I know it’s only theoretical, but why couldn’t something be just slightly smaller?
In: 658
One part of the answer is that our models for understanding of fundamental physics (i.e., relativistic quantum field theory, including the Standard Model) rely on spacetime being flat, or at least flat to a good approximation. In this context, “flat” means basically close enough to what intergalactic space looks like that the difference doesn’t matter, in contrast to near the event horizon of a small black hole, where spacetime is very warped.
What warps spacetime is the presence of energy in some form (usually mass — i.e., the warping of spacetime which is how gravity works). But fundamental particles have mass and energy, and the energy is related to the wavelength through Planck’s constant and the speed of light — E = h c / lambda — at least approximately.
So when you have really small distances being relevant, that means you have really high energies, and that means that ends up meaning that space is warped on a level that is no longer negligible at the distances you are talking about. So the very assumptions that we build relativistic quantum mechanics on no longer work.
To elaborate a little further: The Planck length can be thought of as the wavelength of a photon such that if you convert that photon’s energy into a point mass, the orbital speed at a radius of that wavelength is the speed of light. The actual equations give some factors of small integers and pi and so forth, but the order of magnitude works out.
The reason you can just combine G, h, and c to get this length is because of a strategy of getting approximate answers to physics problems through dimensional analysis — factor out all the dimensionful quantities (in this case, G, h, and c) and you are left with some math equation you need to solve, where the answer is probably close-ish to 1, and so the stuff you factored out is close to your answer. Since you are looking for a length, it *has* to be proportional to sqrt(hG/c^(3)).
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