- #51

JesseM

Science Advisor

- 8,496

- 12

Now suppose at t=0 some light is released at the point where the left end of the horizontal arm meets the bottom end of the vertical arm. Since the light is moving to the right at 1c while the horizontal arm is moving to the right at 0.6c, in this frame the distance between the light beam and the right end will be decreasing at a rate of 0.4c, and since the horizontal arm is 16 light-seconds long, the time for the light to reach the right end will be 16/0.4 = 40 seconds. Then the light is reflected from the right end back at the left end, and now the left end is approaching the light at 0.6c, so the distance between the light and the left end is decreasing at a rate of 1.6c. So, the time for the light to get back to the left end will be an additional 16/1.6 = 10 seconds. So, in this frame the total time for the light to go from the left end to the right end and back is 40 + 10 = 50 seconds.

Now consider the light going up and down the vertical arm. The top end of the vertical arm starts out 20 light-seconds directly above the point where the light is released, then after 25 seconds it is still 20 light-seconds above the release point on the vertical axis, while it has moved (25 seconds)*(0.6c) = 15 light-seconds away from the release point on the horizontal axis. So using the pythagorean theorem, the top end is now a distance of [tex]\sqrt{(20)^2 + (15)^2} = \sqrt{625} = 25[/tex] light-seconds away from the point where the light was released. Since the light travels 25 light-seconds in 25 seconds, this must be the time when the light catches up to the top end. Similar calculations show that 25 seconds after this, the bottom end of the vertical axis will be 25 light-seconds away from the position where the light hit the top end, so this must be the time the light returns to the bottom end. So, you can see that the total time for the light to go from the bottom to the top and back on the vertical arm is 25 + 25 = 50 seconds, exactly the same as the time for the light to go from the left to the right and back on the horizontal arm, so the two light waves will indeed meet at the same point in space and time. As I said before, it is a basic rule in relativity that if one frame says two events happen at the same position and time, this

**must**be predicted in all frames.