# Did Einstein prove $E=mc^2$ correctly? [closed]

In his book "Einstein's mistakes" H. C. Ohanian, the author, holds that Einstein delivered 7 proofs for $E=mc^2$ in his life that all were in some way incorrect. This despite the fact that correct proves had been published and mistakes in his proofs were sometimes pointed out to him.

The first proof e.g. contains a circular line of thought in that it falsely assumes special relativity to be compatible with rigid bodies.
EDIT: Reference: stated by V. Icke, prof. theoretical astronomy at Leiden University in his (Dutch) book 'Niks relatief': "Einstein made an error in his calculation of 1906".

Do you think Ohanian's statement is true or was/is he biased in his opinion?

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Terence Tao has written on Einstein's derivation here terrytao.wordpress.com/2007/12/28/einsteins-derivation-of-emc2 –  j.c. Nov 24 '10 at 22:12
(alert: rhetorical question) do you think Ohanian was right? And more importantly, can you back up your opinion with a reasoned argument? If you have a specific objection to one of Einstein's derivations of $E = mc^2$, you can certainly ask about that here, but this is not the place to poll the community to see who agrees with such-and-such an opinion. Also, asking whether a certain person was biased or not is not a physics question; it strikes me as more of a history question. –  David Z Nov 27 '10 at 4:14
I'm not sure of the merits of this question, but it does seem that the word "arrogant" is being thrown around a lot in the answers. I think we can all agree that all physicists, as a rule of thumb, are arrogant to some degree (or considered to be so by the general public) if only by virtue of the fact that our resumes contains references to such things as grand unification and theory of everything. So it would be nice if we could stick to debating this question on its merits alone. Just my contribution from the peanut gallery. –  user346 Jan 13 '11 at 18:18
I'm downvoting the question. It tells us there's a book that contains a specific argument, and asks us for our opinions on that argument. But we don't have access to the book, and most of us don't speak Dutch, so we wouldn't be able to read the book if we did have access to it. The question can't be answered unless Gerard tells us what the mysterious argument is. In general, I have not been impressed with the material I've seen from the Ohanian book. Specifically, the discussion of length contraction and W.F.G. Swann is totally bogus. –  Ben Crowell Aug 14 '11 at 21:38
There are some strange inconsistencies in the question. "The first proof" would have to refer to the 1905 paper titled "Does the inertia of a body depend upon its energy content?," but this date would contradict the 1906 date in the title of the Icke book. In discussion below, Gerard says, "His proof involves a photon that hits a rigid body." The 1905 paper doesn't involve photons at all, it discusses only emission of light, not absorption, and it never mentions a rigid body. –  Ben Crowell Aug 15 '11 at 0:00

## closed as not constructive by Manishearth♦May 23 '13 at 21:11

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I will exaggerate a bit, but in physics, proof in the sense of mathematical proof is irrelevant. Even if all of Einstein's deductions of the formula were wrong, it still turns out that empirical evidence supports $E=mc^2$.

Now, without the exaggeration, mathematical deduction is important in physical theories because it shows us how conclusions and principles hang together. This can be important when elaborating further theories. Imagine, for the sake of argument, it turns out that the relativity principle is not correct, yet $E=mc^2$. Since usually we deduce the latter from the former, there is something interesting here, it would mean that $E=mc^2$ is more fundamental than the principle of relativity. Special relativity arises itself from this kind of considerations. Realizing that the invariance of the speed of light is more fundamental than the invariance of time, the latter being only approximately true at very low speeds.

EDIT: I still wasn't able to read what Ohanian is saying in particular, but it is no secret that Einstein was not a great mathematician. For instance, if it was not for the help of his friend Marcel Grossmann, Einstein might never have been able to develop the theory of general relativity. From his intuition about the equivalence principle in 1905 to the actual GR in 1915, he had to toil for 10 years with non-Euclidean geometry. In the meantime, he nearly got overtaken by David Hilbert. (See Marek's comment.)

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Corrections: Einstein was far from a lousy mathematician, he just wasn't a great one. Also, Hilbert was nowhere close. He was a (rather arrogant) mathematician, and a somewhat mediocre physicist. –  Noldorin Nov 24 '10 at 19:12
@Raskolnikov: My oh my, what is this with all the anti-Einstein and anti-Hilbert sentiment? Almost nothing that has been said here is true. Einstein wasn't overtaken by Hilbert at all. Hilbert just wrote the E-H action, but this was only in 1915 when he was already familiar with all the Einstein's work. Not only that but he himself admitted all the credit to Einstein. –  Marek Nov 26 '10 at 0:35
@Noldorin: Seem to me that you're quite quick to call people arrogant without actually knowing anything about them. What Hilbert said is actually a common knowledge and he obviously meant only the fact that many physicists are not really able to handle mathematics required for the modern physics and that they don't care for proofs and formal correctness. Actually, it can also be said that "mathematics is too hard for mathematicians" because many of them don't have the physical intuition :-) In any case, take it easy and don't be so quick to judge others ;-) –  Marek Nov 26 '10 at 0:44
@Noldorin: nobody's telling you what to do. I am just saying that you're a little uptight and it wouldn't hurt if you took things a little easier. It is just my advice and you're certainly free to ignore it; but you can count on me arguing with you again simply because I don't like your judgemental behavior. I don't think one has to be a moderator to point out obvious flaws in your argumentation. –  Marek Nov 26 '10 at 17:43
And just a little note, @Noldorin: when I told the Hilbert's statement to my theoretical physics friends some time ago, all of them laughed and agreed. I assume same situation happened long ago with Hilbert's physicist friends. It's quite a pity you see arrogance here. But of course, you are free to hate whomever you want, me and Hilbert included :-) –  Marek Nov 26 '10 at 17:46

The argument is summarized on the Wikipedia page "Mass/Energy equivalence", and it goes like this: imagine a body at rest which then emits two equal photons, one to the right and one to the left. In the rest frame, the body is still at rest because the photons have equal and oppsite momentum.

Shifting to a frame moving to the right. In this frame, the photon moving to the left is blueshifted and carries more momentum, and the photon moving to the left is redshifted and carries less momentum. This means that the object has lost some right momentum after the emission. The velocity is the same before and after, because the velocity is the same in the rest frame, so how could the body lose momentum without changing its velocity? It must have lost mass. If you calculate the lost mass, it equals the energy of the photons divided by c2.

This argument is obviously correct, essentially rigorous (it requires a precise framework to state rigorously, but there are no imprecise assumptions). The bickering came because Einstein demonstrated this and not anyone else, everyone else thought that the mass/energy relation was E=4/3 mc^2. Soon after, Poincare realized what everyone's mistake was.

Planck also discovers E=mc^2, and published after Einstein (but I would bet his work was independent). He just refuses to accept that Einstein's argument is correct, and says his argument is the correct one. This is possibly because of his bitterness at being scooped at such an important result. From this attack come all future error claims.

Ohanian's book in general gets everything wrong. Here is a complete list of Einstein's mistakes (I put an expanded version of this on Wikipedia years ago, but it slowly got reworded, watered down, and moved. That gradual process, of course, was the work of Satan):

### Einstein's mistakes

• 1905: In the original German version of the special relativity pape, Einstein gives the transverse mass as $m/(1 - v^2/c^2)$, while the actual value is $m/\sqrt{1 - v^2/c^2}$ (Max Planck corrected this).
• 1905: In his PhD dissertation, the friction in dilute solutions has an miscalculated numerical prefactor, which makes the estimate of Avogadro’s number off by a factor of 3. The mistake is corrected by Einstein in a later publication.
• 1905: An expository paper explaining how airplanes fly includes an example which is incorrect. There is a wing which he claims will generate lift. This wing is flat on the bottom, and flat on the top, with a small bump at the center. It is designed to generate lift by Bernoulli’s principle, and Einstein claims that it will. Simple action reaction considerations, though, show that the wing will not generate lift, at least if it is long enough.
• 1913: Einstein started writing papers based on his belief that the hole argument made general covariance impossible in a theory of gravity. Einstein realized he was wrong in 1915, and finds General Relativity.
• 1922: Einstein published a qualitative theory of superconductivity based on the vague idea of electrons quantum-mechanically shared in orbits. This paper predated modern quantum mechanics, and is well understood to be completely wrong. Einstein's paper is more of an old-quantum-mechanical version of the modern explanation of ordinary conductivity.
• 1937: Einstein believed that the focusing properties of geodesics in general relativity would lead to an instability which causes plane gravitational waves to collapse in on themselves. While this is true to a certain extent in some limits, because gravitational instabilities can lead to a concentration of energy density into black holes, for plane waves of the type Einstein and Rosen considered in their paper, the instabilities are under control. Einstein retracted this position a short time later, but his collaborator Nathan Rosen maintained that gravitational waves are unstable until his death.
• 1939: Einstein denied several times that black holes could form, the last time in print. He published a paper that argues that a star collapsing would spin faster and faster, spinning at the speed of light with infinite energy well before the point where it is about to collapse into a black hole. This paper received no citations, and the conclusions are well understood to be wrong. Einstein’s argument itself is inconclusive, since he only shows that stable spinning objects have to spin faster and faster to stay stable before the point where they collapse. But it is well understood today (and was understood well by some even then) that collapse cannot happen through stationary states the way Einstein imagined.

There's other mistakes that are not mistakes, but philosophical things:

• In the Bohr-Einstein debates and the papers following this, Einstein tries to poke holes in the uncertainty principle, ingeniously, but unsuccessfully.
• In the EPR paper, Einstein concludes that quantum mechanics must be replaced by local hidden variables. The measured violations of Bell’s inequality show that hidden variables, if they exist, must be nonlocal.

Einstein considered the cosmological constant a mistake, but the cosmological constant is necessary within general relativity as it is currently understood, and it is widely believed to have a nonzero value today.

He had lapses in taste too, usually quickly corrected:

• Einstein briefly flirted with transverse and longitudinal mass concepts, before rejecting them.
• Einstein initially opposed Minkowski’s geometrical formulation of special relativity, changing his mind completely a few years later.
• Based on his cosmological model, Einstein rejected expanding universe solutions by Friedman and Lemaitre as unphysical, changing his mind when the universe was shown to be expanding a few years later.
• Finding it too formal, Einstein believed that Heisenberg’s matrix mechanics was incorrect. He changed his mind when Schrödinger and others demonstrated that the formulation in terms of the Schrödinger equation, based on Einstein’s wave-particle duality was equivalent to Heisenberg’s matrices.
• Einstein rejected work on black holes by Chandrasekhar, Oppenheimer, and others, believing, along with Eddington, that collapse past the horizon (then called the ’Schwarzschild singularity’) would never happen. So big was his influence, that this opinion was not rejected until the early 1960s, almost a decade after his death.
• Einstein believed that some sort of nonlinear instability could lead to a field theory whose solutions would collapse into pointlike objects which would behave like quantum particles. This is impossible by Bell’s inequality.

It is sometimes claimed that the general line of Einstein’s reasoning in the 1905 relativity paper is flawed, or the photon paper, or one or another of the most famous papers. Those claims are all ridiculous.

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Just a small point about wings. Pilots learn to think in terms of the momentum of the downdraft created by a wing. Bernoulli is only the reason why the air over the top of the wing is sucked downward. –  Mike Dunlavey Sep 13 '11 at 13:42
@Mike: yes, of course, this is why Einstein's example is not very good. It is remarkable that he wasn't thinking this way in 1905. –  Ron Maimon Nov 28 '11 at 8:44

Einstein's proof did not rely on having a rigid body. It relies on having a body with mass (obviously). To be more clear.

• The paper only says body.
• It does not rely on any rigid body property (like the size)
• It does not rely on any relativistic speed or condition on the body

The proof merely involve how energy is measured by an observer stationary with the body and one stationary with the emitted waves.

Let there be a stationary body in the system $(x, y, z)$, and let its energy--referred to the system $(x, y, z)$ be $E_0$. Let the energy of the body relative to the system $(\xi,\eta,\zeta)$ moving as above with the velocity $v$, be $H_0$.

In fact the paper is really easy and clear... :-)

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nonsense, the paper is not really easy and clear, hence the controversy. –  Physiks lover Nov 14 '12 at 0:25
Oh, you think it's really easy and clear? I suggest you have a proper read of it! –  Larry Harson Jan 28 '13 at 16:20

In special relativity, it's only accelerating bodies which are not allowed to be rigid. Non-accelerating bodies don't have any forces on them, so there is no obstacle to their retaining the same shape. I haven't checked, but I believe Einstein's intuitive derivation of relativity didn't involve any accelerating bodies.

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His proof involves a photon that hits a rigid body. –  Gerard Nov 25 '10 at 18:18
... which then accelerates it an infinitesimal amount, thus making it infinitesimally non-rigid? If you stick in some $\epsilon$'s and $\delta$'s, this should even be a good enough proof for mathematicians. Of course, he didn't, so I guess it's not quite a rigorous proof, but by physics standards it definitely passes. –  Peter Shor Nov 25 '10 at 19:46
See my edit for a reference from a professor in theoretical astronomy and strong advocate of A.E. Can you prove him wrong? –  Gerard Nov 26 '10 at 8:15
@Gerard : you should either restate the argument of Vincent Icke, and not arrogantly point @Peter to a book written in Dutch, expecting that @Peter learns Dutch (if he doesn't already speaks it), reads the book, find in it the place where you think the argument is stated, and refute it. He already gave you a quite explicit explanation on why the rigid body problem is not so difficult. I don't think your bad usage of the authority argument has place here, and I'm definitely sure it has no place against Peter. –  Frédéric Grosshans Dec 3 '10 at 13:49
@Gerard: "His proof involves a photon that hits a rigid body." You (and Icke) seem to be referring to a different argument, not the 1905 one. See my comments on your question. –  Ben Crowell Aug 15 '11 at 1:09

The answers do not address what Ohanian said. His paper is a free download. As far as I know, Ohanian has not been refuted.

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His paper is so trivial to refute, it is hardly worth the bother: he is saying that Einstein is wrong to use the nonrelativistic expression for energy in his argument, but the velocity of the object that Einstein is considering is entirely due to shifting reference frame, and the velocity of the shift can be infinitesimally small. In addition, while Einstein might have asked "what is the kinetic energy of the body" to determine the loss of mass, the system loses mass when you ask "what is the linear momentum of the body (as explained in my answer). The only relativistic thing is the light. –  Ron Maimon Aug 16 '11 at 4:19
I remember reading Einstein's original paper on this, and it seemed to be argued pretty clearly. I believe he considers a scenario involving the emission and absorption of photons, and uses the length/time dilation factors to get an expression for energy, which hey then takes to the classical (Newtonian) limit and equates with $\frac{1}{2}mv^2$ to show the relation.