- #1

Master Wayne

- 26

- 3

I place these clocks the same distance away from a midway point which contains a light bulb. The clocks travel toward the midway point at the same speed, meeting at that point some time later. Before the clocks start their journey, I set them to show time zero and turn them off. As soon as the clocks start, the light bulb emits a flash that travels in all directions. When that flash hits the clocks, they turn on and start keeping time.

From a reference frame at rest with respect to the midway point, the clocks will always be in sync and when they meet at the midway point the male clock will pass through the female clock with no issue. Therefore, this is the outcome we must expect in all reference frames.

Now, let's analyze things from the reference frame of the male clock. In the image below, I've drawn the axes corresponding to that frame.

As we can see, from the male clock's frame the light flash starts the female clock first and only some time later starts the male clock. Therefore, the female clock gets a head start. However, from the male clock's frame the female clock also runs slow due to time dilation. To avoid a paradox, these two facts must combine in order to allow for a perfect synchronization of the clocks at the midway point.

What I'm having a hard time figuring out is how to make this analysis algebraic in order to demonstrate that this synchronization does happen from the male clock's frame. How can I show that from the male clock's frame, the female clock's head start is compensated by time dilation?

Bonus question (which is also confusing me): how come from the male clock's frame, at any given point in time, the distance D2 between the male clock and the midway point is larger than the distance D1 between the midway point and the female clock? Shouldn't I expect both distances to contract equally? This is the difficulty which has prevented me from drawing a new spacetime diagram from the perspective of the male clock (I can't figure out why the initial conditions would be spatially asymmetric).