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There is a Physics.SE question asking for a proof of $$E=mc^2$$ Multiple users on this forum responded with an argument that is summarized by the user Abhimanyu Pallavi Sudhir as follows:

quote: "The proof is fairly simple. An object releases light $$. in opposite directions, and you consider the rest frame and a frame moving at $$. You consider a low-velocity approximation, where the Doppler factors $\sqrt\frac{1+}{1−}$ and its inverse approach $1+$ and $1−$. 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 $$, but that's just 1), so we argue that the change in momentum due to the light in the moving frame of reference must be $(1+)/2−(1−)/2=$. So this must equal the change in momentum from the decrease in mass $$ (the mass is moving at $$ in the opposite direction in the moving reference frame), and we have $=$, or $=$."

See Debanjan Biswas' answer in the linked forum for a more in-depth explanation of this argument.

I am very new to Physics (I study mathematics in uni), so please forgive me if this is a naïve question. I am a little confused about the logic of this proof in the sense that I do not understand why this shows that $E=mc^2$ in general. I see that it proves $E=mc^2$ in the specific case of an object releasing two identical pulses of light in opposite directions. But Einstein's conclusion is that $E=mc^2$ holds in all scenarios, not just in the one described above.

Was this argument more of a hint, or clue, that motivated Einstein to conjecture that $E=mc^2$ holds in general (which would then need to be verified empirically), as opposed to a full-blown proof? Or does this argument indeed apply more widely without the loss of generality, and I am simply misunderstanding the argument?

Amit
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LYB
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    Welcome Linus. I suggest that you read Dmitry Brant's answer on that same post you linked to. Physical laws can never be proved in the same sense that mathematical theorems can. – Amit Jun 19 '23 at 19:19
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    This argument (and other similar ones due to Einstein) really only shows $\Delta m = \Delta E/c^2$. $E=mc^2$ is the simplest and most convenient choice. But it is a choice, how we define energy of a body at rest. https://hsm.stackexchange.com/questions/2498/why-is-einsteins-mass-energy-relation-usually-written-as-e-mc2-and-not-de – Ján Lalinský Jun 19 '23 at 19:41
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    The experimental "proof" comes in form of e.g. electron-positron annihilation. The total energy of the two resulting gammas matches the mass-energy of the electron and positron. I would note that the general concepts of mass and energy seem to become much more problematic once we admit gravity into the mix, though, so if you are asking in "all generality", then it will probably have to become a lifelong obsession rather than a short answer. – FlatterMann Jun 19 '23 at 20:06
  • Clearly that is not a 'proof', rather a physical argument which secretly assumes a lot, the sort of thing one might put in the intro of a paper to convince people that what you say might be true. See eg the book by Weinberg for a better derivation from first principles. – SvenForkbeard Jun 20 '23 at 07:41

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If experience shows that light energy can be converted into an equivalent amount of other types of energy (kinetic, electrical, thermal), and these energy types are all measured to be consistent – i.e. 100 Joules of electrical power to a lamp is measured to generate 100 Joules of light and heat radiation, and that heats up an object by a temperature corresponding to 100 J increase in thermal energy – then it follows that energy $E$ of light carrying $E/c^2$ of mass must result in the equivalent amount of thermal or electrical energy also carrying that amount of mass. This is essentially a type of transitive property $$A=B~~,~~B=C \implies A=C $$

required in order for existing physics to be consistent.

Also, in Einstein's thought experiment, he makes no assumptions about the nature of the energy emitted, and simply says "a pulse of light (i.e radiation)" that obeys Maxwell's equations by which he derived the

$$\sqrt{\frac{1+\nu}{1-\nu}}$$

factor in an earlier paper. And he uses opposite directions for symmetry so there is no change in the momentum of the emitting object.

But you are right in that the result presented in the original short paper with that example is not intended to be a rigorous physical proof. He's saying, "Here is a noteworthy result that seems to imply all energy corresponds to some amount of mass. It is an area that warrants further study."

RC_23
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One should also note the context in which Einstein is working. He did not just randomly decide, one day, to study the relationship between mass and energy. At the time, one of the most important arguments in physics was how much does electromagnetic mass contribute to the mass of a body. There was 8/3, there was 4/3, the derivation that it is bigger than 1 is already a headache, people were rather annoyed about the situation.

That was the context in which Einstein derived a relation which showed that only 1 is allowed, no other value. While there was not yet the discovery of antimatter, which would allow for the entire mass to be converted to energy and vice versa, it is still a fact that Einstein derived a relation that allows us to simply put the entire mass, or entire energy, into the formula and get the result we want. There was, after all, no restriction on how energetic the photons could be.

Einstein may not yet have derived the deeper relation $$\tag1\left(\frac Ec\right)^2-\vec p\,^2=(m_0c)^2\qquad\implies\qquad E=mc^2$$ but this invariance relation is just like the spacetime invariant interval $$\tag2\mathrm ds^2=-\mathrm dt^2+\mathrm dx^2+\mathrm dy^2+\mathrm dz^2$$ that is all over the place in the Special Theory of Relativity. An argument along this line would also strongly suggest that there is a strict equality, not just a difference equality. But this might be from Minkowski rather than Einstein. I am not a historian.

Still, Einstein tended to think in general terms, so his considerations likely got this far.

Note that in many places in the Special Theory of Relativity whereby the specifically written stuff references light, but the theory really does not care. Light just so happens to be the topic of discussion because it is the first phenomenon that hints to us the correct operation. The actual thing that we are interested, is general and universal to the physics.

naturallyInconsistent
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