The True Twin Paradox.
(Last update : Oct 30th, 2000.)
In the following developments it is accepted that the reader knows this paradox. This page is developed to add some new points of view to the problem.
People from the mainstream of Physics usually consider this paradox as an entertainment or merely as no paradox at all. It is usually said that, as this paradox involves changes in the speed of reference systems, it is outside de scope of special relativity theory. This is not true and there exist a lot of reasons to consider this paradox well inside the logic structure of the special relativity theory:
Einstein, himself, made the first development of this paradox as a direct consequence of his theory in his first 1905 paper.
Other derivations of the special relativity theory, such as the equivalence between mass and energy works very well in accelerated systems, as particles accelerators demonstrate in every experiment at high speeds.
The Twin Paradox is a direct result of the postulate of the constancy of the speed of the light.
In the classic "explanation" or "resolution" it is argued that the twins differentiate one from the other because one of them was subjected to changes in his state of motion while the other is not.
I will show two different analysis of why this explanation is not satisfactory.
First analysis
This paradox can be expressed in different ways that can not be "justified".
Example 1:
Both twins use identical ships that depart in opposed direction and are subjected to the same accelerations and both turn to the origin at a pre-established time (using clocks that were identical at rest). In this case, when faced: Which is the youngest? .During all the travel each one "saw" that the clocks of the other were delaying, but when faced (both at rest) both twins can not claim that the other is the younger because, at rest, an objective measurement can be done.
Example 2:
A very simple variant of the previous case.
It was checked the delay of clocks in movement by means of orbital clocks compared with clocks at rest (fixed to the earth surface).
OK. But what happens if two orbital clocks, turning in opposite direction, are employed (otherwise the orbits would be identical). Well, both clocks must delay with respect to the stationary clock. And must delay in equal quantity because the magnitude of the phenomenon can not rely on the sense of turn (clockwise or counter-clockwise). Nevertheless, in agreement with the special relativity each clock must delay with respect to the other due to the relative translation. Einstein himself accepted during the development of the General Relativity theory, that, as first approximation, the special relativity worked in systems suffering weak gravitatory fields. Additionally the problem is only qualitative. Both clocks are subjected to the same movement, but each one is in relative movement to the other. According to SRT each one must observe that the other delays.
Question: What happens when, completing every turn, one clock face the other?.
Is there coincidence in their respective appreciations?. Any possible answer is in conflict with special relativity theory.
Second analysis
In this case we will accept that the classic explanation is the correct one. It means that if one of the clocks suffers different accelerations than the other, it breaks the equivalence between systems in uniform lineal relative movement. In that case the whole building of Relativity is affected, because the new question is the following:
Suppose two systems actually moving one with respect to the other. If it means that this translation is the result of different acceleration suffered during the previous history of both systems, can not be used this previous history to differentiate the systems?
As a remark, the currently cosmological accepted theories (Big-Bang, Inflation) assumed that the expansion was uniform at the beginning of cosmic history. So that, two systems crossing in the current trajectories, have to have suffered changes of speed in the past.
And appears not to be reasonable to believe that the classical twin paradox is resolved only because the exact previous speed changes are known. I believe that who adhere to the classic explanation accept that the system that does not suffers accelerations is a privileged system (in special relativity theory). And in that case, such way of reasoning would carry us to believe that exist privileged systems in the Universe: Those that do not suffer accelerations from the Big-Bang. And would not have equivalent systems because each one suffers different changes in its history of speeds.
Observation:
As mentioned, the first reference to what currently was known as Twin Paradox was made by Einstein in his article of 1905. It is very illustrative to analyze the original Einstein explanation to this "paradox". In this paper Einstein detailed an experience (only imaginary in that moment) of systems in relative movement, employing terms as the following:
". ..from the previous analysis the following peculiar consequence is derived. If in the points A and B of the K rigid system there are stationary clocks that are synchronous in agreement with realized observations in K, and if the clock of A moves with the speed v toward B, then, on its arrival to B, both clocks will no longer be synchronous. The clock moved from A to B will delay with respect to the clock that remains stationary in B in agreement with the magnitude
1 t v2
----------
2 c2
being t the involved time of the travel.
It appears as reasonable that this result is equally valid if the travel from A toward B proceeded through a polygonal, and also if the points A and B are coincident (departure from A, traveling through a polygonal and return to the same point ). If we assume that the proved result for a polygonal is also valid for a continuously curved line, we obtained the following result: If one of two synchronous clocks is moving in a closed curve with constant speed until it returns to A, with the trip lasting t seconds (in agreement with the indications of the clock that remains at rest in A), the moving clock will be
1 t v2
----------
2 c2
seconds slow at the moment of the return to the point of departure.
Thence we conclude that a clock in the equator must go more slowly (by a very little quantity) that a similar clock situated in a pole, if all the remaining conditions are identical. ...."
Very interesting, I think.
In 1905 there were no clocks that might allow such comparison and was unthinkable to speak of clocks in orbit. But that experience was done actually and the results until where I know were controversial (different results in flights toward East than toward West).
In any way this carries us to my first analysis. If during travels East - West and West - East, the same delay is obtained with respect to the stationary clock, how is it possible that not delays are found between both moving clocks. They are in relative movement (and doubling their relative speed because they are traveling in opposite sense )?
Principals Conclusions of this page:
With any of the two presented analysis it is shown that the Twin Paradox is a not avoidable obstacle for the logic structure of special relativity theory. However a failure in the logic of SRT does not mean a failure in its formulations (in the common usage)
If it is accepted that the Twin Paradox is "resolved" using the "classic" explanation, it would be possible to resolve all the cases of comparison of inertial systems, determining the history of the changes of relative speed between both systems.
Final Note:
If after analyzing the contents of this page, your point of view is something like:
"OK, I can not find the way to justify the resolution of the Twin Paradox in these terms, but it means nothing because the experimental results confirm with no doubt the validity of special relativity theory".
From my own point of view, it means that you still believed that the effect proves the cause. In other words, if the happening of A implies that the result B must be obtained, as B actually happens, it implies that A was its cause. In amused terms, you would be in agreement with the scientist that concluded that "The spider without legs is deaf", when after cutting its octave leg, the spider stopped to respond to the audible stimulus.
A lot of causes (or theories) can drive to the same effect (formulae to describe the physical world).
I tried to reformulate this page on the basis of several received commentaries. I supposed that there were things that were not sufficiently well documented. Usually, I do not reread immediately my own writings. Generally I seat down, write something (without much previous sketch) and I review my documents several months later. In those conditions often I have the sensation to be reading things written by another person (with whom basically I agree in the appreciations) and, sometimes, I am surprised to have made certain "unexpected" commentaries (it is a funny experience).
Well, after rereading this page I am sorry to recognize that I agree one hundred percent with the author. So that, I decided not to change it, but to add a commentary:
I was attending the fifth year of the primary school when the teacher (wonderful "Miss" Rita) requested us a definition of "triangle" with our own words. The result was amused. After trying something like "it is a triangular figure" and other poetic variants, to the fourth or fifth attempt, somebody (Maria Eugenia?) got a reasonable definition saying "is a figure with three straight cutting lines". At this point (tendentiously), the young teacher draw the three lines cutting itself in only one point. She fulfilled the definition but it was not a triangle. But we were approaching to a correct definition and, at this time, several students (including myself) were raising hands desperately to give the answer that each one believed the correct one.
... We remained with the raised hand, because the teacher seeing that we were arriving to a correct definition, decided to give the answer to us. In order to finish the tale, if my memory serves me right, what she said was something like this: "the space (or the figure) geometrically delimited by three straight lines that cross themselves two to two". OK, the official definition were irrefutable. There was no way to make a figure that fulfill the definition and was not a triangle. The objective of the exercise was double. We were about to begin to work in the world of the triangles and needed a suitable definition. And, in addition, she teach us the importance of the correct and no ambiguous or false definitions. But also she showed us (without wanting it) that the authority (she) could establish the limits she wanted we discover with our own deductions. The authority could be diminished if the students reached the correct conclusion without aid. And, please, I am not criticizing the way she organize her class. On the contrary, in my case the objective was completely attained. I only want to establish that the one that is in charge of education also controls most of the variables of the learning process.
And turning back to the Twins Paradox, there is no relativity course where this paradox is not discussed. Impossible not to do it. Einstein himself introduced it in his 1905 paper exemplifying the consequences of his theory. But when the students begin to find faults in the official answer, all the professors do similar things: They say something like: "Well, in fact, this paradox is solved with general relativity (and therefore it is not a paradox), but that is a much more complicated theory. And special relativity is not adapted to solve problems with accelerations ". Full stop.
All of we remained with the raised hand waiting for a better explanation. The great majority of us, never takes a course on general relativity. And those that take it usually are more worried about the equations that by the concepts. So, the Twins Paradox is a always pending subject.
And saying that special relativity does not contemplate accelerations a false argument is used (conscientious or not) because SRT is declared perfectly apt to study particle accelerators, where the results are so good as they can be.
And turning to the version I pointed out with two observers in identical orbits but in opposite way, any effect from previous accelerations must be identical (otherwise the orbits would not be the same). And the gravity is constant and exactly the same over both observers. Therefore the effects of the relative speeds accomplish the requirements of special relativity (it is possible to make the development of Einstein of simultaneity and synchronism for this case). And the answer is incongruent (both clocks cannot be slower one with respect to the other). When both astronauts cross their orbits, both observers can simultaneously see (through the hatchways) both clocks (in the same point of the space).
And the conclusion is simple: A postulate that leads to a real paradox (not an apparent one) must have some failure.
An example of an alternate theory that avoids the twin Paradox can be found in http://www.geocities.com/macpetrol