Paradox of gravitational potential energy

[e2m2a]

The classical paradigm of how an object acquires gravitational potential energy goes like this: First, we take the case where the object is near the surface of the Earth which we define as the zero reference point. The object has a mass of 1 kg. We lift the object by an agent, such as a hand, 1 meter above the surface, and we can account for the 9.8 joules of gpe that object has acquired. It acquired its gpe by the work done on the object by the external agent, the hand. If we release the object, the gpe of the object is converted to kinetic energy on its way down, and by the time it hits the surface of the earth, the object has acquired 9.8 joules of kinetic energy by the conservation of mechanical energy. Thus, all the energy is accounted for, everything balances out, no energy was created out of nothing. The object acquired gpe by the work done on it. Except--take the case where the object is not near the surface of the Earth initially, but at the outer reaches of the universe, such as a cosmic particle heading toward the earth. Here, we must use a different approach, setting infinity as the zero reference point. However, there is no change in the pe to ke conversion paradigm, for as the particle streams toward the earth, the gpe decreases (its negative value increases) and a corresponding increase in kinetic energy must occur in addition to its initial kinetic energy. By the time it hits the surface of the earth, all of the energy can be accounted for. Or can it? What agent "lifted" the cosmic particle out to infinity or did work on the particle to give it its initial increase in gpe (its negative value decreased) in the first place? Work had to be done on the particle by the classical paradigm to account for the particle's initial gpe if the conservation of energy is to remain intact. If no agent can be practically accounted for, then the particle must have acquired its gpe soley by the action of the gravitational field of the Earth on the particle. Does this mean gravity, the curvature of space-time, creates energy out of nothing?

 

[granpa]

actually, if 2 stars merge then their combined gravitational field is larger and containts more (negative) energy than before

 

[MeJennifer]

actually, if 2 stars merge then their combined gravitational field is larger and containts more (negative) energy than before

That is untrue. A reduction in mass-energy distribution reduces both the total mass-energy of all matter and the negative energy of the gravitational field.

 

[granpa]

well, the energy in the field is the integral of the square of the field strength is it not?

in any event is it not obvious that a larger object has more potential energy? between a grain of sand and a planet which has more potential energy?

 

[D H]

Except--take the case where the object is not near the surface of the Earth initially, but at the outer reaches of the universe, such as a cosmic particle heading toward the earth. Here, we must use a different approach, setting infinity as the zero reference point. However, there is no change in the pe to ke conversion paradigm, for as the particle streams toward the earth, the gpe decreases (its negative value increases) and a corresponding increase in kinetic energy must occur in addition to its initial kinetic energy. By the time it hits the surface of the earth, all of the energy can be accounted for. Or can it? What agent "lifted" the cosmic particle out to infinity or did work on the particle to give it its initial increase in gpe (its negative value decreased) in the first place?

There is no paradox. The agent that "lifted" the cosmic particle out to infinity is your initial assumption that the particle started at infinity in the first place. I suspect that, in a roundabout way, you are implying that the expansion of space violates conservation of energy. If this is the case, please say so so that others who have expertise in the subject can address your concerns.

 

[Jonathan Scott]

I don't see any paradox. Basically, when things are far enough apart (as you said "at infinity") so 1/r is small, the gravitational potential tends to a limit, rather than increasing in an unbounded way. That limit is the natural zero point for potential energy. However, when things get close together, the potential energy becomes more negative and the kinetic energy increases.

In a simple relativistic model, the potential energy loss corresponds to time dilation by a factor of (1-GM/rc^2) as the falling object reaches radius r from a central mass M and experiences the distortion of space-time caused by that mass. In a semi-Newtonian approximation, for a freely falling object the mass m decreases by that factor, so the potential energy changes by -GmM/r and the kinetic energy increases by the same amount.

However, that semi-Newtonian approximation only works for a single source and fails quite surprisingly spectacularly for two similar sources (as each one causes the other to be time-dilated in such a way that the total apparent potential energy loss is exactly twice the potential energy loss of the system as a whole). I'm told this is basically because "Newtonian" ideas and "General Relativity" don't work very well together, but if anyone has a more specific explanation I'd like to hear it.

 

[e2m2a]

There is no paradox. The agent that "lifted" the cosmic particle out to infinity is your initial assumption that the particle started at infinity in the first place. I suspect that, in a roundabout way, you are implying that the expansion of space violates conservation of energy. If this is the case, please say so so that others who have expertise in the subject can address your concerns.

Okay, let me rephrase the problem. It is an empirical fact that there are particles at huge (not necessarily infinite) distances from the earth. Not an assumption, a fact. We could assume reasonably that many of those partices never had any "merging with the earth", that is, there was no agent necessary to separate the particles from the Earth to where they are now. Yet, they still have a gpe with respect to the earth, which contradicts the classical paradigm discussed earlier which shows that an object acquires a gpe by the work done on it by an agent.

 

[ZikZak]

Yet, they still have a gpe with respect to the earth, which contradicts the classical paradigm discussed earlier which shows that an object acquires a gpe by the work done on it by an agent.

That is not the classical paradigm. The classical paradigm is that it has potential energy due to its configuration. It doesn't make the tiniest bit of difference how it got into that configuration. Energy is not a fluid that is poured into and out of bodies by external agents. Perhaps you have been studying Aristotle. We don't make a habit of relying on him for physical insight any more.

 

[D H]

Yet, they still have a gpe with respect to the earth, which contradicts the classical paradigm discussed earlier which shows that an object acquires a gpe by the work done on it by an agent.

Given some force $\mathbf F$ that is a function of position, a potential energy function corresponding to that force is any function that satisfies

$$\mathbf F(\mathbf x) = -\nabla \phi(\mathbf x)$$

One consequence is that if $\phi(\mathbf x)$ is a function such its gradient yields the force function, then so is the function $\phi'(\mathbf x) = \phi(\mathbf x) + c$, where c is an arbitrary constant. In other words, potential energy inherently involves an arbitrary constant. This arbitrary constant disappears whenever the potential is used for calculating the relation between kinetic and potential energy because the constant vanishes upon taking the difference between two values of the potential function.

This means that the arbitrary constant can be arbitrarily chosen, and people choose values that make their calculations easier.

 

[granpa]

We could assume reasonably that many of those partices never had any "merging with the earth", that is, there was no agent necessary to separate the particles from the Earth to where they are now. Yet, they still have a gpe with respect to the earth,.

they were merged with the Earth at the moment of the big bang.

 

[JesseM]

Okay, let me rephrase the problem. It is an empirical fact that there are particles at huge (not necessarily infinite) distances from the earth. Not an assumption, a fact. We could assume reasonably that many of those partices never had any "merging with the earth", that is, there was no agent necessary to separate the particles from the Earth to where they are now. Yet, they still have a gpe with respect to the earth, which contradicts the classical paradigm discussed earlier which shows that an object acquires a gpe by the work done on it by an agent.

I think you're conceptualizing potential energy incorrectly, the concept really has nothing to do with whether or not work was done in the past. You can essentially think of it as a bookkeeping device, a function which assigns potential energy to objects based on their position, but not on any other kinematical quantities like velocity or acceleration (though the function can depend on non-kinematical quantities like charge), in such a way that the sum of kinetic energy plus potential energy (plus rest-mass energy in relativity) is always conserved even as kinetic energy is varying due to forces changing the velocity of objects. Another way of stating this is that the potential function has the property that if you start with a collection of objects in certain positions, move them around a bit, and return them to their starting positions, their total potential + kinetic energy will be the same as it was initially. The nontrivial physical aspect of this is that all the known forces have the property that it's possible to construct such a position-based potential function for them such that potential + kinetic is always conserved; it's possible to write down hypothetical force equations where this would be impossible, these are known as nonconservative forces and they are actually sometimes used in Newtonian mechanics, but in the real world it's understood that these forces only appear nonconservative because you are ignoring some "hidden" form of kinetic energy like the increased random motion of molecules in a substance when it is heated up. All the most fundamental forces seem to have the property of being conservative, which guarantees conservation of energy (and by http://en.wikipedia.org/wiki/Noether's_theorem]Noether's[/PLAIN] theorem, conservation of energy is understood to follow from the fact that the laws of physics obey 'time translation invariance', meaning they don't change over time so the equations will be the same in two different inertial coordinate systems which define different moments as t=0)

 

[e2m2a]

they were merged with the Earth at the moment of the big bang.

This would appear on the surface to resolve the paradox, however, if taken literally it leads to other troubling problems. Does this imply that gravitational attraction between two objects is only possible if the two objects were once bound together as a past event in Minkowski 4-space?

 

[ZikZak]

This would appear on the surface to resolve the paradox, however, if taken literally

It wasn't meant to be taken literally (note the smiley). There is no paradox to resolve, only a misconception.

Does this imply that gravitational attraction between two objects is only possible if the two objects were once bound together as a past event in Minkowski 4-space?

No. Gravitational attraction exists between two objects because (classically) they have mass or (relativistically) they have momentum and energy density and flux.

 

[granpa]

an object with mass stretches the space surrounding it. this effect propagates outward forever. we call this the gravitational field. like a stretched spring it holds energy.

 

[JesseM]

an object with mass stretches the space surrounding it. this effect propagates outward forever. we call this the gravitational field. like a stretched spring it holds energy.

This isn't very good as a broad way of explaining potential energy, since there's potential energy in electromagnetism and Newtonian gravity too, and they have nothing to do with stretched space. I have my doubts that this explanation makes sense even in GR--if it were really like a stretched spring, you should be able to define how stretched it is at each point, but space isn't really like a physical material in this way, I doubt that any of the GR tensors would do this.

 

[e2m2a]

I apologize if I am not making the point of my paradox clear. The point is the true origin of the gpe of an object in a gravitational field. Let me restate the problem differently. Suppose a typical high school physics teacher gives a discussion that goes like this, and this is given in all standard high-school textbooks:
Teacher: "If we lift a book with a mass of 1 kg, 1 meter above the surface of the Earth with an infinitesimally small constant velocity, then we have done work on the book equal to 9.8 Newtons times 1 meter. The book at 1 meter above the surface of the Earth will have 9.8 joules of gpe. This gpe acquired by the book did not come from an action by the gravitational field on the book, it came by your applying work to the book to lift it. If the gravitational field did not exist, the work you did on the book would be manifested as some other form of energy, such as 9.8 joules of pure kinetic energy, and of course, the velocity of the book would not be constant as you lifted the book to its 1 meter position above the surface of the earth. Again, the gravitational field did not give the book its 9.8 joules of gpe, you gave it the 9.8 joules of gpe by the work you did on it. Now, if you stop applying a force at the 1 meter position, the book's infinitesimally velocity would go to zero very quickly and the book would fall back to the earth, and before it hits the surface of the earth, assuming no friction, the book will have acquired 9.8 joules of kinetic energy. Hence, all of the energy is accounted for-- 9.8 joules of work equals 9.8 joules of gpe equals 9.8 joules of kinetic energy. The gpe came from the work you did on the book and no where else."
Student: "But what if there was a very tall person who lifted the book with a decreasing force, such that book remained at an infinitesimally small constant velocity, out to an enormous distance beyond our solar system. And suppose the person lifts it to a point where right next to it is another book that did not orginate from our earth. Furthermore, suppose after the person stops applying a force to the book from the earth, the tiny gravitational force of the Earth pulls both books back to the earth. We can account for the kinetic energy of the book from the Earth before it collides at the surface of the earth, according to your explanation, by the work we did on the book to bring it to its final position. But what about the other book, not from the earth, it too will have the same ke as it collides at the surface of the earth, even though we did no prior work on it to bring it to where it was? How did it acquire its ke, if we did no prior work on it? Furthermore, let's assume instead of another book, we let the book from the Earth continue to drift after we stop applying a force to it, such that it eventually falls under the influence of another planet which is 1,000 times the mass of the earth. Clearly, when the book is about to hit the surface of this planet, it will have acquired more ke than if the book came back to the earth. How do we account for the greater ke of the book on this larger planet which will be greater than the initial work we did on it? According to your explanation, the energy accounts would not balance. Perhaps, the reason is that the gpe manifested in the book has nothing to do if there was or was not prior work done on the book. Maybe the gravitational field is the source of the gpe itself. Let me quote a passage from "The Meaning of Relativity", page 83, from none other than Albert Einstein: "It must be remembered that besides the energy density of the matter there must also be given an energy density of the gravitational field, so that there can be no talk of principles of conservation of energy and momentum for matter alone...The gravitational field transfers energy and momentum to the "matter," in that it exerts force upon it and gives it energy..."
Teacher: (Scratches his head.) "Hmmm. Maybe I need to submit this one to the physics forum."

 

[JesseM]

Teacher: "If we lift a book with a mass of 1 kg, 1 meter above the surface of the Earth with an infinitesimally small constant velocity, then we have done work on the book equal to 9.8 Newtons times 1 meter. The book at 1 meter above the surface of the Earth will have 9.8 joules of gpe. This gpe acquired by the book did not come from an action by the gravitational field on the book, it came by your applying work to the book to lift it.

As I already told you, this explanation is completely wrong, potential energy has nothing to do with work done in the past. It is just a sort of bookkeeping device to ensure that the sum (kinetic energy + potential energy) is always constant, in spite of the fact that kinetic energy changes with velocity. For example, if you drop a book from 4.9 meters up, if it starts at rest it will take 1 second to fall this far (since x(t) = 4.9*t^2), and its velocity will have increased to 9.8 m/s (since v(t) = 9.8 * t), so if its mass is 1 kg then it has gained a kinetic energy of (1/2)*m*v^2 = (1/2)*(1 kg)*(9.8 m/s)^2 = 48 kg*m^2/s^2. So in order for energy to be conserved, we must say that its potential energy decreased as it fell by exactly the same amount, which works out if you say that the potential energy at a given height h is h*(9.8 m/s^2). That's all that matters, making sure the sum kinetic energy + potential energy never changes even as each one changes individually. The key, as I said before, is that only for particular types of force laws is it possible to find potential energy functions which only depend on position rather than other kinematical quantities like velocity, and which have the property that if you move some masses around and return them to their starting points, their total energy will be unchanged. It's an interesting and useful property of all the known fundamental force laws that you can construct potential functions of this type for them.

But what about the other book, not from the earth, it too will have the same ke as it collides at the surface of the earth, even though we did no prior work on it to bring it to where it was? How did it acquire its ke, if we did no prior work on it?

Kinetic energy also has nothing to do with "prior work", it is simply a function of an object's velocity and mass at a particular instant. It is true that if you want to increase an object's kinetic energy from its current value by using up some potential energy you have stored up (in your muscles, say), then if you are perfectly efficient at converting that potential energy into kinetic energy of the object (impossible in real life, since some of the potential energy will be converted into kinetic energy in the form of random motion of molecules, i.e. heat), then the increase in the object's kinetic energy will be exactly equal to the decrease in potential energy. But there's no requirement that if you find an object with a given kinetic energy, it must have "acquired" that kinetic energy because some work was done on it, maybe it was moving with that velocity forever.

Furthermore, let's assume instead of another book, we let the book from the Earth continue to drift after we stop applying a force to it, such that it eventually falls under the influence of another planet which is 1,000 times the mass of the earth. Clearly, when the book is about to hit the surface of this planet, it will have acquired more ke than if the book came back to the earth. How do we account for the greater ke of the book on this larger planet which will be greater than the initial work we did on it? According to your explanation, the energy accounts would not balance.

Yes, they'll always balance. The potential fields of both the larger planet and the Earth extend out to infinity, so the book's total potential energy at each moment is a sum of the potential from Earth and the potential from the larger planet. If you do the full calculation, you do find that any increase in kinetic energy from one point to another is always precisely balanced by the decrease in total potential energy from this sum.

Perhaps, the reason is that the gpe manifested in the book has nothing to do if there was or was not prior work done on the book. Maybe the gravitational field is the source of the gpe itself. Let me quote a passage from "The Meaning of Relativity", page 83, from none other than Albert Einstein: "It must be remembered that besides the energy density of the matter there must also be given an energy density of the gravitational field, so that there can be no talk of principles of conservation of energy and momentum for matter alone...The gravitational field transfers energy and momentum to the "matter," in that it exerts force upon it and gives it energy..."

I don't think the concept of "gravitational potential energy" even really makes sense in general relativity, see this page. I assumed your question was about gravitational potential in Newtonian physics.

 

[e2m2a]

Again, the fault is with me. I was not clear on how this paradox could be addressed, not in Newtonian concepts, but in general relativistic concepts. I wanted to address gpe in terms of general relativity. I read of debates on gr on this issue, but to be honest, I don't know the math well enough to see the subtleties. I have heard there are problems with gr because of this issue. Baez does talk about it. I don't think the problem is satisfactorily resolved by all camps in gr. Did Einstein really mean that energy-momentum sources from a gravitational field and only the gravitational field? In the hypothetical discussion I had, the teacher gives a typical explanation of how gpe of an object is acquired. I'm not asserting it, I'm just the messenger. You will find this explanation in a random sampling of textbooks teaching 1st year physics. They do this because the are looking for everything to jive with the conservation of energy, and it must. So, THEY would state that when you lift an obect, there is a decrease in the internal energy of your body by metabolic, chemical processes, etc. and therefore, since the energy can't disappear into nothing, it must be accounted for somewhere else. So, they say the loss in internal energy must add up to the heat manifested, the changes in the chemical states manifested, the ke manifested in the object, and the gpe manifested in the object, and pronounce, we have accounted for all the energy by this convient bookkeeping device. THEY would assert by the kinetic energy-work theorem that any change in ke of an object has to be due to negative or positive work done on the object. Any way, I think gr implies the gravitational field is the source of the energy. No prior work is needed was the point I was trying to make in the problem in the context of gr.

 

[JesseM]

Again, the fault is with me. I was not clear on how this paradox could be addressed, not in Newtonian concepts, but in general relativistic concepts.

Either way, it's not true in Newtonian mechanics that potential energy has anything to do with how much work was done on an object in the past, so why would it be true in GR? Anyway, as the page I linked to mentions, the whole concept of "potential energy", and energy conservation in general, is a bit problematic in GR, although apparently in certain particular metrics like like the Schwarzschild metric, energy conservation can be made to work out:

Light from the Sun appears redshifted to an Earth bound astronomer. In quasi-Newtonian terms, we might say that light loses kinetic energy as it climbs out of the gravitational well of the Sun, but gains potential energy. General relativity looks at it differently. In GR, gravity is described not by a "potential" but by the "metric" of spacetime. But "no problem", as the saying goes. The Schwarzschild metric describes spacetime around a massive object, if the object is spherically symmetrical, uncharged, and "alone in the universe". The Schwarzschild metric is both static and asymptotically flat, and energy conservation holds without major pitfalls. For further details, consult MTW, chapter 25.

Chapter 25 of Gravitation by MTW (Misnter-Thorne-Wheeler) mentions that in the Schwarzschild metric one can define an "effective potential", which is also discussed a bit http://www.bun.kyoto-u.ac.jp/~suchii/eff.potent.html . Apparently the sum of the square of the effective potential divided by m and the square of the radial velocity is equal to the square of the total energy divided by m, so if the effective potential is V, the equation is $$(dr/d\tau )^2 + (V/m)^2 = (E/m)^2$$.

Did Einstein really mean that energy-momentum sources from a gravitational field and only the gravitational field?

I don't understand this sentence, can you elaborate on the question?

In the hypothetical discussion I had, the teacher gives a typical explanation of how gpe of an object is acquired. I'm not asserting it, I'm just the messenger. You will find this explanation in a random sampling of textbooks teaching 1st year physics. They do this because the are looking for everything to jive with the conservation of energy, and it must. So, THEY would state that when you lift an obect, there is a decrease in the internal energy of your body by metabolic, chemical processes, etc. and therefore, since the energy can't disappear into nothing, it must be accounted for somewhere else. So, they say the loss in internal energy must add up to the heat manifested, the changes in the chemical states manifested, the ke manifested in the object, and the gpe manifested in the object, and pronounce, we have accounted for all the energy by this convient bookkeeping device.

Sure, but they're just saying that total energy is always conserved, so the decrease in energy of your body must be compensated by an increase in other forms of energy. That is definitely not the same as saying that potential energy is defined in terms of work done on an object in the past, as you seem to think. In theory, an object can have potential energy even if it has been sitting in the same position for an infinite time, with no work ever having been done on it.

THEY would assert by the kinetic energy-work theorem that any change in ke of an object has to be due to negative or positive work done on the object.

Sure, any change in kinetic energy must be because of a change in potential. But again, this has nothing to do with the idea that an object can only have potential energy if work was done on it in the past, why would you think the first idea would imply the second?

Any way, I think gr implies the gravitational field is the source of the energy. No prior work is needed was the point I was trying to make in the problem in the context of gr.

No prior work is needed in Newtonian physics either, this is just a weird misconception of yours. And when you say "the gravitational field is the source of the energy", what do you mean by "the energy"?

 

[Dale]

I think the problem here is simply a poor pedagogical approach. It appears that e2m2a has been taught that gravitational potential energy is related to work done on an object, rather than simply being related to an object's position in a gravity field. DH's post is probably the best, assuming that e2m2a understands the gradient operator.

 

[Anticitizen]

If one's worldview only extends a few billion kilometers beyond the Earth, then yes, it would appear that objects of free potential energy of mysterious origin were appearing at the edge of the conceptual sphere when they moved into it... however, the universe as a whole has a finite sum of energy within its scope, and that potential energy was 'out there' all along.