# Does energy have a position?

1. Jan 5, 2015

### Stephen Tashi

In classical physics, does energy have a position?

It seems rather obvious that kinetic energy can be regarded as a property of a moving mass.. If carry a cup of hot coffee from one room to another, it seems clear that I have transported heat energy. That indicates that energy can also change position.

However, there is the problem of whether mass has a position. An idealized "point mass" can have a position and move. A real mass needs to have a mass density function. Moving a real mass through space moves its density function. So if I want to talk about the position of kinetic or heat energy in real objects, it appears I need an energy density function.

Then there is the problem of potential energy. If a point mass m is at height h above the surface of the earth we say it "has" potential energy mgh, as if the energy is "in" the mass. But the hope of getting back (m)(g)(h) worth of work if it falls depends on the gravitation field exerting the force (m)(g) throughout the fall. You could say the potential energy depends on the potential of the gravitational field at the specific location of the point mass. However, the fact that a potential function exists depends on the global properties of the gravitational field.

Suppose a mass is at rest in a time-varying graviational field. It seems it ought to have potential energy since a stationary mass in a constant force field does. We could say the mass has a time varying potential energy, but how exactly is its potential energy defined? If its potential energy has a position, where is it? If the potential energy has a density function, what is the extent of it in space?

2. Jan 5, 2015

### ShayanJ

At first you should note that a "real mass" is composed of atoms and molecules so its not a continuum down to any size. Those atoms and molecules are actually carrying the energy whether thermal or kinetic(now these two are somehow the same and somehow different).

About the kinetic energy, when you're moving the mass, of course all of its atoms and molecules are moving with it and all have the same "drift" speed as the speed of the mass as a whole and that "drift" speed can be used to evaluate the kinetic energy of each atom or molecule and the sum of kinetic energies of all of the atoms and molecules becomes the kinetic energy of the mass as a whole. So if your mass is e.g. rotating around an axis, those atoms further from the axis are carrying more of the kinetic energy than the ones nearer the axis.

Now notice I used "drift" speed for kinetic energy by which I meant the speed with which they have a net motion. So you may ask is there another kind of motion? Now this brings us to the thermal energy you asked about. Another kind of motion the atoms and molecules have is called thermal motion and it is the random vibration(of the constituents of a molecule w.r.t. each other) or rotation or oscillation of the atoms or molecules as a whole. If considered for an individual atom or molecule, it is simply called kinetic energy but when considered for a large number of atoms or molecules, it is seen as the random jiggling of them that each of them is doing on its own and has no simple correlation with other ones. This kind of kinetic energy, when considered for the mass as a whole and summed over, is called thermal energy is the kind of energy which we quantify by associating a temperature with the mass.

Here I think you're missing an important point. The potential energy isn't for a particle. It is always for the system as a whole. So if I have the earth and an apple(the only things that exists) then there is a gravitational energy coming from the mutual gravitational attraction between them. But this energy is not just for the apple or just for the earth. Its for both of them as a system. The only fact that causes the confusion that its for the apple, is that we consider earth to be fixed and so in such considerations, only the position and velocity of the apple come into equations.

Now comes the question where the system stores its potential energy. Now that's a tough one. It also arises when we work with electromagnetic fields. We say electromagnetic energy density is given by $u=\frac 1 2 (\vec E \cdot \vec D+\vec B \cdot \vec H)$, but where is this energy stored? I remember reading somewhere that if we can measure the gravitational field of an EM field, then its possible to say where is this energy but the typical EM fields we know and can generate, produce so much of a weak gravitational field that we can't measure.
Anyway, I think about the Newtonian gravitational field, we have no answer to the question of where is its energy. But this shouldn't worry us because its not a correct theory and we have GR.

In GR and classical EM, distortions-and so energy-move with finite velocity and so there are moments where there is energy in spacetime(in GR case) and EM field. So we consider EM field and spacetime as real physical entities(even if emergent ones). So we can say energy is in those two "fields". In the case of classical EM, $u=\frac 1 2 (\vec E \cdot \vec D+\vec B \cdot \vec H)$ tells us how that energy is distributed in the field exactly(at least it seems to me). But in the GR case, there is a peculiarity. Because of the equivalence principle, you can't localize the energy in spacetime and so you can't have something like u that tells you how much gravitational energy is at any "event" of spacetime. But there are may ways of associating energy to a system of masses and the part of spacetime they're living in.

Although I explained things and I think it should be more clear to you now(if its not, tell me where I wasn't good at explaining it), I should confess that what you're asking is actually one of the peculiar things in physics so I don't think I can completely clarify things for you. I think the best you can do is updating your knowledge to the current state of the art(is it what people call it?). But I can't do that too. I just think I gave you a background for others to explain more advanced things or for you to study more.

Last edited: Jan 5, 2015
3. Jan 5, 2015

### Stephen Tashi

You explained things well by going beyond classical physics. I'm curious how far much progress can be made within classical physics. Within classical physics, introducting atoms and molecules would require still defining an energy density for them as material bodies. (Once it was realized that a material body isn't a satisfactory model for atoms, we began to go beyond classical physics).

Last edited: Jan 5, 2015
4. Jan 6, 2015

### ShayanJ

I think I should explain a little bit about the word "classical" and what physicists mean when they use it. Because you said I went beyond classical physics which means, by classical physics, you mean physics before 1900. When physicists say classical physics, they mean pre-quantum physics and so relativity(special or general) is a classical theory too. If you want to talk about physics before both relativity and QM, you should say Newtonian physics I guess.
It seems you're talking about a classical model of atom when we say its just another Keplerian system with gravitational force replaced by electrostatic force and its constituents are a positively charged sphere at the centre and some much smaller negatively charged spheres rotating around the central big sphere. In this model, the atom is just another "real mass". So its energy is its kinetic energy as a whole+all external potential energies+the kinetic energy of its constituents+the potential energy of its constituents regarding its internal interactions. And its kinetic energy as a whole, is just the sum of the kinetic energies of its constituents regarding only that part of their velocity that is the result of the motion of the atom as a whole. And also its external potential as a whole, is just the sum of the potentials of its constituents regarding the external interactions. And again we are led to consider the distribution of energy in those spheres i.e. nucleus and electrons. But if you consider them as fundamental, then it has no meaning to talk about the distribution of energy in them. And if you say they're not fundamental, it means they are again a system with some constituents which are fundamental and so on and so fourth. So there is somewhere that we stop associating an energy density and just say that thing has this much energy.

5. Jan 6, 2015

### Stephen Tashi

My model for classical physics is what is (or used to be) taught in classical mechanics courses and physics classes that rely on a continuum - like elementary theromodynamics. I don't know the history of physics well enough define classical mechanics in historical terms. In my model of classical physics there is no conceptual difficulty thinking of real masses as being defined by a mass density. Perhaps the detailed mechanism for heat can't adequately be explained, but there is no difficulty in thinking of heat within matter being defined by a density function.

There is, as far as I can see, great difficulty in putting potential energy on the same footing other energies. If we think of heat and kinetic energy has being co-located with mass and don't assign potential energy a specific location then we have a rather mysterious picture. "Energy" is conserved, but this involves kinetic energy which has a location changing to potential energy, which doesn't - and vice versa. To make an analogy, it's like a river changing into democracy. Rivers have a location. Democracy seems an abstract property that can't be assigned one.

6. Jan 6, 2015

### ShayanJ

I'm not sure what you mean here. But I think you'll be better if you know that like continuity equation for charge density, mass density and EM energy density, we also have a continuity equation for heat. Then when we add the fact that heat flux is in the opposite direction of the heat gradient, we get the heat equation. But this is, as you mentioned, in the continuum regime of treating real masses. So here we're actually defining a temperature field on the background matter continuum so we have a case similar to GR and classical EM with the difference that temperature field is a scalar field.

Yeah, it seems strange at first but you should consider that in the case of GR and classical EM, because waves have a finite speed, we can think that the fields are physical and real. So as I said, the energy is in those fields and we should accept that the space between the point particles or other masses is also part of the system and energy is distributed in that space(either spacetime or EM field). And GR and Maxwell's equations tell us how energy is distributed in space(time) but with the distinction that in GR, we can only say that that gravitational wave is carrying this much energy and we can't say how much energy is stored at each point. So I guess they are somehow unified but with the difference that one kind of energy is in particles and the other is in fields. But in QFT, we associate fields with even what we previously called particles and so kinetic and potential energy get more unified because they are both now energies of fields.

7. Jan 7, 2015

### Khashishi

No, not really. Maybe with a theory of quantum gravity, it will be possible, but with classical physics, there is no way to assign a position to gravitational potential energy. It is also impossible to totally separate the energy of the electromagnetic field (which covers all space) from the self-energy of a charged particle (which is presumably localized). But we can't really talk about particles in classical physics.

8. Jan 7, 2015

### Khashishi

Maybe I spoke too soon. You can assign a classical energy density of the gravitational field in the same way you can assign an energy density to the electromagnetic field. http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/Mathematical_Thinking/possible_scalar_terms.htm
Of course there are a fair number of problems with this approach. The gravitational self energy of a point particle is infinite, and you need some kind of renormalization to make sense of anything. Also the energy of a massive particle extends over all space due to the gravitational field.

9. Jan 7, 2015

### Stephen Tashi

It's interesting to visualize what happens when two masses move toward each under due to gravitational force. While they are moving, their combined gravitational field should be losing energy as the masses gain kinetic energy. Is there a qualitative way to see why their combined field is losing energy ?

10. Jan 7, 2015

### Khashishi

Because the gravitational potential energy is negative. As the amplitude of the field increases (simple superposition of the fields of two masses), the energy density becomes more negative.

11. Jan 7, 2015

### Stephen Tashi

I can't visualize that yet. Is the combined field weaking at all locations in space? I'd think that if we are at a point in space where one of the moving masses passes close to us that the field would become stronger. [Edit:] and then weaker as the mass moves away.

12. Jan 7, 2015

### Khashishi

I'm not sure what you meant by "weaking". For Newtonian gravity, the gravitational field obeys the superposition principle. The gravitational potential is the sum of the gravitational potential coming from each source. The gravitational field is the vector sum of the field from each source.

13. Jan 7, 2015

### Stephen Tashi

I'm familiar with superposition. I think the increase the kinetic energy of two masses moving toward each other under gravitational attraction should be accompanied by the loss of potential energy in their combined gravitational field. Isn't that the way it should work?

14. Jan 7, 2015

### Matterwave

The gravitational field itself is not an energy, as can be seen by simply looking at the units. The gravitational potential $\phi$ doesn't have units of energy, but energy per mass. In order to actually have a potential energy, you need to have an interaction between masses.

In this case, the gravitational potential energy is $U=-\frac{GM_1M_1}{r}$ and indeed this quantity will become more negative as the masses get closer (and r-shrinks) and the 2 kinetic energies will exactly account for this.

15. Jan 7, 2015

### Stephen Tashi

Yes, but the link in post #8 defines an energy density for the gravitaional field in units of joules per cubic meter.

16. Jan 7, 2015

### Matterwave

There is no need to introduce a field structure for Newtonian gravity since all forces are instantaneous. Newtonian gravitation has no gravitational waves. As such, you will not lose energy to fields when considering the motion of bodies in gravitational fields. I don't see how the fact that you can define the quantity mentioned in post #8 changes anything.

17. Jan 7, 2015

### Khashishi

18. Jan 7, 2015

### Stephen Tashi

It gives potential energy a spatial density, so (with reference to the original post) potential energy has a position - at least it has a position in the same sense as mass (which is also a scalar) has an associated position.

My simplistic reasoning is that if the field has energy density, then it has a total energy found by integrating the density over all space. My further simplistic reasoning is that (perhaps) if two masses are moving and accelerating due to their gravitational fields then their combined field would be losing the potential energy that is converted into kinetic energy.

An alternative view is that the potential energy being used up was "in" the masses themselves. Yet another view is that the potential energy of the bodies is just an scalar that describes an "aspect" of the entire situation, so it has no particular location and density function.

19. Jan 7, 2015

### Khashishi

I have no idea what you are trying to say.

20. Jan 7, 2015

### Matterwave

Like I said, you can't lose energy to these fields since they don't radiate...so what's the point of doing this? Why are you making things 10 times more complicated than you need to? You can solve the Newtonian equations of motions completely by considering just the 2 masses and the force between them. If you want to do energetics, then you can solve the energetics problem completely by considering just the kinetic energies and potential energy that I wrote down in post #14. Why do you want to have to integrate this arbitrary function that was defined in post #8?