Understanding a Paradox in Special Relativity

Abstract

Einstein’s special theory of relativity includes many non-intuitive and apparently paradoxical conclusions about space and time. One of these is time dilation, the fact that a clock moving relative to an observer runs slower than an identical clock at rest in the observer’s reference frame. This is clearly exhibited in the extended mean lifetime of elementary particles moving at high speed, compared to their mean lifetime when at rest, and must be taken into account when processing signals from atomic clocks carried by satellites in the global positioning system (GPS). My intent in this note is not to derive this or other relativistic effects; such derivations may be found in many textbooks. Rather, I would like to share two figures I prepared while teaching elementary physics quite a few years ago, to help understand the apparent paradox that two observers, in relative motion, both measure the other’s clock to run slower than their own. It is my hope that instructors can reproduce these two figures and find them useful in discussions to illustrate these basic concepts. The sketches are only approximately drawn to scale.