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Why Does Time Slow Down as You Approach the Speed of Light?

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I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications, and fun mathematical facts.

Galileo Galilei (1564 - 1642)

Galileo Galilei (1564 - 1642)

Galileo's Principle of Relativity

Before we look at why time appears to slow down as you travel at speeds approaching the speed of light, we need to go back a few hundred years to look at the work of Galileo Galilei (1564 - 1642).

Galileo was an Italian astronomer, physicist and engineer whose incredible body of work is still highly relevant today and set the foundations for much of modern science.

The aspect of his work we are most interested in here however is his 'Principle of Relativity'. This states that all steady motion is relative and cannot be detected without reference to an outside point.

In other words, if you were sitting on a train that was moving along at a smooth, steady rate, you would not be able to tell if you were moving or stationary without looking out of the window and checking if the scenery was moving past.

The Speed of Light

Another important thing we need to know before we begin is that the speed of light is constant, regardless of the speed of the object emitting this light. In 1887 two physicists called Albert Michelson (1852 - 1931) and Edward Morley (1838 - 1923) showed this in an experiment. They found out that it didn't matter if light was travelling with the direction of the Earth's rotation or against it, when they measured the speed of light it was always travelling at the same speed.

This speed is 299 792 458 m/s. As this is such a long number, we generally denote it by the letter 'c'.

Albert Einstein (1879 - 1955)

Albert Einstein (1879 - 1955)

Albert Einstein and His Thought Experiments

At the beginning of the 20th century, a young German called Albert Einstein (1879 - 1955) was pondering about the speed of light. He imagined that he was sitting in a spaceship travelling at the speed of light while looking in a mirror in front of him.

When you look in a mirror, the light that has bounced off you is reflected back towards you by the surface of the mirror, hence you see your own reflection.

Einstein realised that if the spaceship was travelling at the speed of light as well, we now have a problem. How could the light from you ever reach the mirror? Both the mirror and the light from you are travelling at the speed of light, which should mean that the light can't catch up to the mirror, hence you don't see a reflection.

But if you can't see your reflection, this would alert you to the fact that you are moving at light speed hence breaking Galileo's principle of relativity. We also know that the light beam can't speed up in order to catch the mirror as the speed of light is constant.

Something has to give, but what?


Speed is equal to distance travelled divided by time taken. Einstein realised that if the speed was not changing, then it must be distance and time that are changing.

He created a thought experiment (a purely made-up scenario in his head) to test out his ideas.

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A Light Clock

A Light Clock

Einstein's Thought Experiment

Imagine a light clock that looks a little like the picture above. It works by emitting pulses of light at equal time intervals. These pulses travel forward and hit a mirror. They are then reflected back towards a sensor. Each time a light pulse hits the sensor you hear a click.

A Moving Light Clock

A Moving Light Clock

Now suppose this light clock was in a rocket travelling at speed v m/s and positioned so that the pulses of light were sent out perpendicularly to the direction of travel of the rocket. Furthermore there is a stationary observer watching the rocket travel past. For our experiment suppose the rocket is travelling from the observer's left to right

The light clock emits a pulse of light. By the time the pulse of light has reached the mirror, the rocket has moved forward. This means that for the observer stood outside the rocket looking in, the light beam will be hitting the mirror further right than the point it was emitted from. The pulse of light now reflects back, but again the whole rocket is moving so the observer sees the light return to the clock sensor at a point further right of the mirror.

The observer would witness the light travelling in a path like in the picture above.

A Moving Clock Runs Slower Than a Stationary One, But by How Much?

To calculate how much time is changing we will need to do some calculations. Let

v = the speed of the rocket

t' = the time between clicks for a person in the rocket

t = the time between clicks for the observer

c = the speed of light

L = the distance between the light pulse emitter and the mirror

Time = distance/speed so on the rocket t'=2L/c (the light travelling to the mirror and back)

However for the stationary observer we have seen that the light appears to take a longer path.

The Moving Light Clock

The Moving Light Clock

We now have a formula for the time taken on the rocket and the time taken outside of the rocket, so let's look at how we can bring these together.

Deconstructing Pythagorean's theorem

Deconstructing Pythagorean's theorem

Solving for t

Solving for t

How Time Changes with Speed

We have ended up with the equation:

t= t'/√(1-v2/c2)

This converts between how much time has passed for the person on the rocket (t') and how much time has passed for the observer outside of the rocket (t). You can see that as we are always dividing by a number less than one, then t is always going to be bigger than t', hence less time is passing for the person inside the rocket.

© 2020 David

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