The Twin Paradox

Craig Anderson
Math 55
College of the Redwoods
 
Abstract
     This paper will briefly describe the scientific and historical implications of Albert Einstein's special theory of relativity.   A key topic will be it's resultant time dilation and the "twin paradox."  This will be accomplished primarily via a review of an article appearing in the American Journal of Physics co-authored by Richard H. Price and Ronald P. Gruber. 

Introduction


       It was 1905 and an unknown patent clerk named Albert Einstein had just shook the scientific community at its very foundations with five revolutionary papers.  Although each of these papers had a huge impact on physics it was his paper on titled "On the Electrodynamics of Moving Bodies" that introduced us to relativity and created ripples throughout all facets of society.  The basic premise of relativity is that neither time nor space are absolute.  The implications that follow proceeded to raise major ontological and philosophical questions.
     A clock in motion will be found to "run slow" when compared to a clock that is stationary relative to the moving clock. This fact has been tested extensively in laboratories throughout the world with resounding success.  Scientists have extended the definition of a clock to include any measurable periodic device (which is really all a clock is) by accelerating radioactive atoms to near luminary velocities and observing the slowing of their decay rate.  The often troublesome analogy occurs when we can include the life span, heart rate and respiratory rate included in every living creature (including human beings) to be a clock.  This implies that a person in motion will age more slowly relative to a stationary person.  Or, in other words, the "biological clock" of a person in motion at high speeds will appear to slow down.

Background

      In order to bring the reader to an understanding of the mathematics behind this so-called time dilation, I will go briefly through a derivation of the equation that represents the phenomenon.  We can visualize the situation by creating two observers of an event.  Observer S is on the ground and observer S' is on a train car traveling at velocity u with respect to S.  On the train with S' is a special clock shown in the figure to the right.  This clock consists of a light emission source at F a perfectly reflective mirror on the opposite side and a light sensor at D.  We will assume that the angle that the light must travel through is very small compared to the distance Lo and thus may be neglected.  Using our familiar "distance equals rate times time" formula we easily see that S' will measure the time the travel of the light to take

where c is the speed of light.  You should note that this "obvious" time measurement is measured by S' only because the clock is is at rest relative to her.   Both her and the train are both moving at velocity u.  A physicist would describe this by saying that S' and the clock are in the same "inertial reference frame."  This is just a fancy way describing the condition of two items traveling at the same velocity relative to on another. A physicist will refer to the time of an event as measured by an observer in the same inertial reference frame as the event itself (such as S' and the clock) as the "proper time."  Thus Dt is the proper time of the event of the travel of the light beam.
     When we consider the case of S observing the clock as it cruises down the track, we arrive at quite a different conclusion about the time measured.  S is in a different inertial reference frame than S' due the perception that S has of S' traveling down the tracks at speed u. This causes S to view the light emitted at position A, reflected at position B, and finally detected at position C (as shown on the diagram to the right).  If we consider the time that S observes the light to be t then it follows that the train will travel a distance uDt during this interval.  Thus if we use our copious skills of trigonometry we will find that

However, the light will appear to travel a distance 2L according to observer S.  Thus, the total time interval that S will measure for the path of the light is

Now, we can substitute Lo from eq. 1 into eq. 3 and solve  for Dt to arrive at this relationship between proper time and the relative time.
     It should be known that eq. 4 is a less robust form of one of the set of equations referred to as the Lorentz transformation equations.  In fact, the denominator of eq. 4 is often referred to as the "Lorentz factor" which is often represented as simply g.  You should note that as the velocity u in eq. 4 approaches the speed of light c the relative time Dt becomes larger and larger than the proper time Dt.  This is known as time dilation.  To say it differently, the time of an event as measured by an observer in the same reference frame will always be smaller than the time measured by someone in a different reference frame.  Or, more succinctly . . .Moving clocks run slow.  This is the bizarre and counterintuitive fact that special relativity sets on our doorstep. It's predictions have been tested and proven countless times in laboratories throughout the world.  In fact, in one experiment, a highly accurate cesium clock was synchronized with another clock an then flown around the Earth in a jet airliner (these speeds are of course far less than the speed of light).  When the clocks were reunited, the clock that took the journey was found to be slower than the one left on earth by a margin nearly precisely that predicted by eq. 4!
     In order to put eq. 4 in the form to be used later in this article, I will do one more manipulation.  If we allow Dt and Dt to become very small we can arrange eq. 4 into the grandest of all forms . . . a differential equation!

The Twin Paradox

    Let's say that we have two twins.  For a convenient naming convention let's refer to them as Dr. Mills and Dave Arnold.  For an experiment one of these twins will go for a trip at high speed to a distant planet outside of  our solar system and return while the other one will remain on Earth teaching math 376.  Due to seniority, Dr, Mills nominates himself as the traveler.  When they reunite will one of them have aged less or will there be no difference?  Based on the analysis that we outlined above, one might be quick to say that Dr. Mills will have aged less.  However, if you are the ceaseless thinking type, you might think about how Dave's reference frame looks to Dr. Mills during the trip.  If Dr. Mills convinced himself that he was standing still, then he would conclude that Dave was making the trip in the opposite direction but at the same speed.  He would then conclude that Dave will have aged less when they reunite.  Therefore we have a paradox.  They can't both be younger.
    Well, the true answer lies in the fact that Dr. Mills will have to change reference frames in order to turn around and make their second rendezvous.  This change of reference frames makes their roles in the situation asymmetrical.  Dr. Mills will in fact age less than Dave during the trip.  The details of this asymmetry require an analysis of accelerated reference frames as put forth by general relativity.  That analysis is far, far beyond the scope of this article.  The pedagogical problem that arises due to this complexity can lead new comers to relativity to mistakingly make a direct correlation between differential aging and acceleration.  This is the misconception that Gruber and Price attempt to alleviate in their article "Zero Time Dilation in an Accelerating Rocket."

The Misconception Alleviated

    The acceleration misconception has been notorious in the world of relativistic physics education.  Gruber and Price make the logical conclusion that the best way to deal with the dilemma is to design a situation where there is either acceleration or differential aging but not both.  They begin by citing a colleague's previous attempt at this crux. His scenario entails both twins having identical histories of acceleration yet differential aging still occurs.  The authors thought this to be a noble effort but they still saw the allure for students to place the necessity of acceleration on the differential aging.  To dispel the myth they designed a scenario where the traveler is constantly accelerating relative to the earthbound observer yet no differential aging occurs.
    Let's consider a rocket that is demonstrating pseudo-periodic motion (pseudo because of relativistic effects):
where x is the position and t is the time both measured by the earthbound observer.  Now if  we differentiate eq. 6 to arrive at velocity vs. time, substitute the results into eq. 5 for u and finally separate the variables we arrive at this:
Now we can define the duration of the observation to begin at t = 0 and lasting until Dt = n p/w. If we use this for our limits of integration and attempt to solve the ODE we find ourselves here:
If we substitute q for wt and Dt/np for 1/w then divide both sides by Dt and finally integrate over half of the original limits and multiply by two (you will need some paper and a pencil for this one) we can arrive at the following manipulation:
     Because this integral is a wee bit on the difficult side, we can settle for a numerical approximation.
The important thing to glean from the plot is that differential aging can be shown as a function of Vmax only.  Any reference to acceleration would have to include some configuration of Vmaxw.   Indeed Gruber and Price provide a numerical example in their text that demonstrates this point even further.  They provide an example where the time dilation actually contracts when the acceleration increases.  The implication here is that one could arrange a situation with a given time dilation and a given acceleration without considering their correlation.  Pretty snazzy I'd say!

Conclusion

    It is without question that the postulates that Einstein laid out back in 1905 have seriously changed the way we view the world and our perception of it.  The one thing that we all thought we could count on . . . time, is as ambiguous as social interaction.  OK, it's a little bit more quantified than that.  My only concern is that the massive social clamor for the preservation of youth could abuse these great postulates.  Just think what Mary Kay Cosmetics would do if they knew that the secret to increased youth is high speed travel.  This could spawn a whole slew of new industry.  In the future, when someone says that they are going out for a make over, they might not return for 20 years!  Please, I urge of you the reader to get out your pen right now and inform your congressional representative to begin drafting legislation now to nip this ticking bomb in the bud.  Thank you for your continued support.


Works Cited