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Most people know the famous equation:

$$E=mc^2$$

What were his steps of thinking for this equation that helped us discover so much about our world?

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Terence Tao has re-derived $E=mc^2$ here. See also this related Phys.SE post. – Qmechanic♦ Nov 9 '12 at 20:57
A more recent proof by Terence Tao terrytao.wordpress.com/tag/mass-energy-equivalence – John McVirgo Nov 9 '12 at 21:14
This paper by Hecht is required reading about what Einstein actually wrote and did not write. – user11266 Feb 27 at 21:47

4 Answers

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You can find the shortest and easiest derivation of this result in the paper where it was released by Einstein himself (what better reference can you find?) in 1905. It is not the main paper of Special Relativity, but a short document he added shortly afterwards.

A. Einstein,Ist die Trägheit eines Körpers von seinem Energieinhalt Abhängig?, Annalen der Physik 18 (1905) 639. A pdf file of the English translation Does the Inertia of a Body Depend upon its Energy-Content? is available here. (hattip: user53209.)

It is a delightful document to read. There is no dramatic references to huge power release nor anything similar. He simply states after the derivation "If a body gives the energy away $L$ in form of radiation, then its mass decreases in an amount $L/V^{2}$ (...) the mass of a body is a measure for its energy content (...) One can not exclude the possibility that, with the bodies whose energy content changes rapidy, for example radium salts, a proof of the theory will be found (...) If the theory adjusts to the facts, then the radiation transports inertia between emitters and absorbers."

Google for that short paper and see the derivation yourself, it is very easy. The Minkowsky four-dimensional spacetime had not yet been incorporated to special relativity, so the equations are formally very simple, easy to follow with little mathematical training.

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I doubt you've even read the paper and understood it, because if you had, you'd see that it isn't a proof – Larry Harson Jan 28 at 16:16
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@Larry Harson, and I doubt you've even read my answer, because if you had, you'd seen that I make no mention to any proof, but rather I explicitely use the word "derivation", and in the first line. Follow it with your finger, where I say "derivation of this result". – Eduardo Guerras Valera Jan 29 at 2:03
OK, "derivation", but it's not even that, and much has been written on its incorrectness. – Larry Harson Jan 29 at 13:15
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@Larry Harson, please read the Einstein paper, it contains the original, fully correct derivation. Yes, derivation. And that "much" that has been written against 1905 Einstein papers consist on a bunch of pseudo-scientific journalism, mainly from nazi morons. – Eduardo Guerras Valera Jan 29 at 14:40
Special Relativity can be (and will be) super-seeded by new theories, but it is self-consistent, as it is the Einstein derivation (again, derivation) of $E=mc^{2}$ of 1905. – Eduardo Guerras Valera Jan 29 at 14:43
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up vote 4 down vote

Einstein's equation doesn't have a "proof" because it's not a mathematical theorem. It's a physical theory that is overwhelmingly supported by experimental data. So you could say that the "proof" is in the mountains of experimental results that agree with the theory.

To understand Einstein's motivation for developing the theory of relativity, as well as mass-energy equivalence, Wikipedia has an excellent article on the history of relativity.

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Einstein gave an argument which is summarized on Wikipedia, and also regurgitated on Terrence Tao's blog. This answer is not reasonable, physical statements have physical arguments, and these are what people normally mean by "proof" in this context. – Ron Maimon Nov 9 '12 at 21:20
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Ron is out of his mind. It does not matter how beautiful a theory may be nor how neat the derivation, the proof is only in the agreement with the physical world. – dmckee♦ Feb 28 at 5:02
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I will support @dmckee , since this came up again. A theory in physics cannot be proven, only disproven. It can only be validated if experiments agree with it. Even one solid disagreement disproves a theory. The questioner assumes that physical models are the same as mathematical models which end with the QED, but this is not true. The title is misleading, the content of the question is OK and is answerd by Eduardo. – anna v Feb 28 at 5:26
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Yes, Ron is probably out of his mind but he still knows more physics than ten anna vs and four dmckees put together. – Marty Green Feb 28 at 5:28
@MartyGreen Anna V is quite correct. Science doesn't deal with proof. Proof only exists in mathematics and in courtrooms. Even then, it doesn't mean the same thing in both places. Supporting evidence doesn't constitute proof. Evidence accumulates indefinitely while proof connotes finality. – user11266 Feb 28 at 15:11
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Einsteins paper showing proof should help you. The proof is not hard to grasp.

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This paper by Hecht may answer your questions. Hecht takes great measure to emphasize what Einstein actually did and did not write.

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