# Does energy have a position?

bhobba
Mentor
The principle of conservation of energy has a history of problems. Every time energy seems not to be conserved, a new type of energy must be defined to balance the books.

But that was all sorted out once Noether formulated her beautiful theorem. We now know exactly what its is and why its conserved.

GR however is the problem child since it depends on symmetry principles usually enforced by being in an inertial frame which GR only is locally.

Thanks
Bill

bhobba
Mentor
So what about $\frac 1 2 (\vec E \cdot \vec D+\vec B \cdot \vec H)$?

Take the energy in a small volume of the field and take the volume to zero - what is the energy at that point?

The formula for the energy of the EM field is an energy density:
http://www.phy.duke.edu/~lee/P54/Notes/energy.pdf [Broken]

Thanks
Bill

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ShayanJ
Gold Member
Take the energy in a small volume of the field and take the volume to zero - what is the energy at that point?

The formula for the energy of the EM field is an energy density:
http://www.phy.duke.edu/~lee/P54/Notes/energy.pdf [Broken]

Thanks
Bill

Yeah...right. So in classical mechanics potential energy isn't in the fields which we superfluously define and in classical EM and GR, we can say there is energy in the fields but only when we're talking about finite regions! So actually its only kinetic energy that we can exactly associate a position to, the position of the particle carrying it!
And when we talk about QFT, we have HUP and also the particles are treated as fields too and so in QFT even kinetic energy has no exact position.

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bhobba
Mentor
Yeah...right. So in classical mechanics potential energy isn't in the fields

Potential energy in fields - don't know that one.

What energy is and (reasonably) where it resides is answered by Noethers Theorem. For potential energy its a property a particle has by virtue of its position so its associated with the particle and it only seems reasonable to assume its located at the particle - but I suppose its not strictly implied by it.

Thanks
Bill

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bhobba
Mentor
In the non-Newtonian field description, the potential energy of the mass is distributed in the energy density of the gravitational field. As a mass falls, its kinetic energy increases and the total gravitational field energy becomes more negative.

Can you explain that in terms of Noethers Theorem?

As the object falls its potential energy gets converted to kinetic so total energy is conserved. That's what Noether tells us. The reason gravitational energy is negative is a system with an object at infinity should have zero energy. As it accelerates towards the gravitational object it gains positive kinetic energy so for the total energy to be zero the potential energy it has must be negative,

Thanks
Bill

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lavinia
Gold Member
In classical physics, does energy have a position?

I was just reading in Feynmann's Lectures about electromagnetic field energy and whether it can be located in space.

Feynmann starts by writing down the local conservation laws then substitutes terms from Maxwell's equations to come up with a candidate for local conservation of energy in the electromagnetic field.

The conservation equation in general just says that the change in local energy for unit of time is its divergence plus the work done by the field on particles.
So energy is lost or gained by flux across a surface and work done against the field.

Locally the work done is E.J by the Lorenz force law
and the divergence term is the cross product of the electric and magnetic fields time the square of the speed of light(up to an empirical constant).
The remaining term is the change in the localized energy. Up to constants this term is the sums of the squared intensity of the electric and magnetic fields.
This last term suggests how much energy is localized in the region.

He then points out that the local conservation equation can be satisfied in many ways so that the term for localized energy is not uniquely determined. But since energy is equivalent to mass and therefore generates gravity, one could in theory do an experiment that determines the direction of the gravitational field and this would tell you what quantity for the field energy is correct and therefore how it is localized.

So it seems that from Maxwell's equations alone one can not say how energy is localized and one needs relativity to design an experiment to make the exact determination. But Maxwell's equations do suggest that the field energy is localized in some way.

bhobba
Mentor
But since energy is equivalent to mass and therefore generates gravity, one could in theory do an experiment that determines the direction of the gravitational field and this would tell you what quantity for the field energy is correct and therefore how it is localized.

It's picky I know, but strictly speaking energy is not equivalent to mass - mass is a form of energy - not he other way around. The reason energy generates a gravitational field is the source is the stress energy tensor and mass can be part of that.

Gravitational field energy is very problematical with a number of candidates floating about - and if its conserved is even worse. The reason is because the modern definition is via Noethers Theorem and gravity being space-time curvature does not have the symmetries required of the theorem:
http://motls.blogspot.com.au/2010/08/why-and-how-energy-is-not-conserved-in.html
'The main lesson here is that general relativity is not a theory that requires physical objects or fields to propagate in a pre-existing translationally invariant spacetime. That's why the corresponding energy conservation law justified by Noether's argument either fails, or becomes approximate, or becomes vacuous, or survives exclusively in spacetimes that preserve their "special relativistic" structure at infinity. At any rate, the status of energy conservation changes when you switch from special relativity to general relativity.'

Thanks
Bill

It may clarify things to think in terms of the "state" of a physical system and the information needed to specify with "state variables". In Newtonian mechanics, by the usual definition of "state", we can think of a page where each line has entries of mass, time and position, velocity and acceleration at that time. (In Newtonian reality, we need functions that give the corresponding information for mass density functions).

From the concept of "state", we can define "necessary associations" between physical information. For example, in Newtonian mechanics, having mass information like 2 kg and asssociated time and velocity information like "5 m/sec at t = 0" is not a complete set We also need position information and acceleration information. So mass must have associated position and acceleration must have associated velocity, etc. This does not mean that a given numerical value of acceleration (e.g 8 m/sec^2) is always associated with a unique numerical velocity (e.g. 5 m/sec.) It means that in the list of state information, an entry that has an acceleration value must have an associated velocity value.

Theoretically we might define a different set of state variables that are adequate by virtue of the fact that we can deduce the original set of state variables if we know them. For example, we can contemplate whether we could remove the information about masses from a Newtonian system at t = 0 and replace it by specifying the combined gravitational field of the masses at t = 0.

The question of whether one kind of physical quantity must have other associated information depends on how the list state variables is defined and how other physical quantities are computed from them. For example, in dealing with a Newtonian mechanical system, we can define total potential energy as a calculation involving all the state information. If it is defined as some sort of summation of terms and each term has dimensions of energy and each term involves a factor with an (implicit or explicit) position, then it is fair to say that each "piece" of potential energy "has" as position. (We would think of integration and integrands in a continuous case.) If we define "potential energy" as a summation of terms that don't contain a position (meaning none of the variables depends explicitly or implicitly on a specific position ) then potential energy doesn't have position.

It may be that there is one way of computing potential energy at time t= 0 so position (x,y,z) is associated with energy h1 and equivalent way (i.e. getting the same total energy) by a computation that associates position (x,y,z) with a different energy h2. If that happens, then we can say potential energy has position, but not unique position.

If it is impossible to express the computation of potential energy in any way as a sum of terms such that each term involves just a single position then pieces of potential energy have no particular location. This could happen if , each term always depends on several positions. (This outcome would be hard to verify.. For example, if the expression involves terms that use integration over position then that term might be expanded as an integration over single positions.)

One can baulk at the idea that the calculation of a scalar physical quantity can always be viewed as a sum-of-products or integral-of-products. From the point of view of dimensional analysis, I think it is needed to make the physical units come out right.

lavinia
Gold Member
It's picky I know, but strictly speaking energy is not equivalent to mass - mass is a form of energy - not he other way around. The reason energy generates a gravitational field is the source is the stress energy tensor and mass can be part of that.

Gravitational field energy is very problematical with a number of candidates floating about - and if its conserved is even worse. The reason is because the modern definition is via Noethers Theorem and gravity being space-time curvature does not have the symmetries required of the theorem:
http://motls.blogspot.com.au/2010/08/why-and-how-energy-is-not-conserved-in.html
'The main lesson here is that general relativity is not a theory that requires physical objects or fields to propagate in a pre-existing translationally invariant spacetime. That's why the corresponding energy conservation law justified by Noether's argument either fails, or becomes approximate, or becomes vacuous, or survives exclusively in spacetimes that preserve their "special relativistic" structure at infinity. At any rate, the status of energy conservation changes when you switch from special relativity to general relativity.'

Thanks
Bill

Bill

This is above my head. So I don't see what it has to do with Feynmann's argument about the localization of energy in the electromagnetic field. Can you simplify your explanation?

bhobba
Mentor
But since energy is equivalent to mass and therefore generates gravity, one could in theory do an experiment that determines the direction of the gravitational field and this would tell you what quantity for the field energy is correct and therefore how it is localized.

I was referring to the above which looks like a comment on using gravity in some way to measure EM energy. That would require the so called Einstein-Maxwell action:

But once you do that you run into exactly the same problem GR has - namely gravity is curved space-time so the symmetries required by Noethers Theorem breaks down and the modern concept of energy becomes problematical.

If it isn't can you clarify what you were getting at.

Regarding the energy of the EM field not being localised its been ages since I read the Feynman Lectures but in modern times, as per the link I gave, it follows quite naturally from the EM Lagrangian via Noethers theorem. Are you getting at the freedom we have to add a divergence-less quantity? Yes that is an ambiguity but it is usually resolved by requiring the energy momentum tensor to be symmetric.

Thanks
Bill

bhobba
Mentor
IFor example, in dealing with a Newtonian mechanical system, we can define total potential energy as a calculation involving all the state information.

You lost me here.

I cant see how potential energy is a matter of definition - it follows from the Lagrangian.

I know he is a bit terse but Landau examined the whole potential energy thing in the first chapter of Mechanics. Its not a matter of definition - its a consequence of what energy is ie the conserved quantity associated with time symmetry.

I keep getting the impression I am missing something here because its not gelling with me what you guys are getting at - and I suspect conversely as well.

Thanks
Bill