Contradiction in relative simultaneity

[lowerlowerhk]

let's say a train is traveling at relativistic speed along a straight line to right, and a man is standing in the middle of the train.

Both end of the train was installed a gun which aims at the man. The right-ward gun aims at the man's leg and the left-ward gun aims at the man's head.

In the train, each gun shoots a bullet at speed of 0.999c simultaneously. In the train's frame, the bullets will hit his leg and head simultaneously and the man will die.

According to special relativity, however, in the ground's frame, the bullet aiming for the head will travel toward man slower than the other bullet do. Then the man's leg will be hit first, which cause him to fall down (assuming he has enough time to fall), and the head-aiming bullet will then skip the man's head.

Special relativity said both perspective is true. But there are contradictions in the result of events in different reference frames. What is the correct outcome of this thought experiment?

 

[tiny-tim]

welcome to pf!

hi lowerlowerhk! welcome to pf!

In the train, each gun shoots a bullet at speed of 0.999c simultaneously. In the train's frame, the bullets will hit his leg and head simultaneously and the man will die.

According to special relativity, however, in the ground's frame, the bullet aiming for the head will travel toward man slower than the other bullet do. Then the man's leg will be hit first …

no, in the ground's frame, the slower bullet is fired later than the other one

(t' = (t - vx)/√(1 - v²/c²), and (t - vx) is of course different for different x)

 

[K^2]

If the guns fire simultaneously in the train's frame, they don't fire simultaneously in the ground frame. That's kind of a big point in relativity. The net result is that the bullets will reach the man at the same time in all frames.

 

[ghwellsjr]

You have defined two different scenarios that cannot both be true. This is a typical so-called paradox in special relativity. It's similar to if someone were to say something like this:

Suppose two men agree to a duel. Man A is known to be faster than Man B so they agree to wait until 2:00AM to pull the trigger to make things fair, but they inadvertantly decide to do it on the night when we go off Daylight Saving Time and we set our clocks back one hour at 2:00AM, thus repeating 2:00AM one hour later. So then we say that at 2:00AM, they both pull reach for their guns and Man A kills Man B. But then suppose someone who doesn't understand Daylight Savings Time says that 2:00AM, Man A realizes that it's really 1:00AM and waits but Man B pulls his trigger and kills Man A and so he concludes that there is a contradiction in Daylight Saving Time because both men couldn't survive the duel.

Now of course you would laugh and say to anyone that came to that conclusion that they don't understand Daylight Saving Time and that only one of those scenarios could be true.

Well the same thing is true in Special Relativity. You get to pick a scenario and describe it fully according to a frame of reference that you also choose. In this frame, you assign times and locations to the different events, like when guns are fired. Then if you want to see what the times and locations look like in a different frame of reference, you must use the Lorentz Transform to calculate all the values associated with the events. This process will never make the scenario operate any differently, it only changes the numbers of the times and the spatial distances associated with the different events.

But if you don't understand what Special Relativity is all about and you assign what you think are consistent values to the times and locations of the events from the first scenario but you do it incorrectly and inadvertantly create a different scenario, you may end up blaming Special Relativity instead of yourself.