2 minute read

# Special Relativity

## A Pair Of Paradoxes

A paradox is an apparent contradiction that upon closer examination has a noncontradictory explanation. Several paradoxes arise from the special theory of relativity. The paradoxes are interesting puzzles, but more importantly, help illustrate some of the concepts of special relativity.

Perhaps the most famous is the twin paradox. Two twins are initially the same age. One of the twins becomes an astronaut and joins the first interstellar expedition, while the other twin stays home. The astronaut travels at nearly the speed of light to another star, stops for a visit, and returns home at nearly the speed of light. From the point of view of the twin who stayed home, the astronaut was traveling at nearly the speed of light. Because of time dilation the homebound twin sees time as moving more slowly for the astronaut, and is therefore much older than the astronaut when they meet after the trip. The exact age difference depends on the distance to the star and the exact speed the astronaut travels. Now think about the astronaut's reference frame. The astronaut is at rest in this frame. The Earth moved away and the star approached at nearly the speed of light. Then the Earth and star returned to their original position at nearly the speed of light. So, the astronaut expects to be old and reunite with a much younger twin after the trip. The resolution to this paradox lies in the fact that for the twins to reunite, one of them must accelerate by slowing down, turning around and speeding up. This acceleration violates the limitation of special relativity to inertial (nonaccelerating) reference frames. The astronaut, who is in the noninertial frame, is therefore the younger twin when they reunite after the trip. Unlike much science fiction in which star ships go into a fictional warp drive, real interstellar travel will have to deal with the realities of the twin paradox and the speed of light limit.

The garage paradox involves a very fast car and a garage with both a front and back door. When they are both at rest, the car is slightly longer than the garage, so it is not possible to park the car in the garage with both doors closed. Now imagine a reckless driver and a doorman who can open and close both garage doors as fast as he wants but wants only one door open at a time. The driver drives up the driveway at nearly the speed of light. The doorman sees the car as shorter than the garage, opens the front door, allows the car to drive in, closes the front door, opens the back door, allows the car to drive out without crashing, and closes the back door. The driver on the other hand, sees the garage as moving and the car as at rest. Hence, to the driver the garage is shorter than the car. How was it possible, in the driver's reference frame, to drive through the garage without a crash? The driver sees the same events but in a different order. The front door opens, the car drives in, the back door opens, the car drives through, the front door closes, and finally the back door closes. The key lies in the fact that the order in which events appear to occur depends on the reference frame of the observer. (See the section on space-time.)