# Kinetic energy loss in a gravitational field.

If I throw a ball vertically upwards from the surface of the earth, the upward motion slows due to gravity. At the ball's highest point, I have placed a repositioned platform for the ball to rest on; so it sits motionless, perched above the surface of the earth. When the ball began it's flight, it initially had energy of motion (i.e., kinetic). That energy has now vanished as it sits motionless above the earth.

So here's the question: Assuming energy is ALWAYS conserved, where is the (kinetic) energy? Where can I find it? Where precisely is it stored? To say that the energy will be recovered when the ball is allowed to descend back to the earth does NOT answer the question. To say that the energy resides in the (gravitational) field does NOT answer the question. The energy must be PHYSICALLY stored somewhere.

Similarly, a photon looses energy (red shift) as it ascends away from earth's gravity. Where can that energy (loss) be found?

I don't believe there is a simple explanation. I pose the question to explore possible physical model ideas that could conceivably explain the location of the energy.

WannabeNewton
The gravitational field is a physical thing; it can and does store energy. You are entirely mistaken when you think that the gravitational field is not a physical thing. Fields carry energy and momentum and angular momentum analogous to how particles can carry energy, momentum, and angular momentum; this concept is usually first introduced in intermediate texts on electromagnetism.

If I throw a ball vertically upwards from the surface of the earth, the upward motion slows due to gravity. At the ball's highest point, I have placed a repositioned platform for the ball to rest on; so it sits motionless, perched above the surface of the earth. When the ball began it's flight, it initially had energy of motion (i.e., kinetic). That energy has now vanished as it sits motionless above the earth.

So here's the question: Assuming energy is ALWAYS conserved, where is the (kinetic) energy? Where can I find it? Where precisely is it stored? To say that the energy will be recovered when the ball is allowed to descend back to the earth does NOT answer the question. To say that the energy resides in the (gravitational) field does NOT answer the question. The energy must be PHYSICALLY stored somewhere.

Similarly, a photon looses energy (red shift) as it ascends away from earth's gravity. Where can that energy (loss) be found?

I don't believe there is a simple explanation. I pose the question to explore possible physical model ideas that could conceivably explain the location of the energy.

Maybe it would be a good idea to use a gravimeter to figure out the location of the energy.

http://en.wikipedia.org/wiki/Gravimeter

And let's make it a neutron star orbiting satellite-bourne gravimeter, measuring the energy distribution change when a powerful laser pulse is shot from the surface of the neutron star.

I mean: Let's think about what the gravimeter will say, I don't mean we should actually do this experiment.

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Nugatory
Mentor
So here's the question: Assuming energy is ALWAYS conserved, where is the (kinetic) energy? Where can I find it? Where precisely is it stored?

Answering classically - and until we have the classical model down cold, it's premature to consider energy and the gravitational redshift:

Nowhere. Total energy is conserved, but kinetic energy is not the total energy and is not necessarily conserved. Friction turns kinetic energy into heat, the spinning shaft of an electrical generator turns kinetic energy into electrical energy, a moving object sliding into a spring turns kinetic energy into potential energy, throwing an object upwards turns kinetic energy into potential energy... In all these cases the kinetic energy is gone even though the total energy (sum of kinetic energy and the others) is still conserved.

You may object that the potential energy is less "real", just a made-up bookkeeping device, something I added in because otherwise energy wouldn't be conserved. That may be, but the fact remains that the law of conservation of energy says that the conserved quantity includes the potential energy.

I am referring to "the total energy". If the kinetic energy (or red shift energy) is lost, then it must be present elsewhere. Energy of motion is real energy, thus E=mc^2. Said differently, mass (i.e., energy) is lost during the ascension.

As you aptly put it, "potential energy" is a convenient "book keeping device". In my scenario, there is no heat created, no electricity created, and no discernible location of the lost energy. "Potential energy" is another name for "We don't know where the energy is."

PeterDonis
Mentor
2020 Award
Energy of motion is real energy, thus E=mc^2.

This equation is not correct for an object in motion. The full equation, accounting for the motion of the object, is ##E^2 = m^2 c^4 + p^2 c^2##, where ##p## is the object's momentum.

Also, your definition of "real energy" is not a very good one. See below.

Said differently, mass (i.e., energy) is lost during the ascension.

This usage of the term "mass" is not recommended, because it often causes confusion between "relativistic mass" (i.e., energy), and invariant mass (or "rest mass", but the latter term can cause confusion as well). The invariant mass of the object does not change as it rises.

Also, kinetic energy is frame-dependent, so calling it "real energy" doesn't work very well. If I launch myself upward along with the ball, flying in free fall alongside it, its kinetic energy relative to me is zero. So what happened to its "real energy"?

But for free-fall motion (such as a thrown ball, if we ignore air resistance), there is a constant of the motion called "energy at infinity", which appears both in Newtonian physics and in General Relativity. This constant is not frame-dependent, so it's a much better candidate for "real energy" in this scenario. See further comments below.

As you aptly put it, "potential energy" is a convenient "book keeping device". In my scenario, there is no heat created, no electricity created, and no discernible location of the lost energy. "Potential energy" is another name for "We don't know where the energy is."

No, it's another name for "kinetic energy is frame-dependent, but energy at infinity is not". The ball's energy at infinity is the sum of its kinetic energy and potential energy, and it's often helpful to think of it that way. But "kinetic energy" here is defined relative to a static observer, i.e., an observer who is at rest at a constant altitude (which is the way you've implicitly been defining it); so energy at infinity is not frame-dependent, even though kinetic energy itself is.

This brings up another important point: what is special about static observers? Why does kinetic energy relative to them have a special status, whereas kinetic energy relative to me flying along with the ball does not? The answer is that a spacetime with a spherically symmetric gravitating body in it (which we've all been implicitly assuming here) has a time translation symmetry: it looks the same at every instant of time, if "instant of time" is defined relative to the static observers. Also, such a spacetime is "asymptotically flat", which means that the effects of gravity get smaller and smaller the further away you get from the central body, until they go to zero at infinity. This is what enables the definition of energy at infinity as a constant of motion for freely falling objects.

The reason I bring this up is that, in a more general spacetime which does not have a time translation symmetry (such as, for example, an expanding universe), there is no way to define anything like energy at infinity, or any global notion of "energy" that remains conserved. Light traveling to us from distant stars loses energy--we observe the energy loss as redshift--and the energy doesn't "go anywhere"; it just vanishes. So the bottom line is that your assumption that "energy is ALWAYS conserved" is not valid; energy conservation only works in certain special situations, and only if you pick the right definition of energy.

Dale
Mentor
2020 Award
"Potential energy" is another name for "We don't know where the energy is."
Nonsense. We know exactly where it is. Gravitational potential energy is in the gravitational field. Just because you don't like the answer doesn't mean that the answer is unknown.

mfb
Mentor
The gravitational field has an energy. This is similar to electric and magnetic fields. Light is an example how electromagnetic fields can carry energy. Capacitors and coils are other examples where it is clear that the fields have energy.
And your lifted ball is an example where the gravitational field has energy.

I agree with you 100% Wannabe... The gravitational field is a physical thing but the mechanism for storing energy is NOT classical physics. In fact, the mechanism is not discussed at all. So how does this "thing" called a gravitational field store energy? Classically speaking there exists nothing in the vacuum called "space", nothing in the void between stars with which to store energy. Once again the classical perception is TOTALLY incorrect.

The gravitational field is a physical system; it is not magic. So what are the (non-classical) options?

There is energy stored all around us, everywhere in the universe. Imagine if one could tap into that energy! It is apparent that the electromagnetic force (i.e., coulomb potential) has the same energy storage issue. I wonder if it shares the same storage mechanism with gravity?

I think we all agree that energy is somehow stored in the gravitational (and electromagnetic) field. The real issue (for me anyway) is HOW is it physically stored? It IS NOT magic, there exists a tangible mechanism that we can understand. The problem is that it exists in a tiny subatomic realm that we can not easily see or probe. So I'm wondering what possible physical models can exist in the so-called vacuum between the stars.

BruceW
Homework Helper
As you aptly put it, "potential energy" is a convenient "book keeping device". In my scenario, there is no heat created, no electricity created, and no discernible location of the lost energy. "Potential energy" is another name for "We don't know where the energy is."
Well there are two issues here. 1) Is there a common underlying physical mechanism to all kinds of energy? 2) Is energy conservation a tautology?
1) No, and this is why the concept of energy seems somewhat 'ad hoc'. 2) No, as PeterDonis mentioned, we don't necessarily have energy conservation. For example in General relativity we don't necessarily have global conservation of energy.

mfb
Mentor
HOW is it physically stored?
That question is meaningless in physics.
In the same way, you could ask "how are particles "stored?" - with the same result, that is not physics it is philosophy.

The question was "Where is the energy"?

Let's simplify this scenario a bit further to eliminate frame of reference issues. Instead of throwing the ball upwards, I will simply pick up the ball and place it on a perch above the surface of the earth. Simply put, I exert a force through a distance. I have done some work against the force of gravity, expended some energy, and I expect that the energy exerted should be located somewhere.

Maybe the energy is not stored in a (gravitational) field. Maybe the energy flew away as photon or sub atomic particles. Or what??? However, it is so easy to recover the energy that I believe it to be stored in the surrounding field. Maybe the space vacuum between stars is not really empty...?

BruceW has alluded to the idea that maybe energy is not necessarily conserved, and we are chasing a ghost. But in this simplified scenario (without energy of motion), I believe energy is conserved. If so, we should be able to locate it.

BruceW
Homework Helper
if we talk about classical gravitation, the gravitational potential energy is not stored in any place. It is probably best to think of energy as a property of the system. The system obeys certain physical laws such that there is some quantity that does not change with time, that we call energy. In classical gravitation, there can be no meaning attached to 'where' the energy is.

mfb - I think if you are asking "How are particles stored" you are really asking "how is it that particles possess mass-energy"? And that is a very important question in physics.

WannabeNewton
The gravitational field is a physical thing but the mechanism for storing energy is NOT classical physics. In fact, the mechanism is not discussed at all. So how does this "thing" called a gravitational field store energy?
Hi there mate. I see what you're trying to ask but I'm not sure if this can actually be answered by physics. There is a gravitational potential energy associated with the gravitational field and kinetic energies associated with particles and we can track exchanges between the gravitational potential energy and the kinetic energies, with the total sum being constrained by the constant total energy (I'm talking about a system of particles interacting only with the gravitational field here).

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mfb
Mentor
mfb - I think if you are asking "How are particles stored" you are really asking "how is it that particles possess mass-energy"?
No I do not.
Apart from that:
And that is a very important question in physics.
The Standard Model provides a nice mechanism for that.

When you started learning about the concept of energy, you were probably introduced to kinetic energy first and how it's related to the speed of objects. This has perhaps wrongfully given you the idea that energy is something you can locate in space.

Energy is a property of a system and, assuming you accept the common definition of energy as a system's ability to do work, there's really nothing strange at all about where the kinetic energy of your earth+ball system "goes". When the ball comes to rest on your platform, the earth+ball system has the same amount of energy as it did initially, i.e. it has the ability to do the same amount of work, but it's now by virtue of its configuration instead of its speed.

How the energy is stored, i.e. how does the system have the ability to do work, is a much more meaningful question than where is the energy stored.

I think we all agree that energy is somehow stored in the gravitational (and electromagnetic) field. The real issue (for me anyway) is HOW is it physically stored? It IS NOT magic, there exists a tangible mechanism that we can understand. The problem is that it exists in a tiny subatomic realm that we can not easily see or probe. So I'm wondering what possible physical models can exist in the so-called vacuum between the stars.
This is nonsense. Energy is an abstract concept but it's understood perfectly well by many without having to resort to some mysterious realm of science.

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Let's say at time t1 the energy planned to be used to lift a mass, is stored in a gas tank of a tower crane.

At time t2 the gas tank is empty and the mass has been lifted by the crane.

Now the energy is somewhere inside a sphere whose center is the gas tank and whose radius is c*(t2-t1).

BruceW - you make some good clear points. But I disagree. If I exert energy and compress a spring, does it make sense to ask "Where is the energy stored?" Gravitation isn't any different we just can't see what's happening. It's not magic. There is a physical system that absorbs the energy, and allows us to later recover the energy. It's conserved either as a change in mass, a compressed spring, electrical capacitor, or system we have yet to know about.

Which beckons the question "What is the physical mechanism of gravity?" If I knew how gravity works, I could likely explain where the energy is stored.

It's not any different in the case of elastic potential energy.

Energy is a property of a system. Temperature is also a property of a system. You might aswell ask where the temperature is.

Thanks milesyoung. If "energy" is an abstract concept then physics is an abstract concept, because they are inseparable. It turns out the four forces are a "mysterious realm of science".

PeterDonis
Mentor
2020 Award
There is a physical system that absorbs the energy, and allows us to later recover the energy. It's conserved either as a change in mass, a compressed spring, electrical capacitor, or system we have yet to know about.

The intuition you are expressing here is really local energy conservation, not global energy conservation. This is still true in GR, but there is a key difference between gravity and all the other interactions with respect to it.

The way GR expresses local energy conservation is that the covariant divergence of the stress-energy tensor is zero. This just means that, in any small 4-volume of spacetime, the amount of stress-energy going in exactly equals the amount of stress-energy coming out.

For example, take the case of compressing a spring. We take a small 4-volume of spacetime that contains the event of the spring being compressed; you can think of it as a small 4-dimensional hypercube with faces to the past, the future, and each spatial dimension. On the past face of the hypercube, the spring has a certain amount of stress-energy (its rest mass, since it is unstressed), which is coming in to the hypercube (since it's coming through the past face); on the future face of the hypercube, the spring has a larger amount of stress-energy (because it's now compressed, so it has its rest mass plus the stress--note that there are some subtleties here, which I won't go into unless it becomes necessary), which is going out of the hypercube (since it's going through the future face); and on one of the spatial faces of the hypercube, some stress-energy comes in in the form of work done on the spring. So the final tally is:

stress-energy coming in = (rest mass of spring) + (work done) = (rest mass of spring) + (stress in spring) = stress-energy going out

This all works nicely and fits in with the intuition you expressed--as long as gravity is not involved. But when gravity is involved, the tallying works differently from what you would expect from the above (it still works, though, which is an important point).

For example, take the case of slowly raising a ball from the floor to a shelf. Again, we can sum all the incoming and outgoing stress-energy over a 4-volume that encloses the event of raising the ball. On the past face of the 4-volume, we have the ball's stress-energy coming in at a lower height; on the future face, we have the ball's stress-energy coming in at a higher height; and through one of the spatial faces of the 4-volume, we have work coming in.

But now, when making the tally, we have to account for something that didn't come into play in the case of the spring, above: the contribution of a given amount of stress-energy to the tally depends on the gravitational potential. This is because the actual spacetime *volume* of the 4-volume depends on the gravitational potential; more precisely, in order to construct an invariant 4-volume element over which to evaluate the covariant divergence, we have to include the gravitational potential as a factor. So the tally in the case of raising the ball looks like this:

stress-energy coming in = (rest mass of ball) * (potential factor at lower height) + (work done) = (rest mass of ball) * (potential factor at higher height) = stress-energy going out

The potential factor at the higher height is larger, by just the right amount to account for the work done in raising the ball, so the final tally still comes out right: the covariant divergence of the stress-energy tensor is still zero. But the fact that the gravitational potential has to be included to make this work is why people talk about energy being stored "in the gravitational field". Note that the rest mass of the ball does *not* change, and there is no other stored energy in the ball either before or after the process of raising it.

mileyoung I agree with you. It is no different than the case of elastic potential energy. In which case I can tell you precisely where and how the energy is stored. You may be more correct than you realize. Are you suggesting there are elastic bands connecting the ball and earth together?

Energy must always be accounted for. It's ok to ask where is it or where did it go. You asked "Where is the temperature?" If the temperature inside your house suddenly dropped to -100 C, you want to know where the temperature went. Probably worded differently though.

BruceW
Homework Helper
BruceW - you make some good clear points. But I disagree. If I exert energy and compress a spring, does it make sense to ask "Where is the energy stored?" Gravitation isn't any different we just can't see what's happening. It's not magic. There is a physical system that absorbs the energy, and allows us to later recover the energy. It's conserved either as a change in mass, a compressed spring, electrical capacitor, or system we have yet to know about.

Which beckons the question "What is the physical mechanism of gravity?" If I knew how gravity works, I could likely explain where the energy is stored.
http://en.wikipedia.org/wiki/Gravitational_energy
A Newtonian 'energy density of the gravitational field', by analogy to electromagnetism. It can be seen as a term in the Lagrangian density for Newtonian gravitation.
http://en.wikipedia.org/wiki/Lagrangian#Newtonian_gravity
So I guess I was wrong. There is a sense in which the energy (in Newtonian gravity) is localized. Is this the kind of thing you were looking for? p.s. keep in mind that for General relativity, the situation is more complicated. In General relativity, there is no nice idea of 'gravitational energy'.