Gravitational potential energy storage location

[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".

 

[my_wan]

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

I'm referring to adiabatic compression of the gas itself. When you mechanically compress a gas you are adding external energy from the outside. This means not just that the gas volume has been reduced, but that the mean velocity of the particles has likewise been increased.

Hence mechanical compression non-adiabatic involves 2 forms of decreased entropy for the system in question, often treated as just one. One corresponding to the spatial component, container size, and the other a time component, clock time or velocity component of the molecules. A non-adiabatic mechanically compressed gas will then over time lose the increased mean velocity to the environment through heat, leaving behind only the energy from the first adiabatic form to be permanently contained. We lose most of our inefficiencies in heat engines as a result of the second form, particles velocity reduction though heat loss.

An Adiabatic compression only involves the first form such that the mean velocity remains unchanged. It is this first form only, disregarding stacked weight such as Earths center, that the gravitational contraction of a gas entails. The covariant form of the gravitational potential entails that what one person sees as a particle velocity induced entropy reduction another will say no, it appears to be the result a volume reduction. This pair of observables is also covariant with a second set of observables. That is: One will say the mass of the particles apparently increased while another says it remained constant. Yet the quantitative Δstate defined to be a result anyone of these state variables doesn't leave any room for defining any Δstate in terms of any other state variable. Though it makes no difference which state variable you choose.

@PAllen
Seeing how rampant this mixing of these 2 components of Δentropic state is I think I need to be paying a lot more attention to the literature on this topic. I will take my time to go over those documents you linked in detail, and do some document searching on the matter myself. It appears there might possibly even be endemic incongruencies going back to at least Poincare's proof of reversibility.

Consider the claim that Poincare's reversibility proof only applies to enclosed systems. When a gas canister divided between a vacuum and a gas which is then released into the vacuum, Poincare calculated the odds of a spontaneous reversal. This is inherently an adiabatic expansion. Now it is said that reversibility is contingent upon this system being enclosed. But what happens when we let it absorb heat from an external environment keeping the individual particles strictly segregated? If we take the initial state in terms of positions only this increased heat, from a purely mechanical point of view, merely increases the rate at which possible positions are transitioned through. Hence, on its surface, an increased velocity appears to entail that since the position states that occur per unit time has increased then the odds of a reversal occurring (in terms of position space only) in some set unit of time (however large) has increased.

There's another possible issue with Poincare's probabilities. In effect it takes the number of possible equivalent resulting states as a ratio of the number of possible initial states. Yet a direct transition from state A to state B cannot mechanistically involve a single state transition, rather a large number of transitions each with separable odds. In other words it fails to account for the systems mechanistic constraints due to the fact that no two particle can posses the same position moment at the same time. So just because you have odds X of any particle possessing position X does not entail the physical possibility of any two or more particles actually possessing moment state X.

An analogy with dice says that the odds of rolling snake eyes is X. Roll any pair of dice and you have X^2 probability of rolling a pair of snake eyes. Only in the gas law case if the first dice is snake eyes the probability of snake eyes on the second dice goes to zero. So Poincare's reversibility odds is contingent upon how many equal length paths are either non-interfered with by other particles, or contingent upon mutual mechanistic interference, that are not destroyed by even a single rogue particle. A simple count of position state ratios is likely vastly underestimating the number of possible paths to get from state A to state B on simple mechanical interference in the transition sequence. Perhaps infinitely underestimated, given the number of individual states and interactions (collisions) required to actually get from state A to state B. Suppose the distance between two ideal particle during some point in expansion is infinitesimal. What then when a reversal involves only infinitesimal variations? Merely assuming the mechanistic path constraints are linear with the differing ratios of resulting possible states separated by time and space does not constitute a proof.

 

[my_wan]

Wrt the issues I just brought up in terms of Poincare reversibility an empirical justification can be indirectly implicated. What Poincare assumed was the equivalence between classical thermodynamics and statistical mechanics. When Einstein exploited Brownian motion, which for the first time demonstrated the existence of the atom, it explicitly demonstrated the empirical superiority of statistical mechanics in this case. Poincare reversibility theorem came prior to this when they were considered empirically equivalence. I think it is at least possible that mechanistic constraints of reversibility paths could potentially entail another empirical superiority along the lines of that demonstrated by Brownian motion. Might be worth trying to work out a proof for.

 

[my_wan]

I'm still reading and thinking through the paper on the philsci archive site:
Gravity, entropy, and cosmology: in search of clarity

Thanks PAllen, it's telling me a lot about the problematic issues at a far more fundamental level. Yet there is another incongruent issue this article points out very early that disappears when you think of an equilibrium state in purely mechanistic terms rather than 2nd law features, and is directly related to the notion of equilibrium in the early Universe leading to a corresponding equilibrium today.

Start with a spherical ideal gas tank of some volume contained at the center of another such spherical tank containing a vacuum. Assume that the gas is in perfect equilibrium. What exactly does perfect equilibrium mean in purely mechanistic terms? It means that the velocity distribution of the gas molecules are distributed in such a way that any point containing a particle is equally likely to have a particular momentum when those odds are scaled against a Maxwell–Boltzmann distribution. Now what happens when you release the gas inside the smaller internal tank? The faster particles by definition must exist the original volume first. The entire expansion mechanistically must become a non-random stratification of the Maxwell–Boltzmann distribution upon which the random distribution of previously provided the very definition of thermal equilibrium.

This also provides a mechanistic description of group behavior processes that authors generally switch away from mechanical descriptions to provide, and often call the group behavior irreversible. When a uniform molecular gas starts undergoing expansion a lot of lower velocity molecules gets trapped in front of the higher velocity molecules. Hence density bands will form where the bands radiate out as a result of faster bands colliding with slower bands and trading momentum at the leading edge. A more orderly version of a segregation of velocities occurs in an enclosed car when you hit the breaks. To observe this tie a helium balloon to the console floating just short of the ceiling. Now when you hit the breaks, while everything else has a impulse directed toward the front, the helium balloon will dart toward the back. This mechanistically is the direct result of the fact that slower molecules are more subject to the acceleration and crowd toward the front, while the faster less dense molecules over power the acceleration and speed toward the back. This induces a pressure difference between the front and back of the car.

This I hope fully demonstrates that no random distribution, no matter how perfectly uniform the velocity distribution, can possibly maintain a random distribution under free expansion, period. So long as the 2nd law is thought of not as a law but a purely mechanistic consequence all but the reversibility issues go away. I have also previously given fair cause for the reversibility issue to be anything but proven for a mechanistically constrained system in which the 2nd law is not fundamental to the causal factors enforcing the 2nd law. The incongruences only occur when you assume the 2nd law is fundamental then try to apply it to the underlying dynamics where only the end results are required to conform to the 2nd law. With the Hubble expansion the Universe can never reach a high entropic state in the sense that an ideal gas contained in a constant volume must.

 

[my_wan]

From the philsci archive article "Gravity, entropy, and cosmology: in search of clarity" PAllen linked:

Now[/PLAIN] suppose that there is some small heat flow from the core to the envelope. In the immediate term this reduces the kinetic energy of the core, but it also reduces the total energy of the core by the same amount, so in fact it will be entropically favourable for the core to contract (U = 2E and E has just become more negative) and heat up (K = −E). Somewhat counter-intuitively, gravitating systems increase in temperature when they emit heat (for this reason they are often said to have negative heat capacities; see Callender (2009, 2008) for further discussion of this and other anomalous features of gravitational statistical mechanics).

The helium balloon description I provided above explains perfectly well how this occurs mechanistically. Recall that gravity, by the Principle of Equivalence, is nothing more than an inertial acceleration of spacetime. Just like the acceleration of the car I described with the helium balloon. Now we do not have to suppose "some small heat flow" as suggested above, the heat flow is mechanically required exactly as I described in the car causing the helium balloon to accelerate in the same direction as the car in excess of the cars acceleration. The heating results from the stacking effect I mentioned above, when the gas molecules at the center of the mass can no longer mechanically segregate as there is no more space at the center for continued segregation of the molecular velocity components. Hence the entropic description favored in the article is demanded mechanistically even more strongly than the paper suggested without ever referring to entropy specifically.

This kind of paper throws the whole entropy-gravity issue on its head though (Physical Review Letters 2009):
The Einstein equations for generalized theories of gravity and the thermodynamic relation δQ = T δS are equivalent

 

[my_wan]

This is also something I find troubling:
A Collision Between Dynamics and Thermodynamics

All of this is a trivial rehearsal of one of the great episodes of physics. The great puzzle to many philosophers, however, is why so many physicists say and develop theories that seem to deny the lesson of this great episode. Some seek to exorcise the demon, yet Maxwell’s demon is a friend in this story. Maxwell’s thought experiment and subsequent observation of fluctuations destroyed the idea that the second law of thermodynamics is universally valid at all levels. Philosophers also have serious questions about the particular rationale behind some of these exorcisms.

Then farther down:

Zurek (reprinted in [10]) seems to admit that Jauch and Baron’s point holds classically, but insists that a quantum treatment makes it admissible again. So the piston is viewed as admissible when classical mechanical, inadmissible when it conflicts with a higher level theory we know to be approximate, and then admissible again but only due to a lower level treatment! The reasoning in the literature often defies logical analysis; one can find many other examples, but as Feyerabend says about a related problem, “this kind of attack is more entertaining than enlightening” [3].

Even in the context of QM virtual particles strictly depend on the same short term violations.

 

[zonde]

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.

Say we have two identical objects with the same mass. One we call "test mass" but the other one "mass standard". We compare them under different conditions (gravity, velocity, whatever) but conditions are the same for both masses when we compare them.

We would expect that result always will be the same - masses are equal. That's trivial and it gives us no knowledge about conditions where we tested them, right?

 

[PAllen]

Say we have two identical objects with the same mass. One we call "test mass" but the other one "mass standard". We compare them under different conditions (gravity, velocity, whatever) but conditions are the same for both masses when we compare them.

We would expect that result always will be the same - masses are equal. That's trivial and it gives us no knowledge about conditions where we tested them, right?

Yet, if it is impossible, in principle, to measure a test mass higher in gravitational potential as more massive than one lower, how is it scientifically meaningful? All we can say is that a more compact configuration of matter compared to a less compact configuration (all else being the same) measures less massive by a global method (orbits around it). If you try to localize it to the pieces of matter rather than the configuration, you can get pretty silly results. For example, do you just assume that the mass deficit of the bound state is evenly distributed? On what basis? Suppose you try to define the distribution of mass deficit by the amount of work it takes to separate a given chunk from the bound state to infinity. Then, the distribution depends mostly on the order of removal. Whichever piece I pick first has the greatest mass deficit.

It makes more sense to me to think of potential energy, both pre-GR and in GR, as a global function of configuration, not as something you can attach to the mass of some particle based on its position.

 

[zonde]

Yet, if it is impossible, in principle, to measure a test mass higher in gravitational potential as more massive than one lower, how is it scientifically meaningful? All we can say is that a more compact configuration of matter compared to a less compact configuration (all else being the same) measures less massive by a global method (orbits around it). If you try to localize it to the pieces of matter rather than the configuration, you can get pretty silly results.

It is not impossible. It's just that there are more trivial measurements and less trivial. Your example with spring does not compare two masses. But if we assume that physical laws are the same locally at any gravitational potential then your example should give trivial measurement results just the same.

But we can try to come up with some non local measurement setup. Or we can make longer chain of reasoning and try to find less direct measurement "connections". Or we can try to make setup dynamical (mass is falling/rising in gravitational potential) and look for non trivial predictions.

For example, do you just assume that the mass deficit of the bound state is evenly distributed? On what basis?

Ok, just as a starting point.
Say we have system of two objects with the same mass. They are orbiting around common mass center. After some time system has undergone orbital decay and lost some energy (mass deficit increases). If we measure that both masses are still equal (from orbital parameters) I suppose it is logical to say that both masses lost equal amount of mass. Do you agree?

Suppose you try to define the distribution of mass deficit by the amount of work it takes to separate a given chunk from the bound state to infinity. Then, the distribution depends mostly on the order of removal. Whichever piece I pick first has the greatest mass deficit.

I will respond to this later.

 

[zonde]

Suppose you try to define the distribution of mass deficit by the amount of work it takes to separate a given chunk from the bound state to infinity. Then, the distribution depends mostly on the order of removal. Whichever piece I pick first has the greatest mass deficit.

When we separate chunk from larger body we are moving small mass in gravitational potential of large mass and large mass in gravitational potential of small mass. So it seems like half of the energy goes to small chunk and half of the energy goes to large body. Now if we take another chunk of the same size its mass has increased by tiny bit when we took away firs chunk and now it is increasing by tiny bit less than the first chunk as larger mass is bit smaller.

So if we ignore change in gravitational potential due to mass deficit it turns out that order of removal/addition of mass does not matter. If we take into account change in gravity due to mass deficit then last chunk added to larger configuration should be a bit more massive than equivalent first chunks.

It makes more sense to me to think of potential energy, both pre-GR and in GR, as a global function of configuration, not as something you can attach to the mass of some particle based on its position.

I would like to agree that potential energy is non local function rather than real physical quantity. And this function describes change of localized mass into other form of localized energy depending on changes in configuration.

But the problem is that it is common practice to speak about potential energy as real physical quantity. And if you say something like kinetic energy is converted into potential energy then I say it's localized.

 

[Dale]

When we separate chunk from larger body we are moving small mass in gravitational potential of large mass and large mass in gravitational potential of small mass. So it seems like half of the energy goes to small chunk and half of the energy goes to large body. Now if we take another chunk of the same size its mass has increased by tiny bit when we took away firs chunk and now it is increasing by tiny bit less than the first chunk as larger mass is bit smaller.

That is an interesting idea, but I see two problems:
1) it still doesn't allow you to localize the mass deficit within the bound system, i.e. when the objects fall together it is the entire bound system that loses mass, and when the objects are separated one at a time it is the entire bound system than gains mass.
2) why 1/2? Why not M/m or m/M? I would have to see a derivation before I could feel comfortable about that.

 

[zonde]

That is an interesting idea, but I see two problems:
1) it still doesn't allow you to localize the mass deficit within the bound system, i.e. when the objects fall together it is the entire bound system that loses mass, and when the objects are separated one at a time it is the entire bound system than gains mass.

But we can analyze system where two objects are in orbit around common mass center. That is gravitationally bound system and it has mass deficit equal to kinetic energy (according to Virial theorem two times negative average kinetic energy equals average potential energy).
And in that case we can speak about separate masses of two objects.

2) why 1/2? Why not M/m or m/M? I would have to see a derivation before I could feel comfortable about that.

Object with mass m is moving in gravitational potential of object with mass M.
Gravitational potential is proportional to mass of gravitating body: V=kM.
Energy required to move up in gravitational potential is proportional to mass of moving body: E=mV
So we have: E=kMm
But object M is moving in gravitational potential of object m just as well.
If we swap M with m we get exactly the same result. So energy gained by both objects is the same - half of the total energy.

On the other hand momentum should be conserved, but it does not seem to be the case. Hmm, I have to think about this.

 

[Q-reeus]

Object with mass m is moving in gravitational potential of object with mass M.
Gravitational potential is proportional to mass of gravitating body: V=kM.
Energy required to move up in gravitational potential is proportional to mass of moving body: E=mV
So we have: E=kMm
But object M is moving in gravitational potential of object m just as well.
If we swap M with m we get exactly the same result. So energy gained by both objects is the same - half of the total energy.

No. Suppose m is comprised of equal quantities of matter and antimatter, and self-annihilates to release gamma rays. When this occurs with m gravitationally bound to M >> m in potential V (assumed -ve), the escaping gamma rays will be frequency redshifted at infinity by very close to the factor 1+V. Fractional mass deficit of m in the field of M is by this reckoning therefore just that redshift factor. Annihilation of m at infinite separation from M yields unity redshift factor (assuming negligible self-gravity for m), the difference between the two situations is just -V. But -mV is very nearly the total work done in separating. Virtually all of the energy gain is therefore to m. The exact gain to m is slightly less than -mV since M declines slightly to a value somewhat larger than simply (M+mV) during separation.