The Substance and the Fabric

There’s an equation in physics you might have heard of:

I had an idea to see if I could understand this equation more profoundly. I’ve recently been thinking about lots of topics I never imagined I’d be able to think about so deeply, learning more in the last couple of years than I learned in the previous ten combined.

This is all thanks to what I’ve started calling the drunken octopus phenomenon. Thirty years ago, information was hard to come by, and very expensive in terms of time and energy (and sometimes money too!). Today, you can search broadly, deeply, and quickly for just about anything, and if you’re just clever enough, you can find answers to very tough questions.

Because of the ease of access to information, and the ability to process and fact-check ideas in an instant, I started this mental exercise with a bit of a warm-up, where I imagined where basic math principles came to be in the human mind.

Now, with this lengthy preamble out of the way, let’s dive in to the most famous equation in the world.

For starters, E = mc² only has three terms: energy, mass, and the speed of light (c). When Einstein first proposed this equation, the incredible big idea was that mass and energy were just manifestations of the same thing.

This is a little bit like saying that ice is the same thing as water. Both are made up of the same molecules, but the molecules interact a bit differently in each substance. Energy and mass are the same substance, but in different states.

Let’s pause on mass-energy equivalence for a moment. The jargon makes it easy to overlook how wondrous and incredible this implication really is. I don’t want to skate past this part with you.

If you feel warm, that’s energy. That warmth you feel is the same substance as the material you’re made of. One easy way to see this in action is to light a little camp fire—as the wood burns, it turns into light and heat, warming you up and helping you see.

If you punch someone, you’re using the same substance that your fist is made of. The act of punching is mass, in a manner of speaking.

So, matter/energy is the substance of the universe.

I had managed to internalize all of this, but I wanted to know why c should be in there, and then (later on) why it should be squared. The very fact that the amount of energy that you get out of a given amount of matter is multiplied by the speed of light squared, means you’re going to get an absolutely gargantuan amount of energy out of a tiny amount of matter.

This is really easy to understand if you’ve seen footage of a nuclear explosion, or if you can envision how a tiny plutonium pellet can power an entire town for years. Both of these only work if there’s a silly amount of energy released from a tiny amount of matter.

Why c, though? This was a much tougher question, drunken octopus or not.

We need to bring in the fabric to understand the answer. The fabric of the universe is called spacetime.

The first thing to understand about spacetime is that it’s one thing, just like mass and energy are one thing. You can measure the individual components of spacetime as space and time, just as you can measure the length, width, and height of something in space, or the amount of time that has gone by.

However, Einstein showed that if you take away a little space, you gain a little time, and vice versa. In other words, space and time are inextricably tied together, and the way you get from space to time, in a manner of speaking, is through the speed of light.

This is to say that the conversion factor for spacetime is c, the speed of light. You can read a little about how Einstein figured this out by thinking about trains here.

C is the maximum speed anything can go, and no matter who looks at something traveling at c, they’re going to see it as the speed of light. If I am going a billion miles per hour in one direction and you are going a billion miles and hour in the other direction, and I blast a light beam out in front of my ship, you and I will both measure that light beam as going the same speed.

C isn’t just a speed; it’s a fundamental aspect of the universe, just like the substance and the fabric. In fact, it’s what links together the separate components of both the fabric and the substance—by unlocking mass and turning it into energy, for instance, and by transforming distance into seconds.

Maybe you can reason out why c should be the conversion factor for mass-energy equivalence. After all, the fastest anything can go is c, and we define speed as distance divided by time.

If something always goes at c, then when space goes up, time goes down. It is this fabric that bends around c, not the other way around.

But why should the c term be squared? Why should it be multiplied by itself like that?

Spacetime can be thought of as made up of space and time, even though they are the same substance. What I mean is that you need a c for space, and you need a c for time.

It’s better to think of c not as a speed, exactly, but more like a connector for space and time. If you stretch space out by this much, it costs you this much time, divided by c. If you stretch out time, you need to multiply the amount of space by c to figure out how much everything changes.

This is a deeper explanation than I had before, but I have the feeling it’s still more of a surface-level view of what’s really going on. Even still, I wanted to share with you just how wonderful this was to think through!

C is just incredible, isn’t it? It seems to be just a number, but it’s just as important and fundamental to the universe as the substance and the fabric. Not only does c convert mass to energy and space to time, but it also connects the fabric and the substance together.

Upon further consideration, maybe the title of this piece should be The Fabric, the Substance, and the Factor. All three seem equally important.

How’d I do? Was this easy enough to follow along? If you’re a physicist, what did I miss here?