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  • 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 blueit 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 nowlet'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 motionand 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 endRearranging, 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 squaredor, as you've heard before, E equals M C squared!

In mid 1905, Albert Einstein derived what is now the most famous equation in the world:

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B1 中級

アインシュタインのE=mc²の証明 (Einstein's Proof of E=mc²)

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    Aurora Yang に公開 2021 年 01 月 14 日
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