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 $c^2$.
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.
