Sorry I didn't reply to this comment sooner, but I haven't been keeping a very close eye on EvC lately.
You got a lot of replies to this statment that basically said that the math works out, as well as posts that pointed out that your picture implies a 3rd observer. But there is an interesting paradox going on here that is taught to students learning relativity. Of course, it's not a real paradox. Just as the Twins paradox can be resolved, so can this notion of A's time is shorter then B's time while the converse is also true. As long as you know that length contracts at relatavistic speeds just like time slows down it's pretty straighforward. The way it is usually presented is like this-
Suppose the Flash, a man who can run at relatavistic speeds, wants to test relativity. He reasons thus: "If I have a barn that's 10m long and has doors on both ends that can be closed very quickly and a 15m long pole I can show relativity doesn't work. If I run fast enough that length contraction is 50% then I can hold the pole horizontally, run through the barn, and have the farmer try to close both doors on while I'm inside the barn. From the farmers perspective it should work because my pole is length conracted to 7.5 m. So he should be able to close both doors while I'm inside. Of course he'll have to open then again to let me out so I don't undergo acceleration and go into General Relativity's realm.
But for me, the barn is only 5m long, so there is no way I'm going to get my entire pole in there. So when the doors close at the same time they are going to hit my pole. So if the doors are able to close then the farmer's reference frame is the correct one and if they hit my pole on the way down then mine is."
This is basically the same scenario as you proposed, except instead of two times, you are comparing two lengths. So what happens?
From the farmer's reference frame, he closes the doors at the same time and Flash's pole fits inside. From Flash's reference frame, as he is running into the barn the far end closes before he gets there and then opens again to let him pass though. Then, after the entire length of the pole has passed through the near end of the barn, the near end door closes and opens. Thus we have another lesson to learn from relativity.
Simultanius events in one reference frame aren't simultanius in another.
Suppose a farmer is sitting on his porch, over-looking a long stretch of road with syncronized clocks placed every mile. Say the clocks are just striking noon as the Flash wizzes by him. To the Flash, the clocks far in front of him read to be 1pm while the ones far behind him read to be 11 am.
Personally, I think it's one of the wierdest aspects of relativity.
I hope this cleared up the "paradox".
This message has been edited by Raymon, 12-08-2004 07:59 PM