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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 ˝ 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.
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