The so-called Twin Paradox derives from comments made by Einstein in his 1905 paper On the Electrodynamics of Moving Bodies, the paper which introduced his version of relativity. At the end of Section I.4 of that paper, he discusses a "peculiar consequence" which has two clocks experiencing different time rates because of the motion of one of the clocks. An online friend has questioned the logic of that particular section of the paper, and I hope to address all of his questions with this discussion here. I will try to use definitions of concepts as used in Einstein's paper, and parallel the logic of the paper as much as possible, without going into the mathematics. The validity of the theory itself will not be discussed, except peripherally.
The principle of relativity is applied to "all frames of reference for which the equations of mechanics hold good." These frames of reference are inertial reference systems. As is the case in Galilean relativity, the notion of a "stationary" system is not absolute--of two inertial reference systems, one can be assumed to be stationary and the second moving, or vice versa. That is the essential character of the relativity principles.
In Section I.1, Einstein establishes what he means by simultaneity, and the synchronization of clocks in an inertial reference frame. Basically, the clocks must be calibrated and synchronized so that, by the clock measurements, a light pulse that leaves one clock takes the same amount of time to reach the second clock, as it does to return to the first clock. Any number of clocks may be synchronized in this manner, and if one clock is synchronized with two others, then those two will also be synchronized.
In the last paragraph of the section, he points out "It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it 'the time of the stationary system.'" That is, in an intertial reference frame, any point can have a clock which can be synchronized with any other clock, and so we may talk about the actual time of that reference frame--but he limits this notion to inertial reference frames.
In Section I.2, he makes two assumptions. The first assumption is that the principle of relativity, which had been applied by Galileo and Newton to the physics of mechanics, is also applicable to all physical laws. That includes the laws governing electromagnetic phenomenon, such as light, as well. The second assumption is that the velocity of light is a constant, whether it is emitted from a stationary or a moving body. He points out that one consequence of these two assumptions is that different observers who were moving relative to one another could have different opinions about whether a pair of clocks were synchronized or not. He also hints that the two assumptions will allow him to derive formulae later that will show that there is a shortening of the length of a moving rod, as measured in a system that is not moving.
In Section I.3, he derives the mathematical transformations that are usually associated with the theory of special relativity. They follow from his two assumptions in Secion I.2, and basically assert that measurements in one inertial frame of reference will differ from those in another frame of reference that is moving relative to the first frame. As a result, according to the formulas, an observer in each frame of reference sees objects in the other frame as foreshortened, and the time processes as dilated.
In Section I.4, he discusses the physical interpretation of the results that he found in Section I.3. He takes the example of a moving sphere, and points out that as the velocity of the sphere increases, it compacts into more like a disk--viewed from the reference frame in which the sphere is moving. Viewed from the reference frame in which the sphere is not moving, the sphere appears normal--whereas for that reference frame, objects in the first reference frame are "shriveled up into plain figures," or, as he says, "It is clear that the same results hold good of bodies at rest in the 'stationary' system, viewed from a system in uniform motion." This is a restatement of the principle of relativity.
He then derives the familiar formulas for time dilation for reference systems that are moving relative to one another. In the derivation, the difference between the systems is not just an optical illusion. You can position observers and clocks anywhere in each system, and all clocks within a given system can be synchronized--so any observation between a pair of differently moving clocks at one point will be the same as an observation at some other point between two other clocks that have each been synchronized with the first pair.
Now, we come to the "peculiar consequence" that has come to be known as the twin paradox. In Einstein's paper, it is considerably more abstract. In his first example, we can imagine two clocks some distance apart that are synchronized in our reference frame, and a third that is moving past first one and then the other. As the third clock passes the first, we set it so that both clocks agree. When it passes the second clock, it cannot agree, according to the derived formula. The third clock "ages" less than the other two, as observed from our reference frame.
For an observer moving with the third clock, however, it would appear that our clocks are aging less. That is the symmetry of the relativity principles.
We can break that symmetry by turning the clock around. Unfortunately, when a clock is turned around, it no longer remains in an inertial reference frame, and the principles cannot be applied so simply.
Instead, we break up the motion of the third clock into sections, each of which consists of straight uniform motion. Each section is a straight line path, a series of edges of a polygon. As the clock moves along each edge, it matches up with a moving inertial reference system that continues on both before and after the clock has left it. During the time that the clock inhabits the reference frame of each edge, it can be synchronized with the reference frame. At each vertex, the system times can be noted for each of the pair of inertial reference frames that intersect at that vertex, and they can be adjusted to match.
If the clock returns by a closed polygonal path to its starting point, it will have aged less than a clock left at the starting point. Now, any closed path, polygonal or not, can be approximated by a polygon to almost any degree of accuracy--so it would seem that the slowing down of the clock would hold for any path.
There is no longer a symmetrical relationship between the clock that followed the path, and the clock that remained behind. Of course, the clock that followed the path did not remain in any single inertial reference frame. It had to change direction--even if it doesn't change speed--and that is noninertial. The clock has experienced an acceleration, but its motion is analyzed by using unaccelerated reference frames. Even though it has not followed an inertial path, we can still use the relativity principles to derive this "peculiar consequence" of accelerated motion.
Ten years later, in another paper, Einstein would take this particular consequence of accelerated motion, and--assuming that acceleration and gravity were totally equivalent--use it to produce a new theory of gravity. Dirac would use special relativity to successfully modify the equations of atomic chemistry, and put some finishing touches on one of the most successful theories ever--quantum mechanics.
The Shipping Lane Thought Experiment
Let's look at the thought experiment of the twin paradox in a little more detail. Imagine a shipping lane in space, moving at a constant speed, and a parallel one beside it moving in the opposite direction. The outgoing lane has dozens of spaceships, all moving the same speed, as does the incoming lane. Since the ships of each lane are all moving the same speed, their clocks can be syncronized. According to the relativity principles, the folk on the outgoing ships might see the clocks of the incoming ships as they passed, and vice versa. Each ship would see the others as going slow. Someone motionless between the lanes would see the clocks of both lanes as going slow.
What if the motionless observers were to read the clocks of each ship as it passed? "Start the experiment now!" The ship would have a timer, synchronized to their own clock, which is slow relative to the motionless observer. A half a day later, according to their clock, they pass an inbound ship and the clocks of each ship are again read by another observer. The inbound ship would take a half day of their time to return to the first motionless observer, since their clock runs just as slow as the outbound ships. As they pass the observer, their clock will show that it has been a day, total. The observers can than compare notes. According to the observers' clocks, it has been longer.
In that thought experiment, nothing material travels the full round trip. Nothing accelerates, since the ships are moving at a constant speed. But, we could easily imagine that the observer tosses a clock onto the ship. The observer sets it to zero, and throws. As the ship passes, both the ship and the observer see that the timer is set to zero. We can make the transfer as short as possible, so that any discrepancy introduced is small. As the ship goes on its way, it is compared it to the ship's clock to make sure that it is keeping time at the same rate. Then, after half a day, it is passed off to the inbound ship--both ships see, at that moment, that the timer says one half day. When the timer gets back to the observer, it will say one full day roundtrip, even though the observer's own timer says that more time than that has elapsed.
We could do the same with any closed polygonal path--a whole network of intersecting shipping lanes that hand off the clocks. And it doesn't matter whether the clock rides along or not or whether the observers actually watch the clock or not--the timing is the same. The observers are there just to make sure that everything functions properly. There are errors in the timekeeping introduced when the clock is passed between the ships, but we can insure that these are small relative to the overall effect.
The shipping lanes are a more fleshed-out version of Einstein's abstract thought experiment that he described in his paper as a "peculiar consequence". The math is the same, and the result is the same. There is no paradox--both shipping lanes do see each other as aging slower, but the symmetry is broken when the timer changes direction.
Thanks to the members of the old as well as the new Bad Astronomy Bulletin Board, especially SeanF, Wiley, Rosen1, and the Space and Mystery Board (thanks Ratchet) for their comments and critique, and for allowing me to incorporate their suggestions into this webpage. And, thanks, Phil (the Bad Astronomer), you're the worst.