The Twins Paradox: Why Acceleration Is Not Relevant
One of the most baffling and disturbing problems of relativity is the famous thought experiment of the Twins Paradox, in which one of two identical twins leaves Earth in a spaceship on a voyage at relativistic speeds to the nearest star and back. When he returns, he finds that while the trip took him a very short time by his shipboard clocks and he is only a few weeks or months older, many years have passed on Earth, and his identical twin is an old man.
This would seem on the face of it to be a violation of the fundamental principle of relativity, i.e. that motion is relative. If all motion is in fact relative, then why does the twin on earth age and not the one in the spaceship? Why does there seem to be a privileged frame of reference with respect to the passage of time?
It has been argued that the essential asymmetry of the Twins Paradox lies in the fact that the spaceship accelerates, and the Earth does not, and while motion is relative, acceleration is absolute. I intend to show that while this is true, it is only circumstantially relevant to the problem. The true asymmetry of the Twins Paradox, and the one that leads to the age difference of the twins, is the asymmetrical way in which the Lorentz contractions affect the different reference frames. (It might also be pointed out that the Twins Paradox is a Special Relativity problem, and acceleration is only addressed in General Relativity.)
Let us conduct another thought experiment, then, in which the symmetry is preserved, and then I shall attempt to show how the Twin Paradox is a special case of this experiment. Imagine two relativistic passenger trains, the Solar Express and the Proxima Bullet, running on a parallel set of tracks between Alpha Centauri and Earth. Let's say each one is about a light-minute long. These two trains are scheduled to leave their respective stations at around the same time, pass each other about halfway between the two stars, and then arrive more or less simultaneously. Of course, simultaneity is a pretty much meaningless concept in relativity, but exactly when the trains leave and arrive is not important to this problem. We are only interested in the part of the trip, somewhere along the line, when the two trains pass each other.
We don't really know which one has accelerated more than the other, and in any case, it doesn't matter. Let's assume that both have accelerated exactly the same amount since leaving their stations, and so there is complete symmetry between the two trains with respect to speed, length, distance travelled so far, and rate of acceleration. As a result, each engineer will have fairly similar observations of the passage of the other train by him.
From the perspective of the Solar Express, the Bullet's length has contracted to a fraction of the Express' length. The engineer of the Solar Express, then, will see the locomotive of the Proxima Bullet flash by his own locomotive (event A) only seconds before the caboose streaks by (event B). It will, however, take almost a whole minute for the Bullet to reach the caboose of the Express (event C), since the Express is still as long as it ever was, and it will of course take only seconds for the entire length of the Bullet to pass by the caboose (call the moment the cabooses pass event D).
Meanwhile, from the perspective of the Bullet, the Express' length has contracted to a fraction of the Bullet's length. The engineer of the Proxima Bullet, then, will see the locomotive of the Solar Express flash by his own locomotive (event A) only seconds before the caboose streaks by (event C). It will, however, take almost a whole minute for the Express to reach the caboose of the Bullet (event B), since the Bullet is still as long as it ever was, and it will of course take only seconds for the entire length of the Express to pass by the caboose (event D).
Note that while the overall experiences of both engineers are similar, they differ quite dramatically on their measurements of specific intervals. For example, the Bullet engineer will report a much shorter period elapsed between events A and B than the Express engineer will, while the Express engineer will similarly report a shorter elapsed time between A and C than the Bullet engineer will.
Okay, now, here's where the Twins Paradox maps onto our train problem. Say the locomotive of the Solar Express is called Earth, and the caboose is nicknamed Alpha Centauri. In the Twin's paradox, we're only concerned with the time it takes for the Bullet locomotive to go from the Earth to the Alpha Centauri (events A and C), which is of course shorter for the Bullet than it is for the Express. We don't even think about the time it takes for the Bullet's caboose to reach Earth, A-B interval, which is longer for the Bullet than for the Express! The basic asymmetry of the Twins Paradox lies in the landmarks used to define the trip, and not in the acceleration of one or the other reference frames. Indeed, the Twins Paradox would still have the same results if it were the entire Galaxy that accelerated and the spaceship remained still, since the time interval which is of interest remains A-C and not A-B.