Michael Lee's Mathematics and Physics
Length Contraction Equations
Length Contraction Calculations
dt = 0 In order to measure the length of something (a metre stick say) in motion with uniform velocity to the right with respect to our stationary frame, it must be done in an instantaneous moment. That is the ends of the stick must be measured, located on coordinates, at the same time. By "located on coordinates," suppose we are standing on the ground facing a large sheet of graph paper in metric. Suppose the meter stick mentioned above were to fly horizontally to the right past us like a flying arrow. We would need to measure both ends at the same time by marking them on the graph paper. If the right end is measured first and then the left end a few moments later ( dt > 0 ), the length of the stick will measure shorter than it should be; this is due to error and not relativistic length contraction. Likewise if the left end is measured first and then the right end, it will measure longer than it should be and there is no such thing as length dilation. For a metre stick at rest, we can either do that ( dt > 0 ) or measure either end first and then the other over any duration of time ( dt > 0 ) and still get a good measure, but for the stick in motion, both ends must be located simultaneously so ( dt = 0 ). The point here is the length of the stick measured as it is in motion where ( dt = 0 ) ought to be the same as when it is measured at rest. Instead, we find the length of the stick in motion is slightly shorter than one at rest.
