Saturday, November 21, 2009

Irodov Problem 1.356

This is best solved and explained through an example. Say a person located at x=0, fires a gun at time t=0 with respect to frame K. Say the bullet takes time in frame, travels a distance d along the positive x-axis and then hits a board. In other words, the space-time coordinates of the event corresponding to the man firing the gun are (0,0) and that of the event corresponding to the bullet hitting the board are (,d). Now consider a frame K' which is moving at a velocity v along the x-axis. As seen from frame K', using the Lorentz transform, the space-time coordinates of the event of bullet hitting the board are given by . In order for the causality to be disturbed, it should appear to the observer in K' that the bullet hit the board before it was shot. In other words,

Now suppose that the bullet traveled at a velocity vb. Then we have . From equation (1) we have the condition for causality to be contradicted as,

What equation (2) tells us is that in order for causality to be contradicted, at least one among the bullet or the frame K' must be traveling faster that the speed of light! Since nothing can travel faster than the speed of light, causality could never be contradicted.