Interpretation of Potential Energy as Field Property

[zonde]

TL;DR Summary

I am trying to analyze the idea of "potential energy as a property of field".

I will quote this statement from another thread:

Gravitational potential energy (to the extent it can be defined in relativity) isn't a property of the body, but of its interaction with the gravitational field. It's very difficult to pin down "where" gravitational energy is, but you can loosely think of it as a property of the field rather than the body.

So the total energy of the body is higher when it's moving relative to a hovering observer than when it isn't. Probably best not to think of that as mass, though, since GR is formulated in terms of invariants.

In that thread number of other posters seemed to agree with this statement. So I tried to analyze it a bit.

For the sake of my questions let's say we limit GR to Schwarzschild spacetime and if there are problems with gravitational potential energy we can associate it with factual binding energy.
There is a thing that potential energy of bound bodies is negative. So as I see it this negative energy can be interpreted as absence of other energy (rest mass/energy). Alternative would be to consider this negative mass/energy as independent physical phenomena with antigravitating effect. This alternative seems to me much more radical with very unclear way how to incorporate it into some usable physical model.

And so if we think of this absent energy of bound bodies as property of field then (rejecting the alternative interpretation of antigravitating mass/energy) it implies that the rest mass/energy is property of field as well because negative potential energy does not exist by itself but only as an absence of rest mass/energy that was there, before bound state formed.

So that's the question. Do you accept that implication? The argumentation seems quite simple to me.

 

[PeterDonis]

There is a thing that potential energy of bound bodies is negative.

If you adopt the convention that the potential energy is zero for bodies at rest relative to each other at infinite separation, yes.

as I see it this negative energy can be interpreted as absence of other energy (rest mass/energy).

You have just rediscovered the obvious fact that the binding energy of a bound system is the energy that you had to take away from the system to make it bound, assuming you started from a state where all of the bodies in the system were at rest relative to each other at infinite separation. Yes, this is the physical meaning of binding energy and has been known to be so for quite a while now.

if we think of this absent energy of bound bodies as property of field

That's not what @Ibix said. He said it was a property of the interaction of a bound body with the gravitational field of the object that is "binding" it. For example, the binding energy of a satellite in orbit about the Earth is a property of the interaction of the satellite with the Earth's gravitational field: it's the energy you would have to take away from the satellite to put it into its present orbit if you started with the satellite infinitely far away from the Earth and at rest relative to it.

Note, however, that looking at it this way assumes, implicitly, that the satellite itself has negligible mass and does not contribute to the overall gravitational field, so the only field is due to the Earth. This viewpoint breaks down when considering, say, a binary star system. But the more general way of looking at binding energy that I described above still works--you can ask how much energy you would have to take away from the system of two stars to get them from mutual rest at infinite separation to their current orbits, even though both of them contribute to the overall gravitational field that determines their current orbits.

Also note that, in general, there is no way to separate the total binding energy into separate "kinetic" and "potential" energy components. For the case of the satellite orbiting the Earth, you can do it by adopting coordinates in which the Earth is at rest, but that approach does not generalize to cases like a binary star system. For a case like the latter, you can adopt coordinates in which the center of mass of the system as a whole is at rest, defining kinetic energy using that frame, and then subtracting it from the total binding energy to get the potential energy; but doing that means the potential energy won't have the same simple physical interpretation that it does in the Earth-satellite case.

Do you accept that implication?

No, because it's based on your incorrect reading of what @Ibix said. See above.

 

[zonde]

If you adopt the convention that the potential energy is zero for bodies at rest relative to each other at infinite separation, yes.

Yes, of course. But this convention is well motivated - it's easy to specify vacuum. Any configuration with masses would be much more complicated. So it's easy to specify that object has zero potential energy when it's far from any other mass.

That's not what @Ibix said. He said it was a property of the interaction of a bound body with the gravitational field of the object that is "binding" it.

Yes, Ibix said: " Gravitational potential energy isn't a property of the body, but of its interaction with the gravitational field." But he said as well that "but you can loosely think of it as a property of the field rather than the body". So he said two things.

In the same thread Nugatory said:

But note that there is no way of associating this mass increase with the Earth or the object and saying that the mass of of either has increased. It's a property of the system as a whole, the box and its contents.

It is negative PE that has been reduced when some of the binding energy is returned to the system.

In the same thread Dale said:

(The PE does not belong to the rock)

I could say that not all of PE belongs to the rock. But just the same PE is negative, you have to take it from something.

So the feeling is that PE is treated as positive energy that can be viewed as somehow separate from other phenomena, say like kinetic energy. But my point is that PE is negative and therefore it does not work like that.

Also note that, in general, there is no way to separate the total binding energy into separate "kinetic" and "potential" energy components.

This is not quite right. Binding energy plus kinetic energy equals (minus) potential energy. In classical system where virial theorem is applicable it would be that half of PE is converted in KE and other half of PE would be taken away from system as binding energy.
So all the binding energy comes from potential energy. What you might say is that PE of the system can not be separated into binding energy and KE components.

For the case of the satellite orbiting the Earth, you can do it by adopting coordinates in which the Earth is at rest, but that approach does not generalize to cases like a binary star system. For a case like the latter, you can adopt coordinates in which the center of mass of the system as a whole is at rest, defining kinetic energy using that frame, and then subtracting it from the total binding energy to get the potential energy; but doing that means the potential energy won't have the same simple physical interpretation that it does in the Earth-satellite case.

If we ask how some part of bound system contributes to total mass/energy of that system then there is only one reference frame which is meaningful for that question and that is the rest frame of the system (the rest mass of that part of course is defined in that part's rest frame).

 

[PeterDonis]

this convention is well motivated

Agreed.

t's easy to specify vacuum. Any configuration with masses would be much more complicated.

Wrong. The zero point convention for energy has nothing to do with "vacuum". It just means that a configuration with a bunch of masses that aren't interacting with each other at all has zero potential energy. In the case of gravity, "not interacting with each other at all" requires the masses to all be at infinite separation from each other. So that's the zero potential energy configuration.

he said two things.

The second of which had a critical qualifier: "loosely". Which basically means "don't draw any other inferences". But you are trying to draw other inferences.

the feeling is that PE is treated as positive energy that can be viewed as somehow separate from other phenomena, say like kinetic energy

It's not a "feeling". It's a precise mathematical definition that applies under a precisely specified set of circumstances.

the feeling is that PE is treated as positive energy that can be viewed as somehow separate from other phenomena, say like kinetic energy. But my point is that PE is negative and therefore it does not work like that.

PE being negative doesn't mean it "does not work like that". It just means that a bound system has less energy than what would be the sum of energies of its constituents if all of the constituents were free, at rest relative to each other at infinite separation.

Binding energy plus kinetic energy equals (minus) potential energy.

Only if we make a particular choice of coordinates. Kinetic energy is coordinate dependent.

What you might say is that PE of the system can not be separated into binding energy and KE components.

No. The binding energy is a direct observable: how much energy had to escape to infinity to put the system in its bound state.

If we ask how some part of bound system contributes to total mass/energy of that system then there is only one reference frame which is meaningful for that question

Only if you define "total mass/energy" to mean the total rest mass/energy, i.e., the total mass/energy in the frame you say is "meaningful". But nothing requires you to make such a definition. (And, as I pointed out, even if you do make that definition, in the general case the "potential energy" you define won't have the same simple interpretation that it does in the Earth-satellite case.)

 

[zonde]

The second of which had a critical qualifier: "loosely". Which basically means "don't draw any other inferences". But you are trying to draw other inferences.

You are too strict. If you can't draw any inferences then the statement is just useless. Is suppose that "loosely" means "some of the inferences you can draw will be wrong, so be careful".

It's not a "feeling". It's a precise mathematical definition that applies under a precisely specified set of circumstances.

Fair.

PE being negative doesn't mean it "does not work like that". It just means that a bound system has less energy than what would be the sum of energies of its constituents if all of the constituents were free, at rest relative to each other at infinite separation.

Ok, probably my feeling is that I am not satisfied with "shut up and calculate" approach to PE and I am looking for satisfactory interpretation. But I suppose that by itself this feeling is fine - say, a lot of people are not satisfied with "shut up and calculate" QM and spending a lot of time discussing QM interpretations.
But I have to take into account that, some people talk only about "shut up and calculate" part of PE and I shouldn't try to see interpretation part in it, as it is not there.

Only if we make a particular choice of coordinates. Kinetic energy is coordinate dependent.

If we talk about rest mass of an object we use objects rest frame. We could say it's a choice, but then there aren't good alternatives.

Only if you define "total mass/energy" to mean the total rest mass/energy

Let's just say that this is imprecision on my side, I should have said "total rest mass/energy".

 

[PeterDonis]

If we talk about rest mass of an object we use objects rest frame. We could say it's a choice, but then there aren't good alternatives.

Sure there are. The rest mass of an object is an invariant; it's the same in all frames.