Almost every first year student of relativity comes up with the same question. It is so common, it has a special name. It is called "The Twin Paradox". Now the official story is that this is a non-issue. It is not a contradiction or even a paradox (though one could actually call it a paradox in any case). It has been believed for nearly 100 years that relativity is a self consistent theory, but is that assessment true?

First, let's do a quick review:

Imagine there are two astronauts floating by each other in space. One imagines himself to be at rest and the other going by at some velocity v. The other one imagines that he is at rest and the first one is going by at velocity -v. Now according to relativity,only relative velocity exists, therefore both of the observers are correct. This has to be so, otherwise there would have to be some preferencial velocity which would allow them to determine who had the "real" velocity, or if it was distributed between them in some way. Now I have been calling this property 'reciprocity' which means simply that each observer can consider himself to be at rest.

In the twin paradox, one twin leaves earth at a very high speed until he gets to some distant planet. He then decides to return home at a high speed. And this is the basic scenario. The problem arises when we try and determine the time that each twin sees the other one experience according to relativity. According to the stay-at-home twin, the travelling twin ages less,both leaving and returning (since time dilation is dependent on v2). But the first year student will reasonably consider reciprocity and see that the travelling twin would consider himself to be at rest and he would see the stay-at-home as moving away from him (along with the planet) and then returning back to him, thus according to the travelling twin, the stay-at-home is the one who has aged less. This, of course, is a contradiction since they both can't be younger than the other.

The first response a student would probably receive on this question is:

(1) The travelling twin cannot view himself as at rest, because he has accelerated during the trip, and which one has accelerated can be determined by noticeable inertial effects.

However, this argument has several problems. First of all, the Lorentz transformations depend on just distance and time. There is no reference to force at all. Thus while the traveling twin might realize that he is accelerating, the Lorentz transformations do not "realize" it. Furthermore, it is the force difference over the body that one feels. If each atom of the object was being accelerated equally, then there would be no (local) way to determine which twin were accelerating. Furthermore, the traveling twin does not know that he will accelerate until he gets to his destination, thus before that time he clearly sees the stay-at-home aging less. Therefore, the acceleration cannot merely disallow the travelling twin's frame of reference. It must also compensate for the time dilations already observed.

After seeing that this dynamic argument is not sufficient the student might be told that:

(2) Time dilation can only be correctly determined from the point of view of an observer who remains at a constant velocity during the events in question. In other words, you can do the problem in relation to the stay-at-home twin, or you can do it in the frame of reference of the travelling twin while he is leaving, or you can do the problem in the frame of reference of the travelling twin while he is returning, but not both.

This response is inadequate simply because neither of the travelling twin views actually represent the view of the travelling twin over the entire journey. The twin has to be able to view himself at rest in both frames consecutively. A past history of acceleration cannot disallow the current inertial frame. If it could, then the Lorentz transformations could never be applied since all bodies have accelerated in the past at some time. This is an important point and from it one can extrapolate that indeed, even from an accelerating frame the travelling twin must be able to consider itself to be at rest at every instant of the acceleration.



Note that in the above diagram, there are several plots of a piece wise acceleration. Note that for every period of inertial motion, the travelling twin must be able to view himself at rest. Furthermore, each of these inertial frames can be made arbitrarily small in duration. Therefore, even from an accelerating frame the travelling twin must be able to consider itself to be at rest at every instant of the acceleration. Of course, you cannot explicitly use the Lorentz transformations as they are usually written. You would have to work out a differential form. The results of this process though, from just general considerations, would yield a time dilation which would be a type of averaging between the initial velocity and the final velocity of the period of acceleration. This would have to be applicable to both the stay-at-home twin's view of the travelling twin and the travelling twin's view of the stay-at-home twin.

One can see here thus that the contradiction is still not resolved. The student will then be forced to face the dreaded space-time diagram with the argument that:

(3) The space-time diagram has a one-to-one correspondence for each event and therefore cannot be contradictory. It also shows both viewpoints in the same diagram. There is also a specific time called "proper" time which will always yield the true passage of time (in this case for the travelling twin). Furthermore, the proper time is independent of the observer.

The problem here is the diagram does not resolve the contradiction because as soon as it is drawn it biases itself to the viewpoint of the observer at rest in the diagram. It is claimed that all viewpoints are shown in the same diagram, but this can easily be shown to be wrong by merely drawing a new diagram as if the travelling twin were at rest (as shown below). This need only be done for the first part of the journey where the travelling twin has a constant velocity. But such a demonstration shows that both views are not included in a single diagram.



In diagram A, I have set up a typical space-time diagram and have shown the stay-at-home at x=0 and the first leg of the travelling twin's flight with a solid red line. He has an outgoing velocity of v. Since it is constant, the integral for the proper time is easy to do and by the formula we get: T=gt . It is obvious that we will get the same for the return trip (the dotted red line) and we see that the proper time of the travelling twin is indeed less by a factor of gamma. This result plus the claim that the proper time determined by this formula is the same for all observers makes it look like there is no contradiction. This is simply not true. Perhaps the fact that the result is the same as the "point-of-view" of the stay-at-home twin, should have caused some suspicion, but nonetheless:

Imagine that we are the travelling twin and we want to make a space time diagram. We see the stay-at-home travelling away from us at -v. Therefore we would draw diagram B. In this diagram the red line is the stay-at-home twin. Now we determine the proper time again by the formula and we get: T=gt, where T is the proper time of the stay-at-home. This is clearly different from Diagram A.

Drawing a space-time diagram does not resolve the twin paradox, because one diagram does not show the reciprocity required by relativity. There is a sort of pseudo-reciprocity built into the space-time diagrams that works for velocity addition, but with actual reciprocity, the travelling twin makes his own space-time diagram which will obviously disagree with the stay-at-home's diagram.

So the twin contradiction is still not resolved. By this time the professor usually has run out of options, so he just gives the student the line that:

(4) Since relativity is the most experimentally well-supported theory in the history of the world, there cannot be a contradiction.

This is simply irrelevant since no experiment has ever been done that measures mutual time-dilation. That is, no experiment has ever been done where an observer in a moving (in the lab) frame of reference measured time dilation of any object at rest (in the lab frame). Thus the experimental evidence does not even currently support the aspect of relativity that distinguishes it from any theory that has the proper transformations based on an absolute velocity.

In conclusion:
The twin paradox, even after 100 years, is still not resolved. The contradiction is simple, obvious and beyond reconciliation. I cannot fathom why the physics community continues to ignore this problem. They continue to support impossible solutions to the problem and treat with disdain anyone who even brings up the subject.

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