Why are General Relativity and Quantum Mechanics incompatible? Is it because GR field equations allow spacetime to be bent all the way to a singularity, but QM says the quantum of action is the Planck constant (units J x s), and distance is undefined at a singularity?
No. Everyone is comfortable that at a singularity none of the theories make sense. It is partly why it is called a singularity.
The core problem is the reconciliation of locality with quantum physics. This is embodied in the EPR Paradox. (Wikipeadia link)
It depends what you mean. “Relativity” in the sense of the abolition of absolute space and time, and the finite ultimate velocity © which causes all the usual spooky relativistic paradoxes and what not is perfectly consistent with quantum mechanics, and indeed for any high energy phenomenon one routinely uses relativistic quantum mechanics. No problem. The positron comes straight out of that.
On the other hand general relativity describes the motion of spacetime itself which in many cases we must interpret as the motion of matter – and that motion is not quantized, as it should be. It’s purely classical. Roughly speaking, you can calculate the location and momentum of a black hole, say, simultaneously and with infinite precision in GR (e.g. from knowing the orbits of a satellite et cetera). This is a nono from the point of view of quantum mechanics.
Also, visavis the singularity thingy, QM has dick to say about the structure of spacetime, including its allowable curvature. QM is not a theory of structure – of what the universe is made of, or how it’s put together – it is a dynamical theory. It says how things move and change given that they’re made of this and that and structured in this way and this other. In ordinary QM, it’s particles that move, and in relativistic QM, it’s fields of one kind or another that “move” (in the sense that they change amplitude). But in all such cases you need another theory to provide the “input” to QM: the particles or fields that will be moving, and their properties, things like mass and charge and spin (or these days number of dimensions in which they can move). Once you have all that, QM tells you how to construct a dynamical theory and predict motion and change. In principle, GR should be an “input” to QM, much like classical EM was an “input” that gave QED, but alas the theories are so very different no one really knows how to match them up. Either there is some truly clever and nonobvious way to do so that has escaped human ingenuity for almost a century, or else one or the other of these theories is wrong, or incomplete, and given how phenomenally well they both work, that is almost as strange a possibility.
GR says c is a constant. QM says h is a constant. Compromise: let c increase and h decrease in extremely bent spacetime. No singularity necessary
There’s several reasons for the difficulty in marrying the two theories, some of which are intertwined (warning long post ahead!), however the reason given in the OP is not really one of them.
A key difference between the two is that quantum mechanics (QM) takes place against a background of flat 3D space with a distinctly separate time, just as we are used to from Newtonian physics, whereas general relativity (GR) takes place in a background of curved 4D spacetime. However quantum theory (QT) can be carried over to background of geometricallyflat 4D spacetime of special in a satisfactory way by using the more nuanced and complex approach of quantum field theorys (QFTs).
The (very) naive solution is then define quantum fields on a background of curved spacetime and this approach is known by the descriptive, but uninspired title of “quantum field theory in curved spacetime” (QFICS), but there are now two big problems:

Despite QFTs (usually) being done against a background of 4D spacetime they still cling to a vestige of the idea of 3D space and a separate notion of time due to their nonlocal nature. In particular they demand that observers are able to globally separate space and time. This is easily done in flat spacetime, but impossible to do for general curved spacetime. This is known as “the problem of time” and is not thought to be an insurmountable problem and possible solutions have been put forward.

In GR the curvature of spacetime is related to the contents of spacetime by the Einstein field equations (EFEs), which means that the quantum fields themselves must govern the curvature of spacetime. Therefore QFICS can only be nothing more than an approximation with limited applicability, rather than a serious solution to marrying QM and GR (not that it was intended to be a serious solution). The reason for this is you can’t define a quantum field that has a nonnegligible gravitational effect on a predefined background of curved spacetime in a way that is compatible with GR as the quantum field defines the background.
So it seems that we should look at how a quantum field defines a spacetime, which brings us to one of the biggest problems:
In GR the EFEs specifically relate certain properties including the energy and momentum of the contents of spacetime to the curvature of spacetime, however one of the most distinctive features of QT is that properties such as energy and momentum do not always have always welldefined values and are instead governed by probability. Or in other words a quantum field may not have precise values of energy and momentum, but we need precise values to define the spacetime which it occupies!
One way round this is to let the quantum field define the spacetime it occupies without precise values for energy, momentum etc. When dealing with very large quantum systems, such as those which would have nonnegligible gravitational effects, we would expect that measurements of the system would produce a results that are very close to the expectation values for those properties. By far the most obvious solution then is to let the expectation values for the energy, momentum, etc define the curvature of spacetime, this approach is known as semiclassical gravity. However it is easy to show there are always quantum systems where the result of a measurement of a property is never close to the expectation value for that property. This means that semiclassical gravity, like QFICS, is merely an (albeit improved) approximation.
Another way to tackle the problem of quantum fields defining the background is to formulate GR so that the gravitational field doesn’t define the background, rather we predefine the background (this is called background dependence). This is a major breach of the ideas behind GR, which means many view it as heresy, however in fact this can always be done! If we do this what we get is a background spacetime with a classical field which we can think of as the gravitational field, we can then quantitize this field to create a QFT for gravity, a voila! This is called the covariant perturbation method (CPM)
So despite some serious philosophical objections there is a QFT for gravity using the CPM, however the problems start when you try to do anything with it:
Exact solutions are rarer than steak tatare in advanced physics, so for a theory to make physical predictions you have to find ways of approximating that theory. The most pervasive method of approximation in physics is perturbation theory and it is used in GR and famously in (some) QFTs to obtain approximate solutions for problems with no known exact solutions, however a wellknown problem of applying perturbation theory to QFTs is that it that you may have to deal with infinities in your calculations. A set of procedures has been invented to deal with these infinities called renormalization, but this procedure does not work for quantum gravity, so the theory is of highly limited practical use. There may be other ways to get useful results from this approach, but noone has cracked the problem yet.
Many would say though that the above route was not the correct route to quantum gravity anyway. QFTs are properly derived from classical field theories by first casting the latter into Hamiltonian form and indeed GR can be put into such a form and that, they would say, should be the jumping off point. This approach is known as canonical quantum gravity, but as you’ve guessed there’s yet another problem, this time it is that this formulation of GR does not give the correct degrees of freedom. Trying to solve this problem for quantum gravity has led to what was formerly seen as one of the best quantum gravity candidates (Euclidean quantum gravity) and what is currently seen as one of the best quantum gravity candidates (loop quantum gravity). At the moment though noone has cracked the nut of canonical quantum gravity, but it is a very active area.
Here’s how I understand it: there are four fundamental forces in physics: gravity, electromagnetism, the strong and weak nuclear forces. General Relativity really only deals with gravity; the other forces are too weak to be felt at the planetary scale the GR deals with and so GR formulas don’t include them.
QM, on the other hand, deals with the other three forces at a very small scale. At the atomic scale gravity is (usually) too weak to play a part and so it’s left out of the QM formulas.
So in some respects QM is the theory of the very small and GR is the theory of the very large. There is, however, one area were they can interact: in gravity dense situations like black holes and singularities. The gravity is so strong that it can interact with the other forces but we don’t have any formulas that can combine them all together.
Thus the search for the “grand unified theory” which will combine them and let us determine what is going on in black holes.
That would mean that neither is a constant; and you’d get enemies from both camps!
Gravity is by far the weakest of the four fundamental forces. However, the objects giving rise to it are so massive that its effects are grossly obvious at long distance.
Have I commented on how great your username is? It’s delightful.
And appropriate for the topic (in case anyone doesn’t get the pun).
Perhaps I’m missing the pun, but isn’t asymptotically misspelt? Or is “pot” the key word?
Huh, it took me long enough to notice that the second word wasn’t actually “flat”. I’d never even noticed the change in the first word.
[quote=“Asympotically_fat, post:6, topic:747165”]
There’s several reasons for the difficulty in marrying the two theories, some of which are intertwined (warning long post ahead!), however the reason given in the OP is not really one of them.
A key difference between the two is that quantum mechanics (QM) takes place against a background of flat 3D space with a distinctly separate time, just as we are used to from Newtonian physics, whereas general relativity (GR) takes place in a background of curved 4D spacetime. However quantum theory (QT) can be carried over to background of geometricallyflat 4D spacetime of special in a satisfactory way by using the more nuanced and complex approach of quantum field theorys (QFTs).
The (very) naive solution is then define quantum fields on a background of curved spacetime and this approach is known by the descriptive, but uninspired title of “quantum field theory in curved spacetime” (QFICS), but there are now two big problems:

Despite QFTs (usually) being done against a background of 4D spacetime they still cling to a vestige of the idea of 3D space and a separate notion of time due to their nonlocal nature. In particular they demand that observers are able to globally separate space and time. This is easily done in flat spacetime, but impossible to do for general curved spacetime. This is known as “the problem of time” and is not thought to be an insurmountable problem and possible solutions have been put forward.

In GR the curvature of spacetime is related to the contents of spacetime by the Einstein field equations (EFEs), which means that the quantum fields themselves must govern the curvature of spacetime. Therefore QFICS can only be nothing more than an approximation with limited applicability, rather than a serious solution to marrying QM and GR (not that it was intended to be a serious solution). The reason for this is you can’t define a quantum field that has a nonnegligible gravitational effect on a predefined background of curved spacetime in a way that is compatible with GR as the quantum field defines the background.
You seem to know QM well. I’ve been trying to follow the QM’s attempt to understand gravity. I think QM should go back to flat spacetime and describe the part of the universe that it’s calibrated for, and let GR describe black holes. All GR needs from QM is to let us decrease the Planck Constant in extremely bent spactime.