The following statement combined with the application of time dilation principle confused me on how proper length and time are measured and what they represent.
"The speed at which electrons traverse the 3-km beam line of the SLAC National Accelerator Laboratory is slower than c by less than 1 cm/s. As measured in the reference frame of such an electron, the beam line (which extends from the top to the bottom of this photograph) is only about 15 cm long!"
According to the definition of proper length, we know that it is the length measured from the reference frame in which the object is at rest. In this case it makes sense to observe the beam line as shorter from the electron's reference frame, as the electron is at rest in its reference frame, and according to it, the beam line moves with the speed of the electron. On the other hand, we can apply the same reasoning when making observations from the reference frame of the beam line.
However, when a person measures the time it takes an electron to move a certain distance, this is not considered as proper time. Based on the given definitions, proper time is the time between two events measured in a frame where the event occur at the same location. When length is predicted to be relative in Special Relativity, how do we determine the frame where two events occur at the same location?