I've been working through the Feynman Lectures on Physics. I'm currently on lecture 15: The Special Theory of Relativity, specifically 15-5, the section on the deriving the Lorentz Transformation from the idea of length contraction.
I'm confident this question has been asked/answered before, or that my understanding is flawed, but I've been up to this point unable to find an answer to this specific question, so I'd rather just ask.
Feynman considers Joe, with coordinate system $(x,y,z,t)$, and Moe, with coordinate system $(x',y',z',t')$. Moe and Joe are said to be in relative motion, with Moe having some velocity u relative to Joe in the x direction. Both are trying to measure the x coordinate of a point P.
Moe measures the distance to be x'. However, from Joe's perspective, Moe's ruler has been shortened due to length contraction; thus Moe's measurement will be greater than Joe's, by a factor of the Lorentz factor (not yet accounting for the fact that the origin of Moe's coordinate system will be constantly moving).
This assertion makes sense, given the explanation, but I would have tackled it in a different manner, apparently yielding a different result: Point P is in relative motion to Moe, so, from Moe's perspective, the distance between him and P contracts. However, P is at rest from Joe's frame of reference, so Joe will be measuring the proper length. Given this explanation, it would appear that Moe's measurement would be less than Joe's.
Is it that it matters what frame of reference we take to be rest? I'm not naive enough to believe that I've found some "flaw" or paradox in special relativity, but I am ignorant enough to be unable to resolve this issue on my own. Any help will be appreciated, Thanks!