a different speed than you and/or in a different direction than you will experience the shrinking dimension, the slower stopwatches, and the different one-way light trips. Even though this seemed like an impossible explanation, it caught on and so now we have Special Relativity.

So, when you talk about moving toward or away from the lightning flashes, just remember that the one-way speed of light is not something that you can measure, it is something that you assume and in Special Relativity, you can pick any reference frame and in that frame you assume that the one-way speed of light is the same whether you are approaching the source of the light or receding from the source of the light.