Can we say that gravity(indirectly) is responsible for motion of electrons around nucleus?

Question

From Wikipedia

But because general relativity dictates that the presence of electromagnetic fields (or energy/matter in general) induce curvature in spacetime

From Wikipedia

An electromagnetic field (also EMF or EM field) is a physical field produced by electrically charged objects.

From Wikipedia

Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field.

As electrons move around nucleus under the influence of electromagnetic field, and EM field induces curvature in space-time and curved space-time is gravity, So can we say that gravity(indirectly) is responsible for motion of electrons around nucleus?

Note: I'm talking about gravity produced by EM field between nucleus and electrons,not by mass of nucleus and electrons.(which is negligible).

Answer 1

A point to emphasize here is that one cannot separate the rest mass of the nucleons and electrons from that of field: rest masses aren't additive in this way and one can only state the rest mass of a system as a whole. The mass-energy of the fields is already built into the rest mass of the system. The rest mass of an electron includes the energy of the electron's electrostatic field, likewise for the nucleus.

In fact, the total rest mass / energy of a system comprising bound electrons, nucleus and their associated fields is less that that of a system where the nucleus and electrons are separated by infinite distance. We actually lower the system's rest mass by allowing the electrons to become bound - by "sating" the attractive "wish" of the electrostatic field, if you will, in the same way that bound nucleons in an atom have less rest mass as a system comprising them separately, although the binding energy for electrons with their nucleus is much less than the nucleus's binding energy.

Accordingly, the gravitational effects of the electrostatic field will be even less than that of bodies with the rest mass of the electrons and nucleus. And this is utterly negligible. So the curvature of spacetime owing to the atomic system's presence is going to play an utterly negligible role in the attraction.

Another way to come at this problem is to look at what quantum electrodynamics has to say about such problems. When we look at the Dirac equation for the hydrogen atom, for example, and add the effects of the electrostatic field through QED, we get a fantastically accurate calculation of observed energy levels (including the famous Lamb shift) and other atomic behaviors.

This observation tells us that gravity is negligible: our calculations come out correct if we do not include it.

Answer 2

Gravity is very, very weak compared to the electromagnetic force. Whatever curvature of space-time is induced inside an atom would be far too weak to hold mass-energy together at that scale.

The electromagnetic force is 1/137 the strength of the strong nuclear force, while Gravity is 6 * 10^-39 the strength of the strong nuclear force. Here is a link that may help with understanding the four fundamental forces: http://hyperphysics.phy-astr.gsu.edu/hbase/forces/funfor.html.

The logic of the three quotes you put together doesn't seem to me to be valid. Electrons don't orbit the nucleus like particles. An electron exhibits its particle nature only at the instant it's ejected from the atom. Inside an atom, electrons occupy specific energy levels sort of like shells around the nucleus, rather than as particles. Here is a link to an excellent explanation of the atomic model: https://www.chem.wisc.edu/deptfiles/genchem/sstutorial/Text5/Tx53/tx53.html. If you scroll down to the diagrams of electron "orbitals", you'll see that they don't look like the gravitational orbits you may have been thinking of. The locations of electrons within atoms are based on quantized energy levels, and probabilistic locations within those energy levels. Gravity is so inconsequential at that scale that it doesn't enter the picture.

Answer 3

Sorry to be late to the party, I did an answer yesterday and then had some internet glitch and couldn't post it. I shall dust it off:

Can we say that gravity (indirectly) is responsible for motion of electrons around nucleus?

No, I'm afraid not. Note that an electron doesn't actually go round a proton (the simplest nucleus) like a planet round a Sun. Have a look at the Wikipedia atomic orbitals article for that:

"The electrons do not orbit the nucleus in the sense of a planet orbiting the sun, but instead exist as standing waves."

The electron exists as a standing wave. Think standing wave, standing field.

But because general relativity dictates that the presence of electromagnetic fields (or energy/matter in general) induce curvature in spacetime

This is a veiled reference to the wave nature of matter. The thing to note is that it's quantum field theory, not quantum billiard ball theory. The electron isn't some billiard-ball thing that has a field. It is field. We call it matter and say it's made of energy. And general relativity says that any concentration of energy causes gravity. However the force of gravity is extremely weak compared to electromagnetic force. Have a look at hyperphysics for that. The electromagnetic force between the electron and the proton is titanic compared to the force of gravity caused by either or both of them.

As electrons move around nucleus under the influence of electromagnetic field, and EM field induces curvature in space-time and curved space-time is gravity, So can we say that gravity (indirectly) is responsible for motion of electrons around nucleus?

No. But see Einstein talking about electromagnetic and gravitational fields in 1929 here and note this: "It can, however, scarcely be imagined that empty space has conditions or states of two essentially different kinds". According to Einstein, a field is a state of space. And since there's only one state of space where the electron is, the electromagnetic field is some respect the gravitational field too. But if anything the electromagnetic field is responsible for the gravity, not the other way round.