e=mc^2 Why Does E=mc^2?

[Reefa]

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Let's play a game! The speed of light is just a number, right? If you define your units,  for example, what a "meter" and a "second" are, you can say that the speed of light is around 300,000,000 meters per second. Or 670,000,000 "miles" per "hour," whatever those are.

 

What if, instead, we just said the speed of light was equal to…1. Just 1. So, 1 what? I said: just 1. No miles, no seconds, no fortnights, no leagues. Just…1. We're allowed to do it, because it's just a number, and we're picking a system where speed has no units.  In this system, a jet airliner cruises at a snail's pace of 0.000001, or 0.0001 percent of the speed of light. Two of the fastest human-made objects, the Helios probes, zoomed around the solar system at a whopping 0.00025! Look at them go!

 

Now that we've defined the speed of light to be 1, let's look at the most famous equation in physics: E = mc². 

 

We know all the bits, but let's refresh: E is for energy, m is for mass and c is the constant speed of light.  But in our newfangled unit system (called, for the technically minded, geometrized units), c equals 1, and that famous equation boils down to its essence:

 

E = m.

 

I'll even spell it out:

 

Energy = mass.

 

It doesn't get any clearer than that, folks. Energy is mass. Mass is energy. They are equivalent; they are equal. They are the same thing.

 

Wait, wait, wait, you say as you look at me suspiciously. What about light? Photons don't have any mass, but they sure do have plenty of energy. How else do plants eat?

 

You're right, photons don't have mass. But they do have momentum, which is how things like light sails (also called solar sails) get the oomph they need to glide around the solar system: Their propulsion comes from the sun's radiation pressure.  And momentum has energy. But where's the momentum in E=m? It's looking like we don't have enough letters to cram it in.

 

The confusion comes about from the "m" used in E=m. We normally think of "mass" as something concrete and simple. Hold a rock in your hand; it has mass. Throw it, and it has mass and momentum. But that's not the "m" in E=m. Instead, when Einstein wrote that equation down, he meant something different, usually referred to as "relativistic mass."

 

That term isn't used so much nowadays, because it causes so much head-scratching.

 

Let's take a step back and see what Einstein was thinking.

 

You remember kindergarten-level special relativity, and hearing things like "it's impossible to move at the speed of light, because the faster something goes, the more mass it has. To get to the speed of light, it has infinite mass, so it would be impossible to push!" Yeah, well now it's time for first-grade-level special relativity.

 

A fundamental aspect of our universe is that there's a universal (and I really mean universal) speed limit: the same speed that light goes. No matter what, you can never crack that speed. Let's see how that plays out in practice:

 

Let's say I give you a nice, solid shove and send you flying away at 0.9 — that is, 9/10th the speed of light.  What if I catch up to you and give you the exact same shove, again. You won't be going 18/10th the speed of light, because that's not allowed. You'll get closer to the speed of light, but never cross it. So for the exact same force that I impact on your hopeless self, I don't move you as fast. I get less bang for the buck.

 

And the closer you get to the speed of light, the less effective my shoves will be: the first one may get you to 0.9, then the second to 0.99, then 0.999, then 0.9999. Diminishing returns every time.  In fact, it's as if you were getting more massive. That's exactly what more mass means: You get harder to push.

 

So what's going on? The answer is energy. You still have the same old normal, everyday, rest mass that you always had. But you're going really, really fast. And that speed has an energy associated with it — kinetic energy. So it's like all that kinetic energy is acting like extra mass; either way I count it, you get harder to push, because of that fundamental speed limit.

 

In other words, you can say that energy is mass. Huh, whaddaya know.

 

Back to the "m" in E=m. When physicists first started playing with those equations, they were well aware of the universal speed limit and its nonintuitive consequence that you get harder to push the faster you go. So they encapsulated that concept into a single variable: the relativistic mass, which combines both the normal, everyday mass and the "effective" mass you gain from having loads of kinetic energy.

 

When we break up "m" into its different parts, we get:

 

E² = m² + p²

 

Or bringing back our friend c:

 

E² = m²c⁴ + p²c²

 

And we have another character joining the party: p, for momentum. Photons don't have mass, but they do have momentum, so they still get energy.

 

In this view, mass is a kind of energy. But I just said above that energy acts like mass. What's the deal? Are we just talking in circles?

 

No. Mass is energy. Energy is mass. You can count things energywise or masswise. It doesn't matter. They’re the same thing.

 

A hot cup of coffee literally weighs more than a cold cup. A fast-moving spaceship literally weighs more than a slow one. A rock — or an atomic nucleus — is a compact, bundled-up ball of energy, and sometimes we can tease some of that energy out for a big boom.

 

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[Reefa]

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