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But ultimately, in my view the problem is not so much the mathematical modelling, but that in order for gravitational collapse into white dwarfs and neutron stars to work, we need to have the gravitational interaction be stronger when lots of gravitationally-charged material is enclosed in a small volume than when the same material can only be enclosed in a much larger volume. What, if anything, should arrest this strengthening process as the enclosing volume contracts?
Quantum field theory actually coughed up an answer.
Degeneracy pressures -- very much quantum mechanical effects -- keep these compact stars from collapsing further, but those pressures can be overwhelmed (electron degeneracy pressure is overwhelmed in a precursor phase of any neutron star) leading to further collapse. Is there some ultimate degeneracy pressure that sets a minimum volume for an arbitrary amount of matter? We just don't know.
Relatedly, we don't have a general solution to have two spatially separated masses not eventually (after infinite time, even) become one enclosed mass: we can only keep them separated in select families of model universes (in ours, dark energy keeps distant clusters galaxies apart, but not the members of any given cluster) or with selected initial conditions. That is manifested in the formal inability to linearly superpose two gravitational potentials (even though we can do so with other potentials). And we can only handle so much non-linearity with modern tools and techniques (although that keeps improving too).
Another problem is that degeneracy pressure itself becomes a large source of gravitation within compact stars, since it is stress-energy which gravitates [4], rather than individual particles of matter, so a sufficiently strong degeneracy pressure (quark degeneracy, maybe? or something stronger?) spells its own gravitational demise!
So, the tl;dr is that the standard model of particle physics is a linear theory, the standard model of gravitation is a non-linear theory. For astrophysical reasons we can't get rid of the nonlinearities entirely. And we run into problems if gravity's nonlinearities "infect" the linear theories of matter. But we don't know how to avoid that infection for black holes and the very early hot dense universe. QFT approaches give answers, GR approaches give other answers, and our study of nature has not let us decide which answers, if any, are close enough.
Finally, the mathematics of General Relativity are complete and self-consistent, but have a big "box" called the stress-energy tensor. The study of the mechanisms that generate stress-energy is the GR-first approach to strong gravity. Quantum effects can arise in the stress-energy tensor, and degeneracy pressures are an example. There are technical and conceptual problems about quantum mechanical effects on a purely classical background, though. Nevertheless, the straightforward way to avoid infinitely-strong gravity is to block it via the action of the stress-energy tensor. A quantum field theory first approach starts on a different footing in terms of mathematical completeness and/or consistency at all energy ranges, but has different escape valves for strong gravity, precisely because it is no longer a classical background. For instance, gravitons (which are not a feature of classical General Relativity, but are a feature in perturbative quantum gravity [5]) might decay at extraordinary energies (an example lifted out of a real approach: with additional symmetries in a beyond-the-standard-model theory, a Kaluza-Klein type graviton might decay into a pair of Z bosons, which then immediately decay into things that could be spotted by something like ATLAS/CMS (muons, jets, diphoton resonances)).
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[1] Carroll's introductory grad-student course notes (chosen because they're online and because Carroll is in the linked article) at https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll... deals with this briefly; his GR textbook (and others') will supply details if desired, and lead you towards some of the depth, but away from your desire to avoid "reading ... books".
[2] https://blogs.umass.edu/grqft/files/2014/11/DeWitt-Quantum-g... which goes into greater detail than the article linked at the top where Carroll says: "You can have the same classical theory that maps on to two different quantum theories. You can have two different classical theories mapping onto the same quantum theory. So, there’s no direct correspondence and after all, why should there be?
"But again, nevertheless, it has worked for electromagnetism, the nuclear forces and everything else. When you straightforwardly apply that quantization procedure to gravity — we have a classical theory, general relativity, we can quantize it. It just blows up. It just gives us infinite crazy answers."
[3] Kerr lecture on Spinning Black Holes (2016) https://youtu.be/nypav68tq8Q (iirc, around the 51 minute mark, although he raises the point several times in the second half of the lecture, particularly highlighting that the Kerr solution is strictly a vacuum solution).
[4] one can analogize: just as the motion of sub-proton particles within a proton gives it a large fraction its rest mass, exceeding that of the sum of the rest-masses of those internal particles, the motion of particles within a compact star (neutron star, white dwarf) generates a large fraction of the gravitational mass of the compact star, which is more than that which would be generated by its components if they were spread out within a larger enclosing volume. The analogy is deeply interesting, quoting Strassler: "... there is positive motion energy from all those particles running around in there, as well as some amount of positive mass energy, and then there is also a very negative potential energy from the fact that all those particles are tightly bound in there. We do not have a simple description of a proton analogous to a hydrogen atom, where you can work out where all the energy comes from. It’s a big complicated mess, but in the end the sum of the energies for a proton at rest is 0.938 GeV. Yes, highly relativistic bound states are a lot more complicated than nice simple non-relativistic atoms." (a comment in the excellent <https://profmattstrassler.com/articles-and-posts/largehadron...>).
Quantum field theory actually coughed up an answer.
Degeneracy pressures -- very much quantum mechanical effects -- keep these compact stars from collapsing further, but those pressures can be overwhelmed (electron degeneracy pressure is overwhelmed in a precursor phase of any neutron star) leading to further collapse. Is there some ultimate degeneracy pressure that sets a minimum volume for an arbitrary amount of matter? We just don't know.
Relatedly, we don't have a general solution to have two spatially separated masses not eventually (after infinite time, even) become one enclosed mass: we can only keep them separated in select families of model universes (in ours, dark energy keeps distant clusters galaxies apart, but not the members of any given cluster) or with selected initial conditions. That is manifested in the formal inability to linearly superpose two gravitational potentials (even though we can do so with other potentials). And we can only handle so much non-linearity with modern tools and techniques (although that keeps improving too).
Another problem is that degeneracy pressure itself becomes a large source of gravitation within compact stars, since it is stress-energy which gravitates [4], rather than individual particles of matter, so a sufficiently strong degeneracy pressure (quark degeneracy, maybe? or something stronger?) spells its own gravitational demise!
So, the tl;dr is that the standard model of particle physics is a linear theory, the standard model of gravitation is a non-linear theory. For astrophysical reasons we can't get rid of the nonlinearities entirely. And we run into problems if gravity's nonlinearities "infect" the linear theories of matter. But we don't know how to avoid that infection for black holes and the very early hot dense universe. QFT approaches give answers, GR approaches give other answers, and our study of nature has not let us decide which answers, if any, are close enough.
Finally, the mathematics of General Relativity are complete and self-consistent, but have a big "box" called the stress-energy tensor. The study of the mechanisms that generate stress-energy is the GR-first approach to strong gravity. Quantum effects can arise in the stress-energy tensor, and degeneracy pressures are an example. There are technical and conceptual problems about quantum mechanical effects on a purely classical background, though. Nevertheless, the straightforward way to avoid infinitely-strong gravity is to block it via the action of the stress-energy tensor. A quantum field theory first approach starts on a different footing in terms of mathematical completeness and/or consistency at all energy ranges, but has different escape valves for strong gravity, precisely because it is no longer a classical background. For instance, gravitons (which are not a feature of classical General Relativity, but are a feature in perturbative quantum gravity [5]) might decay at extraordinary energies (an example lifted out of a real approach: with additional symmetries in a beyond-the-standard-model theory, a Kaluza-Klein type graviton might decay into a pair of Z bosons, which then immediately decay into things that could be spotted by something like ATLAS/CMS (muons, jets, diphoton resonances)).
- --
[1] Carroll's introductory grad-student course notes (chosen because they're online and because Carroll is in the linked article) at https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll... deals with this briefly; his GR textbook (and others') will supply details if desired, and lead you towards some of the depth, but away from your desire to avoid "reading ... books".
[2] https://blogs.umass.edu/grqft/files/2014/11/DeWitt-Quantum-g... which goes into greater detail than the article linked at the top where Carroll says: "You can have the same classical theory that maps on to two different quantum theories. You can have two different classical theories mapping onto the same quantum theory. So, there’s no direct correspondence and after all, why should there be?
"But again, nevertheless, it has worked for electromagnetism, the nuclear forces and everything else. When you straightforwardly apply that quantization procedure to gravity — we have a classical theory, general relativity, we can quantize it. It just blows up. It just gives us infinite crazy answers."
[3] Kerr lecture on Spinning Black Holes (2016) https://youtu.be/nypav68tq8Q (iirc, around the 51 minute mark, although he raises the point several times in the second half of the lecture, particularly highlighting that the Kerr solution is strictly a vacuum solution).
[4] one can analogize: just as the motion of sub-proton particles within a proton gives it a large fraction its rest mass, exceeding that of the sum of the rest-masses of those internal particles, the motion of particles within a compact star (neutron star, white dwarf) generates a large fraction of the gravitational mass of the compact star, which is more than that which would be generated by its components if they were spread out within a larger enclosing volume. The analogy is deeply interesting, quoting Strassler: "... there is positive motion energy from all those particles running around in there, as well as some amount of positive mass energy, and then there is also a very negative potential energy from the fact that all those particles are tightly bound in there. We do not have a simple description of a proton analogous to a hydrogen atom, where you can work out where all the energy comes from. It’s a big complicated mess, but in the end the sum of the energies for a proton at rest is 0.938 GeV. Yes, highly relativistic bound states are a lot more complicated than nice simple non-relativistic atoms." (a comment in the excellent <https://profmattstrassler.com/articles-and-posts/largehadron...>).
[5] 't Hooft Lecture - Perturbative QUANTUM GRAVITY - Universiteit Utrecht https://webspace.science.uu.nl/~hooft101/lectures/erice02.pd...
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