Is special relativity relevant to understand nuclear bombs? It is often claimed that Special Relativity had a huge impact on humanity because its understand enabled the invention of the nuclear bomb. Often, the formula $E = mc^2$ is displayed in this context, as if it would explain how nuclear bombs work. I don't understand this line of reasoning.

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*The key to understanding nuclear fission is understanding neutron scattering and absorption in U-235. For this, quantum mechanics is key.

*Bringing quantum mechanics and special relativity together correctly requires quantum field theory, which wasn't available in the 1940's.

*When U-236 breaks up, it emits neutrons at 2 MeV (says Wikipedia), which is a tiny fraction of their rest mass, that is to say, far below light speed. Meaning hopefully that non-relativistic QM calculations should give basically the same answers as relativistic ones.

So it seems to me that in an alternate universe where quantum mechanics was discovered and popularised long before special relativity, people would still have invented fission chain reaction and the nuclear bomb, without knowing about special relativity.
Or, more precisely: You don't need to understand special relativity to understand nuclear bombs. Right?
Of course it's true that you can apply $E = mc^2$ and argue that the pieces of the bomb, collected together after the explosion, are lighter than the whole bomb before the explosion. But that's true for a TNT bomb as well. And what's more, it's an impractical thought experiment that has no significance for the development of the technology.
 A: "Often, the formula $E=mc^2$ is displayed in this context, as if it would explain how nuclear bombs work. I don't understand this line of reasoning."
Neither do I. $E=mc^2$ provides a neat way of calculating how much energy is released in nuclear fission, but to understand why this energy is released I need an explanation in terms of nuclear forces.
A: The fact that a chain reaction can continue with a neutral particle (neutron) causing fission, and the tremendous release of energy from the decrease in rest mass from the fission reaction, were very quickly recognized.  It was then obvious that fission could quite likely be utilized for a massively powerful bomb.
"Release of energy" (from chemical reactions) was long recognized and called "energy (or enthalpy) of formation".  Calorimetry measurements were used to evaluate the energy release for chemical reactions, such as combustion.
Later, special relativity explained this energy release as a change in rest mass. So, even without special relativity the concept of a reaction releasing energy was known.  The key point for fission is how an estimate of the energy release was estimated. Fission was discovered by observing fission products (e.g. barium). Special relativity had been developed (1905) when fission was discovered (1938), so the equivalence of energy and mass was understood. Using $E = mc^2$ allowed an estimate of the tremendous amount of energy released per fission reaction (about 200 MeV) compared to a chemical reaction (about 10 eV).  I do not know if there was another way to estimate this energy release based on the early observations of fission; I do not think early experiments with fission created enough reactions to utilize calorimetry, but I could be wrong.
A: in 1938 Lise Meitner was instrumental in linking the energy gained from the fission reaction to the drop in mass. As soon as this was noticed by the scientific community a number of weapons programs were started (in secret of course). The formula showed that the result was not a mistake, and that an enormous amount of energy could be tapped at a time when war in europe was looming.
While you could probably find a way to ignore $E=mc^2$ it is a very simple equation that is well recognised by the public, I think this is the main reason it gets quoted so often.
A: Promoted from a comment by Andrew:

I would actually take the opposite point of view. TNT releases energy in chemical bonds. In principle, there is some tiny amount of mass associated with the chemical bonds because of $E=mc^2$. This mass is lost in TNT. Nuclear binding energy is much larger than chemical binding energy, so while you can get away with saying mass is conserved with chemical reactions because you can't measure the change in mass, you cannot ignore the change in mass in nuclear reactions. (Keep in mind that since $c^2$ is huge in SI units, apparently tiny changes in mass correspond to huge amounts of energy).

Furthermore, the reply

Indeed, nuclear Q values are no different from Q values for chemical reactions. Knowing that Q values corresponds to rest mass energy differences adds understanding, but is not needed to work out the physics of fission, which is basically non-relativistic.

A: It's certainly relevant. Mass is very measurably non-conserved in nuclear reactions. Using special relativity allows us to determine the potential energy release of a given nuclear reaction just by directly measuring the masses of the nuclei involved and using $E=mc^2$ to convert that to an energy.
For instance, consider the reaction:
$$ \rm ^{235}U+n\to {^{140}Xe}+ {^{94}Sr}+2n $$
The masses on the left add up to about 236.05 atomic mass units, while the masses on the right add up to about 235.85 atomic mass units. Multiplying the difference by $c^2$ gives an energy of $185~\rm MeV$.
In other words, special relativity allows us to take measurements of a few hundred masses and to determine fission candidates that way rather than having to painstakingly attempt to experimentally determine it for every possible isotope. It also gives us a way to measure the energy release independent of having to actually precisely measure the kinetic energy of all the daughter particles (which is hard), and instead only requires us to identify the daughter isotopes and measure their masses (which is less hard).
A: In Serber's book The Los Alamos Primer, he explains the energetics of fission bombs and asserts that the actual fission process (i.e., the impact of a neutron with a certain amount of incident energy upon a metastable nucleus, causing it to deform and neck down into two primary fission fragments which start off almost touching and then electrostatically repel one another with tremendous force and thereby accelerate to great speed) is entirely nonrelativistic.
(Nuclear forces enter the picture in the sense that in the uranium nucleus, the attractive nuclear force is just barely capable of holding the nucleus together against the mutual repulsion of all the protons in it, and when furnished with the right activation energy, the nucleus flies apart.)
But to calculate the energy yield of the fission process, one easily notes the resulting mass deficit among the primary and secondary fragments and uses Einstein's formula to equate that to a net energy release per fission.
A: Without knowing anything about relativity, a good guess would have been $E = 1/2 mc^2$ - that is the kinetic energy of a mass accelerated to the speed of light, ignoring relativity. That's half as much as the correct formula, but still plenty to give an almighty explosion and plenty to start trying to build a nuclear bomb.
And while we can determine quite precisely how much mass is lost by the fission of a single uranium atom, we don't actually have any idea about what percentage of the uranium will become part of the explosion. 30% fissioned with $E = mc^2$ or 60% fissioned with $E = 1/2 mc^2$ would give exactly the same explosion.
So I very much doubt that relativity was of any importance to building and understanding of nuclear bombs. With my alternative formula, our estimates of what percentage of uranium is part of the explosion would be off by a factor two, but it wouldn't make any difference. Only if the percentage is suspiciously high physicists would say "it's hard to imagine how we managed to make 90% of the uranium part of the fission process", or worse "we calculated that 110% of the uranium fissioned - clearly something is going on that we don't understand".
