35.8 - Interactive problem: Experiment with the light clock
In this simulation, you experiment with a light clock. The professor rolls across a
basketball court with a light clock on his skateboard, while Katherine watches from
the side. You will view the light clock in operation from the professor’s reference
frame, and from the student’s reference frame. You are asked to calculate the time
measured by each observer for the professor to cross the basketball court.
The simulation launches in the professor’s reference frame, where the light clock is
stationary. Press GO and watch the basketball court and the background pass by. A
counter will record the number of light clock cycles, the number of round trip
journeys made by the light pulse. The clock is 3.0 m tall.
Then press the tab labeled “Student’s reference frame” and press GO again. You
will see the same series of events from Katherine’s reference frame. The simulation
displays the path of the light and indicates some key distances in a fashion similar
to the derivation of the prior section.
If you asked the professor how long it took him to cross the basketball court, what
would he say? What if you asked Katherine the same question? Check your
answers by entering them in the simulation.
Now a thought question for you, foreshadowing a future topic. Katherine measures the length of this court as 24.0 m (alas, she does not have
an NBA standard court). You can use this length and the time she measures to determine the professor’s speed. This is the same speed the
professor would measure of the ground moving beneath him.
Let’s say the professor decided to use the time he measures and this speed to determine how long the court is. How long does he think the
court is? Does he think it is shorter or longer than 24.0 m?
35.9 - Length contraction
Length contraction: The length of an object is
less when it is moving relative to an observer
than when it is stationary.
Einstein’s postulates require that time dilates í observers measure different time
intervals between two events when their reference frames are in relative motion. The
postulates also require that length “contract”. The length of an object will be less when it
is moving relative to an observer than when it is stationary. (You will see that the effect
is far too small to measure for everyday speeds, which is why you do not notice this
effect.)
Consider the train scenario on the right. The train is moving rapidly past an observer at
0.8c. This is not an everyday speed for even a bullet train. The professor is onboard the
train, at rest relative to it. He measures the length of his car as 10 meters. This is the
car’sproper length, the length measured by an observer stationary relative to the length
being measured. The use of proper here is analogous to the use of proper in “proper
time”.
Sara, standing on the ground, measures a different value. She measures the train car
as being just six meters long. The relative motion of the two observers causes the
differing measurement.
The equation on the right quantifies length contraction. This equation converts the
proper length measured by an observer who is stationary relative to the length being
measured, to the length observed by an observer who views the object as moving.
As with time dilation, you face two challenges: First, believing it to be true, since it
defies your intuition and experience, and second, understanding the notation for the
equation. We are not sure how much we can change your intuition, but at the least we
can work on the notation. To stress the notation once more: The proper length (L 0 ) on
the right side of the equation is the length measured in a reference frame stationary
relative to the object being measured. In the illustration, it is the professor measuring
the length of the car.
The quantity L on the left-hand side of the equation represents the length that will be measured by a person who views the object (and its
reference frame) as moving. This person observes the object as moving at velocity v.
Length contraction occurs only along the direction of the relative motion. In the scenario in this section, only the car’s length changes, not its
height nor depth. If the train were carrying a light clock, the vertical dimension of the clock, used by onboard travelers to measure time
intervals, would not be affected by length contraction.
Does the moving object “really” contract? Well, to pose an analogous question from everyday life, do faraway objects really get smaller, as they
Length contraction
Observer stationary with respect to
object measures one lengthLength contraction
Observer viewing moving object
measures shorter (contracted) length