Understanding special relativity

Or, a very cool idea that lets you do nothing at all.

Just a warning – this post will use some scientific language that might be a little incomprehensible. But I’ll do my best to explain it, and this idea is so cool that you should really try to get through it.

Right. Special relativity. I am now going to explain special relativity in 2 sentences. When you travel really fast, time slows down. And when you travel really fast, lengths become shorter.

Before I start any of that, I will explain how speed, and relative speed, usually works.

Imagine a car, car A moving forwards at 50km/hr. (For any Yankees, a kilometre is like a mile, but a bit shorter). Then imagine another car, car B moving in the same direction at 60km/hr. Then the driver in the slower car (car A) looks out of the window at car B. Car B will appear to be moving forwards at … 10km/hr! This is the kind of physics we are used to. You can simply add and subtract speeds, and find the relative speed.

But things get tricky when you start looking at light. The speed of light in a vacuum is a whopping 299,792,458 km per second, but for interests of ease, I’m going to call it 300,000,000 km/s.

So imagine a spaceship that can travel at the speed of 295,000,000 km/s. (By the way, the speed record for any human object is 70 km/s, so we have never got anywhere near close to this speed.) That spaceship will then race at top speed in the same direction as a beam of light. So if we call the speed of light 300,000,000km/s, what speed will the light be moving relative to the ship? Conventional physics says 5,000 km/s.

This is wrong. In 1887, two physicist named Michelson and Morley conducted an experiment that proves the speed of light is equal in all directions, for all observers. In other words, that spaceship from before? It will see the light beam move forwards at 300,000,000 km/s. The light doesn’t speed up – an observer standing still will see the light move at the same speed. Something very weird is happening.

This may be a little confusing, so here is a simplification. Even if you are travelling at huge speeds towards or away from a light beam, it will move at the same speed relative to you.

(If at this point you are confused and not in the mood for maths, skip forwards to the bit of the post that looks like this: ^^^^^)

Understandably, this is hard to wrap your head around. The entire world of physics was equally confused. Until a chap called Hendrik Lorentz showed up, and decided to make a formula.

He created the Lorentz factor, a number that can be used to recalculate both time and distance. It goes a little like this.

CodeCogsEqn (5)

Don’t worry, the equation is more afraid of you than you are of it.

This formula defines the Lorentz factor, gamma, using several variables. You are probably familiar with the number 1. You are probably also familiar with the square root and the power of two. That leaves and cis the velocity of an object relative to an observer. And c is the speed of light. And once we have the Lorentz function, we can make two more formulas.

CodeCogsEqn (3)

CodeCogsEqn (4)

Again, do not be scared. I will walk you through these. The top formula says that as you approach the speed of light, the length L will multiply by 1 divided by the Lorentz factor. And as you approach the speed of light, the time t will multiply by the Lorentz factor. So time slows down, and lengths become shorter.

Now, when he published these results, Lorentz publicly declared that he had no idea why these numbers worked. He had just made a pretty formula that allows physics to resume – plug in the numbers so that everything works and pretend nothing happened. Until Einstein came along.

This is how maths works at the speed of light

What Einstein said was simple, and was the basis for the Theory of Special Relativity. He said (and I am paraphrasing here):

The Lorentz factor isn’t just a nice trick. I think the universe uses the Lorentz factor as well.

Einstein said that the effects that the Lorentz factor described were more than just ways to make the maths work. Einstein thought that when you approached the speed of light, these things actually happened. And he was right.

(You can resume here, mathophobes – ^^^^^^^^^^)

Can you think about how crazy that is? When you go really fast, the speed of time will actually change. Or that a spaceship journey could actually make you slimmer! Again, this is real physics. Time and space themselves are warped at high speeds.

I can now give you an example. It involves a very fast spaceship and a barn.

Ok. We have a spaceship that can travel at 86.6% of the speed of light and is 18 metres long. We have a 10 metre long barn with two open doors at each end, but at the press of a button they will snap closed and then open immediately. For obvious reasons, our 18 metre rocket will not fit in a 10 metre barn. But with the help of some special relativity, we can make it fit. Let’s fill in the equation.
CodeCogsEqn (5)

Now v is 0.886 of c. That means we can fill in the v and the c.

CodeCogsEqn (6)

And oh look! There is a  on the top and bottom of the fraction. And we all know what that means! We can cancel the ! Yippee! Us physicists live for the small pleasures in life, like being able to cancel s. So in other words,

CodeCogsEqn (7)

And that can be simplified into:

CodeCogsEqn (9)

And then, if we skip out some of the maths,

CodeCogsEqn (11)

Gamma – the Lorentz factor – is equal to 2. So we can bung that into the other equation, the one that finds the new lengths.

CodeCogsEqn (12)

Isn’t that crazy! By accelerating the spacecraft up to 86.6% of the speed of light, it has halved in length! Which means it can be flown through the barn, the doors can blink shut and for a tiny amount of time, we have put a 18 metre long rocket into a ten metre long barn. Ta-da.

There is one other thing to point out. These equations break down once you get to the speed of light. You end up having to calculate 1/∞, and maths doesn’t cover that. You can’t use special relativity to travel back in time. And there is another part of the theory, one that Einstein did come up with himself. It’s called the relativity of simultaneity, and basically says that when you travel close to the speed of light, space doesn’t just get warped. The front of the object travelling at that speed will actually get shoved into the future.

The universe is weird, weirder than we can possibly imagine, and physics is perhaps the only way we can look at it in a clear light.


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