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.
- 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.
Best Answer
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).