\magnification 1200
\centerline {\bf Developments in Normal and Gravitational Thermodynamics}
\vskip 0.5cm
\centerline {{\bf by Geoffrey L. Sewell}\footnote*{Partially
supported by European Capital and Mobility Contract No. CHRX-
Ct.92-0007}}
\vskip 0.5cm
\centerline {\bf Department of Physics, Queen Mary and Westfield
College}
\centerline {\bf Mile End Road, London E1 4NS, U.K.}
\vskip 1cm
\centerline {\bf Abstract}
\vskip 0.3cm
We review developments in three areas of thermodynamics,
resulting from advances in statistical mechanics and relativity
theory. These concern (a) the resolution of some basic questions,
concerning the thermodynamic variables and the phase structure
of normal matter, (b) thermodynamical
instabilities in non-relativistic gravitational systems, and (c)
Black Hole thermodynamics, as formulated in terms of strictly
observable quantities, and thus not involving any BH entropy
concept.
\vskip 1cm
\centerline {\bf 1. Introduction}
\vskip 0.3cm
The object of this article is to discuss advances in three areas
of thermodynamics, that have stemmed from progress in quantum
statistical mechanics and general relativity.
I shall keep the mathematics here very simple, even though the
advances that I am going to discuss depend largely on results
obtained by rather abstract arguments.
\vskip 0.2cm
The three areas of thermodynamics I shall discuss are those of
normal systems, i.e. ones whose energies are extensive variables;
non-relativistic gravitational systems, whose energies are not
extensive, because of the long range of the Newtonian
interactions; and Black Holes, which arise as a consequence of
relativistic gravitational collapse of stars.
\vskip 0.2cm
I shall start, in ${\S}2,$ with an expose' of developments in the
statistical thermodynamics of normal systems, that have been
achieved within the framework in which a macroscopic system is
represented as an infinitely extended one of finite density
[1-3]. This amounts to an idealisation, which serves to
reveal intrinsic bulk properties of macro-systems, that are
otherwise masked by boundary and other finite-size effects. In
particular, by contrast with the traditional statistical
mechanics of finite systems, it accommodates the {\it phase
structure} of matter, as manifested not only by the singularities
in thermodynamic potentials, but also by the coexistence of
equilibrium states with different microstructures. Here, I shall
sketch two basic and relatively new results, obtained
within this framework [3, Ch.4], that go beyond those of
traditional statistical, as well as classical, thermodynamics.
The first provides an answer to the fundamental question of what
comprises a complete set of macroscopic observables,
corresponding to the thermodynamic variables of a system. The
second establishes a statistical mechanical basis for the
empirically known connection between two {\it a priori} distinct
characteristics of phase transitions, namely thermodynamic
singularities and phase coexistence.
\vskip 0.2cm
I shall then provide a brief note, in ${\S}2,$ on the
thermodynamics of non-relativistic gravitational systems,
consisting of fermions of a single species that interact via
Newtonian forces. Here, the main result [4] is that these
systems have negative {\it microcanonical} specific heat in a
certain regime, and consequently are unstable there with respect
to energy exchanges with thermal reservoirs.
\vskip 0.2cm
In ${\S}3,$ I shall pass to the thermodynamics of Black Holes.
This poses rather special conceptual problems because, by its
very nature, a Black Hole has no observable microstructure.
Hence, the standard concept of entropy as a structural property
of an operationally determinable microstate [5, Ch.5] is
inapplicable to it; and therefore the proposed extension [6,7]
of this concept to non-observable objects has a subjective
component. In order to restore objectivity to the theory, I have
made a different approach to the thermodynamics of processes
involving the exchange of matter with Black Holes [8]. This is
based exclusively on observable quantities and thus involves no
BH entropy concept. Here, I shall provide a simple and, I
believe, improved treatment of the argument of [8], which leads
to Bekenstein's formula [6] for a generalised second law of
thermodynamics, but now interpreted within a framework built on
the observables of the exterior region of the Black Hole.
\vskip 0.5cm
\centerline {\bf 2. Normal Systems}
\vskip 0.3cm
A normal system is one whose energy is
an extensive variable. At the quantum mechanical level, this
means that the potential energy of interaction between the
particles in two disjoint regions is a 'surface effect': for
precise mathematical specifications of this, see [3, Ch.4]. In
fact, the extensivity condition is fulfilled by systems with
suitably short-range interactions or even by electrically neutral
Coulomb systems, because of Debye screening [9].
\vskip 0.2cm
As is well known, the extensivity property permits one to
represent normal macroscopic systems in the so-called
thermodynamical limit, where they are idealised as infinitely
extended assemblies of particles [1-3]. This is the model
we shall employ, since it is needed for the mathematically sharp
characterisation of macroscopic phenomena such as thermodynamic
singularities, phase coexistence and irreversibilty.
\vskip 0.2cm
{\bf Thermodynamic Variables.} In Classical Thermodynamics (CT),
the macrostate of a system is represented by a set $q=(q_{1},..
\ ,q_{n})$ of intensive variables, that are global densites of
extensive conserved quantities, such as energy, magnetic moment,
etc. The entropy density is then a function, $s,$ of $q,$ whose
form governs the thermodynamics of the system. We note here that,
for any given system, {\it CT provides no specification of the
variables, $q,$ or even of their number, $n.$} For that, as well
as for the form of $s,$ we need to pass to the statistical
mechanical model, ${\Sigma},$ of the system.
\vskip 0.2cm
There, we introduce the concept of {\it thermodynamic
completeness}, as applied to a set ${\hat Q}=({\hat Q}_{1},.. \
,{\hat Q}_{n}),$ of extensive conserved observables of
${\Sigma},$ in the following way [3, Ch.4]. Denoting by ${\hat
q}=({\hat q}_{1},.. \ ,{\hat q}_{n})$ the intensive observables,
given by the global densities of ${\hat Q},$ we term ${\hat Q}$
thermodynamically complete if
\vskip 0.2cm
(C.1) {\it ${\hat Q}_{1}$ is the energy;}
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(C.2) {\it $({\hat Q}_{1},.. \ ,{\hat Q}_{n})$ are linearly
independent; and}
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(C.3) {\it if ${\hat s}({\rho})$ is the
entropy density}\footnote*{This is defined [1, Ch.7] as a natural
generalisation, to infinite systems, of the density of the Von
Neumann entropy, $-kTr({\rho}{\log}{\rho}),$ of a microstate
whose density matrix is ${\rho}.$} {\it of a mixed state ${\rho}$
of ${\Sigma},$ then
\vskip 0.2cm
(a) for any given expectation value $q$ of ${\hat q},$ there is
precisely one state, ${\rho}_{q},$ that maximises ${\hat s},$ and
\vskip 0.2cm
(b) the same is not true for any proper subset of $({\hat
q}_{1},.. \ ,{\hat q}_{n}).$}
\vskip 0.2cm
In other words, ${\hat Q}$ is thermodynamically complete if the
value of its density, ${\hat q},$ determines the equilibrium
microstate, without redundancy. For example, in the case of a
ferromagnetic system, such as the Ising model, ${\hat q}$
comprises the energy density and the polarisation.
\vskip 0.2cm
We assume, as the condition for ${\Sigma}$ to support a
thermodynamics, that it possesses a complete set of extensive
conserved observables, ${\hat Q},$ that is unique, up to linear
combinations. The intensive thermodynamical variables of the
system are then the expectation values, $q,$ of the global
densities, ${\hat q},$ of ${\hat Q}.$
\vskip 0.3cm
{\bf Thermodynamic Potentials.} The equilibrium entropy density
of ${\Sigma},$ as a function of the thermodynamical variables $q$
may be seen from the condition (C.3a) to be
$$s(q){\equiv}{\hat s}({\rho}_{q})\eqno(2.1)$$
In fact, this is a generalisation of the standard definition of
microcanonical entropy [10], and reduces to that in the case
where ${\hat Q}$ consists of the energy only.
\vskip 0.2cm
We can also represent the thermodynamics of the system in the
canonical description, by the Legendre transform, $p,$ of $s:$
this is just the pressure, in units of $k_{B}T.$ Thus, denoting
by ${\theta}=({\theta}_{1},.. \ ,{\theta}_{n})$ the thermodynamic
conjugates of $q,$
$$p({\theta})={\max}_{q}(s(q)-{\theta}.q); \ {\theta}.q{\equiv}
{\sum}_{j=1}^{n}{\theta}_{j}q_{j}\eqno(2.2)$$
{\bf Note.} The maximisation of $s-{\theta}.q$ corresponds to the
minimisation of the Gibbs free energy and thus occurs precisely
when $q$ takes an equilibrium value, ${\overline q}.$ By the
completeness condition (C3), this determines the corresponding
equilibrium microstate ${\rho}_{\overline q}.$ Thus, the
condition for {\it phase coexistence}, for given ${\theta},$ is
that $s-{\theta}.q$ attains its maximum at more than one value
of $q.$
\vskip 0.2cm
The above definitions of $s$ and $p$ imply [3, Ch.4]) that the
former is concave and the latter convex, i.e., for
$0<{\lambda}<1,$
$$s({\lambda}{\theta}+(1-{\lambda}){\theta}^{\prime})
{\ge}{\lambda}s({\theta})+(1-{\lambda})s({\theta}^{\prime})
\eqno(2.3)$$
and
$$p({\lambda}{\theta}+(1-{\lambda}){\theta}^{\prime})
{\le}{\lambda}p({\theta})+(1-{\lambda})p({\theta}^{\prime})
\eqno(2.4)$$
These are thermodynamic stability properties. They imply
[11] that the functions $s$ and $p$ are continuous, except
possibly at the boundaries of their domains of definition. On the
other hand, they do not guarantee differentiability; and, in
fact, $p$ can support singularities, as given by points,
${\theta},$ where it is not differentiable [1,3]. Such points
correspond, of course, to phase transitions.
\vskip 0.2cm
{\bf Equilibrium States and Phase Structure.} This last
observation brings us to the point that, in Classical
Thermodynamics, there are two characterisations of phase
transitions, namely,
\vskip 0.2cm
(a) singularities in thermodynamic potentials; and
\vskip 0.2cm
(b) coexistence of different equilibrium states.
\vskip 0.2cm\noindent
Here, we define phase coexistence as in the Note following equn.
(2.2).
\vskip 0.2cm
In fact, although (a) and (b) are {\it a priori} distinct from
one another, it is generally assumed, on empirical grounds, that
they always arise together. It is therefore natural to ask
whether there is any {\it neccessary} connection, imposed by the
underlying quantum statistical structure, between these
characterisations. The answer to this question is provided by the
following Proposition, which was proved [3, Ch.4] on the
basis of the concavity/convexity properties (2.3) and (2.4).
\vskip 0.3cm
{\bf Proposition.} {\it Under the above definitions and
assumptions, phase coexistence occurs at precisely those values
of ${\theta}$ where the potential $p$ is not differentiable.}
\vskip 0.2cm
In other words, the structures imposed by the quantum statistical
model render the characterisations (a) and (b) equivalent.
\vskip 0.5cm
{\bf 3. Non-relativistic Gravitational Systems}
\vskip 0.3cm
We consider now a system, ${\Sigma}_{N},$ of $N$ electrically
neutral, massive fermions of one species, interacting via
Newtonian forces. This is a model of a neutron star, and its
Hamiltonian takes the form
$$H_{N}=-{{\hbar}^{2}\over 2m}{\sum}_{j=1}^{N}{\Delta}_{j} \
-{\kappa}m^{2}{\sum}_{j,k({\neq}j)=1}^{N}r_{jk}^{-1}\eqno(3.1)$$
where ${\kappa}$ is the gravitational constant, ${\Delta}_{j}$
is the Laplacian for the $j'$th particle, and $r_{jk}$ is the
distance between the $j'$th and $k'$th particles.
\vskip 0.2cm
Because of the long range of the interactions, the energy of the
system is {\it not} an extensive variable, and so the theory of
${\S}2$ is inapplicable to this model. In fact [4], for a
sequence of systems ${\Sigma}_{N},$ occupying geometrically
similar spatial regions, the volume now scales as $N^{-1},$
instead of $N:$ this results from the competition between the
Newtonian attraction, which favours implosion, and the Fermi
pressure. Correspondingly, the energy, $E_{N},$ and the
microcanonical entropy, $S_{N},$ scale as $N^{7/3}$ and $N,$
respectively, and the system enjoys the following thermodynamic
properties [4].
\vskip 0.2cm
(G1) {\it If $N^{-7/3}E_{N}, \ NV_{N}$ converge to $e, \ v,$
respectively, as $N{\rightarrow}{\infty},$ then the
microcanonical specific entropy $N^{-1}S_{N}$ converges to a
function, $s,$ of $(e,v).$}
\vskip 0.2cm
(G2) {\it There is a regime in which $s$ is convex w.r.t. $e,$
i.e. where the system is unstable against energy exchanges with
a thermal reservoir.}
\vskip 0.2cm
(G3) {\it The system undergoes a phase transition, of the Van der
Waals type, when $s$ changes from concave to convex w.r.t. $e.$}
\vskip 0.2cm
For further properties of the model, see Refs. [2, Section 4.2]
and [12-14]. In particular, when $N$ becomes sufficiently large,
${\approx}10^{60},$ the non-relativistic model becomes
unphysical, since it implies that the mean particle velocities
are comparable with the speed of light.
\vskip 0.5cm
\centerline {\bf 4. Black Hole Thermodynamics}
\vskip 0.3cm
According to Classical
General Relativity, the collapse of a star, due to gravitational
implosion, results in the formation of a Black Hole [15, Pt.7],
i.e. a bounded spatial region, from which no light can escape.
Thus, the only observables of the BH that can be perceived from
the outside are its mass, electric charge and angular momentum,
these being the ones that are registered by the external
gravitational and electromagnetic fields [15, P.876]. This is
Wheeler's famous principle that "Black Holes have no Hair", the
word 'hair' here meaning 'microstructure observable from
outside'. It
has dramatic consequences for thermodynamics, since it implies
that a BH has no operationally determinable microstate. The
concepts of statistical mechanics, including that of entropy, are
therefore inapplicable to it, and, consequently, the standard
form of the Second law of Thermodynamics has no predictive value
for processes in which observable systems discharge matter into
the Hole.\footnote*{This remark was credited to J. A. Wheeler by
Bekenstein [6].}
\vskip 0.2cm
On the other hand, the relativistic mechanics of Black Holes
[16] exhibits remarkable analogies with classical
thermodynamics, with the surface area of a Hole playing the role
of entropy. Bekenstein [6] argued that these analogies carried
physical content by showing that, in certain Gedankenexperiments,
in which matter was dropped into a Black Hole, the sum of the
entropy, $S,$ of the exterior region and the area, $A,$ of the
BH, multiplied by a certain constant, ${\lambda},$ never
decreased, i.e.,
$${\Delta}S+{\lambda}{\Delta}A{\ge}0,\eqno(4.1)$$
On this basis, he proposed that the BH had entropy ${\lambda}A,$
and that processes involving BH's conformed to a Generalised
Second Law (GSL), represented by equn. (4.1). This proposal was
supported [7,17] by Hawking's subsequent argument that Black
Holes act as sources of thermal radiation, of quantum mechanical
origin, with temperature corresponding to the assumed form,
${\lambda}A,$ of the BH entropy. At the quantum statistical
level, Bekenstein introduced a {\it subjective} element into the
theory, suggesting that this entropy was an information-theoretic
quantity, that represented the external observer's ignorance of
the state of the BH.
\vskip 0.2cm
In order to recast the theory in purely objective terms, that
retain the standard connections between thermodynamic variables
and underlying microstructures, we have provided a different
treatment of processes involving Black Holes [8]. This is based
on the statistical thermodynamics of the exterior, observable
region, and does not involve any concept of a BH entropy. As we
shall see, it leads to a version of the GSL (4.1), as adapted to
open systems, but with the last term there representing
mechanical work on the BH, not entropy.
\vskip 0.2cm
We formulate the theory on the basis of a thermodynamics of a
test-body, ${\Sigma},$ which is placed in the Hawking radiation
of a Black Hole, $B,$ and exchanges mass, electric charge and
angular momentum with both $B$ and the radiation. ${\Sigma}$ is
thus an {\it open system}. Accordingly, we base our treatment on
the following general principles.
\vskip 0.2cm
(I) {\it The Second Law of Thermodynamics, which tells us that
the Gibbs potential, ${\Phi},$ of an open system cannot
spontaneously increase. Equivalently [18, Ch.2], if mechanical
work, $W,$ is done on the system, then}
$$W{\ge}{\Delta}{\Phi}\eqno(4.2)$$
\vskip 0.2cm
(II) {\it The laws of Black Hole Mechanics [16], which ensue
from General Relativity. These are that
\vskip 0.2cm
(BH1) the total energy, electric charge and angular momentum of
the BH and the exterior system are conserved in any process;
\vskip 0.2cm
(BH2) the surface gravity, ${\sigma},$ i.e. the (proper)
acceleration of a freely infalling body at the surface of $B,$
is uniform over that surface;
\vskip 0.2cm
(BH3) the surface area, $A,$ of $B$ is a function of its energy,
$E_{B},$ charge, $Q_{B},$ and angular momentum, $J_{B},$ and
satisfies the differential relation
$${\sigma}c{^2}dA/8{\pi}{\kappa}=dE_{B}-{\phi}dQ_{B}
-{\omega}.dJ_{B}\eqno(4.3)$$
where ${\phi}$ is the electric potential and ${\omega}$ the
angular velocity of $B;$ and
\vskip 0.2cm
(BH4) $A$ increases in irreversible processes, and remains
constant in reversible ones. Here, it is the the origination of
the BH from the collapse of a star that is the source of the
irreversibility.}
\vskip 0.2cm
(III) {\it The Hawking thermal radiation phenomenon [17], which
we have shown [19] to be a general, model-independent
consequence of the basic principles of quantum theory, relativity
and statistical thermodynamics. The temperature of this radiation
is
$$T={\hbar}{\sigma}/2{\pi}kc\eqno(4.4)$$
and its electric potential and angular velocity are those of $B,$
namely ${\phi}$ and ${\omega},$ respectively.}
\vskip 0.3cm
{\bf Note.} The laws of Black Hole Mechanics, (BH1-4), have an
obvious analogy with classical thermodynamics, with ${\sigma}$
and $A$ playing the roles of temperature and entropy,
respectively. However, we do {\it not} interpret $A$ as an
entropy, for the following reasons.
\vskip 0.2cm
(1) In the case of a normal physical system, a microstate
corresponds to a density matrix, ${\rho},$ whose explicit form
may be operationally determined, and the entropy is a function,
$-kTr({\rho}{\log}{\rho}),$ of this state, which provides a
measure of its disorder [2, P.57]. Thus, its essential
significance is not merely that it has an information-theoretic
form, but that it represents a {\it structural property} of the
microstate. In the case of a Black Hole, however, no similar
quantity could exist, since the BH has no observable
microstructure.
\vskip 0.2cm
(2) One sees immediately from (4.3) that the last term in (4.1)
l.h.s. corresponds to the mechanical work required to effect
changes ${\Delta}E_{B}, \ {\Delta}Q_{B}, \ {\Delta}J_{B}$ in
$E_{B},\ Q_{B}, \ J_{B},$ respectively. Hence, apart from the
considerations of (1), there is no call for it to be regarded as
an entropy.
\vskip 0.2cm
(3) Whereas thermodynamic irreversibility arises as a result of
'phase mixing', there is no such mechanism operating in (BH4).
\vskip 0.3cm
{\bf Derivation of the GSL.} We now pass to the thermodynamics
of a body, ${\Sigma},$ placed in the Hawking radiation of a Black
Hole, $B.$ Thus, by (III), ${\Sigma}$ is immersed in a heat bath
of temperature $T,$ electric potential ${\phi}$ and angular
velocity ${\omega}.$ Its Gibbs potential is therefore
$${\Phi}=E-TS-{\phi}Q-{\omega}.J\eqno(4.5)$$
where $E, \ S, \ Q$ and $J$ are its energy, entropy, charge and
angular momentum, respectively.
\vskip 0.2cm
Suppose now that ${\Sigma}$ is placed just outside $B$ and that
it then discharges some of its matter into this Black Hole. Then
it follows from (I) and equn. (4.5) that
$${\Delta}S{\ge}T^{-1}({\Delta}E-{\phi}{\Delta}Q-
{\omega}.{\Delta}J)\eqno(4.6)$$
$$ \ \ \ \ =-T^{-1}(({\Delta}E_{B}-{\phi}{\Delta}Q_{B}-
{\omega}.{\Delta}J_{B}), \ by \ (IIa)\eqno(4.6a)$$
$$ \ \ {\ge}-T^{-1}{\sigma}c{^2}dA/8{\pi}{\kappa}, \ by \
(IIc,d)\eqno(4.6b)$$
Hence, by (4.4),
$${\Delta}S+{\lambda}{\Delta}A{\ge}0; \ with \ {\lambda}
=kc^{3}/4{\kappa}{\hbar}\eqno(4.7)$$
This is precisely the GSL proposed by Bekenstein, though now we
interpret the area term as representing mechanical work, rather
than entropy.
\vskip 0.2cm
We can easily extend our treatment to cover the situation where
${\Sigma}$ is transported by some machine ${\cal M}$ from an
exterior position to the boundary of $B$, then discharges matter
into this BH, and is finally brought back by ${\cal M}$ to its
starting point. Thus, denoting the increments in an arbitrary
thermodynamic variable, $V,$ of ${\Sigma}$ for the first, second
and third phase of this operation by
${\Delta}_{1}V,{\Delta}_{2}V$ and ${\Delta}_{3}V,$ respectively,
it follows from (4.2) that the total work, $W,$ done on the
system by ${\cal M}$ satisfies the inequality
$$W{\ge}{\Delta}_{1}{\Phi}+{\Delta}_{3}{\Phi};\eqno(4.8)$$
while it follows from (4.5)-(4.7) that
$${\Delta}_{2}{\Phi}+{\Delta}E_{B}-T{\lambda}{\Delta}A-
{\phi}{\Delta}Q_{B}-{\omega}.{\Delta}J_{B}{\le}0\eqno(4.9)$$
Hence, denoting by ${\Delta}V$ the change in a variable $V$ over
the whole operation, we infer from (4.4), (4.8) and (4.9) the
following extended form of the GSL.
\vskip 0.3cm
{\bf Proposition.} {\it In any operation involving the exchange
of matter between a system, ${\Sigma},$ and a Black Hole, $B,$
the total mechanical work done on ${\Sigma}$ by external agencies
satisfies the inequality}
$$W{\ge}{\Delta}{\Psi}\eqno(4.10)$$
{\it where}
$${\Psi}=E+E_{B}-T(S+{\lambda}A)-{\phi}(Q+Q_{B})-
{\omega}.(J+J_{B})\eqno(4.11)$$
\vskip 0.3cm
{\bf Comment.} This is a generalised version of the Second Law,
as represented by (4.2). Here, ${\Psi}$ is evidently the
relevant thermodynamic potential for the system composed of
${\Sigma}$ and $B.$
\vskip 0.5cm
\centerline {\bf References}
\vskip 0.2cm\noindent
1. D. Ruelle: "Statistical Mechanics", W. A. Benjamin, Inc., New
York, 1969
\vskip 0.2cm\noindent
2. W. Thirring: "Quantum Mechanics of Large Systems", Springer,
New York, Vienna, 1980
\vskip 0.2cm\noindent
3. G. L. Sewell: "Quantum Theory of Collective Phenomena", Oxford
University Press, Oxford, New York, 1989
\vskip 0.2cm\noindent
4. P. Hertel and W. Thirring: Pp. 310-323 of "Quanten und
Felder", Ed. H. P. Durr, Vieweg, Braunschweig, 1971
\vskip 0.2cm\noindent
5. J. Von Neumann: "Mathematical Foundations of Quantum
Mechanics", Princeton University Press, 1955
\vskip 0.2cm\noindent
6. J. Bekenstein: Phys. Rev. D {\bf 7}, 2333 (1973); and Pp. 42-
62 of "Jerusalem Einstein Centenary Conference", Ed. Y. Neeman,
Addison-Wesley, Reading, MA, 1981
\vskip 0.2cm\noindent
7. S. W. Hawking: Phys. Rev. D {\bf 13}, 2188 (1976)
\vskip 0.2cm\noindent
8. G. L. Sewell: Phys. Lett. A {\bf 122}, 309 (1987); Erratum,
Phys. Lett. A {\bf 123}, 499 (1987)
\vskip 0.2cm\noindent
9. J. L. Lebowitz and E. H. Lieb: Adv. Math. {\bf 9}, 316 (1972)
\vskip 0.2cm\noindent
10. R. B. Griffiths: J. Math. Phys. {\bf 1447}, (1965)
\vskip 0.2cm\noindent
11. R. T. Rockafellar: "Convex Analysis", Princeton University
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\vskip 0.2cm\noindent
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\vskip 0.2cm\noindent
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\vskip 0.2cm\noindent
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\vskip 0.2cm\noindent
15. C. W. Misner, K. Thorne and J. A. Wheeler: "Gravitation", W.
H. Freeman, San Francisco, 1973
\vskip 0.2cm\noindent
16. J. M. Bardeen, B. Carter and S. W. Hawking: Commun. Math.
Phys. {\bf 31}, 161, (1973)
\vskip 0.2cm\noindent
17. S. W. Hawking: Commun. Math. Phys. {\bf 43}, 199, (1975)
\vskip 0.2cm\noindent
18. L. D. Landau and E. M. Lifschitz: "Statistical Physics",
Pergamon Press, Oxford, 1959
\vskip 0.2cm\noindent
19 G. L. Sewell: Ann. Phys. {\bf 41}, 201, (1982)