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If I walk from the back to the front of a train at 3 m.p.h., and the train is traveling at 60 m.p.h., then common sense tells me that my speed relative to the ground is 63 m.p.h. As we have seen, this obvious truth, the simple addition of velocities, follows from the Galilean transformations. Unfortunately, it can't be quite right for high speeds. We know that for a flash of light going from the back of the train to the front, the speed of the light relative to the ground is exactly the same as its speed relative to the train, not 60 m.p.h. different. Hence it is necessary to do a careful analysis of a fairly speedy person moving from the back of the train to the front as viewed from the ground, to see how velocities really add.
We consider our standard train of
length L moving down the track at steady speed v, and equipped with synchronized clocks at
the back and the front. The walker sets off from the back of the
train when that clock reads zero. Assuming a steady walking speed
of u meters per second (relative to the train, of
course), the walker will see the front clock to read L/u seconds on arrival there.
How does this look from the ground? Let's assume that at the instant the walker began to walk from the clock at the back of the train, the back of the train was passing the ground observer's clock, and both these clocks (one on the train and one on the ground) read zero. The ground observer sees the walker reach the clock at the front of the train at the instant that clock reads L/u (this is in agreement with what is observed on the train - two simultaneous events at the same place are simultaneous to all observers), but at this same instant, the ground observer says the train's back clock, where the walker began, reads L/u + Lv/c2. (This follows from our previously established result that two clocks synchronized in one frame, in which they are L apart, will be out of synchronization in a frame in which they are moving at v along the line joining them by a time Lv/c2.)
Now, how much time elapses as measured by the ground observer's clock during the walk? At the instant the walk began, the ground observer saw the clock at the back of the train (which was right next to him) to read zero. At the instant the walk ended, the ground observer would say that clock read L/u + Lv/c2, from the paragraph above. But the ground observer would see that clock to be running slow, by the usual time dilation factor - so he would measure the time of the walk on his own clock to be:
How far does the walker move as viewed from the ground? In the time tW, the train travels a distance vtW, so the walker moves this distance plus the length of the train. Remember that the train is contracted as viewed from the ground! It follows that the distance covered relative to the ground during the walk is:
The walker's speed relative to the ground is simply dW/tW, easily found from the above expressions:
This is the appropriate
formula for adding velocities. Note that it gives the
correct answer, u
+ v, in the low
velocity limit, and also if u or v equals c, the sum of the velocities is c.
Actually, the first test of the addition of velocities formula was carried out in the 1850s! Two French physicists, Fizeau and Foucault, measured the speed of light in water, and found it to be c/n, where n is the refractive index of water, about 1.33. (This was the result predicted by the wave theory of light.)
They then measured the speed of light (relative to the ground) in moving water, by sending light down a long pipe with water flowing through it at speed v. They discovered that the speed relative to the ground was not just v + c/n, but had an extra term, v + c/n - v/n2. Their (incorrect) explanation was that the light was a complicated combination of waves in the water and waves in the aether, and the moving water was only partially dragging the aether along with it, so the light didn't get the full speed v of the water added to its original speed c/n.
The true explanation of the extra term is much simpler - velocities don't simply add. To add the velocity v to the velocity c/n, we must use the addition of velocities formula above, which gives the light velocity relative to the ground to be:
(v + c/n)/(1 + v/nc)
Now, v is much smaller than c or c/n, so 1/(1 + v/nc) can be written as (1 - v/nc), giving:
(v + c/n)(1 - v/nc)
Multiplying this out gives v + c/n - v/n2 -v/n×v/c, and the last term is smaller than v by a factor v/c, so is clearly negligible.
Einstein, in 1911, predicted that: "If we placed a living organism in a box ... one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations. For the moving organism the lengthy time of the journey was a mere instant, provided the motion took place with approximately the speed of light."
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 a fig 1 below.
Fig1
In the first part of fig 1 you can see a stationary light source "S" 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 ˝ the source value. From our discussion above, ˝ 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 ˝ 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.
simultaneity (or lack thereof) is a terrific tool for understanding many of the paradoxes associated with SR. And, if I am to be thorough, simultaneity must be considered for all SR events between separate frames of reference. Let's re-visit the twin paradox (John travels out 12 hours at 60% the speed of light and returns at the same speed). Basically, there are three frames of reference to consider. First, the twins are on the earth with no relative velocity between them. Second, John embarks on the outgoing leg of his trip. Thirdly, John (after instantaneously turning around) embarks on his return leg of his trip. I am using the same example as before, except I am using numbers from the Lorentz Transforms as opposed to the Relativistic Doppler Shift to explain the observed phenomena.
1st frame:
Hunter and John each agree on everything they observe. This
should be easy to understand since there is no relative velocity
between the two twins. They are in motion together.
2nd frame:
John travels out 12 hours by his clock. With the two postulates
in mind, we realize that Hunter observes time dilation for John's
outgoing trip. Thus, if John records 12 hours, Hunter will record
15 hours. Remember that at 60% the speed of light, the time
dilation will be 80%. Therefore, if John records his time to be
12 hours, this is 80% of what Hunter records - 15 hours. But what
does John observe for Hunter's time? He observes the time
dilation as effecting Hunter; therefore, he measures his trip to
be 12 hours, but he observes 9.6 hours (80% of his clock's time)
for Hunter's time.
2nd frame totals:
Hunter measures his time to be 15 hours, but John's time to be 12
hours. John measures his time to be 12 hours, but Hunter's time
to be 9.6 hours.
Obviously, the event, which is the end of the outgoing trip, is not simultaneous. John thinks Hunter's time is 9.6 hours but Hunter thinks his time is 15 hours. On top of that, they both think that John's time is 12 hours, which doesn't agree with either of the first two times.
3rd frame:
From Hunter's perspective, nothing new has happened. He remained
in his initial frame of reference and John returned at the same
velocity he left with. Therefore, Hunter measured the return trip
to take 15 hours for his frame (same as the outgoing trip) and
observes the trip to take 12 hours for John. From John's
perspective, he encountered a major change. He actually changed
frames from one of traveling out to one of traveling back. Now,
at the start of the return trip, when John looks at his clocks,
he observes his clock to read 12 hours and Hunter's clock to read
20.4 hours. Think about this. John now shows that Hunter's clock
has jumped ahead from 9.6 hours to 20.4 hours. How can this
be???? When John changed from the 2nd frame to the 3rd frame, the
established symmetry between Hunter and John was broken. Thus,
each views their own time as having no change. And since John was
the one that actually changed frames, he showed more elapsed time
for Hunter. From here on out, it is business as usual. The return
trip is clocked at 12 hours by John, but he observes 9.6 hours
for Hunter. Again, let's clean this up…
3rd frame totals:
Hunter measures his time to be 15 hours, but he measures John's
time to be 12 hours. John measures his time to be 12 hours, but
he measures Hunter's time to be 9.6 hours. Remember, this 9.6 is
only for the return trip after the frame change.
Trip totals:
Hunter measured his time to be 15 hours for the outgoing trip +
15 hours for the return trip…30 hours.
Hunter observed John's time to be 12 hours outgoing + 12 hours
return …24 hours.
John measured his time to be 12 hours outgoing + 12 hours return…24
hours.
John observed Hunter's time to be 20.4 hours (after outgoing trip
and frame change) + 9.6 hours for the return trip…20.4 + 9.6 =
30 hours.
Can you find any events in which both John and Hunter agree on the time for both themselves and the other? No, you can't. The lack of simultaneity is the key to the paradox. Both twins are measuring and observing. Unfortunately, they are not measuring and observing the same events. It is impossible for them to consider something like the end of the first leg as simultaneous when they each view it occurring at different times for Hunter. It's interesting to note that the results are the same as the Relativistic Doppler shift results. Is there a pattern here? SR allows for various methods to be employed to resolve the problems. For this case, use of space-time diagrams (there's those words again) would clearly show every point that we have talked about. I have merely used the Lorentz transforms in combination with the Relativistic Doppler effect.
Many people have trouble with the twin paradox because of the way in which the frame change is handled. In this case, the jump on John's clock for Hunter after the frame change (9.6 to 20.4 hours) is the problem. There really is no problem here. If you want to integrate the acceleration to use various inertial frames during the turn around, it can be done (with the same results). Another common approach is to imagine someone else in space that passes John just when he reaches the point of his turnaround. This person is heading towards Hunter at the same speed that John was travelling, so there is no need to consider John any further. The key fact is that if we then went back in the substitute's frame and looked at his clock for Hunter, it would show that some amount of time had already been recorded when the substitute began his trip towards Hunter. How far back should we go? Since John traveled out 12 hours on the outgoing trip, we should go back 12 hours in the substitute's frame. At this starting point for the substitute, his clock for Hunter would read 10.8 hours. This is extremely important. It clearly shows that both twins or the twin and the substitute observe the other as having slower times. The big shift occurs when the frame of reference is changed. This means that both observe the other to have a slower time during the actual outgoing and return trips, but there is a shift during the frame change that more than makes up for John's account of Hunter's slowly running clock. After the frame change, the damage has been done. John will still observe Hunter's clock to run slow, but it will never slow down enough to compensate for the 10.8 hours that were perceived during the frame change. Is this time jump a physical occurrence? No. The time jump occurs because when John changes frames, he is no longer using the same event as a reference. When John made his turnaround, the event in Hunter's frame that John thought was simultaneous with his turnaround changed. John's frame change caused this confusion because his new frame uses a different time for the event in Hunter's frame. More clearly, the turnaround event in Hunter's frame has a different time value for the outgoing leg and the return leg, as perceived by John. Keep in mind that in the above references to Hunter's frame, I'm really talking about what John thinks Hunter's frame time would be. This time difference is only apparent to John because it is his frame change that causes the discrepancy. In Hunter's frame, nothing changes for Hunter when John changes frames. Here again, by realizing that the two events are not simultaneous, the paradox is resolved. The point I am trying to emphasis is that there are a variety of ways to handle the paradox. All of the methods yield the same result, but if you actually consider the simultaneity of the situation, then the how's and why's become more clear.
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