# I Potential Energy for "stone + field" physical system

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1. Nov 23, 2018

### cianfa72

Consider the classical scenario a stone falling in the Earth gravitational field.
Classically we attach a Potential Energy to the stone and using the law of conservation of (mechanical) energy we are able to evaluate the dynamic of the falling stone.

This model assume a stone in a "external" field (due to the Earth) with potential energy attached to the stone itself (here the physical system to take in account is actually "the stone").

On the other hand I was thinking we can attach a Potential energy to the field itself considering as physical system the union "stone + field": this way potential energy of the system is basically the energy attached to the field while the kinetic one is that of the stone.

Does it make sense ?

2. Nov 23, 2018

### PeroK

It's a good question.

If you move to the theory of GR, then the potential energy as a concept disappears (as does the concept of force), except in Newtonian approximations. What you have in GR is the concept of the energy of a particle measured locally (by a local observer). This energy is the kinetic energy (plus the rest-mass energy). The particle has no potential energy, as such. Instead, the different measurements of energy depend on the motion of the particle and the observer through spacetime. (Technically, these are four-velocity of the particle and observer.)

If you take a step further and consider an expanding universe, then you lose the principle of energy conservation. Or, if you want to retain energy conservation, then some energy has to be stored in the gravitational field. There's a good piece about that by Sean Carroll here.

http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

3. Nov 23, 2018

### lightarrow

With "Classically" what you mean? Classical mechanics?
E. g. when you study the hydrogen energy levels in QM you don't say "the electron energy levels", exactly because this energy must be ascribed to the whole system "atom". Even in classical electrodynamics you usually don't ascribe potential energy to a single charged particle, but to the system of interacting ones.

--
lightarrow

4. Nov 23, 2018

### PeroK

Whatever was meant by "classical", it was not QM!

5. Nov 23, 2018

### cianfa72

Sure I mean not QM nor GR
Just to be more clear: at basic (undergrad) level the model above is based on an external (let me say assigned) field described by a Potential function (note "Potential" and not "Potential Energy"...maybe here is the root of my confusion). Where is "stored in a form of energy" the work done against the field when the stone changes position in the external field ? We can assume it is "stored" into the Potential Energy of the stone in the field I guess, and here there exist no way to store it in the field itself (the external field is assigned and cannot change).

In the other scenario that also includes in the model the "source" of gravitational field (the Earth body) the physical system to consider is basically "Earth body + field + stone" and this time we are able to "store" the work done in the field itself (now the field is not "assigned" but reflects the interaction between bodies).

Last edited: Nov 23, 2018
6. Nov 23, 2018

### Mister T

Many authors of introductory textbooks and curricula have taken the approach that the potential energy is a property of the Earth-stone system. And moreover that it must be done this way to develop a concept of energy that is consistent across both newtonian mechanics and thermodynamics.

7. Nov 24, 2018

### cianfa72

Going further and considering SR preventing us to have infinite interaction speed (bound to the speed of the light) I believe also in the simple "Earth+stone" system we are "forced" to introduce the concept of "field" in order to mediate interactions between bodies. This way Earth actually interacts with field as the stone too. Therefore the physical system to take in account is formed actually by "Earth + field + stone" and we can assign to the current field distribution a (potential) energy density in which we can image "stored" the work done to bring the system in the current configuration.

Coming back to the model "stone in an external assigned field", it surely simplify the model to calculate the stone dynamic but from Energy point of view I believe it is poor, what do you think about ?

8. Nov 27, 2018

### cianfa72

Or, maybe, it is better to say in that model the system is actually made of "stone + field" with potential energy a property of that system (basically here "the field" acts let me say as a body or "thing")

9. Nov 29, 2018

### cianfa72

Could you elaborate more this point ? Thanks

10. Nov 29, 2018

### PeroK

Do you have a specific question?

11. Nov 29, 2018

### cianfa72

In the context of SR even with finite interaction speed (basically limited by the speed of the light - c) I believe we can still introduce the concept of "field" including the associated potential energy as in the case of "classic" physics. Consider for instance the system made of 2 electric charges: the work done on them by external forces basically increases the "electrical potential energy" of the system that we can image "stored" in distribution of the electric field itself.

Does it make sense ?

12. Nov 29, 2018

### PeroK

There were attempts, I believe, to reconcile Newtonian gravity with SR. But, I don't know much about them, except they didn't succeed.

13. Nov 30, 2018

### cianfa72

ok but in this example we are talking about electrical (or electromagnetic) field in the context of SR neglecting GR (for instance spacetime curvature)

14. Nov 30, 2018

### PeroK

I thought we were talking about gravity. Maxwell's theory of EM was/is compatible with SR. For example, you have the four-potential:

https://en.wikipedia.org/wiki/Electromagnetic_four-potential