# Does energy have a position?

• Stephen Tashi
that is definitely not wrong ... what you showed was just a simplification assuming that there is only one object in this universe ... o GPE can be found within infinity range ...yes .. it's completely true that every object is bound in the gravitational field as GPE is always negative ... and this also throws aside the question of interaction time difference

The "speed of gravity" in Newtonian physics is infinite in the following sense: the vector force between two objects will always point along the line connecting the center of mass of both objects, regardless of either object's trajectory, speed, or, most importantly, acceleration. In this sense, the vector force compensates instantaneously when either object accelerates, even if the two objects are very very far apart. I can thus construct a method of instantaneous communication the following way:

Imagine we have two objects which are very far apart from each other, which are not allowed to accelerate towards each other, but only parallel to each other (e.g. I constrain these objects on parallel tracks). If I am to accelerate one object into a simple harmonic motion, the other object will instantaneously be subjected to a harmonic force. By viewing the second, far away, object, you can instantaneously know that I am moving my object. Therefore, I have created an instantaneous form of communication this way, and what we call "the speed of gravity" is said to be infinite.

The "speed of gravity" in Newtonian physics is infinite in the following sense: the vector force between two objects will always point along the line connecting the center of mass of both objects, regardless of either object's trajectory, speed, or, most importantly, acceleration. In this sense, the vector force compensates instantaneously when either object accelerates, even if the two objects are very very far apart. I can thus construct a method of instantaneous communication the following way:

Imagine we have two objects which are very far apart from each other, which are not allowed to accelerate towards each other, but only parallel to each other (e.g. I constrain these objects on parallel tracks). If I am to accelerate one object into a simple harmonic motion, the other object will instantaneously be subjected to a harmonic force. By viewing the second, far away, object, you can instantaneously know that I am moving my object. Therefore, I have created an instantaneous form of communication this way, and what we call "the speed of gravity" is said to be infinite.
i understand what you are trying to state ... it is very explanatory ... but yes we all know now that gravity propagates at the speed of light ... if we even let alone GR and work with Newtonian physics then we can see that gravitational force acts instantaneously ( nowhere i heard Newton say about infinite speed ) ... instantaneous reflex and infinite speed has different meanings ... one can be considered within approximation and the other one is throwing a value beyond appoximation

i understand what you are trying to state ... it is very explanatory ... but yes we all know now that gravity propagates at the speed of light ... if we even let alone GR and work with Newtonian physics then we can see that gravitational force acts instantaneously ( nowhere i heard Newton say about infinite speed ) ... instantaneous reflex and infinite speed has different meanings ... one can be considered within approximation and the other one is throwing a value beyond appoximation

Of course, mathematically speaking, infinity is not a real number so if you like, we can call "the speed of gravity in Newtonian mechanics" undefined, unbounded, or divergent. But most physicists will not be so rigorous when making off-the-cuff remarks.

But there is no need for "approximation" in the mathematical model for Newtonian gravity. That the vector changes direction instantaneously is a rigorously defined statement since simultaneity slices in Newtonian space time are absolute. In other words, let's assume our object is at the origin, then we can say "given position function ##\vec{r}(t)##" the force felt by our object is ##\vec{F}(t)=\frac{Gm_1m_2}{|\vec{r}(t)|^3}\vec{r}(t)## regardless of the form of ##\vec{r}(t)##. The fact that ##t## is the same on the left and right sides of the equation gives us a rigorous definition of what "instantaneously acts" means.

It is still meaningless, even given a reference frame, for one to say "I have 10 Joules of energy", because the 0 is still arbitrary.

I'd count establishing a reference for zero potential energy as part of defining a reference frame adequate to talk about energy. The existence of certain amount of energy is no more (and no less) meaningless that the existence of other measurements that depend on establishing a reference. It may be true that all measurements (velocity, acceleration, momentum) that depend on reference frames are, in some sense, meaningless, but I don't see singling out energy to be worse off than the others.
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Of course, your line of reasoning will quickly run into the fundamental problem, which is how we actually define energy. Not just different types of energy, but really what is energy. This is a very deep question in physics.

I agree. We run into a problem merely asking if energy has position. 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.

Mass is a scalar that not only may have an associated position, it must have position [in classical mechanics] in order for a physical description to be complete. It would be disconcerting to find a problem in a mechanics textbook that began:

Problem 3. A 2.5 kg mass, that is initially at rest has no definite position. ...

Other things can inherit an association with position by their association with mass. Forces act on masses. Velocities are velocities of masses, etc. When we disassociate force or velocity from a particular mass, we create force fields or velocity fields and these have specific values at specific positions. If we speak of a force or velocity without an association of position, we have an incomplete description of a physical situation. (e.g. "There is a velocity of 83 m/sec" isn't a complete description.) So it is indeed mysterious why some forms of energy would have no association with position..

It's not only that you need not define a field in order to do problems and predict E.O.M., it's also that this field, whatever it is, can't steal energy from your system.

This point has been brought up several times. I agree that in classical physics, the energy books balance without our having to say moving bodies radiate energy. But if the moving bodies are using up potential energy "from the field" and returning potential energy "to the field" so the total energy remained constant then we don't have to account for energy lost by radiation. The question is whether the mathematics of that model would work out.

I didn't yet get an answer to whether the energy density of the combined gravitational field of two moving masses is independent of their relative position. If the energy density at a given location varies with the position of the masses, does the total energy in the field remain constant?

Unlike in the E&M field where energy (and momentum) can be lost to the field (through EM radiation), the gravity field in classical mechanics CAN NOT radiate.

This brings up the question of whether potential energy has a position in the E&M field (After all the thread is about "energy", not just gravitational energy. )

If I can't "give energy to the field", and really all that this field can do is mediate transfer of energy to OBJECTS, which by the way is done fully correctly by simply considering the potential energy function ##E=-\frac{GM_1 M_2}{r}##, then why am I considering this field at all when I am considering energetics? What is the point other than making things harder on yourself?

I find it interesting to ask theoretical questions. Regarding simplicity, there are aesthetic arguments both ways. One argument is "It's adequate never to associate a position to gravitational energy. Hence theory should proceed along the simplest lines". Another argument is "It's better to have a theory that assigns the same positional features to all forms of energy because this is conceptually simpler than making gravitational potential energy a special case."

So let's say we give the following different answers, and consider ONLY classical mechanics. How would you propose we differentiate between the situations?

1) The gravitational interaction energy between two objects is stored in neither object alone, but is a property of the system (2 objects) as a whole.

This is what I would call a "gestalt" approach. It's analagous to saying "No individual line in the picture represents an elephant, you have to appreciate at the whole thing to see the animal." If this theory is "correct" in the sense that it says all that can be said correctly about potential energy then it should be impossible to be more specific about energy. To test it experimentally and theoretically, you could try to make more detailed statements about energy and see if they work. If they work, this theory is not "incorrect", but it is incomplete.

2) The gravitational interaction energy between two objects is stored in the object with smaller mass. In the case of equal mass, it is stored equally between the two objects.
I see nothing logically inconsistent about that theory. It's consistent with the idea that potential energy can be assigned an arbitrary position. It raises the question of whether gravitation potential energy is unique in this respect.

3) The gravitational interaction energy between two objects is stored in the gravitational field as defined in post #8.

Assuming conservation of energy, this approach can be falsified for particular energy density functions. For example, if we have two masses moving soley due to gravitational attraction, the total potential energy "in the field" plus the kinetic energy of the masses should be conserved. If it isn't, the proposed energy density function for the field doesn't work.

The falsifiability of this approach is an argument in favor of investigating it.

we can say that the potential energy is distributed among particles, maybe we can say particles with larger mass,carry more of the energy and vice versa. So we can say each particle has a "gas tank" with two parts: 1) My kinetic energy 2) My share of potential energy. So whenever the configuration is changing, the particles are just exchanging energy with each other from their gas tanks now either from the kinetic part or from the potential part. Now because interactions are instantaneous, at all moments, energy is in the particles and there is no time that energy is out of the particles and so its wrong to define a potential energy density function for space.

I understand this point of view. One interpretation of it is that gravitation potential energy is a property of mass.. Another interpretation of it is that the association of gravitational potential energy with position is arbitrary, so we might as well think of gravitation energy as being "in" mass. By the first interpretation it is wrong to think of gravitational energy being distributed in empty space. By the second interpretation, it is not anymore wrong to think of gravitational energy being distributed in space than thinking of it being distributed in mass.

Better try the other way round: "Does position of something have an energy?".
:D

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A major revolution in the transition from Newtonian physics to special and general relativity is the idea that all physical transactions are local--there is no action at a distance. When you have action at a distance, as in the theory of Newtonian gravity, energy can be instantly transmitted over distances, so it makes no sense to assign a position to the energy. In relativity, there is no action at a distance, so when two objects interact, energy must be carried by a field between them. In this case, energy always has a position because energy is locally conserved at any point in space. So if it is possible to assign an energy to gravity, it must be associated with the gravitational field and be carried by gravitational waves (which should travel at c).

I don't know enough general relativity to talk about gravitational energy in GR, but it is possible to use a hybrid Newtonian-relativistic approach, which uses the classical Newtonian field, but assumes that it propagates at speed c (rather than infinite). Then the physics are analogous to that of electromagnetic waves, and we can assign an energy density to gravity. We can use the theory of gravitomagnetism to account for the dynamics of the gravitational field. http://en.wikipedia.org/wiki/Gravitoelectromagnetism
This theory is a good approximation when gravity is weak. In this case, the total energy includes a contribution from the gravitic E field and the gravitic B field (which is small for slow moving masses).

In relativity, there is no action at a distance, so when two objects interact, energy must be carried by a field between them. In this case, energy always has a position because energy is locally conserved at any point in space. So if it is possible to assign an energy to gravity, it must be associated with the gravitational field and be carried by gravitational waves (which should travel at c).

Two strong currents in this thread are:
1) Forget about an energy being present in the Newtonian gravitational field
2) Energy is present in the real gravitational field, which is described by other theories.

I posted in the Classical Physics section to test the extent of a Newtonian approach, but, Ok, I'll bend to the irresistible digression toward reality.

Accepting the principle that Newtonian physics is an approximation or limiting case for situations described by more elaborate physical theories, it should be possible to view the nonexistent distribution of gravitational energy in Newtonian physics as an approximation for the actual distribution of gravitational energy in some situation. How can we do this? (The crude thought is "let the speed of light c approach infinity". I don't know whether that makes sense.)

Let's talk about an object of mass m initially at rest at a height h above the surface of the Earth in the non-Newtonian gravitational field where effects are propagated at a finite speed. Is it meaningful to ask "Where is the potential energy of the mass?". If we release the mass and it falls, how do we describe the change of potential energy into kinetic energy? Is energy at certain positions in the field transported to kinetic energy "in" the mass?

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.

I feel like you are talking in circles here.

Two strong currents in this thread are:
1) Forget about an energy being present in the Newtonian gravitational field
2) Energy is present in the real gravitational field, which is described by other theories.

I posted in the Classical Physics section to test the extent of a Newtonian approach, but, Ok, I'll bend to the irresistible digression toward reality.

Accepting the principle that Newtonian physics is an approximation or limiting case for situations described by more elaborate physical theories, it should be possible to view the nonexistent distribution of gravitational energy in Newtonian physics as an approximation for the actual distribution of gravitational energy in some situation. How can we do this? (The crude thought is "let the speed of light c approach infinity". I don't know whether that makes sense.)

Let's talk about an object of mass m initially at rest at a height h above the surface of the Earth in the non-Newtonian gravitational field where effects are propagated at a finite speed. Is it meaningful to ask "Where is the potential energy of the mass?". If we release the mass and it falls, how do we describe the change of potential energy into kinetic energy? Is energy at certain positions in the field transported to kinetic energy "in" the mass?
I know, you're seeing some light here but I have to disappoint you because GR is actually delicate at the linearized level. I think it was around 1950s or 1960s, that some people attempted to formulate gravitation as a linear theory on SR. But it turned out that this theory is actually inconsistent because it doesn't account for the energy in the gravitational field itself. So people tried to make it consistent and it turned out that this process actually gives GR! So you can't use linearized GR for what you want. Maybe some other approximation may work, or maybe you need full GR. But I don't think I can help further here.

As a mass falls, its kinetic energy increases and the total gravitational field energy becomes more negative.

Does the field energy becomes more negative in the vicinity of the falling mass before this effect is propagated to the distant parts of the field? I'm curious whether the model involves a flow of energy. Or does it just say the energy of the total field decreases without specifying any sort of flow?

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When the masses are accelerated, it takes some time for the change in the mass distribution to propagate over the gravitational field. The propagation is at c.

The "speed of gravity" in Newtonian physics is infinite in the following sense: the vector force between two objects will always point along the line connecting the center of mass of both objects, regardless of either object's trajectory, speed, or, most importantly, acceleration. In this sense, the vector force compensates instantaneously when either object accelerates, even if the two objects are very very far apart. I can thus construct a method of instantaneous communication the following way:

Actually interactions occurring instantaneously is foundational to classical non relativistic mechanics (even though its seldom emphasised) - see page 8 Landau - Mechanics.

But getting back to the original question - energy comes in a number of different forms - a point particle is one form and it has position - an EM field is another form and it doesn't.

Thanks
Bill

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an EM field is another form and it doesn't.
So what about $\frac 1 2 (\vec E \cdot \vec D+\vec B \cdot \vec H)$?

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

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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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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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

• THE HARLEQUIN
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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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.

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.

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?

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

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 can't 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