Conceptual question about Lorentz transformations

[tnedde]

I am a newcomer to relativity, currently studying the subject on my own, via Modern Physics by Bernstein et al. I have a question based on pgs 57-58 of the text.

Suppose that two reference frames S and S' are similarly oriented, and S' is moving with constant velocity v in the positive x-direction relative to S. Let their origins coincide at time t = 0, and suppose that at that moment, a light pulse is sent out from the origin. After time t', an observer in S' sees the light pulse hit both the point (ct', 0, 0) and the point (0, ct', 0), simultaneously. An observer in S also watches these events unfold, but they are not simultaneous to her. My question is: which happens first for her?

Let t_x be the time in S of the pulse reaching the point (ct', 0, 0) in S'; likewise let t_y be the time in S of the pulse reaching (0, ct', 0) in S'. The Lorentz transformations seem to indicate that t_x > t_y, but my intuition says it should be the other way around. My reasoning is thus: In the S frame, the S' frame appears Lorentz-contracted in the x-direction. Thus the point (ct', 0, 0) in S' should appear closer to the origin of S' than the point (0, ct', 0). But because the observer in S (and indeed, in any inertial reference frame) observes the speed of light to be the same in all directions, the pulse should reach the (seemingly closer) point on the x'-axis before it reaches the (seemingly farther) point on the y'-axis.

Can anyone clear this up?

 

[Doc Al]

My reasoning is thus: In the S frame, the S' frame appears Lorentz-contracted in the x-direction. Thus the point (ct', 0, 0) in S' should appear closer to the origin of S' than the point (0, ct', 0). But because the observer in S (and indeed, in any inertial reference frame) observes the speed of light to be the same in all directions, the pulse should reach the (seemingly closer) point on the x'-axis before it reaches the (seemingly farther) point on the y'-axis.

It's certainly true that the x' axis appears contracted in the S frame, but what you've forgotten is that S' is moving. Figure out the path that the light takes along the x' and y' axes as seen in S, then you can compute the travel time for each.