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Everybody knows
what
the classical black holes Are. The main feature that characterizes
black
holes and distinguishes them from other physical objects is their
universality.
Such an universality is widely known as the "no hair conjecture''. It
means
that everything that can be radiated away does radiate during the
gravitational
collapse, and the resulting black hole is described by only few
parameters,
namely, by its mass, angular momentum and gauge charges. The universal
character
of black holes allowed J.Bekenstein to put forward an analogy between
black
hole physics and thermodynamics. According to Bekenstein, the black
hole
is endowed by some temperature and entropy, the value of entropy being
proportional to the area of the event horizon. The rigorous proof of
four
laws of thermodynamics for the most general stationary black holes was
given by Bardeen, Carter Hawking. And at the top of these is the
Hawking's
discovery that a black hole does radiate as a black body with this very
temperature. S.Hawking considered a quantized scalar field on the given
Schwarzschild background and showed that it is nontrivial causal
structure
of the black hole metric that is responsible for the black body
radiation.
Such an effect is purely quantum, and the Hawking's temperature is a
generalization
of the Unruh's temperature seen by an uniformly accelerated observer in
the flat spacetime. The analogy between these two effects illustrates
the
famous equivalence principle. The discovery of the black hole
evaporation opened, in a sense, the quantum era in the black hole
physics. The matter is that the very definition of the black hole
involves the notion of the so called event horizon (which is the
boundary between geodesics that can escape to infinity and those that
cannot). The event horizon can be determined only globally, and the
procedure requires knowledge of the whole history. Since, due to
evaporation, the black holes disappear (at least, if do not take into
account the back reaction of the radiation on the spacetime metric) the
very notion of the black hole becomes only approximate. Why should we
study quantum black holes? First, this problem is interesting by itself
and it could provide additional links between General Relativity and
Quantum
Theory. Second, at the final stage of evaporation black holes are so
small
that quantum effects can no more be ignored. Third, small black holes
can
be formed by large enough fluctuations of both matter fields or
spacetime
metric in the Very Early Universe (the so called Primordial Black
Holes)
or in the course of vacuum phase transitions. But how small should be a
black hole in order to be considered as a quantum object? On purely
dimensional grounds using Newton's constant G, Planck's constant and
velocity of light c, we are able to construct two different quantities
with the dimension
of length for some object of mass m, namely, the Compton's length and
the
gravitational (Schwarzschild) radius r_g = 2 G m/c^2. For masses much
smaller
than the Planckian mass m_{Pl} = \sqrt{\hbar c/G} the gravitational
radius
is much smaller than the Compton's length and the object is purely
quantum
(this also indicates that there may be no black holes with so small
masses).
If the mass is much higher than the Planckian mass, the Compton's
length
is well inside the black hole horizon and in this case we are already
have
purely classical black holes (if it is possible, of course, to ignore
the
Hawking radiation). Thus, the range of quantum black hole masses is
somewhere
in-between. The first attempt to obtain the quantum black hole mass
spectrum
was due to J.Bekenstein. He noticed that an event horizon area for
slowly
evolving black holes is an adiabatic invariant. So, the usual
quasiclassical
quantization leads to an equidistant spectrum for a black hole surface
area.
In the case of the Schwarzschild black hole this results in the now
famous
square-root mass spectrum m_{BH} \sim \sqrt{n}, where n is an integer
quantum
number. The same type of spectrum was then advocated by J.Bekenstein
and
S.Mukhanov later on by many others. It was shown also that such a
spectrum
is compatible with the Hawking radiation. In what follows we confine
ourselves
with consideration of the Schwarzschild (neutral, nonrotating) black
holes
only. The very fact that the mass (total energy) of black holesdepends
on
only one quantum number n can be viewed as a generalization of the "no
hair''
conjecture and confirms the universal character of black holes also on
the
quantum level. But we should pay some price for such a universality.
And
this price is that the every energy level is highly degenerate. Indeed,
since
the black hole entropy is proportional to the area of the horizon, and
the
latter has (approximately) an equidistant spectrum, the number of
quantum
states with the same total energy (mass) grows exponentially with the
quantum
number n. The real physical explanation of this phenomenon is still an
open
problem. In our opinion, this is because we do not yet know what is an
object
that could be called a quantum black hole. In other words, we need a
definition.
It seems it is the universality that could become the crucial feature
which
would distinguish quantum black holes from any other quantum object. It
is worth noting that in the recent paper by G.Gour the equidistant
spectrum
for the black hole area and the exponential degeneracy of the energy
levels
have been taken as postulates. The author constructed the Hamiltonian
operator
and the algebra of observables for quantum Schwarzschild black holes
and,
thus, showed the selfconsistency of these two postulates. The
abovementioned
attempts to construct a theory of quantum black holes can be called
phenomenological.
The equidistant (or any other) black hole area spectrum which leads to
a
discrete mass spectrum poses one serious problem. The classical, in our
case - Schwarzschild, black hole state is described by only one
parameter,
its mass, irrespective of how this black hole has been formed. Let us
suppose
that the black hole is formed by the collapse of gravitating particles.
The
motion of these particles can be either bound or unbound, such
qualitative
difference in the history of the constituents in no way reflected in
the
mass of the resulting black hole. This is not surprising in classical
theory.
But in quantum theory it does makes a difference. The bound motions
give
rise to discrete mass spectra while the mass spectra for unbound
motions
are continuous. And if the black hole mass spectrum is just the mass
(energy)
spectrum of this system of particles (what is the case in classical
theory)
we can easily distinguish the black holes formed due to bound motion
from
that formed due to unbound motion. But this contradicts the principle
of
universality (quantum ``no-hair'' conjecture). Then, what is it that
gives
the discrete mass spectrum for quantum black holes? Clearly, we need a
deeper
insight into the nature of these objects.
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