The first direct quantum calculation of black hole entropy was
given by Gibbons and Hawking [50] in the context of Euclidean quantum gravity. They started with
a formal, functional integral expression for the canonical
ensemble partition function in Euclidean quantum gravity and
evaluated it for a black hole in the ``zero loop'' (i.e,
classical) approximation. As shown in [100], the mathematical steps in this procedure are in direct
correspondence with the purely classical determination of the
entropy from the form of the first law of black hole mechanics. A
number of other entropy calculations that have been given within
the formal framework of Euclidean quantum gravity also can be
shown to be equivalent to the classical derivation (see [61] for further discussion). Thus, although the derivation of [50] and other related derivations give some intriguing glimpses
into possible deep relationships between black hole
thermodynamics and Euclidean quantum gravity, they do not appear
to provide any more insight than the classical derivation into
accounting for the quantum degrees of freedom that are
responsible for black hole entropy.
It should be noted that there is actually an inconsistency in the use of the canonical ensemble to derive a formula for black hole entropy, since the entropy of a black hole grows too rapidly with energy for the canonical ensemble to be defined. (Equivalently, the heat capacity of a Schwarzschild black hole is negative, so it cannot come to equilibrium with an infinite heat bath.) A derivation of black hole entropy using the microcanonical ensemble has been given in [29].
Another approach to the calculation of black hole entropy has
been to attribute it to the ``entanglement entropy'' resulting
from quantum field correlations between the exterior and interior
of the black hole [24,
31,
57]. As a result of these correlations across the event horizon,
the state of a quantum field when restricted to the exterior of
the black hole is mixed. Indeed, in the absence of a short
distance cutoff, the von Neumann entropy,
, of any physically reasonable state would diverge. If one now
inserts a short distance cutoff of the order of the Planck scale,
one obtains a von Neumann entropy of the order of the horizon
area,
A
. Thus, this approach provides a natural way of accounting for
why the entropy of a black hole is proportional to its surface
area. However, the constant of proportionality depends upon a
cutoff and is not (presently) calculable within this approach.
(Indeed, one might argue that in this approach, the constant of
proportionality between
and
A
should depend upon the number,
N, of species of particles, and thus could not equal 1/4
(independently of
N). However, it is possible that the
N
-dependence in the number of states is compensated by an
N
-dependent renormalization of
G
[87] and, hence, of the Planck scale cutoff.) More generally, it is
far from clear why the black hole horizon should be singled out
for a such special treatment of the quantum degrees of freedom in
its vicinity, since similar quantum field correlations will exist
across any other null surface. It is particularly puzzling why
the local degrees of freedom associated with the horizon should
be singled out since, as already noted in section
2
above, the black hole horizon at a given time is defined in
terms of the entire future history of the spacetime and thus has
no distinguished local significance. Finally, since the
gravitational action and field equations play no role in the
above derivation, it is difficult to see how this approach could
give rise to a black hole entropy proportional to Eq. (8
) (rather than proportional to
A) in a more general theory of gravity. Similar remarks apply to
approaches which attribute the relevant degrees of freedom to the
``shape'' of the horizon [81] or to causal links crossing the horizon [41].
A closely related idea has been to attribute the entropy of
the black hole to the ordinary entropy of its thermal
atmosphere [88]). If we assume that the thermal atmosphere behaves like a free,
massless (boson or fermion) gas, its entropy density will be
(roughly) proportional to
. However, since
T
diverges near the horizon in the manner specified by Eq. (12
), we find that the total entropy of the thermal atmosphere near
the horizon diverges. This is, in effect, a new type of
ultraviolet catastrophe. It arises because, on account of
arbitrarily large redshifts, there now are infinitely many modes
- of arbitrarily high locally measured frequency - that
contribute a bounded energy as measured at infinity. To cure this
divergence, it is necessary to impose a cutoff on the locally
measured frequency of the modes. If we impose a cutoff of the
order of the Planck scale, then the thermal atmosphere
contributes an entropy of order the horizon area,
A, just as in the entanglement entropy analysis. Indeed, this
calculation is really the same as the entanglement entropy
calculation, since the state of a quantum field outside of the
black hole is thermal, so its von Neumann entropy is equal to its
thermodynamic entropy (see also [69]). Note that the bulk of the entropy of the thermal atmosphere
is highly localized in a ``skin'' surrounding the horizon, whose
thickness is of order of the Planck length.
Since the attribution of black hole entropy to its thermal
atmosphere is essentially equivalent to the entanglement entropy
proposal, this approach has essentially the same strengths and
weaknesses as the entanglement entropy approach. On one hand, it
naturally accounts for a black hole entropy proportional to
A
. On the other hand, this result depends in an essential way on
an uncalculable cutoff, and it is difficult to see how the
analysis could give rise to Eq. (8) in a more general theory of gravity. The preferred status of
the event horizon and the localization of the degrees of freedom
responsible for black hole entropy to a ``Planck length skin''
surrounding the horizon also remain puzzling in this approach. To
see this more graphically, consider the collapse of a massive
spherical shell of matter. Then, as the shell crosses its
Schwarzschild radius, the spacetime curvature outside of the
shell is still negligibly small. Nevertheless, within a time of
order the Planck time after the crossing of the Schwarzschild
radius, the ``skin'' of thermal atmosphere surrounding the newly
formed black hole will come to equilibrium with respect to the
notion of time translation symmetry for the static Schwarzschild
exterior. Thus, if an entropy is to be assigned to the thermal
atmosphere in the manner suggested by this proposal, then the
degrees of freedom of the thermal atmosphere - which previously
were viewed as irrelevant vacuum fluctuations making no
contribution to entropy - suddenly become ``activated'' by the
passage of the shell for the purpose of counting their entropy. A
momentous change in the entropy of matter in the universe has
occurred, even though observers riding on or near the shell see
nothing of significance occurring.
Another approach that is closely related to the entanglement
entropy and thermal atmosphere approaches - and which also
contains elements closely related to the Euclidean approach and
the classical derivation of Eq. (8) - attempts to account for black hole entropy in the context of
Sakharov's theory of induced gravity [47,
46]. In Sakharov's proposal, the dynamical aspects of gravity arise
from the collective excitations of massive fields. Constraints
are then placed on these massive fields to cancel divergences and
ensure that the effective cosmological constant vanishes.
Sakharov's proposal is not expected to provide a fundamental
description of quantum gravity, but at scales below the Planck
scale it may possess features in common with other more
fundamental descriptions. In common with the entanglement entropy
and thermal atmosphere approaches, black hole entropy is
explained as arising from the quantum field degrees of freedom
outside the black hole. However, in this case the formula for
black hole entropy involves a subtraction of the (divergent) mode
counting expression and an (equally divergent) expression for the
Noether charge operator, so that, in effect, only the massive
fields contribute to black hole entropy. The result of this
subtraction yields Eq. (9
).
More recently, another approach to the calculation of black hole entropy has been developed in the framework of quantum geometry [3, 10]. In this approach, if one considers a spacetime containing an isolated horizon (see section 2 above), the classical symplectic form and classical Hamiltonian each acquire an additional boundary term arising from the isolated horizon [9]. (It should be noted that the phase space [8] considered here incorporates the isolated horizon boundary conditions, i.e., only field variations that preserve the isolated horizon structure are admitted.) These additional terms are identical in form to that of a Chern-Simons theory defined on the isolated horizon. Classically, the fields on the isolated horizon are determined by continuity from the fields in the ``bulk'' and do not represent additional degrees of freedom. However, in the quantum theory - where distributional fields are allowed - these fields are interpreted as providing additional, independent degrees of freedom associated with the isolated horizon. One then counts the ``surface states'' of these fields on the isolated horizon subject to a boundary condition relating the surface states to ``volume states'' and subject to the condition that the area of the isolated horizon (as determined by the volume state) lies within a squared Planck length of the value A . This state counting yields an entropy proportional to A for black holes much larger than the Planck scale. Unlike the entanglement entropy and thermal atmosphere calculations, the state counting here yields finite results and no cutoff need be introduced. However, the formula for entropy contains a free parameter (the ``Immirzi parameter''), which arises from an ambiguity in the loop quantization procedure, so the constant of proportionality between S and A is not calculable.
The most quantitatively successful calculations of black hole
entropy to date are ones arising from string theory. It is
believed that at ``low energies'', string theory should reduce to
a 10-dimensional supergravity theory (see [67] for considerable further discussion of the relationship between
string theory and 10-dimensional and 11-dimensional
supergravity). If one treats this supergravity theory as a
classical theory involving a spacetime metric,
, and other classical fields, one can find solutions describing
black holes. On the other hand, one also can consider a ``weak
coupling'' limit of string theory, wherein the states are treated
perturbatively. In the weak coupling limit, there is no literal
notion of a black hole, just as there is no notion of a black
hole in linearized general relativity. Nevertheless, certain weak
coupling states can be identified with certain black hole
solutions of the low energy limit of the theory by a
correspondence of their energy and charges. (Here, it is
necessary to introduce ``D-branes'' into string perturbation
theory in order to obtain weak coupling states with the desired
charges.) Now, the weak coupling states are, in essence, ordinary
quantum dynamical degrees of freedom, so their entropy can be
computed by the usual methods of statistical physics. Remarkably,
for certain classes of extremal and nearly extremal black holes,
the ordinary entropy of the weak coupling states agrees exactly
with the expression for
A
/4 for the corresponding classical black hole states; see [58] and [74] for reviews of these results. Recently, it also has been
shown [32] that for certain black holes, subleading corrections to the
state counting formula for entropy correspond to higher order
string corrections to the effective gravitational action, in
precise agreement with Eq. (8
).
Since the formula for entropy has a nontrivial functional dependence on energy and charges, it is hard to imagine that the agreement between the ordinary entropy of the weak coupling states and black hole entropy could be the result of a random coincidence. Furthermore, for low energy scattering, the absorption/emission coefficients (``gray body factors'') of the corresponding weak coupling states and black holes also agree [66]. This suggests that there may be a close physical association between the weak coupling states and black holes, and that the dynamical degrees of freedom of the weak coupling states are likely to at least be closely related to the dynamical degrees of freedom responsible for black hole entropy. However, it remains a challenge to understand in what sense the weak coupling states could be giving an accurate picture of the local physics occurring near (and within) the region classically described as a black hole.
The relevant degrees of freedom responsible for entropy in the
weak coupling string theory models are associated with conformal
field theories. Recently Carlip [33,
34] has attempted to obtain a direct relationship between the
string theory state counting results for black hole entropy and
the classical Poisson bracket algebra of general relativity.
After imposing certain boundary conditions corresponding to the
presence of a local Killing horizon, Carlip chooses a particular
subgroup of spacetime diffeomorphisms, generated by vector fields
. The transformations on the phase space of classical general
relativity corresponding to these diffeomorphisms are generated
by Hamiltonians
. However, the Poisson bracket algebra of these Hamiltonians is
not isomorphic to the Lie bracket algebra of the vector fields
but rather corresponds to a central extension of this algebra. A
Virasoro algebra is thereby obtained. Now, it is known that the
asymptotic density of states in a conformal field theory based
upon a Virasoro algebra is given by a universal expression (the
``Cardy formula'') that depends only on the Virasoro algebra. For
the Virasoro algebra obtained by Carlip, the Cardy formula yields
an entropy in agreement with Eq. (9
). Since the Hamiltonians,
, are closely related to the corresponding Noether currents and
charges occurring in the derivation of Eqs. (8
) and (9
), Carlip's approach holds out the possibility of providing a
direct, general explanation of the remarkable agreement between
the string theory state counting results and the classical
formula for the entropy of a black hole.
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The Thermodynamics of Black Holes
Robert M. Wald http://www.livingreviews.org/lrr-2001-6 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |