It is well known that bodies may suffer a gravitational collapse, providing their mass is sufficiently large, their dimensions are sufficiently small and their measurable internal motion ceased [1,2]. In such a state they are black holes. We cannot have any information about their internal state. Any mass or radiation signal falls in the infinite space-time singularity of the black holes. In order to get a qualitative criterion of the black-hole condition we use
for the gravitational energy of a spherical mass M with radius R, where
is the gravitational constant; if
(where
is the speed of light in vacuum), i.e. if
, the mass collapses; the condition may also be written as
, where
is close to the Schwarzschild radius
. We take
as the radius of any black hole with mass M.
The radiation inside a black hole (of any kind), being delocalized, moves in the highly curved space-time of the black hole. Consequently, there appear quantum-mechanical transitions [3] and, near the black-hole horizon (radius), radiation quanta may escape, for a while, from the black hole. This is related to the so-called Hawking fluctuating radiation [4]. If the black hole fluctuates, i.e. if its mass M and radius
fluctuate, we may think that the black hole has an internal statistical motion and a thermodynamics. The current description of the statistical motion and the thermodynamics of a black hole raises serious questions [2]. This description assumes usually that the area of the black hole can only increase [5], as a consequence of accretion, so this area divided by the Planck area is a dimensionelss parameter which can only increase. Therefore, the reasoning goes further, it is an entropy, which may be equalized with the ratio of any two energies, one being viewed as heat, the other as temperature. This argumentation is insufficient to admit that the area of the black holes is proportional to their entropy.
The way to the statistical physics and the thermodynamics of a black hole is provided by the quantum motion of the radiation inside the hole. Indeed, the space quantization requires
where n is any positive integer and
is the radiation wavelength; on the other side, the time quantization requires
for the energy
of the black hole, where h is Planck’s constant and
is a positive integer. Inserting equation (1) in equation (2) we get
whence we get
obviously, these equations can only be satisfied if
and
We can see that the mass is an integral multiple of the Planck mass
and the radius
is an integral multiple of the Planck length
(up to a
factor). The energy of a black hole
is an integral multiple of the fundamental Planck energy
(Planck “temperature”), which corresponds to the Planck wavelength
. This particular circumstance of the quantization of the motion arises from the black hole condition
.
The frequency
(the reciprocal of the Planck time) is very high; The Planck energy is of the order
, or
, or
. Because of the extremely high frequency of these quanta we may call them black quanta.
The statistical physics of the quantum-mechanical motion described above is immediate; it corresponds to a single quantum-mechanical state of energy
occupied by n black quanta; the number n is the (main) statistical variable. The thermodynamic potential
(free energy
) is
where
is the temperature (the chemical potential is zero and we include
in summation); the mean occupation number is
the mean energy is
and the entropy is
we can see that the black hole has a mean mass
and a mean radius
. Also, we can see that
for
(according to the third principle of thermodynamics). The relative fluctuation in the occupation number is
. This is also the relative fluctuation in energy, mass, radius; it can be related to the entropy fluctuation by using
(which follows from equations (7) and (8). This latter formula shows that at equilibrium, for a given mean energy, the maximum of the entropy gives the mean occupation number
in equation (7); and any change out of equilibrium decreases the entropy, as expected (in agreement with the law of increase of entropy - the second principle of thermodynamics).
At eqilibrium the thermodynamic potential is stationary (
); since, on one hand, in equilibrium transformations,
and, on the other,
, we get
, i.e. the change in energy is, in fact, a change in the amount of heat. The temperature T is a measure of the internal energy E of the black hole (proportional to the mean mass
, or mean size
); according to equation (7), for
no Planck mass (black quanta) is excited inside the black hole and
; on the contrary, for
a large number of black quanta are excited and
The black holes may have an electric charge and, also, they may rotate; these external parameters may be included in the thermodynamics by adding to the energy
the energies
and
, where
is the electric charge,
is its electric potential,
is the angular frequency and
is the angular momentum; since
, the charge contribution is proportional to
; similarly, since
, the rotational term brings a contribution proportional to
. The full expression of the total energy, which is a function of n, is introduced in equation (6), which gives the thermodynamic potential. The contributions brought by electric charges or rotations to the thermodynamic properties are very small, such that they may be treated as corrections to the potential given by equation (6).
In conclusison, we may say that the quantization of the motion inside a black hole identifies elementary excitations called black quanta, which correspond to Planck mass, wavelength, energy, frequency, time, etc. The statistics and the thermodynamics of these elementary excitations are computed explicitly in this paper.