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Is there any way for someone with an undergrad level background in special relativity and QM to understand why GR and QM don't work together? I've heard someth

Is there any way for someone with an undergrad level background in special relativity and QM to understand why GR and QM don't work together? I've heard something like "the Feynman path integrals spit out infinities" many times but I'm wondering if there is a deeper explanation I could grasp without reading several books.

Koshkin 5y ago
Why should they? Newton's mechanics and electrodynamics don't work together, either. They are simply different theories - both conceptually and mathematically.
- civilized 5y ago
  Physical theories are supposed to work together, and classical mech and electrodynamics mostly do. Electrodynamics is expressed in the language of Newtonian mechanics (e.g. fields represent force per charge), and the experiments that established the laws of electromagnetism used Newtonian mechanics as their framework. There are specific inconsistencies between Newtonian mechanics and electrodynamics, inconsistencies which spurred the development of special relativity. Physics theories are not supposed to be mutually incommensurable, like the ideas of two philosophers from different schools who don't understand each other. Physics doesn't work that way, and physicists have never been satisfied with that.
akomtu 5y ago
Because QM needs absolute time and space coordinates?
wanda 5y ago
In GR, gravity is an emergent property of mass interaction with the geometry of 4D spacetime. The more mass, the more spacetime is curved by the massive object. In QM, gravity is a force, and efforts to express gravity in a quantum framework use the graviton, a massless force carrier for gravitation. We don't necessarily expect this to exist, the graviton is simply a mathematical tool if you don't like the idea. In relativity, it makes no intuitive sense to have a force carrier for gravity, as gravity is not a force in the same way the others are, it's a property of the geometry of spacetime. In GR, spacetime is continuous, there is no minimum or maximum length. In QM, spacetime is discrete. There is a minimum quantity. So already, you can see some incompatibility, because both GR and QM make many predictions that have proven both models to be pretty good descriptions of nature. GR has been more or less proven by experiments with time dilation, gravitational lensing, and more recently gravitational waves. QM has been supported by many experiments too, double slit experiment, experiments with quantum entanglement, you name it, QM has predicted it and with extreme accuracy. So you have two descriptions of reality that work on two different scales, which is sort of acceptable. Use the two at the appropriate scale and you have the Standard Model of physics, and that mostly works. Except when you have to tackle something like a black hole or the big bang where the scale is quantum level, but the mass-energy is relativistic. So we want a quantum theory of gravity, and the infinities emerge when trying to formulate a quantum theory of gravity. Specifically, gravitation is non-renormalisable, i.e. at high enough energies, our usual methods to deal with divergences toward infinity doesn't work and itself results in problems with infinities. Physicists have been working for the best part of a century to unify the fundamental forces, well what does this mean? It means that if you put enough energy into a system, like the amount you'd have at the big bang, it is possible to demonstrate that the electromagnetic force and the weak nuclear force are one force, and "grand unified theories" refer to theories in which the strong nuclear force and gravitational force are similarly found to be all part of one force at the unification energy. Unification is far from complete, but we have at least a decent picture of the electromagnetic, weak and strong forces individually, in quantum mechanics. The problem with unification is primarily gravitation, which is extremely weak compared to the other fundamental forces, and we don't know how to describe it in quantum mechanics without using a representative particle. And even then, as you can tell, we haven't gotten all that far. Most efforts have produced purely mathematical extensions to the Standard Model, like String Theory. Problem with string theories is that they do not make predictions that we can check, and thus the theories are not falsifiable. And there could probably be n different string theories that all work and stitch GR and QM together, but no way to determine which string theory us right. A lot of people dislike the idea that string theories and some other alternative quantum gravity theories propose which is that there are more than 4 dimensions, anywhere from 5 to 26 to n dimensions. The reason why people don't like that is that we have no idea what these extra dimensions would look like, we wouldn't even see them, they're basically just more co-ordinates but it doesn't seem very satisfactory just to say there's extra dimensions and that gravity leaks out that way.
- civilized 5y ago
  This sounds like pop physics. Even with my undergrad background I know that spacetime isn't discrete in QM.
  - wanda 5y ago
    Sorry, I meant to say: for theories of quantum gravity, spacetime is discrete. I got ahead of myself there. Even then it's not every theory of quantum gravity, but certainly those with any traction have tended to discretise spacetime (e.g. loop quantum gravity). And yes I was trying to keep it simple/accessible. Given the parent question, why would I word my response inaccessibly?
raattgift 5y ago
1/2 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 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%93Newton_equation 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. 1/2
raattgift 5y ago
2/2 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)). - -- [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/Carroll4.html 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-gravity-1.pdf 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 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/largehadroncolliderfaq/whats-a-proton-anyway/ https://profmattstrassler.com/articles-and-posts/largehadron...>). [5] 't Hooft Lecture - Perturbative QUANTUM GRAVITY - Universiteit Utrecht https://webspace.science.uu.nl/~hooft101/lectures/erice02.pdf https://webspace.science.uu.nl/~hooft101/lectures/erice02.pd... 2/2
deleted 5y ago
[deleted]

Koshkin 5y ago

Why should they? Newton's mechanics and electrodynamics don't work together, either. They are simply different theories - both conceptually and mathematically.

civilized 5y ago
Physical theories are supposed to work together, and classical mech and electrodynamics mostly do. Electrodynamics is expressed in the language of Newtonian mechanics (e.g. fields represent force per charge), and the experiments that established the laws of electromagnetism used Newtonian mechanics as their framework. There are specific inconsistencies between Newtonian mechanics and electrodynamics, inconsistencies which spurred the development of special relativity. Physics theories are not supposed to be mutually incommensurable, like the ideas of two philosophers from different schools who don't understand each other. Physics doesn't work that way, and physicists have never been satisfied with that.

akomtu 5y ago

Because QM needs absolute time and space coordinates?

wanda 5y ago

In GR, gravity is an emergent property of mass interaction with the geometry of 4D spacetime. The more mass, the more spacetime is curved by the massive object. In QM, gravity is a force, and efforts to express gravity in a quantum framework use the graviton, a massless force carrier for gravitation. We don't necessarily expect this to exist, the graviton is simply a mathematical tool if you don't like the idea. In relativity, it makes no intuitive sense to have a force carrier for gravity, as gravity is not a force in the same way the others are, it's a property of the geometry of spacetime. In GR, spacetime is continuous, there is no minimum or maximum length. In QM, spacetime is discrete. There is a minimum quantity. So already, you can see some incompatibility, because both GR and QM make many predictions that have proven both models to be pretty good descriptions of nature. GR has been more or less proven by experiments with time dilation, gravitational lensing, and more recently gravitational waves. QM has been supported by many experiments too, double slit experiment, experiments with quantum entanglement, you name it, QM has predicted it and with extreme accuracy. So you have two descriptions of reality that work on two different scales, which is sort of acceptable. Use the two at the appropriate scale and you have the Standard Model of physics, and that mostly works. Except when you have to tackle something like a black hole or the big bang where the scale is quantum level, but the mass-energy is relativistic. So we want a quantum theory of gravity, and the infinities emerge when trying to formulate a quantum theory of gravity. Specifically, gravitation is non-renormalisable, i.e. at high enough energies, our usual methods to deal with divergences toward infinity doesn't work and itself results in problems with infinities. Physicists have been working for the best part of a century to unify the fundamental forces, well what does this mean? It means that if you put enough energy into a system, like the amount you'd have at the big bang, it is possible to demonstrate that the electromagnetic force and the weak nuclear force are one force, and "grand unified theories" refer to theories in which the strong nuclear force and gravitational force are similarly found to be all part of one force at the unification energy. Unification is far from complete, but we have at least a decent picture of the electromagnetic, weak and strong forces individually, in quantum mechanics. The problem with unification is primarily gravitation, which is extremely weak compared to the other fundamental forces, and we don't know how to describe it in quantum mechanics without using a representative particle. And even then, as you can tell, we haven't gotten all that far. Most efforts have produced purely mathematical extensions to the Standard Model, like String Theory. Problem with string theories is that they do not make predictions that we can check, and thus the theories are not falsifiable. And there could probably be n different string theories that all work and stitch GR and QM together, but no way to determine which string theory us right. A lot of people dislike the idea that string theories and some other alternative quantum gravity theories propose which is that there are more than 4 dimensions, anywhere from 5 to 26 to n dimensions. The reason why people don't like that is that we have no idea what these extra dimensions would look like, we wouldn't even see them, they're basically just more co-ordinates but it doesn't seem very satisfactory just to say there's extra dimensions and that gravity leaks out that way.

civilized 5y ago
This sounds like pop physics. Even with my undergrad background I know that spacetime isn't discrete in QM.
- wanda 5y ago
  Sorry, I meant to say: for theories of quantum gravity, spacetime is discrete. I got ahead of myself there. Even then it's not every theory of quantum gravity, but certainly those with any traction have tended to discretise spacetime (e.g. loop quantum gravity). And yes I was trying to keep it simple/accessible. Given the parent question, why would I word my response inaccessibly?

raattgift 5y ago

1/2 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 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%93Newton_equation 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. 1/2

raattgift 5y ago

2/2 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)). - -- [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/Carroll4.html 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-gravity-1.pdf 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 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/largehadroncolliderfaq/whats-a-proton-anyway/ https://profmattstrassler.com/articles-and-posts/largehadron...>). [5] 't Hooft Lecture - Perturbative QUANTUM GRAVITY - Universiteit Utrecht https://webspace.science.uu.nl/~hooft101/lectures/erice02.pdf https://webspace.science.uu.nl/~hooft101/lectures/erice02.pd... 2/2

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