I had the following question I was working from a book:
Question: A physics professor runs across the hallway covering 120 ns of distance in 150 ns of time as measured in the frame of the earth. Assume the professor travels at a constant velocity, how much time does the professor's watch indicate has elapsed during the trip?
Solution: Let A be the event that the professor enters the hallway and B be the event that he leaves the hallway. The metric equation is $\Delta s^2 = \Delta t^2 - \Delta d^2$. If you plug in the given data you will get $\Delta s$ = 90 ns. This is the correct solution as given in the book.
However, I'm confused why I can't analyze the situation in the frame of the professor. I cannot say that from the Professor's frame, events A and B have separation $\Delta d$ and take place $\Delta t$ apart so that the time passing in the earth frame is $\Delta s$ = 90 ns. These would contradict each other, but I cannot see why. It seems to me that the symmetry implies that I should be able to analyze this in either frame since they are both inertial. Furthermore, doesn't the fact that less time ticks off the professor's clock indicate a violation of relativity in that it distinguishes the results in the earth frame from the professor's frame which are both taken to be inertial in the problem?