Gravitational potential energy storage location

[jpo]

Hello,

I thought of asking would someone know what is the correct thinking here...
A body is thrown upwards, its kinetic energy gradually transforms into gravitational potential energy until it is entirely absorbed.

At this moment, where is the potential energy? According to wikipedia (N. Umov)
"Umov considered potential energy as kinetic energy of some environments "imperceptible for us". From this hypothesis, he made a conclusion: it is always possible to specify a place where the energy is in."

Perhaps general relativity gives the answer... Sadly, I hardly know anything about GR

Thanks in advance.

 

[Dale]

It is not always possible to localize energy in GR. In fact, it can become difficult to even define it:
http://www.desy.de/user/projects/Physics/Relativity/GR/energy_gr.html

 

[jpo]

How should one think of the gravitational potential energy location then...

With a moving body it seems easier, because wherever the body is, at this same locale is the capability of collision and consequently doing work. With waves, the Poynting vector shows the direction of energy flow, which still gives an idea of "energy location"

Potential energy seems very confusing; the body at height h does not have energy yet; it has the "potential" to have it. Where , in what locale did the energy go?

Another confusing issue with the thrown up body is the loss of kinetic energy. Can we say its kinetic energy has been absorbed by the gravitational field?

 

[Dale]

How should one think of the gravitational potential energy location then...

One shouldn't think of the location of the potential energy. Just think of it as a property of the system as a whole. Don't try to attribute it to one part of the system.

 

[jpo]

What about the energy storage? If a body has been lifted up and put on a shelf indefinitely, this should mean that the gravitational field has stored its kinetic energy (needed to have it lifted), correct?

 

[Dale]

Yes, the energy is stored, but that doesn't mean that there is a need to localize.

 

[jpo]

DaleSpam, thank you for your replies. Is there a text one can read about how gravitational field stores energy? Esp. a text for those without GR knowledge :)

I'd like to read more on the storing mechanism... Or perhaps you can share more on this subject?

 

[zonde]

Hello,

I thought of asking would someone know what is the correct thinking here...
A body is thrown upwards, its kinetic energy gradually transforms into gravitational potential energy until it is entirely absorbed.

At this moment, where is the potential energy?

It goes into mass of the body.

It can be easily seen for reverse scenario. When body falls down on other mass there is mass deficit after energy equivalent of kinetic energy is radiated out.
You can read about mass deficit in wikipedia - http://en.wikipedia.org/wiki/Mass_deficit#Mass_deficit

 

[pervect]

DaleSpam, thank you for your replies. Is there a text one can read about how gravitational field stores energy? Esp. a text for those without GR knowledge :)

I'd like to read more on the storing mechanism... Or perhaps you can share more on this subject?

If you can get a hold of MTW's gravitation, https://www.amazon.com/dp/0716703440/?tag=pfamazon01-20

you'll find one whole chapter (a short one) that explains why you can't localize the energy in a gravitational field, which is what people have been telling you here and where your question started out.

Offhand I'd say that this fact about gravity is incompatible with your idea of a "mechanism", because I would think any concept of a "mechanism" as being basically some mechanical analogue, would localize the gravitational energy.

Perhaps I'm wrong, as it's not totally clear what sort of "mechanism" you're asking for.

Unfortunately, localizing the energy of the gravitational field is not possible, at least not within the context of GR. There are some other theories of gravity that might allow such a localization, most of these theories predict and require extra fields that nobody has been able to observe to date.

Classical GR, however, does NOT allow such a localization of gravitational energy.

 

[jpo]

I am trying to understand what people have posted above (for which posts I am thankful)...

The problem is that the conversion of kinetic energy of a thrown up body into gravitational potential energy leaves the said body at height h. It has lost its kinetic energy and has gained nothing, except the potential to produce back the lost kinetic energy. In the meantime, if put on a shelf at height h, its temperature has not risen; its internal pressure has not risen etc; there are no methods to prove it has energy, unless it is dropped back down.

zonde, do I understand you correctly? You wrote that the gained gravitational potential energy actually adds mass to the body and it becomes m + dm, where dm is a result of the binding energy, e.g. when the body is dropped down, its mass will be reduced back to m and dm will be lost through heat etc which is the binding energy

Do I understand this correctly?

Many thanks.

 

[zonde]

zonde, do I understand you correctly? You wrote that the gained gravitational potential energy actually adds mass to the body and it becomes m + dm, where dm is a result of the binding energy, e.g. when the body is dropped down, its mass will be reduced back to m and dm will be lost through heat etc which is the binding energy

Do I understand this correctly?

Yes, that's correct.

 

[my_wan]

The whole notion of the localization of energy ill conceived to me. This is not unique to either kinetic or potential energy. Starting with kinetic energy, if two asteroids have kinetic energy $E_k$ with respect to each other then asteroid at asteroid A it is said that asteroid B possesses $E_k$. At asteroid B it is said asteroid A possesses $E_k$. Neither A nor B, which may be separated by millions of km, alone possesses $E_k$. So where is $E_k$ really at? The question itself is meaningless.

In the potential energy $E_p$ case the $E_p$ is not defined by a property of the mass, but rather by the fact that there happens to be another lower energy state available nearby. Hence it's relational in a similar way to the relational character of $E_k$. Remove this nearby lower energy state and the $E_p$ of that mass disappears without making any changes to the state of that mass alone. Asking where $E_k$ or $E_p$ is at is like an American asking a Chinese why they call up down.

This same issue comes up over and over again which even the basics of relativity illustrate how pointless it is. Yet relativity tends to be falsely thrown out only in the context of disavowing realism while the location of purely relational properties we associate with such objects is debated as if even the basics of relativity do not so obviously disavow the meaning of the question. We have even formalized this reality of objects = reality of relational properties locally inhere to said object in many of our no-go theorems. You can instantly increase $E_k$ of an object many light years away simply by walking toward it.

 

[jpo]

the gravitational potential energy adding a mass dm to the lifted body...
I am assuming this has been verified experimentally, (specifically for gravity)?

 

[PAllen]

the gravitational potential energy adding a mass dm to the lifted body...
I am assuming this has been verified experimentally, (specifically for gravity)?

So far as I know, the only thing measured is lower mass in a bound state. However, the relation between this and purported higher mass from simply moving something 'up' a potential well is tricky.

Consider gravity and collapsing dust: as the dust collapses, the dust gains KE, thus heats up. The excess heat is radiated away (in the typical case where it started in thermal equilibrium with its surroundings). Before radiating away heat, mass has not changed - both rest mass and KE contribute to inertial and gravitational mass (per relativity). As thermal equilibrium is re-gained, the mass of the collapsed dust is lower, but only by virtue of radiating away random KE = heat as EM radiation.

Now imagine trying to reverse the process. Energy must be expended pulling the dust apart again. Assuming KE of dust particles is unchanged (in, e.g. COM frame of the system), the mass of the dust farther apart is unchanged. What has decreased in mass is whatever provided the energy separating the dust back to greater distance from COM.

I think my_wan's post is thus particularly apt - the whole issue of localizing energy misguided or at least frame dependent and convention dependent.

 

[russ_watters]

How should one think of the gravitational potential energy location then...

With a moving body it seems easier, because wherever the body is, at this same locale is the capability of collision and consequently doing work.

That isn't correct/the same problem exists for KE as PE. Consider a car on a road. You can't attribute a single KE to it: it has a different KE to different observers. For example, it could collide with a wall or a moving car and have two different energies. As said, the energy is a property of the system, not the car.

 

[my_wan]

the gravitational potential energy adding a mass dm to the lifted body...
I am assuming this has been verified experimentally, (specifically for gravity)?

Yes it has been verified directly through momentum, i.e., relativistic mass, gains in photons traveling up and down the side of a tall building using interferometers. The frame and convention dependence of these descriptions can be quantitatively justified by the fact that given any set of these frame dependent descriptions can be be converted to any other via affine transforms. So anyone is a valid description of all other valid frames. It is only when you start with questions about which affine space of the system is the "real" affine space that you get into trouble with trying to localize observables that do not have locations.

 

[PAllen]

Yes it has been verified directly through momentum, i.e., relativistic mass, gains in photons traveling up and down the side of a tall building using interferometers. The frame and convention dependence of these descriptions can be quantitatively justified by the fact that given any set of these frame dependent descriptions can be be converted to any other via affine transforms. So anyone is a valid description of all other valid frames. It is only when you start with questions about which affine space of the system is the "real" affine space that you get into trouble with trying to localize observables that do not have locations.

Can you indicate what experiments you're referring to? Things like Pound-Rebka do not, in any way, demonstrate a kg at the top of building is more massive that a kg at the bottom. They measure the anolog for photons of the following: drop a mass from tall building, and at bottom it has more total energy (mass + KE) than it does locally at the top. If it bounces inelastically back to the top, it again has the original total energy (same mass, no KE). What Zonde seemed to be proposing is some sense in which (in principle) a stationary kg at the top of a building is more massive than a stationary kg at the bottom. This runs up against the fact that there is really no way to measure 'mass at a distance'. The closest I can come to a coordinate framework in which you can claim to measure this effect is as follows:

At the top of the building, have a measuring instrument that is moving upwards at a speed such that its clock is in synch with the ground, and light it emits would not be blueshifted (upward motion red shift balancing gravitational blue shift). Then, that instrument, as it approaches the top of the building, would see light emitted from the top of the building blueshifted the same as a ground observer would; further, a stationary kg will have more total energy than one moving with the instrument - a truism because it is moving relative to the instrument. So setting up coordinates in just this special way, you could claim to demonstrate the high kg having more energy but it would look like extra KE not extra rest mass.

Can you suggest a more direct way to measure, in principle, a higher mass for the higher kg block?

[Edit: removed edit. Just confused matters. ]

 

[PAllen]

Re #17, I thought of another example of how you try to measure mass change of an 'identical' object at different altitudes. It shows just how difficult it is to physically justify the claim of that a higher kg is has more mass.

Consider an ideal spring oscillator running horizontally (so gravity not directly involved) on a frictionless surface. Spring fixed at one end, attached to frictionless 1kg block on the other. Set up e.g. 10 cm oscillation. Its period depends on the mass, for a given ideal spring (longer period, greater mass). Then transport your apparatus to the top of a tall building. Ask someone remaining at the top to repeat the experiment while you go back down and watch. Compared to your original timing, it runs fast. So you conclude, measured this way, the 1kg at the top must be less massive. Of course, if you then say that's stupid, let me adjust for time dilation, you find the mass is the same.

 

[my_wan]

Can you indicate what experiments you're referring to?

I tried looking it up but it was before the internet though after Pound-Rebka. At least the 70s. From the best of my recollection lasers pointed down a building and reflected back up compared to a reference reflected parallel to the ground was employed, with the results I think obtained by a phase shift in the interference. Though I don't have a clue at this point how they got a reference. Given the time period it was likely either Discover or Scientific America, more likely the later, where I read the description.

This runs up against the fact that there is really no way to measure 'mass at a distance'.

Even the setup I read had to calculate the effective mass. The effective mass variance of Mercury also plays a role in the procession.

Can you suggest a more direct way to measure, in principle, a higher mass for the higher kg block?

The most relevant scheme I have previously thought about was not explicitly to measure mass variances at different heights, or even mass itself as such, but rather a scheme to measure more directly the variance in the spacetime metric as the result of a gravitational field. Like a GR based version of a gravimeter. This consisted of a large wheel, or magnets on the end of a spinning wheel or spokes, passing a magneto. The output of the magnetos would be input such that the voltages cancelled. In this way it would only read the voltage difference, which could be arbitrarily amplified within noise constraints. To obtain a reference where cancellation of voltages occur you turn the magneto pairs so they're parallel to the gravitational and adjust out the voltage differences. To insure this cancellation is not biased do the same with the magneto positions swapped. Now when the magnetos are set top to bottom there should be a voltage difference as a result of the different spacetime metrics at the two magnetos. This difference should also vary depending on the height of the difference meter relative to the two magnetos, due to the relativistic character of the difference. At least within the min/max heights defined by the difference in magneto distances. Noise would be a particularly troublesome issue but it may at least be possible to verify the effect with enough noise control and shielding on magnetic bearings. This explicitly depends on the spacetime metrics covariant effects on all physical phenomena, not just mass.

I'll look some more and/or try to come up with other schemes. My memory of the experiment I mentioned is likely faulty in significant ways.

 

[PAllen]

The most relevant scheme I have previously thought about was not explicitly to measure mass variances at different heights, or even mass itself as such, but rather a scheme to measure more directly the variance in the spacetime metric as the result of a gravitational field. Like a GR based version of a gravimeter.

That should be possible, but that has nothing to do with Zonde's claim that there is physically meaningful sense in which an object maintaining constant temperature, density, etc. gains mass by moving up a gravity well. That is, that there is a well defined way to localize potential energy to an object and measure it as mass increase. I dispute this. I would be really interested in a measurement that claimed to do this (even in principle; esp as I've proposed two thought experiments that suggest it is not so).

 

[my_wan]

I feel like the location issue was answered. The notion of whether mass is really added gravitationally or not is as real as any measure of kinetic and potential energy in general. Saying when and where (as in a particular mass), and how much if you are thinking in terms of a single state variable, is as frame dependent as the asteroid analogy. In the case where one frame defines a particular state variable as lessened or absent it still entails that another state variable must contain the equivalent quantity, even if that is somewhere else completely. Affine spaces are isomorphic this way.

The notion that the rest mass of an object can go to zero by separating it far enough from another mass is only technically true in the limit. But for practical purposes it is false. In essence to accomplish this you have to separate the the entire mass out to infinity. Not very practical. Even separating two intact parts to infinity does not reduce the masses to zero, only their gravitational potential energy. Almost a tautology when you think about it.

----
I thought of another possibility for measuring the relative spacetime metrics in a gravitational field. In the case where you change the gravitational depth of a standard power meter reading of some electrical output the total power will not change. Much the same reason it is so hard to measure a metric that is not local to the measuring device. Instead what should happen is the voltage to amperage ratio should inversely vary. So perhaps take a tiny constant power output in watts but with an extremely high volt to amp ratio. Now any tiny variation in amps should be highly magnified by the voltmeter. Basically a magnifier for the inverse relation between space and time as measured at different altitudes. It would also be much easier to control noise in this setup.

 

[zonde]

Consider an ideal spring oscillator running horizontally (so gravity not directly involved) on a frictionless surface. Spring fixed at one end, attached to frictionless 1kg block on the other. Set up e.g. 10 cm oscillation. Its period depends on the mass, for a given ideal spring (longer period, greater mass). Then transport your apparatus to the top of a tall building. Ask someone remaining at the top to repeat the experiment while you go back down and watch. Compared to your original timing, it runs fast. So you conclude, measured this way, the 1kg at the top must be less massive. Of course, if you then say that's stupid, let me adjust for time dilation, you find the mass is the same.

Period of oscillator depends from mass and properties of spring (spring constant). So it's hard to say why it runs faster.

But consider another example. We have big gravitating mass and small test mass orbiting big mass. Orbital period of test mass depends from mass of gravitating body and radius of orbit.
If we increase gravitating mass leaving radius the same orbital period decreases i.e. system is "oscillating" faster.

 

[zonde]

the gravitational potential energy adding a mass dm to the lifted body...
I am assuming this has been verified experimentally, (specifically for gravity)?

It does not seem feasible for gravity.
But it certainly is common knowledge for nuclear reactions that bound system weights less by dm that is equivalent to binding energy. There it is much easier because lost energy is quantized (as single photon) and easily can be measured.
Maybe it can reach detectable levels in chemistry as well.

 

[PAllen]

Period of oscillator depends from mass and properties of spring (spring constant). So it's hard to say why it runs faster.

This is a thought experiment. We've raised the apparatus up a tall building. Nothing can change about it for an observer right next to it. It oscillates faster as observed by someone remaining at the bottom purely due to time dilation deeper in the gravity well. Then, compensate for this, and mass is the same. No reasonable method gives you mass greater. See esp. my post #14 explaining how gravitational binding energy (mass deficit) physically comes about.

 

[jpo]

thank you all for your input; the additional mass dm added to m to explain the potential energy is an interesting concept and ties up well with the concept of binding energy; albeit one should in all fairness remain skeptical without verifiable experimental observation

 

[Naty1]

PAllen posts:

Consider gravity and collapsing dust: as the dust collapses, the dust gains KE, thus heats up. The excess heat is radiated away (in the typical case where it started in thermal equilibrium with its surroundings). Before radiating away heat, mass has not changed - both rest mass and KE contribute to inertial and gravitational mass (per relativity). As thermal equilibrium is re-gained, the mass of the collapsed dust is lower, but only by virtue of radiating away random KE = heat as EM radiation.

This seems incorrect:

Before radiating away heat, mass has not changed - both rest mass and KE contribute to inertial and gravitational mass (per relativity)

Seems like rest mass HAS changed due to the increased thermal motion of particles while KE does NOT contribute to gravitational mass...since in the frame of motion
invarient rest mass remains constant...but likely frame dependent secondary curvature is affected:

In this recent discussion, very different conclusions were reached, or I do not understand something:

https://www.physicsforums.com/showthread.php?t=558535&page=7

I apologize for referring you to such a long winded thread but as it unfolds PeterDonis,Pervect,and Atyy seem to provide opposite conclusions.
A quick summary of one conclusion might be this:

...For a system of particles moving rapidly in different directions, all frames will show a system of rapidly moving particles; which is moving which way will change, but you can't transform away the fact that total KE of the system is greater than for a slow moving system of similar particles.

(and my comment in that discussion:)
And the old example of a compressed spring in a jack in the box also results in greater gravitational attraction: you can't transform away the compressed spring energy either.

I'd like to know what you think.

edit: Here is another post from that long one above:

"...The "amount of gravity produced" by the object is not a function of its energy alone, it's a function of its stress-energy tensor [ SET] of which energy is only one component. In a frame in which the object is moving, there will be other non-zero components of the SET as well as the energy, and their effects will offset the apparent "effect" of the increase in energy, so the final result will be the same as it is for a frame in which the object is at rest..."

 

[PAllen]

PAllen posts:This seems incorrect: Seems like rest mass HAS changed due to the increased thermal motion of particles while KE does NOT contribute to gravitational mass...since in the frame of motion
invarient rest mass remains constant...but likely frame dependent secondary curvature is affected:

In this recent discussion, very different conclusions were reached, or I do not understand something:

https://www.physicsforums.com/showthread.php?t=558535&page=7

I apologize for referring you to such a long winded thread but as it unfolds PeterDonis,Pervect,and Atyy seem to provide opposite conclusions.
A quick summary of one conclusion might be this:

I'd like to know what you think.

edit: Here is another post from that long one above:

"...The "amount of gravity produced" by the object is not a function of its energy alone, it's a function of its stress-energy tensor [ SET] of which energy is only one component. In a frame in which the object is moving, there will be other non-zero components of the SET as well as the energy, and their effects will offset the apparent "effect" of the increase in energy, so the final result will be the same as it is for a frame in which the object is at rest..."

What I said is correct and consistent with that thread. For collapsing dust, the total momentum remains zero, therefore all of the KE adds to the invariant mass. Also note that thermal motion = KE in COM frame (which is what was obviously being used for collapsing dust).

[Edit: Then, after radiating excess heat, the mass decreases. The lower mass after thermal equilibrium is then what we call the mass deficit due to gravitational binding energy.]

 

[zonde]

This is a thought experiment. We've raised the apparatus up a tall building. Nothing can change about it for an observer right next to it. It oscillates faster as observed by someone remaining at the bottom purely due to time dilation deeper in the gravity well. Then, compensate for this, and mass is the same.

Yes, you can use viewpoint like that but this is no argument against my viewpoint.

No reasonable method gives you mass greater.

I gave my example.

See esp. my post #14 explaining how gravitational binding energy (mass deficit) physically comes about.

That is all clear. Except maybe that in reverse case you have not said at what moment mass deficit disappears. Instead you say "the mass of the dust farther apart is unchanged".
But I think it's clear that taking it in symmetric fashion mass deficit disappears when external energy is added to the system.

 

[PAllen]

I gave my example.

I don't understand the relevance of your example. It was just the obvious statement that orbital period at a given radius decreases as the central mass increases. I don't follow any relation to showing greater mass for a given object when it has higher gravitational potential energy. You may well have some argument in mind, but you didn't state it at all.

That is all clear. Except maybe that in reverse case you have not said at what moment mass deficit disappears. Instead you say "the mass of the dust farther apart is unchanged".
But I think it's clear that taking it in symmetric fashion mass deficit disappears when external energy is added to the system.

I thought I said quite explicitly. The mass of whatever provides the energy to separate the dust decreases. So the total mass of dust plus whatever provides the energy to separate it remains unchanged. You can say the mass deficit was transferred to the supplier of energy to separate the dust without changing its temperature or state. If you imagine repeating this process, you end up converting mass to radiation. The only time total the mass in the local area ever changes is when it decreases by radiation.

[Edit: Ok, I think I see your way of looking at it, and I agree it is a valid way of looking at it. If you imagine N dust particles at a given temperature and state collapsed, they definitely appear less massive from a distance than the same N particles, the same temperature and state, when they are farther apart. This is certainly true, even if it doesn't happen 'magically' just by moving the dust. It happens because to achieve same state after collapse, energy must be radiated; and to achieve same state on expansion, energy must be supplied. ]

 

[Dale]

But it certainly is common knowledge for nuclear reactions that bound system weights less by dm that is equivalent to binding energy. There it is much easier because lost energy is quantized (as single photon) and easily can be measured.

Yes, but note that even for nuclear reactions, where the mass deficit is easily measurable, it is not possible to localize the deficit to a specific nucleon. E.g. A helium nucleus is less massive than two protons and two neutrons, but we cannot say that it is the protons which lost the mass.

Nobody is disputing the mass deficit when it applies, merely the ability to localize it.

 

[zonde]

I don't understand the relevance of your example. It was just the obvious statement that orbital period at a given radius decreases as the central mass increases. I don't follow any relation to showing greater mass for a given object when it has higher gravitational potential energy. You may well have some argument in mind, but you didn't state it at all.

Sorry, I made it too short.
You proposed to use oscillating spring to test mass of the object. I propose setup where massive object is orbited by test mass and we find out mass of the object from period and radius of test mass orbit. Well, of course my proposed setup should be a bit bigger but on the other hand it's just a thought experiment after all.

Then like in your case we observe this system in lower gravitational potential then transfer it to higher gravitational potential and observe again locally and from original location.
After system is transferred local observer sees it the same way as it was before. But observer in original location sees it as faster than before and so he calculates bigger mass for the object i.e. mass has increased.

Or alternatively we can leave that system where it is but instead move observer up and down in gravitational potential. And then observer uses that system as reference to calibrate his mass standard (instead of clock).

Ok, I think I see your way of looking at it, and I agree it is a valid way of looking at it. If you imagine N dust particles at a given temperature and state collapsed, they definitely appear less massive from a distance than the same N particles, the same temperature and state, when they are farther apart. This is certainly true, even if it doesn't happen 'magically' just by moving the dust. It happens because to achieve same state after collapse, energy must be radiated; and to achieve same state on expansion, energy must be supplied.

Yes, exactly. Certainly nothing 'magical'.

 

[PAllen]

Sorry, I made it too short.
You proposed to use oscillating spring to test mass of the object. I propose setup where massive object is orbited by test mass and we find out mass of the object from period and radius of test mass orbit. Well, of course my proposed setup should be a bit bigger but on the other hand it's just a thought experiment after all.

Then like in your case we observe this system in lower gravitational potential then transfer it to higher gravitational potential and observe again locally and from original location.
After system is transferred local observer sees it the same way as it was before. But observer in original location sees it as faster than before and so he calculates bigger mass for the object i.e. mass has increased.

Or alternatively we can leave that system where it is but instead move observer up and down in gravitational potential. And then observer uses that system as reference to calibrate his mass standard (instead of clock).

Ah, this is where the localization issue comes into play. I think we all agree that if I test, via orbital period at a given radius, the mass of configuration of matter in a contracted state versus expanded state (all temperature, internal KE, etc. being the same), the former will measure less massive. But take a piece away from that configuration, and measure it at different places in the ambient 'potential' of massive body, using the same test, and I believe the mass will come out the same everywhere. The mass deficit is a property of the gravitating system as whole, and cannot actually be measured in any way as a property of each separate piece of matter.

 

[my_wan]

The heat resulting from the gravitational collapse of a gas is misleading. When you compress an ideal gas a large part of the energy used to compress it is lost when it cools. Let this heat dissipate and you effectively lose the compression energy used to heat loss. You can still regain it because when a compressed ideal gas is released it will reabsorb that heat from the environment. Normal atmosphere is pretty far from an ideal gas and interacts even without collisions.

When an ideal gas is gravitationally compressed (no mechanical compression) the heat gain is only apparent due to reduced clock rates at lower altitudes, or conversely the increased spatial metric making the collisional length of the gas appear shorter. In higher gravity it can also be viewed as if you are measuring more space with the same yardstick. Same effect whether you look at it from a space or time metric. This is why you actually lose energy as you gain gravitational depth, because it is not contributing to heat though normal mechanical energy input for compression. Any apparent increased thermal motion is strictly due spacetime dilation effects on measurement, not actual heat increases.

For the Earth's center or thick atmospheres the extra heat is generated by the weight of the material above it, not simply by the local gravitational compression alone.

I would have to break out the calculator for more detail but it seems to me that the assumption of increased thermal motion is misplaced.

 

[PAllen]

The heat resulting from the gravitational collapse of a gas is misleading. When you compress an ideal gas a large part of the energy used to compress it is lost when it cools. Let this heat dissipate and you effectively lose the compression energy used to heat loss. You can still regain it because when a compressed ideal gas is released it will reabsorb that heat from the environment. Normal atmosphere is pretty far from an ideal gas and interacts even without collisions.

When an ideal gas is gravitationally compressed (no mechanical compression) the heat gain is only apparent due to reduced clock rates at lower altitudes, or conversely the increased spatial metric making the collisional length of the gas appear shorter. In higher gravity it can also be viewed as if you are measuring more space with the same yardstick. Same effect whether you look at it from a space or time metric. This is why you actually lose energy as you gain gravitational depth, because it is not contributing to heat though normal mechanical energy input for compression. Any apparent increased thermal motion is strictly due spacetime dilation effects on measurement, not actual heat increases.

For the Earth's center or thick atmospheres the extra heat is generated by the weight of the material above it, not simply by the local gravitational compression alone.

I would have to break out the calculator for more detail but it seems to me that the assumption of increased thermal motion is misplaced.

I disagree with a number of your statements, and stand by what I wrote earlier. However, I don't have time to respond in detail right now. These issues have been subject to dispute even among experts. For now, I provide two links whose main focus is entropy, but they relate also to the issue under discussion.

http://philsci-archive.pitt.edu/4744/1/gravent_archive.pdf

http://math.ucr.edu/home/baez/entropy.html

 

[Dale]

When an ideal gas is gravitationally compressed (no mechanical compression) the heat gain is only apparent due to reduced clock rates at lower altitudes

What do you mean by this? Specifically, "no mechanical compression".