1.3 Time Dilation and Length Contraction Effects (Special Relativity)
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
1.3 Time Dilation and Length Contraction Effects (Special Relativity)
Now, I give an example of how time dilation can help explain a
peculiarity that arises from the above concept. Again we consider a case
where I am on a train and you are outside the train, but let's give the
train a speed of 0.6 c with respect to you. (Note that c is generally used
to denote the speed of light which is 300,000,000 meters per second. We can
also write this as 3E8 m/s where "3E8" means 3 times 10 to the eighth). Now
I (on the train) shine a small burst or pulse of light so that (to me) the
light goes straight up, hits a mirror at the top of the train, and bounces
back to the floor of the train where some instrument detects it. Now, in
your point of view (outside the train), that pulse of light does not travel
straight up and straight down, but makes an up-side-down "V" shape because
of the motion of the train. This is not just some "illusion", but rather it
is truly the way the light travels RELATIVE TO YOU, and thus this is truly
the way the situation must be considered in your frame of reference. Below
is a diagram of what occurs in your frame:
Diagram 1-1
/|\
/ | \
/ | \
light pulse going up->/ | \<-light pulse on return trip
/ | \
/ | \
/ | \
/ | \
---------|---------->train's motion (v = 0.6 c)
Let's say that the trip up takes 10 seconds in your frame of reference.
The distance the train travels during that time is given by its velocity
(0.6 c) multiplied by that time of 10 seconds:
(Eq 1:1)
(0.6 * 3E8 m/s) * 10 s = 18E8 m
The distance that the light pulse travels on the way up (the slanted line to
the left) must be given by its speed with respect to you (which MUST be c
given our previous discussion) multiplied by the time of 10 seconds:
(Eq 1:2)
3E8 m/s * 10s = 30E8 m
Since the left side of the above figure is a right triangle, and we know the
length of its hypotenuse (the path of the light pulse) and one of its sides
(the distance the train traveled), we can now solve for the height of the
train using the Pythagorean theorem. That theorem states that for a right
triangle the length of the hypotenuse squared is equal to the length of one
of the sides squared plus the length of the other side squared. We can thus
write the following:
(Eq 1:3)
Height^2 + (18E8 m)^2 = (30E8 m)^2
so
Height = [(30E8 m)^2 - (18E8 m)^2]^0.5 = 24E8 m
(It is a tall train because we said that it took the light 10 seconds to
reach the top, but this IS just a thought experiment.) Now we consider my
frame of reference (on the train). In my frame, the light is truly traveling
straight up and straight back down to me. This is truly the way the light
travels in my frame of reference, and so that's the way we must analyze the
situation relative to me. Again, according to our previous discussion, the
light MUST travel at 3E8 m/s as measured by me as well. Further the height
of the train doesn't change because relativity doesn't affect lengths
perpendicular to the direction of motion. Therefore, we can calculate how
long it takes for the light to reach the top of the train in my frame of
reference. That is given by the distance (the height of the train) divided
by the speed of the light pulse (c):
(Eq 1:4)
24E8 m / 3E8 (m/s) = 8 seconds,
and there you have it. To you the event takes 10 seconds, while according to
me it must take only 8 seconds. We measure time in different ways.
You see, to you the distance the light travels is longer than the
height of the train (see the diagram). So, the only way I (on the train)
could say that the light traveled the height of the train while you say that
the SAME light travels a longer distance is if we either (1) have different
ideas for the speed of the light because we are in different frames of
reference, or (2) we have different ideas for the time it takes the light to
travel because we are in different frames of reference. Now, in Newton's
days, they would believe that the former were true. The light would be no
different from, say, a ball, and observers in different frames of reference
can observer different speeds for a ball (remember our first "train" example
in this introduction). However, with the principles of Einstein's
relativity, we find that the speed of light is unlike other speeds in that
it must always be the same regardless of your frame of reference. Thus, the
second explanation must be the case, and in your frame of reference, my
clock (on the fast moving train) is going slower than yours.
As I mentioned in the last part of the previous section, length
contraction is another consequence of relativity. Consider the same two
travelers in our previous example, and let each of them hold a meter stick
horizontally (so that the length of the stick is oriented in the direction
of motion of the train). To the outside observer (you), the meter stick of
the traveler on the train (me) will look as if it is shorter than a meter.
One can actually derive this given the time dilation effect (which we have
already derived), but I wont go through that explanation for the sake of
time.
Now, DON'T BE FOOLED! One of the first concepts which can get into the
mind of a newcomer to relativity involves a statement like, "if you are
moving, your clock slows down." However, the question of which clock is
REALLY running slowly (yours or mine) has NO absolute answer! It is
important to remember that all inertial motion is relative. That is, there
is no such thing as absolute inertial motion. You cannot say that it is the
train that is absolutely moving and that you are the one who is actually
sitting still.
Have you ever had the experience of sitting in a car, noticing that you
seemed to be moving backwards, and then realizing that it was the car beside
which was "actually" moving forward. Well, the only reason you say that
"actually" the other car was moving forward is because you are considering
the ground to be stationary, and it was the other car who was moving with
respect to the ground rather than your car. Before you looked at the ground
(or surrounding scenery) you had no way of knowing which of you was "really"
moving. Now, if you did this in space (with space ships instead of cars),
and there were no other objects around to reference to, and neither space
ship was accelerating (they were moving at a constant speed with respect to
one another) then what would be the difference in saying that your space
ship was the one that was moving or saying that it was the other space ship
that was moving? As long as neither of you is undergoing an acceleration
(which would mean you were not in an inertial frame of reference) there is
no absolute answer to the question of which one of you is moving and which
of you is sitting still. You are moving with respect to him, but then again,
he is moving with respect to you. All motion is relative, and all inertial
frames are equivalent.
So what does that mean for us in this "train" example. Well, from my
point of view on the train, I am the one who is sitting still, while you zip
past me at 0.6 c. Since I can apply the concepts of relativity just as you
can (that's the postulate of relativity--all physical laws are the same for
all inertial observers), and in my frame of reference you are the one who is
in motion, that means that I will think that it is YOUR clock that is
running slowly and that YOUR meter sticks are length contracted.
So, there is NO absolute answer to the question of which of our clocks
is REALLY running slower than the other and which of our meter sticks is
REALLY length contracted smaller than the other. The only way to answer this
question is relative to whose frame of reference you are considering. In my
frame of reference your clock is running slower than mine, but in your frame
of reference my clock is running slower than yours. This lends itself over
to what seem to be paradoxes such as "the twin paradox" (doesn't it seem
like a paradox that we each believe that the other person's clock is running
slower than our own?). Understanding these paradoxes can be a key to really
grasping some major concepts of special relativity. The explanation of these
paradoxes will be given for the interested reader in Part II of this FAQ.
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