Web Design by US!!!    all rights reserved 2002

The Twin Paradox using Simultaneous Events >>>here

The Infamous Twin Paradox

Since SR dictates that two different observers each have equal right to view an event with respect to their frames of reference, we come to many not-so-apparent paradoxes. With a little patience, most of the paradoxes can be shown to have logical answers that agree with both the predicted SR outcome and the observed outcome. Let's look the most famous of these paradoxes - The Twin Paradox.

Suppose two twins, John and Hunter, share the same reference frame with each other on the earth. John is sitting in a spaceship and Hunter is standing on the ground. The twins each have identical watches that they now synchronize. After synchronizing, John blasts off and speeds away at 60% the speed of light. As John travels away, both twins have the right to view the other as experiencing the relativistic effects (length contraction and time dilation). For the sake of simplicity, we will assume that they have an accurate method with which to measure these effects. If John never returns, there will never be an answer to the question of who actually experienced the effects. But what happens if John does turn around and return to the earth? Both would agree that John aged more slowly than Hunter did, thus time for John was slower than it was for Hunter. To prove this, all they have to do is look at their watches. John's watch will show that it took less time for him to go and return than Hunter's watch shows. As Hunter stood there waiting, time passed faster for him than it did for John. Why is this the case if both were traveling at 60% the speed of light with respect to one another?

The first point to understand is that acceleration in SR is a little tricky (it's actually handled better in Einstein's Theory of General Relativity - GR). I don't mean to say that SR can't handle acceleration, because it can. In SR, you can describe the acceleration in terms of locally "co-moving" inertial frames. This allows SR to view all motion to be uniform, meaning constant velocity (non-accelerating). The second point is that SR is a "special" theory. By this, I mean that it is applicable in situations where there is no gravity, hence where space-time is flat. In GR, Einstein unifies acceleration and gravity so actually my previous statement is redundant. Anyway, the lack of gravity in SR is why it is called "Special Relativity". Now, back to the paradox… While both did view the other as shrinking and slowing down, the person that actually underwent the acceleration to reach the high speed is the one that aged less. If you dig deeper into the world of SR, you will realize that it's not really the acceleration that is important; it's the change of frame. Until John and Hunter returned to a frame of reference where their relative motion was zero (where they are standing beside each other) they would always disagree with what the other said he saw. As strange as this seems, there really isn't a conflict - both did observe that the other was experiencing the relativistic effects. One technique that is used to show the dynamics of the Twin Paradox is a concept is called the Relativistic Doppler Effect.

The Doppler Effect basically says that there is an observed frequency shift in electromagnetic waves due to motion. The direction of the shift is dependent on whether the relative motion is traveling towards you or away from you (or vice versa). Also, the amplitude of the shift is dependent on the speed of the source (or the speed of the receiver). A good place to start in understanding the Doppler effect would be to first look at sound waves. There is a Doppler Shift associated with sound waves that you should recognize easily. When a sound source approaches you, the frequency of the sound increases and likewise, when the sound source moves away from you, the frequency of the sound decreases. Think about an approaching train blowing its whistle. As the train approaches, you hear the whistle tone as a high note. When the train passes you, you can hear the whistle tone change to a lower note. Another example occurs when cars race around a racetrack. You can hear a definite shift in the sound of the car as it passes where you are standing. One last example is the change in tone you hear when a police car passes you with its siren on. I'm sure that at some point in our lives, all of us have imitated the sound of a passing car or passing police car; we imitated the Doppler Shift. This Doppler shift also affects light (electromagnetic radiation) in the same manner with one critical exception; the shift will not allow you to determine if the light source is approaching you or if you are approaching the source and vice versa for moving away. This being said, let's look at the figure below.  

In the top part of the figure you can see a stationary light source is emitting light in all directions. In the second part, you can see that source "S" is moving to the right and the light waves are shifted (they look as though they are being compressed in the front and dragged in the rear). If you approach the light source or the light source approaches you, the frequency of the light will appear to increase (notice that the waves in the front are closer together than in the rear). The opposite is true for a light source that is moving away from you or that you are moving away from. The importance of the frequency change is that if the frequency increases, then the time it takes for one complete cycle (oscillation) is less. Likewise, if the frequency decreases, the time it takes for one complete cycle is more.

Now let's apply this information to the Twin Paradox. Recall that John sped away from Hunter at 60% the speed of light. I picked this speed, because the corresponding relativistic Doppler shift ratio is "2 times" for an approaching source and "1/2" for a source that is moving away. This means that if the source is approaching you, the frequency will appear doubled (time is then halved) and if the source is moving away from you, the frequency will appear halved (time is then doubled). (similarly I could have used any speed for the paradox; for example, 80% the speed of light would have led to a Doppler shift of "3" and "1/3" for approaching and moving away respectively). Remember, the direction of the shift is dependent on the direction of the source, while the amplitude of the shift increases with the speed of the source.

Let's take another trip with the twins, but this time John will travel 12 hours away and 12 hours back, as measured by his clock. Every hour he will send a radio signal to Hunter telling him the hour. A radio signal is just another form of electromagnetic radiation; therefore, it also travels at the speed of light. What do we get as John travels away from Hunter? When John's clock reads "1 hour" he sends the first signal. Because he is moving away from Hunter at 60% of the speed of light, the relativistic Doppler Effect causes Hunter to observe John's transmission to be ½ the source value. From our discussion above, ½ the frequency means the time it takes is twice as long, therefore, Hunter receives the John's "1 hour" signal when his clock reads "2 hours". When John sends his "2 hour" signal, Hunter receives it at hour 4 for him. So you can see the relationship developing. For every 1-hour signal by John's watch, the elapsed time for Hunter is 2 hours. When John's clock reads "12 hours" he has sent 12 signals. Hunter, on the other hand, has received 12 signals, but they were all 2 hours apart…thus 24 hours have passed for Hunter. Now John turns around and comes back sending signals every hour in the same manner as before. Since he is approaching Hunter, the Doppler shift now causes Hunter to observe the frequency to be twice the source value. Twice the frequency is the same as ½ the time, so Hunter receives John's "1 hour" signals at 30min intervals. When the 12-hour return trip is over, John has sent 12 signals. Hunter has received 12 signals, but they were separated by 30 minutes, thus 6 hours have pasted for Hunter. If we now total up the elapsed time for both twins, we see that 24 hours (12 + 12) have elapsed for John, but 30 hours (24 + 6) have elapsed for Hunter. Thus, Hunter is now older than his identical twin, John. If John had traveled farther and faster, the time dilation would have been even greater. Look at the twins again, but this time let John travel 84 hours out and 84 hours back (by his clock) at 80% the speed of light. The total trip for John will be 168 hours, and the total time elapsed for Hunter will be 280 hours; John was gone for 1 week by his clock, but Hunter waited for 1 week 4 days and 16 hours by his clock. Remember that Hunter will receive John's outgoing signals at half the frequency which means twice the time. Therefore, Hunter receives John's 84 hourly signals every 3 hours for a total of 252 hours (3 is the Relativistic Doppler shift for 80% the speed of light). Likewise, Hunter receives John's return trip 84 hourly signals every 20 minutes for a total of 28 hours (20 minutes is the 1/3 Relativistic Doppler shift for the return). Now you know the total round trip from Hunter's perspective, 252 + 28 = 280 hours or 1 week 4 days and 16 hours. John, on the other hand, traveled 84 hours out and 84 hours back for a total of 168 hours or 1 week.

Now let's look at the twins again, but this time Hunter will send a signal every hour by his clock. What will John see? When Hunter sees the outgoing leg of John's trip end, his clock reads 15 hours and he has sent 15 signals. John, however, will say that he received 6 signals separated by 2-hours (relativistic Doppler shift) for a total of 12 hours. What happened to the other 9 signals? They are still in transit to John. Therefore, when John changes to his return leg, he will now encounter the missing 9 signals plus the 15 signals Hunter sent for the 15 hours his clock recorded for the return leg. So John receives 24 signals that are 30 minutes apart for a total of 12 hours. Like the previous example, these 24 signals have all been doppler shifted to a higher frequency because John is now approaching them. Now if we total the whole trip, Hunter sent one signal every hour for thirty hours, but John received 6 signals that were 2 hours apart and 24 signals that were 30 minutes apart. Hunter sent 30 signals in 30 hours; John received 30 signals in 24 hours. The result is the same as before, but the twins do not agree on when the first leg ended and the last leg began. So from this we can conclude that the change of frame for John (from outgoing to return) is what distinguishes him from Hunter. For Hunter, nothing changes at all. Anyway you look at it; he waits 30 hours without a change. John, however, does change. He changes from a frame in which he is moving away to a frame in which he is moving back. It is this change that breaks the symmetry between John and Hunter, thus removing the paradox as well.

Before going on to the next concept, I want to make sure that a couple things about SR and the speed of light are properly understood. First, SR predicts doom for anything with mass approaching the speed of light from a slower speed due to length contraction and time dilation, but it does allow for speeds greater than the speed of light. Consider the speed of light as a barrier. SR allows for existence on both sides of the barrier, but neither side can cross over to the other. As of yet, nothing has been discovered on the faster-than-light side, and all that we have are theories on particles (tachyons) that may have the ability to exist there. Maybe one day someone will discover their existence.

Secondly, velocities from a different frame of reference can not be summed. For example, if I run 5 miles/hour and at the same time, throw a rock 5 miles/hour, the only reason you (standing still) can say the rock is travelling 10 miles/hour is because the speed is so small with respect to the speed of light. We use the Lorentz Transformations to transform from one frame to another using the relative velocity of the frames. These transformations tell us mathematically that while at slow speeds the error in straight addition is much too small for us to detect, at very fast speeds, the error would become quite large. So classical mechanics, which teaches us to sum these velocities, is actually incorrect. We can do it, but it's a case of getting the right answer for the wrong reason.