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A black hole has a radius of $R = \frac {2Gm}{c^2}$, in this context, if we take a single proton and neutron as a black hole, its Schwarzschild radius will be near about $4.8 \times 10^{-52} \mathrm{m}$. 

Now quantum mechanics says that both particles can stay together if they satisfy the uncertainty relation as $$\Delta P \Delta R \geq \hbar$$ by taking mass as the mass of the proton and space as the Schwarzschild radius; this uncertainty relation is definitely violated.

However, in this scenario, a black hole is not consistent with the quantum theory. Please answer my question.

Níck Aguiar Alves
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Flynn Ryder
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1 Answers1

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Black holes are a prediction of classical general relativity, which completely ignores any sort of quantum effects. At small scales, this is a very bad approximation and most physicists (perhaps all of them) believe general relativity will fail to provide an accurate description and we will need something else, such as a full theory of quantum gravity.

A possible solution would be the quantization of the gravitational field. In this case, the uncertainty on the particle's position would also lead to an uncertainty on the gravitational field itself, and even on the black hole itself. Through some mechanism like this, you might be able to recover Heisenberg's principle. (This paragraph is speculative: I'm merely illustrating a possible solution, not claiming it is the correct one. No one knows what happens with gravity at such small scales).

Shortly, black holes are classical solutions to general relativity. They do not take quantum effects into consideration.

Níck Aguiar Alves
  • 24,192