Five-minute Explainer: Special Relativity

This is going to be the the briefest history of time ever. When I’m done, my goal is for you to understand not only that motion changes time and space but how and why.

Einstein created the theory of special relativity to answer these questions, and it does so in a very satisfying and complete way which physicists still haven’t improved on. It may surprise you to know that the paper in which he originally described it was only thirty-one pages long.

Postulates

Before we start, we need just the slightest background here on what Einstein had to work with when he came up with special relativity. Namely, he used two assumptions.

  • The laws of physics are the same for any point of view which seems stationary to the observer. This is known as the principle of relativity. It means that it’s physically impossible to distinguish between whether I’m moving or the world is moving if I were to jump up and down. Both are true at the same time. You can take whichever point of view you like. That stationary point of view is called an inertial frame of reference.
  • The speed of light is always the same in all inertial frames of reference, regardless of the speed of whatever is emitting the light.

There are some additional assumptions you have to bake in, like conservation laws and so on, but none of those would strike you as terribly strange, and they’re subtle enough points that they don’t bear discussing here.

From those two assumptions, which are today called postulates, the rest of the theory emerges. Remember that carefully—every strange part of special relativity is an emergent consequence of those two postulates. No alternative to the theory could work without some change to the postulates, and at the time Einstein was working, he was almost certain they were true. Einstein merely carried the assumptions to their logical conclusion: special relativity.

The Principle of Relativity

Einstein loved to visualize things, and he used trains to illustrate his theory originally because of the train station in Bern, Switzerland. We’ll use trains, too. To make things even easier to visualize, we’ll not deal with light directly but instead use a thought experiment involving a thrown ball.

Imagine a person standing still on the ground can throw a ball at 100 kilometers per hour. The ball crosses a distance of 27.78 meters in 1 second. The thrower can always throw at that speed, so over the course of 1 second, the distance traversed is always 27.78 meters.

Now imagine this same person is standing on a train car. The train is moving at 30 kilometers per hour. They throw the ball in the direction of travel at 100 kilometers per hour. How far does the ball travel?

For an ordinary ball, the distance traveled depends on where you stand. The person throwing the ball on the train sees nothing out of the ordinary because they too are moving along. They see the ball travel at 100 kilometers per hour, and so they likewise see the ball cross 27.78 meters in 1 second. This accords with the first postulate, the principle of relativity. No matter that the train is moving. To the person on the train, it might as well be still while the rest of the world moves backward.

However, a person standing stationary on the ground next to the train would see the ball thrown at 130 kilometers per hour because the train’s motion adds onto the ball’s motion. Therefore, the ball travels 36.11 meters over 1 second. The velocities add together.

A Constant Speed of Light

So far, I have described the ordinary, intuitive behavior you would expect in this situation. Now let’s change things up: The ball can never travel at any other speed than 100 kilometers per hour in any direction, regardless of who is standing where or who is in motion compared to whom. This is similar to how light behaves, according to the second postulate.

The person on the train throws the ball, and it travels at 100 kilometers per hour relative to them, crossing 27.78 meters in 1 second. So far, so good! But the person on the ground also sees the ball travel 100 kilometers per hour, despite the velocity of the train being 30 kilometers per hour. The velocity of the train no longer adds onto the velocity of the ball, yet the ball is still in motion when it is thrown.

How far does the ball travel now?

Here, the universe encounters some very awkward bookkeeping problems. If the person on the ground also saw it travel 27.78 meters in 1 second at 100 kilometers per hour, that ball would land somewhere other than where the person on the train sees it land. This train is in motion, along with the ball. We expect it to cover a distance in 1 second equivalent to the motion of the train in addition to the 27.78 meters which the person on the train sees. That’s 36.11 meters. Yet it cannot cross that distance in 1 second because the ball cannot go faster than 100 kilometers per hour.

For the ball to go different places for different people violates causality itself—it would mean cause and effect were broken. Einstein assumed cause and effect would work out because without them, science wouldn’t do us much good for describing the universe. At the same time, however, he had this very annoying issue of how to solve this problem of reconciling time and distance under these conditions. He thought about it until he understood that some of the assumptions he held about what the universe would keep constant were not, in fact, fixed.

The way to resolve the problem above is that the ball does travel only 27.78 meters. How can that be? Because the train itself gets shorter. The person on the ground, looking up at the train, would see the train squished in the direction of travel. The universe solves the bookkeeping problem around the speed of light by altering space itself! In point of fact, those 27.78 meters over which the ball travels would look different to a person standing on the ground compared to someone on the train.

This is not just a coincidence. It is the way the universe must work because of the finite and invariant speed of light—or, in this case, the invariant speed of the ball thrown. It simply cannot be any other way without breaking causality or changing one of the postulates.

There is one additional problem though. If a ball is moving at 100 kilometers per hour, and the distance it’s moving over has just shortened, wouldn’t it arrive in less than a second now? This problem really bothered Einstein until he let go of the assumption that the universe kept an absolute time clock somewhere—in other words, that there’s no one real, absolute time. This let the universe bend time in order to make the math come out right.

This means that the solution to the problem is for the ball to slow down, along with everything else, when it’s in motion relative to an observer. The sum total of these effects leads to the stationary person on the ground seeing the ball move through a shorter distance while also moving more slowly. There can be no distortion in space without a matching distortion in time, and it appears that space and time are so inextricably bound that it’s easier to deal with them as a single thing called spacetime.

For the person on the train, they see everything happening there normally, but due to the principle of relativity, they see things on the ground also squished and in slow motion.

Cause and Effect

By changing the rules the ball followed in our thought experiment, some very unintuitive consequences emerged. As it happens, light really does behave the way the ball does. Because time and space warp for light, they warp for anything moving at any speed, though the speed limit is so high that the effects aren’t obvious in ordinary life.

It was only necessary to change the behavior of the ball—and hold everything else equal—to see the effect of special relativity on time and space. It caused the everything in motion to squish (length contraction) and begin moving in slow motion (time dilation) when seen by the person on the ground next to the train.

In the next five-minute explainer, I’ll describe even more incredible effects which emerge from the invariant speed of light which Einstein and others found later.

I am grateful again to Zuzu O. for her thoughtful suggestions on improving the readability of this post.