What was Albert Einstein's proof for $E=mc^2$?

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?

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.

• @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 '13 at 2:03
• @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 '13 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 '13 at 14:43
• @Qmechanic, thanks for the nice edit with the link to the english version. I've just added the link to the original one too. What does that "hattip: userXXXX" means? – Eduardo Guerras Valera Feb 28 '13 at 3:53
• The proper spelling is in two words: hat tip. – Qmechanic Feb 28 '13 at 7:03

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.

• 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
• 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 --- ex-moderator kitten Feb 28 '13 at 5:02
• 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 '13 at 5:26
• 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 '13 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 '13 at 15:11

It is incorrect to say that $E=m$ cannot be proven, it is a mathematical theorem and can be proven within the axiomatic system of special relativity. The point of experiment is to verify this axiomatic system or theory (or more precisely, to alter our confidence in this axiomatic system), but "it's all experiment" is an unsatisfying and useless answer when it comes to understanding the math. It's particularly bad in this case, because it doesn't teach you what energy is, so it's silly to say you can experimentally prove things about it.

The assumptions you need to prove this (and other relativistic dynamics) are: (1) the postulates of special relativity (2) de Broglie's relation (3) conservation laws (this along with 1 also imply the Newtonian approximation at low speeds, but it's better to list it directly, since it's where the definitions of energy and momentum come from). I mean, in principle you could use some other set of results as your axioms, but that's useless and boring, don't do it.

The proof is fairly simple. An object releases light $E$ in opposite directions, and you consider the rest frame and a frame moving at $v$. You consider a low-velocity approximation, where the Doppler factors $\sqrt{\frac{1+v}{1-v}}$ and its inverse approach $1+v$ and $1-v$. The total energy is clearly conserved anyway, so we look at momentum conservation instead -- the momentum is equal to the energy in magnitude (by a factor of $c$, but that's just 1), so we argue that the change in momentum due to the light in the moving frame of reference must be $(1+v)E/2-(1-v)E/2=vE$. So this must equal the change in momentum from the decrease in mass $m$ (the mass is moving at $v$ in the opposite direction in the moving reference frame), and we have $vE=mv$, or $E=m$.

You can also use this to prove other dynamical results, see e.g.

This is Einstein's derivation -- there's a lot of crap online, such as from minutephysics, claiming to be Einstein's original derivation, and if you believe them, I have a mass-loss bridge to sell you.

• In this derivation you assume Einstein's relation $E=h\nu$ for the light pulses, which is unnecessarily special assumption. The formula at question is valid for any energy loss via EM radiation, not just those that conform to that idea of quantization. What is really needed are the transformation properties of momentum and energy of a light pulse (the relation $p=vE$ you have used above). These follow directly from EM theory and Lorentz transformations, no quantum theoretical assumptions are needed. – Ján Lalinský Jul 5 '18 at 22:13
• @JánLalinský You're right -- we just need $E=pc$ for light. Deriving it from $p=vE$ just seems un-motivated (and also trivial). – Abhimanyu Pallavi Sudhir Jul 7 '18 at 4:23