Professor Dave here, let's learn about time dilation.
We went over
the postulates of special relativity, and
we saw that in order for the second one
to be true, we have to rethink how time
works. If the speed of light is the same
in all inertial reference frames, then
someone standing on earth and someone
moving in a very fast spaceship must be
experiencing the passage of time at
different rates. This concept is called
time dilation, and it tells us that the
faster you move relative to the speed of
light, the slower time will pass for you.
This concept is best demonstrated
through a thought experiment. Let's say
that there is an observer on earth and
another in a spaceship moving with
constant velocity relative to the earth.
On the spaceship there is a device that
emits a light pulse, which hits a mirror
and bounces back to a detector that sits
right by the device. Each observer has a
clock, and we pretend that the observer
on earth can see what's going on in the
spaceship. For the astronaut it's easy to
tell how long it takes for the light to
make the trip. If speed is distance over
time, then time is distance over speed, so
it's just twice the distance between the
device and the mirror to get there and
back divided by the speed of light, or
2D over c. Let's call this time interval
delta T zero. For the person on earth
it's a little different, because the
spaceship is moving relative to earth
and therefore the light pulse has a
horizontal component of motion in
addition to the vertical one. The light
follows this diagonal path up to the
mirror and back, which is much longer
than the other path. If this horizontal
distance is L then we use the
Pythagorean theorem to get s, which is
root D squared plus L squared, so twice
this term will be the distance traveled
by the light pulse according to the
observer on earth. But the speed of light
is always the same, so if the light
travels a greater
distance at the same speed, the observer
on earth must register a greater time
interval, delta t, for this event to occur
than the person on the spaceship. This is
why this phenomenon is called time
dilation, because to dilate is to expand
and the time interval delta t
has expanded relative to delta t 0. This
means that the clock on the spaceship is
running more slowly than the one on
earth, as this is the only way for both
observers to get the same result for the
speed of light. We can calculate exactly
how much slower in fact. Let's take this
expression and realize that 2s, the
distance travelled by the light pulse
according to the person on earth, is
equal to c delta t, since distance equals
speed times time. That also means that
the distance L, being half the distance
traveled by the ship, is equal to the
speed of the ship v times the time
interval delta t over 2, since L is only
half the total horizontal distance. So L
equals v delta t over 2. If we do a bunch
of algebra, we arrive at this expression
relating delta t zero, the time interval for
an observer at rest with respect to the
events in question, also called the
proper time interval, and delta t, the
dilated time interval for an observer in motion
with respect to the events. Remember that
although the spaceship is clearly in
motion, it is the observer on earth that
is in motion relative to the events in
question, as the light is inside the
spaceship. This equation allows us to
plug in some velocity for the ship and
calculate the precise time dilation that
will result, or calculate the proper time
given the dilated if we rearrange
slightly. Let's try an example. Say an
astronaut is in a ship moving at 80
percent light speed. If they move at this
constant speed for what they measure as
one year, how long will that be for
someone on earth? Well let's take the
equation and put one year for the proper
time interval. Then we can plug in 0.8c
for the velocity, since that is equal to
80 percent the speed of light. We square that
and find that the c squared terms cancel.
We subtract from one and take the square
root, so dividing the proper time
interval of one year by 0.6, we get
1.67 years as the dilated
time interval. This concept, as abstract
as it seems, actually has practical
applications. GPS satellites in orbit
around the Earth are moving so fast that
there are relativistic effects. Because
of this, the clocks on board have to be
programmed with correction factors so
that they report time the same way as
clocks on earth, otherwise they would be
out of sync and could not perform their
intended function. We use the time
dilation equation to do this to an
incredible level of precision. Of course,
satellites move very fast, going around
the earth in a couple hours, but even for
an object moving more slowly, like a
plane, time dilation still occurs and has
been measured with ultra precise clocks.
If one clock stays on the ground and
another one goes in a plane that flies
around the earth, these two clocks will
show a discrepancy after the long flight
of a few billionths of a second. Not a
big difference, but we can still measure
it and see that it precisely matches the
prediction made by special relativity.
In the time dilation equation, as v gets
very small relative to the speed of
light, this term approaches 0. 1 minus 0
is 1, and root 1 is 1, so for objects
moving nowhere near the speed of light
the dilated time interval is essentially
identical to the proper time interval.
That's why we don't notice time dilation
on earth as we walk and run around, but
for something like space travel it
becomes significant. The best example of
this is something called the twin
paradox. Let's say there is a set of
twins and one goes on a journey through
space while the other remains on earth.
The traveler gets in a super fast ship
and travels for 10 years, then turns
around and comes back. Time dilation
doesn't only affect mechanical clocks
but biological clocks as well, so upon
returning to earth, they are no longer the same age.
The supposed paradox arises when we
consider inertial reference frames.
To the twin on earth, the other one sped
away and was moving very fast for 20
years, so the traveler should have aged
less. But to the one in the ship, it is
the earth that sped away and came back
20 years later, so the twin on earth
should have aged less. But when they
reunite, one must be younger than the
other, so which one is it? The paradox is
resolved when we realize that the ship
was the one doing all the accelerating
and decelerating, making it a non
inertial reference frame, where
relativity does not apply. So we must
treat the earth as the inertial
reference frame. This means that the twin
who left on the ship will indeed return home
younger than the one who stayed.
By precisely how much will depend on how
fast the ship was moving. As strange as
this all sounds, time dilation is only
one of the many ramifications of special
relativity, which leaves us much more to
discuss, but first let's check comprehension.
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