I'm thinking about a very basic scenario in special relativity, but I've got something backwards, and I could use help understanding why it's wrong. Suppose a rod R is moving past an observer O at speed v. The observer wants to know how long the rod is, so he starts his stopwatch as soon as the front of the rod reaches him, and stops his watch as soon as the end of the rod passes him by. The observer measures To seconds. Knowing the rod is moving at v, he calculates its length as he sees it, Lo, as: [itex]L_o = v T_o[/itex] From the rod's perspective, the rod is sitting still and the observer is moving. If the rod measured the same way, starting a stopwatch as soon as the front edge reached the observer and stopping when the rear edge passed the observer, the rod would say it took Tr seconds to pass, making the rod's length [itex]L_r = v T_r[/itex] The observer should find the duration of the crossing to be inflated by gamma, [itex]T_o = \gamma T_r[/itex] Substituting yields: [itex]L_o = \gamma L_r[/itex] ... that's not right, though. The time should increase by gamma, but the length should decrease by gamma. But length and time are proportional to each other. What's wrong here?