Firstly, I'll note that I read your comment about another answer here seeming like pop sci to you. I will also not leave the answer at only my next two words.
> deeper explanation I could grasp without reading several books
Probably not, and I'm not sure undergrad SR helps enough, although I'll assume you encountered quantum electrodynamics, and so guess that maybe we can slyly work out a weak comma-goes-to-semicolon approach[1], replacing partial derivatives in Schrödinger's i\hbar \partial \psi /\partial t=-(\hbar^2/2m)\nabla^2 \psi with covariant derivatives and ignore that we have only Galilean invariance in the time-dependent Schrödinger eq (TDSE), rather than full Lorentz covariance. Not ignoring that turns into a course in differential geometry and curved spacetime. In fact, I will even trim the "explanation" to "we can adapt TDSE to some distributions on some spacetimes, changing its partial derivatives to covariant derivatives, which looks really easy when you just write it (i\hbar \partial \psi /\partial t=-(\hbar^2/2m)\nabla^2 \psi), but making that typographical trick work is where you find the 'deeper explanation'". But the takeaway is that doing weakly-relativistic quantum mechanics in a weakly-curved spacetime (like our solar system, or even the overwhelming majority of the central parsec of our galaxy) is tractable, and we can leave "the problem of time" in a box to be opened at leisure by philosophers.
Now we've taken an initial step towards QM on a more general spacetime than that of the very special one of SR.
Let's step back a bit.
If you can cope with Gauss and Poisson laws for electromagnetism (http://web.mit.edu/6.013_book/www/chapter4/4.3.html and surrounding sections in that chapter), and the usual Schrödinger equation, you can take a step towards a Gauss law for gravity (\Phi_g = -4 \pi Gm_{enc}, where we have substituted the constant 1/ε_0 with -4 \pi G, and the enclosed electric charge q_{enc} becomes the enclosed "gravitational charge" m_{enc}). As Carroll says in the fine article at the top, we can quantize this just fine, given at least the conditions in my first paragraph (and typically several others too).
Purely electromagnetically, one might set out a number of sources, with each source's generating a Poisson potential. Those sum linearly through the principle of superposition, so this is just fine. We can then perform a "second quantization" or "canonical quantization". This goes outside your implicit scope, but essentially we convert a classical electromagnetic system into a quantum field theory by this procedure. From that we can recover our usual TDSE. This is a grad topic, books required.
Gravitationally, we can follow the same approach. We have to be careful with our distributions of sources (mass, rather than electric charges), but if they are far enough apart (the idea being that the Poisson potential for any source asymptotes to zero at infinity) and not moving relativistically (i.e., compared to the speed of light), we can do a "second quantization" of gravitation. This is called Canonical Quantum Gravity, and works well for many applications. I think CQG is beyond an undergrad background without resort to books. Try to see how much of DeWitt you can follow at [2].
Now let's return to "my" elephant in the room: the superposition principle. The matter we describe with either form of TDSE above superposes linearly: we can take a distribution of charged sources and completely determine their interactions, and we can drop in a test particle that feels these interactions, so can probe them by being attracted to our various sources, even as we strongly densify sets of sources. This works for all the charges in the Standard Model, with some effort. Unfortunately, this fails for our notion of the gravitational charge m_{enc}: as we densify m, we can get to a runaway collapse, which we capture in the nonlinearity introduced in Schrödinger-Newton. Worse, even without the runaway, we can't in general linearly superpose the gravitational potential from significant sources (for less significant sources we can pretend that instead of asymptoting to zero at infinity, the potential reaches zero closer to the source; we do this for other long-range potentials too, in practice). So we can't with generality solve the whole Schrödinger-Newton calculation: we need to be careful about the distribution of mass.
We already have this problem in classical General Relativity, where the Raychaudhuri equations lead to caustics, so it's not exactly surprising that a flavour of that appears in a system which arose from quantizing classical General Relativity. Dealing with those takes some mathematical work classically and in QFT approaches like the one I've very briefly sketched above, and there is no approach known to be fully general. Caustics associated with colliding gravitational waves is something we can't deal with satisfactorily, and we have reasons to suspect those may occur in a relevant way in the very early history of our universe. Singularities in the deep interior of black holes are something we can't deal with satisfactorily either. However, even in classical General Relativity practically nobody trusts the interior solution of the Kerr metric (not even Kerr himself! [3]). And of course, classically, black holes do not evaporate. They likely do in most QFT approaches and certainly should in the approach I sketched above. But we also don't know what goes on in the black hole interior in practically any QFT approach that has a universe like ours (i.e., no extra dimensions we fail to notice all the time; an expanding rather than collapsing universe; things like that; and formal descriptions things that are like black holes but also formal descriptions that recover the motions of objects in our solar system). All of that arises from the nonlinearities in the Einstein Field Equations of General Relativity; those nonlinearities have real astrophysical significance (multi-pulsar systems are excellent testbeds for nonlinearities), so we can't just wave them away.
So wherever we add a nonlinear term to a quantum field theory, as done above, or in several alternative approaches, we are doing the right thing to model natural phenomena we observe. But we also encounter mathematical difficulties. These may be technical (there may be non-perturbative renormalization procedures that don't rely on power counting yet to be invented by mathematicians and/or mathematical physicists). These may be solved by nature itself (there may be some UV fixed point, where UV is "extremely short distances" in essence, coming from ultraviolet, which has extremely short wavelengths, and a fixed point is something that saves perturbative renormalization for gravitation, as it did with quantum chromodynamics (2004 Nobel Prize in Physics)). Nonlinearity problems might still be found in the non-gravitational sector (the standard model of particle physics is not completed into the UV), and if so, maybe there's a way to kill two birds with one stone. Just nobody knows for sure if there are zero, one, or two (or more) birds; and if you don't know what the birds are like (or if they even exist), it's hard to plan seriously about what stone is the right tool.
Firstly, I'll note that I read your comment about another answer here seeming like pop sci to you. I will also not leave the answer at only my next two words.
> deeper explanation I could grasp without reading several books
Probably not, and I'm not sure undergrad SR helps enough, although I'll assume you encountered quantum electrodynamics, and so guess that maybe we can slyly work out a weak comma-goes-to-semicolon approach[1], replacing partial derivatives in Schrödinger's i\hbar \partial \psi /\partial t=-(\hbar^2/2m)\nabla^2 \psi with covariant derivatives and ignore that we have only Galilean invariance in the time-dependent Schrödinger eq (TDSE), rather than full Lorentz covariance. Not ignoring that turns into a course in differential geometry and curved spacetime. In fact, I will even trim the "explanation" to "we can adapt TDSE to some distributions on some spacetimes, changing its partial derivatives to covariant derivatives, which looks really easy when you just write it (i\hbar \partial \psi /\partial t=-(\hbar^2/2m)\nabla^2 \psi), but making that typographical trick work is where you find the 'deeper explanation'". But the takeaway is that doing weakly-relativistic quantum mechanics in a weakly-curved spacetime (like our solar system, or even the overwhelming majority of the central parsec of our galaxy) is tractable, and we can leave "the problem of time" in a box to be opened at leisure by philosophers.
Now we've taken an initial step towards QM on a more general spacetime than that of the very special one of SR.
Let's step back a bit.
If you can cope with Gauss and Poisson laws for electromagnetism (http://web.mit.edu/6.013_book/www/chapter4/4.3.html and surrounding sections in that chapter), and the usual Schrödinger equation, you can take a step towards a Gauss law for gravity (\Phi_g = -4 \pi Gm_{enc}, where we have substituted the constant 1/ε_0 with -4 \pi G, and the enclosed electric charge q_{enc} becomes the enclosed "gravitational charge" m_{enc}). As Carroll says in the fine article at the top, we can quantize this just fine, given at least the conditions in my first paragraph (and typically several others too).
Purely electromagnetically, one might set out a number of sources, with each source's generating a Poisson potential. Those sum linearly through the principle of superposition, so this is just fine. We can then perform a "second quantization" or "canonical quantization". This goes outside your implicit scope, but essentially we convert a classical electromagnetic system into a quantum field theory by this procedure. From that we can recover our usual TDSE. This is a grad topic, books required.
Gravitationally, we can follow the same approach. We have to be careful with our distributions of sources (mass, rather than electric charges), but if they are far enough apart (the idea being that the Poisson potential for any source asymptotes to zero at infinity) and not moving relativistically (i.e., compared to the speed of light), we can do a "second quantization" of gravitation. This is called Canonical Quantum Gravity, and works well for many applications. I think CQG is beyond an undergrad background without resort to books. Try to see how much of DeWitt you can follow at [2].
We can then combine these two paragraphs following the <https://en.wikipedia.org/wiki/Schr%C3%B6dinger%E2%80%93Newto...> approach, extracting the adapted TDSE decorated with partial derivatives (commas -> semicolons ~ d -> \partial).
Now let's return to "my" elephant in the room: the superposition principle. The matter we describe with either form of TDSE above superposes linearly: we can take a distribution of charged sources and completely determine their interactions, and we can drop in a test particle that feels these interactions, so can probe them by being attracted to our various sources, even as we strongly densify sets of sources. This works for all the charges in the Standard Model, with some effort. Unfortunately, this fails for our notion of the gravitational charge m_{enc}: as we densify m, we can get to a runaway collapse, which we capture in the nonlinearity introduced in Schrödinger-Newton. Worse, even without the runaway, we can't in general linearly superpose the gravitational potential from significant sources (for less significant sources we can pretend that instead of asymptoting to zero at infinity, the potential reaches zero closer to the source; we do this for other long-range potentials too, in practice). So we can't with generality solve the whole Schrödinger-Newton calculation: we need to be careful about the distribution of mass.
We already have this problem in classical General Relativity, where the Raychaudhuri equations lead to caustics, so it's not exactly surprising that a flavour of that appears in a system which arose from quantizing classical General Relativity. Dealing with those takes some mathematical work classically and in QFT approaches like the one I've very briefly sketched above, and there is no approach known to be fully general. Caustics associated with colliding gravitational waves is something we can't deal with satisfactorily, and we have reasons to suspect those may occur in a relevant way in the very early history of our universe. Singularities in the deep interior of black holes are something we can't deal with satisfactorily either. However, even in classical General Relativity practically nobody trusts the interior solution of the Kerr metric (not even Kerr himself! [3]). And of course, classically, black holes do not evaporate. They likely do in most QFT approaches and certainly should in the approach I sketched above. But we also don't know what goes on in the black hole interior in practically any QFT approach that has a universe like ours (i.e., no extra dimensions we fail to notice all the time; an expanding rather than collapsing universe; things like that; and formal descriptions things that are like black holes but also formal descriptions that recover the motions of objects in our solar system). All of that arises from the nonlinearities in the Einstein Field Equations of General Relativity; those nonlinearities have real astrophysical significance (multi-pulsar systems are excellent testbeds for nonlinearities), so we can't just wave them away.
So wherever we add a nonlinear term to a quantum field theory, as done above, or in several alternative approaches, we are doing the right thing to model natural phenomena we observe. But we also encounter mathematical difficulties. These may be technical (there may be non-perturbative renormalization procedures that don't rely on power counting yet to be invented by mathematicians and/or mathematical physicists). These may be solved by nature itself (there may be some UV fixed point, where UV is "extremely short distances" in essence, coming from ultraviolet, which has extremely short wavelengths, and a fixed point is something that saves perturbative renormalization for gravitation, as it did with quantum chromodynamics (2004 Nobel Prize in Physics)). Nonlinearity problems might still be found in the non-gravitational sector (the standard model of particle physics is not completed into the UV), and if so, maybe there's a way to kill two birds with one stone. Just nobody knows for sure if there are zero, one, or two (or more) birds; and if you don't know what the birds are like (or if they even exist), it's hard to plan seriously about what stone is the right tool.
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