The definitions of temperature and entropy are related to the concept of "thermodynamic equilibrium". In the absence of equilibrium, these concepts cannot be introduced. A black hole in combination with radiation presents a closed system, one of the subsystems of which (radiation) is close by its parameters to equilibrium, but other subsystems (black hole itself) do not. I.e., if the emitted radiation carries away entropy, this does not mean that in the remaining part it (being additive) is reduced by the same amount, because the very notion of a black hole cannot be introduced.
See our paper
Melkikh A.V., Melkikh E.A. Can we use thermodynamics in the system with gravity? 2017. Modern Physics Letters B.
Article Can we use thermodynamics in the systems with gravity?
Dear Dr. Gert Van der Zwan!
Of course thermodynamics was established before statistical mechanics but it was established for the systems (mainly - for gases) with local equilibrium. This equilibrium was implicitly assumed, although this concept itself arose later.
But now it is clear (in our paper we give arguments in favor of this) that without equilibrium there is no thermodynamics at all.
What about the black hole - it is impossible to introduce equilibrium for it, as a system. Thus, it is impossible to introduce thermodynamics for the black hole at all.
But Carnot cycle implicitly assumes equilibrium. For the systems with long-range interaction (for example - gravity) thermodynamic equilibrium is impossible. Thus Carnot cycle cannot be performed for such a systems.
Long-range interactions is a very special case of statistical physics (see, for example Levin et al, Physics Reports, 2014 and references in our paper). In particular, numerical calculations show that there is no equilibrium in such a system.
Wnat about a black hole with the temperature 2.73 K - I think that radiation of such black hole can be at this temperature, but black hole itself can not be attributed temperature.
The difference is simple: bodies which radiate black body radiation and are in equilibrium itself can have a temperature. Bodies which radiate black body radiation and are NOT in equilibrium itself cannot have temperature. The black holes belong to the second class.
By the way according to the classical results of mechanics the planetary orbits close to circular with sufficiently weak interaction between them remain on their initial orbits for arbitrarily long period of time, only slightly deviating from initial values, so demonstrating weak chaos behavior. This result does not depend on the number of bodies - it is valid for macroscopic system also.
ref: http://www.scholarpedia.org/article/Bekenstein-Hawking_entropy
These similarities do not allow us to introduce entropy of a black hole. The main condition for thermodynamics - equilibrium - is absent in black holes.
Problem with thermal equilibrium is addressed by Hawking in his 1976 paper, see section III, which deals with this,
"Although the canonical ensemble does not work for black holes, one can still employ a micro-canonical ensemble of a large number of similar insulated systems each with a given fixed energy E. Each of these systems will have a number of different configurations compatible with the given energy. These configurations will form a surface in the configuration space of the system."
S. Hawking: Phys. Rev. Lett. 13, 191 (1976)
Instability of canonical distribution cleary indicates that systems with long-range interaction do not come to equilibrium.
If a black hole at 2.73K very far away from other masses - this change nothing in the problem of equilibrium, because there is no equilibrium inside the black hole.
Hawking and Bekenstein results indicated that radiation of a black hole is close to radiation of a black body. They concluded that black body itself also has an entropy. I think that the last statement is wrong.
We also should to differ results for 1D, 2D and 3D gravity. The first and second cases are not real for our world, but in the third case such a sustem will evaporate. This means that it will not reach equilibrium.
Black body radiation of the black hole tell us only about ratiation itself but not about black hole. For statement that "black hole is in equilibrium" we need model of black hole with it's internal structure. If this model indicates that black hole is in equilibrium, we musy agree with this. Without such a model we cannot speak about equilibrium of a black hole.
I think that canonical ansamble is the main basis of equilibrium. Microcanonical distribution is the consequence of energy conservation law and don't connected directly with thermodynamics.
If the canonical ansamble is unstable there is no equilibrium and there is no heat capacity.
The term "heat capacity" connected with the tern "heat'. Heat can be determined as dQ=TdS. If there is no equilibrium - T and S are not determined. That is heat capacity does not exist.
To Melkikh: This is a really wrong statement
"Without equilibrium, there is no thermodynamics at all"
Today, we have a well developed Non-Equilibrium Thermodynamics, for disscrete
systens and also in field formulation. Because Gravitation Theory is a field theory, you
cannot use terms as dQ, dT, dE.. and so on, and also the reversible Carnot device
has nothing to do with General Relativity. You have to apply thermodynamics in field
formulation. Of course, black holes are thermodynamical objects, because otherwise
you would state that phenomenological thermodynamics is a restricted theory. But it
may be that --as in Stochastic Thermodynamics-- the Second Law is not strictly
valid. Clear is, that the thermodynamics of black holes need an update with
respect to the Non-Equilibrium Thermodynamics which we know today.
Yes, there is non-equilibrium thermodynamics but it exists only when equilibrium also exists. Otherwise, it is impossible to enter the criterion of nonequilibrium state.
All nonequilibrium thermodynamics is built on the equation of the entropy balance. However, if entropy can not be introduced, then nonequilibrium thermodynamics as such can not be formulated.
Of course, you have to introduce the phenomenological non-equilibrium quantities,
such as energy density, heat and power exchange, contact temperature, entropy density production and supply. All these quantities are measurable ones, may be with
a statistical or/and quantum-theoretical background. If you deny their existence, a
description in the framwork of physics is not possible, but we do not believe that.
There is nothing surprising in the fact that some quantities can not be introduced, since they have their own application area. This does not mean that physics ceases to work. Of course, mechanics continue to operate.
This is a rather interesting problem. I suggest you should have a look at:
Brit. J. Phil. Sci. 61 (2010), 513–540
Long range interacting systems, as any other physical system evolves towards thermod equilibrium, but after a violent relaxation (LindenBell theory) they get trapped in quasi stationary states (out of equilibrium states), and the way the go to equillibrium diverges with the system size.
Astrophysical systems are, among others, particularly a very good example of these systems.
Thank you for the reference! By the way I think that these systems do not come to equilibriun at all. Our models confirm this conclusion.
If you are interested to go deep into this problem, take a look at:
D. Benest, C. Froeschle, E. Lega (Eds.), Topics in Gravitational Dynamics, in: Lecture Notes in Physics, 729 (2007).
Entropy: A concept that is not a physical quantity
https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity
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