Necessity of Quantum Gravity given Planck scales for nuclear physics

[Phys12]

TL;DR

Given that the Planck mass is about $10^-8$ kg and Planck length is about $10^-35$ m, do we need to understand quantum gravity to understand nuclear physics?

In the solutions (page 6, points ii) and iii)), https://ocw.mit.edu/courses/physics...pring-2013/assignments/MIT8_04S13_ps1_sol.pdf, it is mentioned that given that the Planck mass is about 20 orders of magnitude larger than a proton and that the Planck length is about 20 orders of magnitude smaller than nuclear radius, we do not need a theory of quantum gravity to understand nuclear Physics. However, it seems like the part about the Planck mass being 20 orders of magnitude larger than a proton suggests that you do need a theory of quantum gravity to study nuclear Physics, but the part about the Planck length being 20 orders of magnitude smaller than a nuclear suggests that you do not need a theory of quantum gravity to study nuclear Physics. How do they reconcile the two and suggest that ultimately, we can ignore quantum gravity for nuclear Physics?

 

[Dale]

Summary:: Given that the Planck mass is about $10^-8$ kg and Planck length is about $10^-35$ m, do we need to understand quantum gravity to understand nuclear physics?

How do they reconcile the two and suggest that ultimately, we can ignore quantum gravity for nuclear Physics?

There is nothing to reconcile. Before Plank scale effects would become relevant you would need a roughly a Planck mass inside a Planck radius. A proton is nowhere near that.

 

[Phys12]

There is nothing to reconcile. Before Plank scale effects would become relevant you would need a roughly a Planck mass inside a Planck radius. A proton is nowhere near that.

So because the proton is not inside a Planck radius (but it is inside Planck mass), you don't need quantum gravity? You need both the mass and radius of the particle to be under the Planck mass and radius for the effects of quantum gravity to have any contribution?

 

[Dale]

You need both the mass and radius of the particle to be under the Planck mass and radius

No, you need the mass over and the radius under. A proton has neither a large enough mass nor a small enough radius

 

[Phys12]

No, you need the mass over and the radius under. A proton has neither a large enough mass nor a small enough radius

So only for a black hole would the effects of quantum gravity be relevant, right? Or are there other cases as well where the effects would matter?

 

[PeterDonis]

You need both the mass and radius of the particle to be under the Planck mass and radius for the effects of quantum gravity to have any contribution?

Not quite. The way to think about it is as a density: you need a density of the same order of magnitude as one Planck mass per Planck volume, i.e., one Planck mass per Planck length cubed. The mass of the proton is 20 orders of magnitude larger than the Planck mass, but the volume of the proton is 60 orders of magnitude smaller, so the net effect is a density 40 orders of magnitude smaller than the Planck density.

The reason density is the important parameter is that, according to the Einstein Field Equation, energy density (and more generally density of stress-energy) is the source of spacetime curvature. So to get spacetime curvature intense enough that quantum gravity effects are expected to be significant, you need a density large enough to cause that order of magnitude of spacetime curvature. In fact, if you use "geometric" units where mass has the same units as length, density and curvature have the same units--inverse length squared. So another way of thinking of the above is that you need spacetime curvature of order one inverse Planck length squared, which is about 40 orders of magnitude larger than the spacetime curvature caused by a proton.

 

[Dale]

Oops, yes, I made a mistake above