Binding energy in QM and in GR

[zonde]

I would like to ask rather general question.
Can a binding energy of some QM process at the same time be binding energy of gravity?
I am just trying to find overlap between QM and GR and I have thought about this question but I'm not sure how to tackle it.

 

[PeterDonis]

Can a binding energy of some QM process at the same time be binding energy of gravity?

Can you give an example of "binding energy of some QM process"? I'm not sure what you mean by that.

 

[dextercioby]

The "binding energy" in the H-atom is of electromagnetic nature and as of such can be seen as a source of gravity for the spacetime in which the proton and the electron "live".

 

[zonde]

Can you give an example of "binding energy of some QM process"? I'm not sure what you mean by that.

Say binding energy of H-molecule.
And I suppose that black body radiation emitted by cooling body is another example. But in this case it's probably a question if black body radiation can be regarded as binding energy at QM level.

 

[zonde]

The "binding energy" in the H-atom is of electromagnetic nature and as of such can be seen as a source of gravity for the spacetime in which the proton and the electron "live".

Sure, but I was not asking about that.
Let's say that binding energy of H-atom is radiated away while the atom remains as a part of larger body. Say I am looking from far way and I don't use spectrometer. Can I say it's gravitational binding energy that was radiated away? Would it be correct?

 

[PeterDonis]

Say binding energy of H-molecule.

Ok. Then the answer is that, since the binding energy contributes to the externally measured mass of the H molecule (the "contribution" is actually negative), it will affect the H molecule's behavior as a source of gravity. In practical terms this won't be observable for a single H molecule, but if you imagine, say, a large planet whose atmosphere is made up largely of hydrogen, like Jupiter, if you try to estimate its mass by adding up the masses of all of the H molecules, you will need to take into account the electromagnetic binding energy of each H molecule (i.e., the fact that its mass is slightly less than the sum of the masses, considered in isolation, of the protons and electrons making it up).

However, considering a case like the planet Jupiter also brings up another point. Jupiter's externally measured mass also has an additional "binding energy" contribution (which is negative) because it is a gravitationally bound system. In other words, suppose we have, say, $10^{60}$ or so isolated hydrogen molecules and we want to make a planet out of them. The mass of those molecules when we start out, since they are all isolated and their mutual gravity is negligible, will be $10^{60}$ times the mass of an individual hydrogen molecule--which is less than the mass of the same number of isolated protons and electrons, because of the binding energy we talked about above. But if we now take those molecules and form them into a planet, the final mass of the planet will be smaller--we will have to extract energy from the system in the process, and this extracted energy is the gravitational binding energy of the planet. So gravitational binding energy for a large, gravitationally bound system is not quite the same as the binding energy of individual molecules.

Let's say that binding energy of H-atom is radiated away while the atom remains as a part of larger body.

Binding energy is negative, so it won't be radiated away. You would have to add energy to an H atom to separate its proton and electron (the amount of energy required, in the case of an H atom in its ground state, is 13.6 electron volts).

If we suppose that the H atom in question is somewhere in the atmosphere of Jupiter, say, and we add energy to it to ionize it (separate the proton and electron), we now have a system consisting of a free proton and a free electron, whose mass is larger (by the aforementioned 13.6 electron volts) than the mass of the H atom was. But the system as a whole is still bound gravitationally to Jupiter and contributes to Jupiter's externally measured mass. The question is, where did the 13.6 electron volts come from that were used to ionize the atom? If it came from some source internal to Jupiter, then this whole process would have no effect on Jupiter's externally measured mass. But if it came from somewhere outside Jupiter, then the whole process, seen from the outside, amounts to adding 13.6 electron volts to Jupiter's externally measured mass.

 

[zonde]

Ok, this makes sense. Gravitational binding energy is different than molecular binding energy.

However, considering a case like the planet Jupiter also brings up another point. Jupiter's externally measured mass also has an additional "binding energy" contribution (which is negative) because it is a gravitationally bound system.

Well, that additional binding energy can still be radiated away as photons, right? (when matter becomes gravitationally bound) Say two large but solid bodies gravitationally accelerate towards each other and then collide plastically. So they become bound but they heat up (and expand a bit). So the gravitational binding energy should be radiated away as black body radiation to restore both chunks of matter in a similar (thermal) state they had before collision.

Binding energy is negative, so it won't be radiated away.

I meant when proton and electron forms an atom binding energy is radiated away.

 

[PeterDonis]

Say two large but solid bodies gravitationally accelerate towards each other and then collide plastically. So they become bound but they heat up (and expand a bit). So the gravitational binding energy should be radiated away as black body radiation to restore both chunks of matter in a similar (thermal) state they had before collision.

Ok, yes, this is correct: if the initial state is two large bodies at temperature T, and the final state is one large body at the same temperature T, then energy has to be radiated away during the process, and that energy corresponds to the gravitational binding energy of the final state.