What's Wrong With Black Hole Thermodynamics?

[juanrga]

An interesting review of usual claims done in black hole literature by an expert in thermodynamics.

 

[PAllen]

An interesting review of usual claims done in black hole literature by an expert in thermodynamics.

Can you reference the abstract rather than the PDF.

I've seen some other critical reviews of this. If I can find them, and they are in similar spirit, I'll post them here.

[EDIT: Here is the abstract link: http://arxiv.org/abs/1110.5322

 

[pervect]

http://adsabs.harvard.edu/abs/1980PhLA...78..219L "Entropies need not to be concave"
seems to be in some disagreement about one of the major premises of the author. I stumbled over this while trying to see if the original paper was peer reviewed - I see other peer reviewed papers by the author, but I haven't found that the arxiv paper was ever published. Unfortunately the published papers mostly seem to require subscriptions to access.

 

[Naty1]

yea, well Hawking thought Beckenstein was wrong as well...until Hawking arrived at Beckenstein's answer using a completely different approach.

Besides:

Although Hawking's calculations gave further thermodynamic evidence for black hole entropy, until 1995 no one was able to make a controlled calculation of black hole entropy based on statistical mechanics, which associates entropy with a large number of microstates. In fact, so called "no hair"[7] theorems appeared to suggest that black holes could have only a single microstate. The situation changed in 1995 when Andrew Strominger and Cumrun Vafa calculated the right Bekenstein-Hawking entropy of a supersymmetric black hole in string theory, using methods based on D-branes. Their calculation was followed by many similar computations of entropy of large classes of other extremal and near-extremal black holes, and the result always agreed with the Bekenstein-Hawking formula.

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

It would be fun, though, if the cited paper above added new insights...Anything there??

 

[PAllen]

http://adsabs.harvard.edu/abs/1980PhLA...78..219L "Entropies need not to be concave"
seems to be in some disagreement about one of the major premises of the author. I stumbled over this while trying to see if the original paper was peer reviewed - I see other peer reviewed papers by the author, but I haven't found that the arxiv paper was ever published. Unfortunately the published papers mostly seem to require subscriptions to access.

I also noticed that only a few of the early paper by this author were peer reviewed, but not this one. "Expert on Thermodynamics" seems a little overblown relative to the published history. But I didn't want to bring this up until I had tried to digest the paper for its content.

 

[juanrga]

http://adsabs.harvard.edu/abs/1980PhLA...78..219L "Entropies need not to be concave"
seems to be in some disagreement about one of the major premises of the author. I stumbled over this while trying to see if the original paper was peer reviewed - I see other peer reviewed papers by the author, but I haven't found that the arxiv paper was ever published. Unfortunately the published papers mostly seem to require subscriptions to access.

The word «entropy» is used with different meanings by many people; very often those meanings are not compatible with the well-established concept of thermodynamic entropy.

 

[juanrga]

I also noticed that only a few of the early paper by this author were peer reviewed, but not this one. "Expert on Thermodynamics" seems a little overblown relative to the published history. But I didn't want to bring this up until I had tried to digest the paper for its content.

Curious, instead reviewing the work cited, people here goes on reviewing the author.

Well, he is a well-known expert in thermodynamics, their works are cited by other thermodynamicians and his book in thermodynamics of irreversible processes is published by Dover classics.

He has received Galilei Gold Medal 2009

For his work on irreversible thermodynamics and contributions to many areas of physics including that of Brownian motion, and in the establishment of the statistical basis of thermodynamics, and his contributions in astrophysics/cosmology.

And with a simple search in google scholar I can find

The thermodynamics of endoreversible engines
BH Lavenda - American Journal of Physics, 2007 - link.aip.org

Mean entropies
BH Lavenda - Open Systems & Information Dynamics, 2005 - Springer

High temperature properties of the MIT bag model
BH Lavenda - Journal of Physics G: Nuclear and Particle …, 2007 - iopscience.iop.org

...

It is difficult to believe that are not peer reviewed...

 

[twofish-quant]

The paper seems to be gooble-gook. C_p is a derivative and can be be negative or non-existent. He does a lot of equations based on the behavior of classical ideal monatomic gases, which I suppose merely shows that black holes are not made of classical ideal monatomic gasses.

 

[twofish-quant]

Well, he is a well-known expert in thermodynamics, their works are cited by other thermodynamicians and his book in thermodynamics of irreversible processes is published by Dover classics.

And I've known extremely brilliant people in one field that were cranks when they were in another one. Roger Penrose is an example. One thing about the author is that he seems to have no experience dealing with objects in which the gravity field makes a considerable contribution to the system, which is not good when you are dealing with black holes. He seems to miss completely the point about polytropes.

He might be brilliant in thermodynamics in other fields, but the arguments that he is giving in that paper seems to be total non-sense.

I may not be an "expert in thermodynamics" but I do know a thing or two about collapsed systems. His arguments make absolutely no sense because in any sort of stellar object, you are moving energy back and forth between the material object and the gravity field, and you can't just take an object and consider only the themodynamic energy. If you want to do your bookkeepping right, you have to consider the energy that is in the gravity field, which he doesn't do.

Since he isn't including the energy in the gravity field, all of his other arguments fall apart. If you include gravity, you get the results in the first section, which he doesn't seem to understand.

 

[jambaugh]

I note in the paper he also invokes a strong form of the 3rd law, \lim_{T\to 0}S = 0. This form ignores residual entropy due to a degenerate ground state. He is here ignoring degeneracy.

His reasoning may be implicitly the "no hair" theorem but that doesn't apply. The very debate, whether BH's evaporate, is a question of observing "internal" degrees of freedom in the configurations of emitted thermal radiation (here "internal" to the surface configuration?).

Reading the short paper, he is invoking analogue physical systems (partitioned volumes, ideal gasses) without any direct thought experiments about a BH per se. I don't see the paper pointing out any physical contradictions, only the exceptional behavior of BH thermodynamics.

 

[juanrga]

The paper seems to be gooble-gook. C_p is a derivative and can be be negative or non-existent. He does a lot of equations based on the behavior of classical ideal monatomic gases, which I suppose merely shows that black holes are not made of classical ideal monatomic gasses.

C_p is not a mere derivative, but a physical quantity with determined properties. Your 'argument' could be re-used to say that (4) "is a derivative and can be be negative or non-existent", but it is difficult to believe someone would accept negative or non-existent mass for a black hole or an imaginary speed of light or similar nonsense...

Effectively, nowhere he says or even suggests that black holes are made of "classical ideal mono-atomic gasses". He uses the simple case of an ideal gas for illustrating the difference between c and C.

 

[juanrga]

And I've known extremely brilliant people in one field that were cranks when they were in another one. Roger Penrose is an example. One thing about the author is that he seems to have no experience dealing with objects in which the gravity field makes a considerable contribution to the system, which is not good when you are dealing with black holes. He seems to miss completely the point about polytropes.

He might be brilliant in thermodynamics in other fields, but the arguments that he is giving in that paper seems to be total non-sense.

I may not be an "expert in thermodynamics" but I do know a thing or two about collapsed systems. His arguments make absolutely no sense because in any sort of stellar object, you are moving energy back and forth between the material object and the gravity field, and you can't just take an object and consider only the themodynamic energy. If you want to do your bookkeepping right, you have to consider the energy that is in the gravity field, which he doesn't do.

Since he isn't including the energy in the gravity field, all of his other arguments fall apart. If you include gravity, you get the results in the first section, which he doesn't seem to understand.

It is evident that this kind or argument can be inverted. People as Hawking, experienced in black holes and general relativity, can say nonsense when entering in the field of thermodynamics.

Even if we ignore now that «the energy that is in the gravity field» is not well-defined in general relativity, what you say about thermodynamic energy and gravitation seems to be without any basis.

Already many ordinary textbooks explain how gravitational energy M\phi contributes to thermodynamic energy. Of course, in a BH the situation is more complex and M\phi is not enough, but thermodynamics in presence of gravitation continues to hold and I fail to see your point.

 

[twofish-quant]

C_p is not a mere derivative, but a physical quantity with determined properties. Your 'argument' could be re-used to say that (4) "is a derivative and can be be negative or non-existent", but it is difficult to believe someone would accept negative mass for a black hole or imaginary speed of light or nonsense as that...

In the context of black holes and stars, it isn't. You add energy, the gravitational field rearranges itself and you get a different temperature. C is a quantity that includes both the effects of gravity and the physical characteristics of the object.

Effectively, nowhere he says or even suggests that black holes are made of "classical ideal monoatomic gasses". He uses the simple case of an ideal gas for illustrating the difference between c and C.

And in the current situation "c" is irrelevant. What matters is C.

And in the case of black holes the physical material gets crushed to a singularity in a finite time leaving behind only the gravitational field whose thermodynamic properties are not constrained by the limits that constrain physical objects. Black holes are dominated by the gravitational field so if you add energy, the field will reconfigure itself, and that's what you are observe.

The problem is that the author is used to laboratory thermodynamics in which you don't have to worry about the energy of the gravitational field, which works very badly when you figure out the thermodynamics of objects which are dominated by gravity. So in doing the energy calculations, he is completely ignoring gravity, which results in conclusions that are ridiculous.

 

[juanrga]

I note in the paper he also invokes a strong form of the 3rd law, \lim_{T\to 0}S = 0. This form ignores residual entropy due to a degenerate ground state. He is here ignoring degeneracy.

His reasoning may be implicitly the "no hair" theorem but that doesn't apply. The very debate, whether BH's evaporate, is a question of observing "internal" degrees of freedom in the configurations of emitted thermal radiation (here "internal" to the surface configuration?).

Reading the short paper, he is invoking analogue physical systems (partitioned volumes, ideal gasses) without any direct thought experiments about a BH per se. I don't see the paper pointing out any physical contradictions, only the exceptional behavior of BH thermodynamics.

I was said in my thermo course that residual entropies are the result of ignoring some interaction that breaks the degeneracy. I.e. that those degeneracies are fictitious. In any case I cannot see how substituting the strong form by the weak form \lim_{T\to 0}S = S_0 changes anything for BH thermo.

How do you adjust the 'exceptional behaviour' of evaporating BHs with the thermodynamic properties of the supposedly emitted thermal radiation?

 

[juanrga]

You add energy, the gravitational field rearranges itself and you get a different temperature. C is a quantity that includes both the effects of gravity and the physical characteristics of the object.

If you confound internal energy with rest energy or with some other kind of energy then C can contain everything you want, but then better calls it X.

And in the current situation "c" is irrelevant. What matters is C.

But as the author emphasizes C is not c. The non-numbered equation before (6) implies dE=CdT, but that is different from (7). Many people believes that C is the heat capacity, but for an open system dE ≠ dQ.

And in the case of black holes the physical material gets crushed to a singularity in a finite time leaving behind only the gravitational field whose thermodynamic properties are not constrained by the limits that constrain physical objects. Black holes are dominated by the gravitational field so if you add energy, the field will reconfigure itself, and that's what you are observe.

The problem is that the author is used to laboratory thermodynamics in which you don't have to worry about the energy of the gravitational field, which works very badly when you figure out the thermodynamics of objects which are dominated by gravity. So in doing the energy calculations, he is completely ignoring gravity, which results in conclusions that are ridiculous.

Not only it seems that you have not studied enough thermo, but you did not even read #12, where such claims were corrected.

 

[twofish-quant]

If you confound internal energy with rest energy or with some other kind of energy then C can contain everything you want, but then better calls it X.

This gets into definitional issues, but when astrophysicists talk about black holes and stars or whatever, they are indeed talking about C.

But as the author emphasizes C is not c. The non-numbered equation before (6) implies dE=CdT, but that is different from (7). Many people believes that C is the heat capacity, but for an open system dE ≠ dQ.

When you dump energy into a black hole or star, some of that energy goes into the gravitational field that effects the thermodynamics of the object. You can define a "heat capacity" based the interaction of the system to additional energy.

Also, a non-radiating star or black hole is a closed system.

Not only it seems that you have not studied enough thermo, but you did not even read #12, where such claims were corrected.

And that "correction" is wrong. The basic problem I have with the entire paper is that he is using energy balance arguments. When you use energy balance arguments in astrophysical objects, you have to take into account the interaction of the gravity field. If you don't, then none of your results make any sense. Essentially, he writes an entire paper about black holes, and not once does he mention the word "gravity", and nowhere does anything approaching gravity enter into any equations.

If you can come up with an argument in which you can argue that it's possible to talk about black holes while ignoring gravity, I'd like to hear it.

 

[twofish-quant]

It is evident that this kind or argument can be inverted. People as Hawking, experienced in black holes and general relativity, can say nonsense when entering in the field of thermodynamics.

Exactly. That's why argument to authority fails. In this situation, I think Hawking is right.

Even if we ignore now that «the energy that is in the gravity field» is not well-defined in general relativity, what you say about thermodynamic energy and gravitation seems to be without any basis.

It's pretty simple. Gravitational potential energy makes up a substantial amount of energy in astrophysical objects. Hence arguments based on energy conservation that ignore interactions of the gravity field just don't work. If you introduce the gravity field, the situation that Hawking finds is pretty standard.

Already many ordinary textbooks explain how gravitational energy M\phi contributes to thermodynamic energy. Of course, in a BH the situation is more complex and M\phi is not enough, but thermodynamics in presence of gravitation continues to hold and I fail to see your point.

My point is that adding gravitation introduces additional terms into the energy equation, and once you introduce those terms, the arguments in the paper break down. The paper makes arguments that ignores the impact of gravity on the thermodynamic equations, and you just can't do that.

Gravity changes everything. Once you have a gravitational field, then adding or removing energy from the system will cause interactions with the gravitational field, and *that's* what gives you heat capacities that you don't see in non-self gravitating objects.

Black holes are not monoatomic ideal gasses. Because of self-gravitation, black holes are different enough so that you can't even use monoatomic ideal gasses as an analogy.

 

[PAllen]

Of interest to discussion of entropy of self gravitating systems is the following essay. Of particular interest is the discussion of gravothermal catastrophe - that self gravitating systems really have no equilibrium point short of a black hole.

This essay is not relevant to the core question of the validity of Bekenstein-Hawking entropy. It takes that as a given, and argues that this is exceptional and not some natural limit of ordinary gravitational collapse processes.

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

 

[twofish-quant]

I've been trying to think of a "thought experiment" that illustrates what happens so that we can argue about actual science rather than personalities.

Here's an attempt...

You have a satellite that is in orbit around the Earth in a distant orbit. Because it is in a distant orbit, it orbits slowly. Now you take energy away from the satellite. What happens to it? Well, because the satellite has less energy, it's going to start falling into earth. Once it gets to a lower orbit, it's going to orbit faster.

Now imagine a cloud of atomic sized satellites in orbit around the earth. You take energy away from them. They'll drop to lower orbits. Once they drop to lower orbits, they will move faster. If you measure the temperature of the cloud, you'll find that it has increased because faster atoms = higher temperature.

So if you have a self-gravitating object, pumping energy into the object will boost things into higher orbits which slows things down and makes things cooler. Taking energy out will put things into lower orbits which speeds things up and make things hotter. In other words the object has negative heat capacity.

What's wrong with this picture? (My answer is that nothing is wrong with this picture and this is exactly what happens).

 

[PAllen]

I've been trying to think of a "thought experiment" that illustrates what happens so that we can argue about actual science rather than personalities.

Here's an attempt...

You have a satellite that is in orbit around the Earth in a distant orbit. Because it is in a distant orbit, it orbits slowly. Now you take energy away from the satellite. What happens to it? Well, because the satellite has less energy, it's going to start falling into earth. Once it gets to a lower orbit, it's going to orbit faster.

Now imagine a cloud of atomic sized satellites in orbit around the earth. You take energy away from them. They'll drop to lower orbits. Once they drop to lower orbits, they will move faster. If you measure the temperature of the cloud, you'll find that it has increased because faster atoms = higher temperature.

So if you have a self-gravitating object, pumping energy into the object will boost things into higher orbits which slows things down and makes things cooler. Taking energy out will put things into lower orbits which speeds things up and make things hotter. In other words the object has negative heat capacity.

What's wrong with this picture? (My answer is that nothing is wrong with this picture and this is exactly what happens).

Right, I completely agree.

 

[juanrga]

Also, a non-radiating star or black hole is a closed system.

Reading again stuff that was corrected before is one thing, but your claim that a black hole is a closed system is too nonsensical to continue.

 

[juanrga]

Of interest to discussion of entropy of self gravitating systems is the following essay. Of particular interest is the discussion of gravothermal catastrophe - that self gravitating systems really have no equilibrium point short of a black hole.

This essay is not relevant to the core question of the validity of Bekenstein-Hawking entropy. It takes that as a given, and argues that this is exceptional and not some natural limit of ordinary gravitational collapse processes.

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

The entropy of gas ideal is function of its internal energy E_int.

Its equation 5 reads E=K+U, but the thermodynamic expression in presence of fields is E=U+K+E_int.

 

[juanrga]

I've been trying to think of a "thought experiment" that illustrates what happens so that we can argue about actual science rather than personalities.

Here's an attempt...

You have a satellite that is in orbit around the Earth in a distant orbit. Because it is in a distant orbit, it orbits slowly. Now you take energy away from the satellite. What happens to it? Well, because the satellite has less energy, it's going to start falling into earth. Once it gets to a lower orbit, it's going to orbit faster.

[...]

The problems with your "thought experiment" already start here. What energy? The total energy is E=K+V+U, where K is kinetic energy, V is potential energy due to gravitational field and U is internal energy.

If I take away internal energy dU<0 (example via cooling the satellite), but maintain constant its mechanical energy (K+V), satellite will continue to orbit exactly in the same orbit.

If I take away kinetic energy dK<0 (example via decelerating the satellite), but maintain constant its internal energy (U), satellite will start falling into Earth... but this have nothing to see with thermodynamics or with negative heat capacities (indeed in this case Q=0).

 

[PAllen]

The entropy of gas ideal is function of its internal energy E_int.

Its equation 5 reads E=K+U, but the thermodynamic expression in presence of fields is E=U+K+E_int.

In the context of discussion in that section, internal energy is held constant and can be ignored. For that section the author is explicitly treating the gas particles has incapable of changes of state. Later sections broaden that to discuss fusion and other process that do change internal energy.

For example, just a few lines before the equation you criticize is:

"Consider an ideal, uniform gas of
total energy E and kinetic energy K, containing N particles whose internal
degrees of freedom I will assume to be either nonexistent or irrelevant at the
system’s current temperatures,"

 

[juanrga]

In the context of discussion in that section, internal energy is held constant and can be ignored. For that section the author is explicitly treating the gas particles has incapable of changes of state. Later sections broaden that to discuss fusion and other process that do change internal energy.

For example, just a few lines before the equation you criticize is:

"Consider an ideal, uniform gas of
total energy E and kinetic energy K, containing N particles whose internal
degrees of freedom I will assume to be either nonexistent or irrelevant at the
system’s current temperatures,"

The internal degrees of freedom he is referring are essentially atomic/molecular variables (an example is molecular dipole orientation). This has nothing to see with the concept of internal energy E_int. The internal energy of an ideal gas is computed assuming that atoms/molecules do not have internal degrees of freedom.

The entropy of an ideal gas is a function S=S(U,V,N)

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

Notice that internal energy is usually denoted by U and potential energy by V so that total energy is

E=K+V+U

The author that you cite use U for potential energy (although in a latter section uses U supposedly for internal energy, which he calls "some energy") and confounds internal energy with total energy.

If composition is constant and V=V(R) this reduces to S=S(U,R). He confounds internal energy with total energy, among other mistakes, and writes S(E,R).

One does not need to appeal to fusion to change the internal energy U of an ideal gas (and would be ridiculous because if exists fusion then the gas is NOT ideal).

 

[twofish-quant]

Reading again stuff that was corrected before is one thing, but your claim that a black hole is a closed system is too nonsensical to continue.

Why? You can treat stars and black holes are something very close to closed thermodynamic systems. The only thing that makes them non-closed is radiation and 1) you can include radiative transfer in the book keeping, and 2) it doesn't make a huge difference. The total energy that is radiated at a given point is much, much smaller than the total internal energy.

Considering stars and black holes as something close to closed thermodynamic system is standard basic astrophysics. If you want something to be taken seriously by astrophysicists, and you think that this is ridiculous, you are going to have to argue the point since it's a standard approximation and a very good one.

There's also the issue of professional courtesy. I think part of the issue is that astrophysicists may be using terminology in ways that are different and perhaps bizarre to people in other areas of physics, but there are usually reasons behind this. If you go in with the attitude "you idiot astrophysicists don't understand anything about stars or self-gravitating objects" it's not going to end well nor should it.

And yes this works both ways... If a condensed matter physicist nicely e-mailed me a copy of that paper asking for feedback, and we were talking over lunch, I'd explain the flaws in a very nice way. The context is a paper posted to LANL, and once someone does that, I'm expecting that he wants it to be peer reviewed, at which point I go into "sadistic attack dog" mode, and rip it to shreds with strong language if I think the arguments are weak.

I can say with very strong certainty, that I'm making the points that anyone with astrophysics background would make, and that if his goal is to get the paper published in Ap.J., A.A., MNRNAS or Physics Review, that the reviewers are doing to toss it out with perhaps even stronger language that I've been using. If it tries to publish in some journal that is not astrophysics, what is likely to happen is that the editor will send the paper to someone with astrophysics background, and then they'll say the same thing.

Now if the author goes back, takes some of the points I've made seriously and tries to rewrite the paper, and then comes up with the same result, *that* would be interesting.

 

[twofish-quant]

If I take away kinetic energy dK<0 (example via decelerating the satellite), but maintain constant its internal energy (U), satellite will start falling into Earth... but this have nothing to see with thermodynamics or with negative heat capacities (indeed in this case Q=0).

It has *everything* to do with thermodynamics and negative heat capacities. Once you have enough satellites, it becomes a gas, and once you have a gas, you can do thermodynamics on it. The system of Earth and satellites is a closed thermodynamic system and you can define temperatures and heat capacities.

This is all pretty standard stuff in astrophysics. It's also pretty important. Stars lose energy, and the heat capacity tells you what happens to the temperature of the star as it radiates. As stars radiate, the core temperatures increase which eventually causes fusion reactions that don't occur at lower temperatures.

 

[juanrga]

It has *everything* to do with thermodynamics and negative heat capacities.

If internal energy U of the satellite is held constant and only the mechanical energy (K+V) is varied, then you do not need thermodynamics, but only mechanics. Indeed, the laws of mechanics alone explains, perfectly, why the satteliite modifies its orbit and falles when you take away kinetic energy.

It is evident that you do not need thermodynamics for this. Gibbs and heat capacities are not even mentioned.

Once you have enough satellites, it becomes a gas, and once you have a gas, you can do thermodynamics on it. The system of Earth and satellites is a closed thermodynamic system and you can define temperatures and heat capacities.

Taking an enough collection of satellites does not magically convert a mechanical problem into a thermodynamic one. Moreover, even assuming that the system of satellites can be considered a thermodynamic system, still there is none need to confound internal energy U with total energy E (including potential energy due to Earth) as done in previous posts. Similar remarks about stars.

The equation

\frac{\partial u}{\partial t} = c_V \frac{\partial T}{\partial t} + u_k \frac{\partial n_k}{\partial t}

in presence of field with potential \psi changes to

\frac{\partial u}{\partial t} = c_V \frac{\partial T}{\partial t} + (u_k + \tau_k \psi) \frac{\partial n_k}{\partial t}

with \tau_k the coupling factor. The heat capacity c_V is the same, in presence or absence of field, this is well-know, at least for thermodynamicians.

Of course if you want to confound rest energy with internal energy U and with total energy E, if you pretend that a black hole is a closed thermodynamic system, etc. then you can obtain anything that you want, but that is 'astrophysics', not thermodynamics.

 

[twofish-quant]

If internal energy U of the satellite is held constant and only the mechanical energy (K+V) is varied, then you do not need thermodynamics, but only mechanics. Indeed, the laws of mechanics alone explains, perfectly, why the satteliite modifies its orbit and falles when you take away kinetic energy.

OK. Suppose the satellite is a hydrogen atom, and you are trying to calculate the properties of a large number of "satellites". At this point, the kinetic energy of the satellites *becomes* the internal energy of gas of satellites.

There are several other ways of calculating this. You can use the virial theorem or the equipartition theorem. See

http://en.wikipedia.org/wiki/Equipartition_theorem
http://en.wikipedia.org/wiki/Virial_theorem

The thing about self-gravitating objects is that you *can't* separate mechanics and thermodynamics. Every thermodynamic equation in astrophysics has to include mechanical interactions.

Taking an enough collection of satellites does not magically convert a mechanical problem into a thermodynamic one.

Yes it does. You change those satellites into atoms and then have several gazillion of them. At which point you have a gas.

Moreover, even assuming that the system of satellites can be considered a thermodynamic system, still there is none need to confound internal energy U with total energy E (including potential energy due to Earth) as done in previous posts. Similar remarks about stars.

Except that you don't. You have the internal energy of the gas which consists of the kinetic energy of the gas molecules (H), and then the potential energy of the gravitational field (U). The relationship between the two are defined by the equipartition theorem and the virial theorem. Because the sign of the potential energy is negative, what happens is that as you remove total energy from the system, the kinetic/internal energy of the gas increases. (It's annoying here, because we are running out of letters.)

The virial theorem says that the internal energy is going to be -1/2 the potential energy. You can get a similar relationship through the equipartition.

This poses an interesting problem. If you just use classical mechanics, then you could in principle collapse the star to zero radius, and extract infinite energy which then violates all sorts of thermodynamic principles. What keeps this from happening is that if you collapse the star enough, it becomes a black hole that that sets a bound for the amount of energy you can extract. With some rather simple arguments you can convert those bounds into entropy bounds, which is what Hawking and Berkenstein have done.

\frac{\partial u}{\partial t} = c_V \frac{\partial T}{\partial t} + (u_k + \tau_k \psi) \frac{\partial n_k}{\partial t}

the heat capacity is the same, as is well-known.

You are missing a term. The problem is that the field is not an external field but the result of self-interactions. What that means is that you have an extra \partial (\tau_k) / \partial t term which includes how the gravitational potential changes in response to external interactions.

This is off the top of my head so don't shoot me if there is a problem (and I'm getting confused with all of the letters meaning different things), but if you add energy into a gravitationally bound system then the \partial (\tau_k) / \partial t is going to be strongly positive which means that once you add that term then c_V is going to turn negative.

I'm not surprised that someone that isn't familiar with stars would make that mistake, since in laboratory experiments interacting with the system doesn't change the potential, but in self-gravitating systems, it does.

One other wrinkle is that hydrodynamics beats thermodynamics. A star takes a few minutes to reach hydrodynamic equilibrium, but several thousand years to reach thermodynamic equilibrium. Because of these time scale differences, the hydrodynamics drives the thermodynamics. What that means is that if you add energy to the system, the hydro will cause the change to happen immediately that create temperature gradients that cause the system to go out of general thermodynamic equilibrium.

So you start with an isothermal gas, and add energy to it. Rather than staying in thermodynamic equilibrium, the energy will get distributed hydrodynamically, which will throw the system out of thermo equilibrium.

Of course if you want to confound rest energy with internal energy U and with total energy E, if you pretend that a black hole is a closed thermodynamic system, etc. then you can obtain anything that you want.

No confusion. You have total energy which consists of internal energy + potential energy. Internal energy consists of kinetic energy of the gas molecules + internal degrees of freedom. Equipartition and virial theorems give you very simple relationships between these quantities. So once you have figured out the total energy, the virial theorem gives you the relationship between that and the internal energy and the potential energies.

Of course if you want to confound rest energy with internal energy U and with total energy E, if you pretend that a black hole is a closed thermodynamic system, etc. then you can obtain anything that you want.

I don't get the problem with considering a static black hole (or star) a closed thermodynamic system. It's just sitting there in empty space. The only thermodynamic interaction it has with the rest of the universe is radiation and you can add that to the bookkeeping.

And it turns out that you can mostly ignore the radiation when calculating stellar structure. The amount of energy that a star radiates over a year is an insignificant amount of the total internal energy. If you look at nuclear energy, the star can shine for billions of years, and if you look at gravitational energy, you get enough energy to shine for tens of millions. So if you are looking at the dynamics of a star over a day, you can consider the system to be adiabatic.

And you can't get anything you want. People (Chandrasekar) worked out how all of these things interact years ago, and it's in every first year astrophysics graduate text.

Again it's a problem of tone. If someone asked "it's very odd that self-gravitating bodies have negative heat capacity, why is that?" that's one thing. If the implied message is "astrophysicists are being obvious idiots for ignoring something stupidly obvious" that's not going to provoke a friendly reaction. Also, uploading a paper to LANL implies that you want people to tear it to shreds.

 

[twofish-quant]

To summarize the special thermodynamic properties of astrophysical self-gravitating systems:

1) you cannot separate the mechanics from the thermodynamics. If you have a hydrogen gas in orbit around the earth, you have to take into account orbit mechanics in looking at the behavior of said gas.

2) you cannot treat external potentials as a constant addition to the thermo equations. Adding and removing energy from the system will change the interaction of the system with the field and that needs to be taken into account. Even if the field is constant (i.e. hydrogen atoms surround the earth) the fact that the atoms will wind up in different orbits means that you have to add extra terms in the thermo equations.

3) unlike laboratory systems, the hydrodynamic timescales of astrophysical systems are much smaller than the global thermodynamic time scales. This means that arguments based on global thermodynamic equilibrium will not work. If you have a isothermal laboratory system and add energy to it, you can expect the system to quickly move to isothermal. This isn't true with astrophysical systems where the thermal time scales are huge in comparison to the hydro scales, and also long in comparison to the observational time scale (i.e. the thermal time scale of stars are in the thousands of years). There are in fact astrophysical objects (white dwarfs) whose thermal equilibrium time is longer than the age of the universe. For stars you can assume local thermal equilibrium, but for atmospheres, even LTE is not true.

This means that if you dump energy into an astrophysical system that's in thermo equilibrium, it will quickly reach hydro equilibrium but that will drive it out of thermal equilibrium, which means that arguments based on thermal equilibrium won't work.