Before trying to understand special relativity, you should ask yourself the following question: what problem does special relativity solve? Put another way, why was this theory invented in the first place?

Like many theories in physics, special relativity was invented to fit empirical data. In particular, experiments had shown that light did not behave like a particle, nor did it behave like a wave. Let’s take a closer look.

Does light behave like a particle?

Let’s say I can throw a baseball at a top speed of 50mph. If I’m standing on the pitcher’s mound and you’re standing on home plate and I throw the baseball towards you at my top speed, ignoring things like air resistance, the baseball will move towards you at a speed of 50mph.

How might I increase the baseball’s speed relative to you? One way would be to stand on the back of a pickup truck that was moving towards you at a speed of 30mph. If I throw the baseball at you now, the baseball will move away from me at my top speed (50mph) but towards you at a speed of 80mph. The speed of the pickup truck and the baseball add together to produce a higher speed relative to you.

But there’s nothing special about a baseball in this example. All particle-like things act this way. I could change the example such that I’m now firing a bullet at you. The numbers would change, but the general point would stay the same: the bullet would travel faster, relative to you, when I fire it from the back of a moving pickup truck.

How about light? Instead of firing a bullet, let’s say I “fired” a photon (a particle of light) towards you by turning on a flashlight in your direction. Classical Newtonian mechanics would suggest the same result: the photon would travel faster, relative to you, when fired from the back of a moving pickup truck.

However, this is where the behavior of light deviates from the behavior of other particles. Experiments in the late nineteenth and early twentieth centuries strongly indicated that this effect does not occur with light. One such experiment was made by Russian astronomers in 1955 who measured the speed of light from two opposite sides of the rotating sun. One side is rotating towards us while the other is rotating away from us, so one would expect to measure different light speeds from each side. However, it was found that the speed of light was the same from both sides.

In short, the speed of light did not seem to depend on the motion of its source.

Does light behave like a wave?

Now, before you go off and think how strange (as I did), let me try to convince you that this is, in fact, not strange at all when it comes to waves! Maybe there is nothing mysterious about light; maybe it just behaves like all other waves.

Take sound waves as an example. The speed of a sound wave does not depend on the speed of the source of the sound. In fact, all sound waves travel at roughly the same speed1.

Why doesn’t it depend on the speed of the source? I don’t really know but… <digression> I think it has something to do with the fact that - in a wave - no single physical object is traveling with the wave. Instead, the wave is causing very local perturbations in the particules that make up whatever medium it’s traveling in (e.g. water for water waves, air for sound waves). Consider a single water wave propagating to the right, and specifically think about a water molecule that is currently on the crest of that wave. Now consider that same particle one second later. How far to the right has it moved? Not at all. It has moved down, not to the right (even though the wave crest is moving right). Not all waves work this way. Transverse waves cause the medium to move perpendicular to the direction of the wave; longitudinal waves cause the medium to move parallel to the wave. But in either case, particules are moving back and forth in a relatively small region; they are not tracking the motion of, say, the crest of the wave. </digression>

I bet you already knew that waves don’t behave like particles, even if you hadn’t thought too deeply about it. Have you ever heard of a “sonic boom”? That occurs when an aircraft travels faster than the sound waves that it is producing. This sort of thing makes no sense when talking about the motion of particles. If I throw a baseball from a vehicle moving at a constant velocity, I’m not going to somehow pass that baseball. The speed is, by definition, my speed plus whatever speed I can throw a baseball. But supersonic aircrafts do in fact pass the sound waves that they themselves produce. Waves are weird.

So we’ve established that the speed of waves are independent of the speed of their source. Experiments suggested that the speed of light is also independent of the motion of its source. So maybe light is just a wave?

Sure, let’s continue thinking about light as a wave. If it’s a wave, then it travels by disturbing some medium. Let’s call this medium the “luminiferous aether” as they did in the nineteenth century, or “ether” for short. We’ve established that the speed of the wave doesn’t depend on the speed of the source, but does the speed of the wave relative to an observer depend on the speed of the observer relative to the ether? In most waves, it does.

Here’s an example. Consider a sound wave that travels through the air at some constant velocity \(v\). The speed of the sound wave relative to me is also \(v\) if I’m standing still (or more precisely, if I’m not moving relative to the air). But it’s more than \(v\) if I run through the air towards the oncoming sound wave, and less than \(v\) if I run through the air away from the sound wave.

Is the same true of light? Does the speed of light depend on the motion of the observer? Put another way, does it depend on the motion of the observer relative to the ether (the suggeted medium for light)?

Enter the Michelson–Morley_experiment. If light moves at a constant speed through the ether and we move at different speeds relative to the ether, then light should move at different speeds relative to us! So they measured the speed of light (relative to us) in many different directions hoping that they would get different answers. From this, they would be able to infer our motion relative to the ether. Alas, as you can probably predict, the experiment found no detectable differences in the speed of light in any direction. This was the first strong evidence against the existence of a light ether.

The real problem

This brings us to the real problem that special relativity is designed to solve. Not only does the speed of light not depend on the speed of the light source, but it also doesn’t depend on the speed of the observer. Now we are in truly unfamiliar territory. Particles nor waves behave in this manner.

The strangeness can be described as follows. Let’s say there are three things A, B, and C, arranged in that order from left to right, and they are all moving at constant speed to the right. A is moving the fastest, B the second fastest, and C the slowest. A will pass B at some speed, relative to B. A must then pass C at a different speed, relative to C, right? Since B and C are travelling at different speeds, then A must pass B and C at different (relative) speeds, right?

Our every day experience suggests yes, but that’s not how light works. If A was a light beam, it would somehow pass B at a speed of \(c\), relative to B, and then also pass C at a speed of \(c\), relative to C. Again, this is assuming constant speeds for all three objects. This defies intution.

Let’s consider one last example to drive the aburdity home. There’s a race between a sports car, a human, and a tree. Since, generally speaking, sports cars are faster than humans who are in turn faster than trees, we give them different starting points for this race. The race begins! The race car quickly acheives its top speed of 200mph. The human quickly acheives their top speed of 10mph. The tree doesn’t move. Before long, the race car passes the human. What is the race car’s speed relative to the human? 190mph. Later, it passes the tree. What is its speed relative to the tree? 200mph. It has to be different because the human and the tree aren’t moving at the same speed. Well, not so for light beams. Replace the race car with a beam of light and the answer to both questions is \(c\).

Something has to give.

Special Relativity

Einstein’s theory of special relativity assumes the following two postulates to be true and follows them through to their (fairly absurd) logical conclusions:

  1. There is no way to tell whether an object is at rest or in uniform motion relative to a fixed ether. Translation: light is unlike other waves in that there is no detectable medium.
  2. Regardless of the motion of its source, light always moves through empty space with the same constant speed. Translation: light is unlike particles in that its velocity does not add with the velocity of its source.

Other physicists had considered these two postulates, but most thought they violated common sense so much that they preferred to believe that one of them must be wrong. Einstein accepted them and, from them, deduced that there must be no meaning to the concepts of absolute length or time.

  1. This isn’t strictly true. It depends on certain properties of the medium, which is air in this case. Temperature, pressure, humidity, etc. can affect the speed of a sound wave. But the important point remains, which is that the speed of the source is not a factor.