35.7 - Exploring and deriving time dilation
We will use the experiment portrayed on the right to derive the equation for time
dilation. Following the steps of the derivation may also help you to understand how
Einstein’s postulates lead inescapably to the conclusion that time dilation occurs.
The experiment uses a light-based clock mounted on a high-speed skateboard.
Ordinary clocks use periodic mechanical or electric processes, such as the oscillation of
a pendulum or a timing circuit, to establish a unit of time. A light clock uses the amount
of time it takes a pulse of light to travel a particular distance. The light clock is
convenient to use in this scenario, but any clock would record the same result.
A light pulse is emitted from the base of the clock. The pulse reflects off the top of the
clock and returns to the bottom. The clock measures time by using the relationship of
time to distance and speed. The elapsed time for the up-and-down journey equals the
distance the light pulse travels, divided by the speed of light.
In our experiment, let’s consider one “tick” of the clock. The light rises from the bottom
of the clock, reflects off the top, and returns to the bottom. The professor, who is also on
the skateboard, sees the light pulse moving straight up and down, and he measures the
distance it travels as being twice the height of the clock.
Now, let’s consider what another observer sees. Katherine is standing on the ground
and watches the professor and the clock pass by. She also watches the light pulse as it
moves. However, she measures a different value for the distance traveled by the light
pulse. She sees it not only moving up and down, but also forward. She measures the
light pulse moving through the distance indicated by the two lines labeled s in Concept
3.
If this seems confusing, just think of a friend riding a train, throwing a ball straight up
and down. From your friend’s perspective, the ball just travels up and down. From a
vantage point on the ground outside, you would see the ball moving horizontally at the
same time it is moving up and down.
The clock uses light, which according to Einstein’s second postulate has a constant
speed independent of any frame of reference. Einstein’s first postulate states that the
laws of physics are the same in any inertial reference frame. That is, in our scenario,
both observers can use the same equation: Time equals the distance traveled by the
pulse, divided by the speed of light.
Having explained the experiment, we will now analyze it algebraically, calculating the
distance traveled by the clock and the time interval required for one tick of the clock in
each frame of reference. This will enable us to derive the equation for time dilation
shown in Equation 1.
Variables
We will use the triangle shown to the right in Equation 1 to relate displacement, speed
and a time interval. The triangle reflects half of one tick of the light pulse, as it moves
from the bottom to the top. The clock moves L horizontally during half of one tick, and
Katherine observes the light moving a distance s.
Light clock
Light flash bounces up and down
Professor’s reference frame
Light moves strictly up and down
Katherine’s reference frame
Light moves up, down, horizontally
Time measurements differ
Time = distance / speed of light
Katherine, professor measure different
distances
Speed of light is constant
Katherine, professor measure different
timemeasured by
Katherinemeasured by
professor
clock’s horizontal displacement, half tick L 0 mdistance light pulse moves, half tick sh
clock’s speed v 0 m/s
height of clock hh
elapsed time t t 0
speed of light c = 3.00×10 (^8) m/s