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# Einstein's Proof of E=mc²

In mid 1905, Albert Einstein derived what is now the most famous equation in the world:
E equals M C squared. But he didn't just write this down out of the blue – it followed
directly from his paper on special relativity that we talked about in last week's video…
and here's how he did it:
Suppose you're watching a cat float freely in empty space, when suddenly it emits a flash
of light in all directions. The light carries away some energy, we'll call it "E", so by
conservation of energy the cat must have lost energy E… but since the light was emitted
symmetrically in all directions, it won't have changed the cat's velocity. So where
did the energy for the light come from?
Never mind that now… let's imagine you get bored and zoom off in a spaceship in the middle
of the experiment. But from your new perspective, you're sitting still in your spaceship and
the cat is the one moving past outside the window! Therefore you'll calculate that the
cat has some kinetic energy, that is, energy of motion… and when you see the cat emit
the flash of light, you'll again measure that its energy decreases by the energy of the
light.
Except now that you're moving, special relativity tells us that time passes at different rates
for you and the cat, so you'll measure a different value for the frequency, and thus energy of
the flash of light. This is the relativistic doppler effect, and for our purposes, it amounts
to multiplying the energy of the light by one plus your velocity squared divided by
twice the speed of light squared.
So to recap, if you take off at velocity v, you'll see the cat gain some kinetic energy
KE1, then at the flash you'll see the cat's energy decrease by E times one plus v squared
over two c squared. On the other hand, if you wait, you'll see the cat's energy decrease
by E, and now when you take off you'll see it gain kinetic energy KE2.
But this is silly! You never touch or otherwise influence the cat in either case, so you should
get the same total energy at the end… Rearranging, we see that the kinetic energy before and
after the flash must be different! And the kinetic energy of an object is one-half of
its mass times velocity squared, but we know that the velocity was the same in both cases…
so in order to account for the difference, the cat's mass must change when it emits the
flash of light!
Now if we cancel things out, you can see that the change in mass of the cat must be equal
to the energy divided by c squared – or, as you've heard before, E equals M C squared!