Measuring the Speed of Light: A Mirror, a Train, and an Observer


By Don Lincoln, Ph.D., Fermi National Accelerator Laboratory (Fermilab)

There are only two basic postulates of relativity. The first one is that both primed and unprimed observers can correctly consider themselves to be the unmoving center of the universe. The second is that the speed of light is the same for all observers. So, what are the implications of these assumptions?

Beam of light striking an observer's eye.
Einstein’s theory of relativity says that the speed of light does not change with the direction of motion of the observer. (Image: Tatiana Popova/Shutterstock)

A Person Shining a Light on a Train

We’ll start by figuring out how the perception of time depends on whether you’re moving or not. We’ll start with a simple example, which requires a train car, a mirror, a flashlight, and a very accurate stopwatch.

Let’s put a person, who we will denote unprimed, in the train car. On one side of the car is a flashlight, and on the other side is a mirror. The width of the train car is denoted d.

The unprimed person shoots a pulse of light from the flashlight toward the mirror, which then reflects it back to him. Because the pulse of the light travels to the mirror and back, the distance travelled is 2d.

If we denote the speed of light as c and the travel time for the pulse of the light as t, we see that the time it takes for this to happen is just the distance divided by velocity, or symbolically, t=2d/c.

So that’s the time experienced by the unprimed observer to see the flash of light cross the train and back.

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The Moving Light on the Train

Now, let’s see how a person outside the train, who is seeing the train moving, experiences the same event.

Let’s say that the train is moving to the right at a velocity v according to a second observer, who we will now call the primed observer.

In this case, since both the mirror and the flashlight are moving to the right along with the train car, the light doesn’t just zoom across the train and back. You have to take into account the motion of the train. We see that the light travels along the path of a triangle.

It’s the classical and simple way of thinking. The train is moving to the right at a velocity v, and the light is moving across the train at a velocity c. Those two directions form the sides of a right triangle. The combined velocity of the light moving along the hypotenuse, which is the path it has to follow, is easily found using the Pythagorean theorem from introductory geometry. We could write that as (vhypotenuse)2 = v2 + c2.

Blurred streaks of light for an outside observer viewing a moving train at night.
For an outside observer, the light inside a moving train appears to move, too. (Image: IM_photo/Shutterstock)

Now, that’s pretty straightforward, but let’s see why it’s wrong. It’s wrong because it says that light travelling along the hypotenuse is travelling at a speed that’s greater than the speed of light that everyone sees, which is c. Thus, this approach fails Einstein’s second postulate.

Learn more about Einstein’s general theory of relativity.

Slowing Down Time

Illustration of Pythagoras' theorem.
The theorem of Pythagoras as applied to the train experiment seems to imply that light can be faster than itself. (Image: Oleg_Yakovlev/Shutterstock)

Let’s redo the calculation—this time, let’s respect the second postulate. The situation is basically the same. You have the train moving to the right at velocity v, and you have light moving along the hypotenuse.

Since the primed observer sees the light moving at c, we can again use the Pythagorean theorem to determine the component of the velocity of light across the train. In order for the speed of light along the hypotenuse to be c, the speed of light across the train according to the primed observer must be found by the formula (vcrossing)2= c2-v2.

Now, if that’s the component of the velocity that the primed observer sees for light crossing the train, we can easily calculate the time experienced by the primed observer, which we call tprime. Since he sees the light travel the same distance, which is 2d, we find that tprime is equal to 2d divided by the square root of (c2-v2).

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Time for Primed and Unprimed Observers

All this means that the relationship between the time experienced by the primed observer and the unprimed observer is just tprime = γ*t.

Now, gamma is always greater than or equal to 1. Gamma is never less than 1, nor is it negative. And as an object’s velocity approaches c, gamma approaches infinity.

So, with this knowledge that gamma is between 1 and infinity, we see that ‘tprime‘ is always greater than or equal to ‘t’. This means that the primed observer always experiences more time than the unprimed observer.

Or, described another way, a person seeing a clock move experiences more time than the same person that sees the clock as being stationary. This effect is called time dilation.

Common Questions about Measuring the Speed of Light

Q. What does the theory of relativity say about the speed of light?

The theory of relativity postulates that the speed of light is the same for all observers, whether primed or unprimed.

Q. What experiment may be devised to mimic the observation of the speed of light for an unprimed observer?

A person can be placed with a flashlight in a train. On one side of the car is a flashlight, and on the other side is a mirror. The unprimed person shoots a pulse of light from the flashlight toward the mirror, which then reflects it back to him. This mimics the unprimed person’s perception of being unmoving.

Q. In the train experiment, what path does light take when returning from the mirror to the unmoving primed observer?

When reflecting back from the mirror, the light must take a path along the hypotenuse of an imaginary triangle to reach the primed observer.

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