|AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy||Gravity, black holes and physics|
B L A C K H O L E S
4.1. The role of gravity in the formation and evolution of stars
4.2. The final stages of stellar evolution. Gravitational collapse
4.3. Schwarzschild static black holes
4.4. Rotating and electrically charged Kerr-Newman black holes
4.5. The "black hole has no hair" theorem 4.6. Laws of black hole dynamics
4.7. Quantum radiation and thermodynamics of black holes
4.8. Astrophysical significance of black holes
4.9. Total gravitational collapse - the biggest catastrophe in nature
4.7. Quantum radiation and thermodynamics of black holes
phenomena in a black hole
In the previous §4.6 we showed a very close analogy between the laws of black hole mechanics and the laws of thermodynamics. But if we wanted to on the basis of this analogy in the context of classical (non-quantum) physics black holes attributed to the actual temperature (proportional to the surface gravity) and the actual entropy (proportional to the surface horizon) - to build thus a sort of "thermodynamics of black holes" - this thermodynamics would not be consistent. According to classical physics, a black hole can absorb bodies and radiation, but it cannot emit anything; its temperature must therefore be absolute zero regardless of the magnitude of the surface gravity. Similarly, the entropy of a black hole according to classic physics should be infinite regardless of the size of the horizon surface, because a black hole can theoretically be formed by the collapse of an infinitely large number of infinitesimal particles. In classical physics, therefore, the analogy between the laws of black hole mechanics and the laws of thermodynamics would remain only an external (and essentially random) similarity without a deeper physical meaning.
So far, however, we have investigated the properties of black holes only in terms of the classical general theory of relativity, without taking into account quantum phenomena . A completely consistent quantum approach is not yet feasible, because a complete quantum theory of gravity has not yet been developed. We must therefore be content with a kind of "semi-quantum" approach: we will leave gravity unquantited, we will only quantize other physical fields. In other words, we will use quantum physics to describe the motion of matter and the behavior of non-gravitational fields in curved spacetime (against the background of unquantited geometry of spacetime - in the unquantited gravitational field). Because the effects of quantition of spacetime itself can be significantly applied in microscales of the order of ~ 10 -33 cm (see §B.3), while in the scales of ordinary physics (ie above 10 -13 cm) they are very small, it can be expected that this semiquantum approximation will be to describe reality well in practically all situations, except for phenomena in the immediate vicinity of singularities.
Before proceeding to
Hawking's own effect of quantum evaporation of black holes, let
us illustrate the contribution of the quantum approach to the
thermodynamics of black holes in the imaginary experiment
according to Fig. 4.22 (designed by J. Bekenstein ) . Let
us have an empty box thermally insulated from the surroundings,
whose inner walls are heated to a temperature T m and the
inside of the box contains thermal radiation of a temperature Tm . Let a
black hole (for simplicity Schwarzschild's) of mass M be
at a sufficient distance
, on which we
will slowly lower the box with thermal radiation (so that there
is no noticeable radiation of gravitational waves) on a perfectly
strong rope *) using a winch. The potential energy of the box in
the gravitational field of a black hole turns into work spinning
a winch; by means of a dynamo, it is converted into electrical
energy, which is stored in the accumulator. We will assume that
the efficiency of conversion and accumulation is 100%.
*) This is an imaginary experiment ! A rope of any material known in physics would necessarily have to break quite far from the horizon; the same goes for the locker.
It is not a question of materials engineering, but the limitations are imposed by the laws of nature themselves. The strength of all known (also in the future) materials is formed by the electromagnetic interaction between the electronic shells of the atoms of the material; and it is completely insufficient in the gravitational field near the horizon. Even hypothetical materials held, albeit by strong interaction, would not work: the basic physical boundary is determined by the laws of the special theory of relativity. The tension of the rope can never be greater than the mass per unit length multiplied by c 2 .
Note also that the whole analysis of the imaginary experiment has only a heuristic character and does not serve to accurately derive the quantum-thermodynamic laws of black holes.
Fig.4.22. An imaginary experiment for studying the thermodynamic properties of a black hole - a heat engine with a black hole as a cooler.
a ) Slow lowering of the thermal radiation box to the black hole; the energy obtained is stored in the battery.
b ) At the horizon, part of the thermal radiation is absorbed by a black hole -> cooling of the box.
c ) The cooled and unloaded box is pulled back from the battery using energy, the remaining energy coming from the thermal radiation absorbed by the hole.
In order to be able to pull the box back again, we can lower it as deep as possible, so that its bottom just touches the horizon. Thus, its center of gravity can reach a maximum distance of L ( half the height of the box) from the horizon of the black hole. The total work that is released and stored in the battery when lowering the box from infinity to a distance L from the horizon is equal to the binding energy
|E 1 = m (1 - k .L),||(4.57)|
where m is the mass of the box and k is the surface gravity on the horizon of the black hole. Now we open the bottom of the box, so that part of the thermal radiation of mass d m escapes and is absorbed by the black hole. Close the bottom again and slowly pull the lightweight and cooled box (weighing m- d m) back by replacing the dynamo with a motor powered by battery energy, previously accumulated when the box was started. It will take work to pull the lightweight box back to infinity
|E 2 = (m - d m) (1 - k .L).||(4.58)|
The energy E 1 - E 2 remains in the accumulator , so we can say that the amount of heat d m was converted into the work E1 -E2 . The arrangement in Fig. 4.22 is thus, from the point of view of thermodynamics, a " heat engine " converting heat in the cabinet into work, the role of the "cooler" being played by a black hole. The efficiency h of heat conversion d m for the obtained work E 1 -E 2 is equal to
|h = (E 1 - E 2 ) / d m = 1 - k .L.||(4.59)|
This efficiency is
always less than one; approaching the ideal unit efficiency would
only be possible if the box size L was close to zero.
And here comes the word quantum physics , according to which this is not possible, because the box must be greater than the characteristic wavelength of thermal radiation corresponding to a given temperature T m (in order for this thermal radiation to "fit" into box) : L ³ h / kT m , where h is the Planck constant ("crossed out": h = h / 2 p ) and k is the Boltzman constant. The efficiency of heat conversion at work is therefore limited by inequality
|h £ 1 - ( k . h / k) / T m .||(4.60)|
According to the 2nd Act of Thermodynamics, the well-known Carnot theorem applies to the thermal-energy process in Fig. 4.22
|h £ 1 - T coolers / T heaters ,||(4.61)|
where equality applies only to the reversible action. If we compare both inequalities (4.60) and (4.61), where the T of the heater is equal to the original temperature T m of the box and the cooler is a black hole, we see that from a thermodynamic point of view the black hole should have a thermodynamic temperature
|T H = k . h / k||(4.62)|
(T H = h / kr g for the Schwarzschild black
hole), which is proportional to the surface gravity k . Let's look further at the entropy balance
in this process according to Fig. 4.22 ; let us admit that a black hole
has some finite entropy in the usual way related to mass and
temperature, even if it is not in accordance with classical
physics. The entropy that the box loses when it passes part of
its thermal radiation to the black hole is equal to d S of the box = d m /
T m . The black hole acquires an entropy d SH = d
M / T H
, where the weight gain of the black hole dAccording
to the law of conservation of energy (1st law of black hole
mechanics), M is equal to the difference between the transferred
mass d m and the energy remaining in the
accumulator : d M = d m -
(E 1 -E 2 ) = d m - d m (1 - k
.L) = d m. to .L. The entropy of a black hole
thus increases by d SH = d m. K. L / T H ; if we substitute here for T H
z (4.62) and if we use quantum conditions L ³ h /
kT m for the box , we get an inequality for the
total entropy balanced S H ³ d S cabinets . Thus, in accordance with the
2nd theorem, a thermodynamic black hole gains more entropy than a
box loses. Substituting into the relation d S H
= d M / T H to change the entropy of a black
hole for d M from the first law of black
hole mechanics (4.50) and for T H z (4.62), we get
d S H = (k / 8 p h ) d A;
since it is natural to assume that a non-existent black hole of zero mass has zero entropy, ie lim A ® 0 S H (A) = 0, integration leads to the relation
|S H = (k / 8 p h ) A.||(4.63)|
We get the result, according to which the entropy of a black hole is proportional to the area of ??its horizon; it is in full accordance with the analogy between the laws of black hole mechanics and the laws of thermodynamics (discussed in the previous §4.6 " Laws of black hole dynamics ") .
At the beginning of this
paragraph, we said that according to classical physics, the
entropy of a black hole should be infinite, because a black hole
can theoretically be caused by the collapse of an infinitely
large number of particles of infinitely small mass. According to
quantum physics (the uncertainty principle), however this is not
possible, the Compton length h/mc
of black hole-forming a black hole must be limited by gravitational
radius of the horizon 2M, so that the number of
configurations, from which a black hole of
a given mass can form is finite. According to quantum mechanics, the entropy of a black hole is very large, but finite .
Loss of information in a black hole. Holographic principle.
From wiewpoint of statistical physics (and information theory) is the entropy degree of disorder of the system  . When a black hole is formed by gravity collapse, all information about the individual properties of the collapsing mass is lost to the outside world except for the total mass, charge and rotational momentum - "a black hole has no hair", a black hole has no microscopic structure. This unusual situation is referred to as the " information loss paradox " - see also the discussion in the final section of this chapter, " Quantum Evaporation: The Return of Matter from a Black Hole? ". This vast amount of lost information is then a measure of the entropy of a black hole. Due to the proportionality between the surface of the horizon and the entropy of the absorbed matter, the somewhat misleading name " holographic principle " was later adopted :
Holographic image in optics it is created by dividing a beam of coherent light (from a laser) into two parts, one of which impinges on the photographic layer directly, the other part after reflection from the displayed object. Both of these beams interfere, and a structure of thin interference fringes is formed on the photographic emulsion, carrying information about the phase differences of the two beams. If we then illuminate this two-dimensional image with coherent light (again from a laser), the reflected rays reconstruct the same phase differences as the image they created - the impression of a three-dimensional image of the original object is created. The holographic image has the interesting property that even from a fragment of the hologram we can see the whole three-dimensional image, albeit with a lower resolution.
The two-dimensional surface of the horizon of the black hole carries everything (correspondingly reduced - "black hole has no hair")information about three-dimensional configurations of absorbed matter in a black hole, similar to a two-dimensional hologram carrying information about a three-dimensional object. However, this ends the similarity with holography, as detailed information about the absorbed matter (except M, J, Q) is lost and cannot be reconstructed in any way.
The "holographic principle" was further generalized in connection with the construction of quantum theories of gravity: "Information (degrees of freedom) about the system within the volume V can be located (encoded) on the surface ¶ In this volume, the density of information does not exceed one bit per Planck surface l p2 ".
The holographic principle is generally the statement that information about the N-dimensional region (its interior) is encoded onN-1 dimensional boundary of this area. Theoretically, it is justified only on the horizons of black holes. In other cases, these are just extrapolations and analogies - unconfirmed hypotheses ....
When capturing an elementary particle with a black hole to an outside observer loses the ability to identify this particle. Expressing ourselves in computer terminology, one "bit" of information has been lost: the yes-no particle. The entropy of the black hole thus increases by a value proportional to the Boltzman constant k ( d S H = k.ln2) and at the same time the area of the horizon A of the black hole increases accordingly . Bekenstein ,  calculated that when a mass of mass m (m << M) and size d is absorbed by a black hole, the area of its horizon increases by at least 8 pmd. In quantum mechanics, each particle has an effective diameter of the order of the Compton wavelength, so if we take the size of an elementary particle as its Compton diameter h / mc (but we work in units where c = 1), the area of the horizon increases by at least 8 p h . The ratio between the increase in entropy of a black hole and the increase in the area of its horizon when capturing an elementary particle is thus
d S H / d A ~ k / 8 p h .
According to quantum
mechanics and statistical physics, we again get the relation
(4.63), according to which the entropy of a black hole is
proportional to the area of ??its horizon. The coefficient of
this proportionality then follows from the relation (4.62) for
the temperature of a black hole under the Hawking effect of
quantum evaporation and from the 1st Act of Black Hole Mechanics
(4.50), see below.
In the thermodynamics of gases, liquids and solids, the entropy is directly proportional to the number of particles, ie the volume filled with matter. In the quantum physics of gravity, however, the entropy is directly proportional to the surface , so that significantly less information can be "encoded" into a given volume than would correspond to the classical idea. Equation (4.63) for black hole entropy can be rewritten in equivalent form
|S H = k. A / 4 l p 2 ,||(4.63´)|
where l p = Ö (G h c -3 ) is the Planck length (B.10) introduced in §B.4. We see that the entropy of a black hole is given by the number of Planck surfaces l p 2 , which can cover the horizon of a black hole (with a coefficient of 1/4). Black hole entropy (4.63) is also the maximum entropy could be "squeezed" into the volume enclosed inside the area sizes A . In other words, a black hole represents the object that most effectively concentrates the entropy - the area of its horizon A is the smallest possible surface of the spatial area in which the mass of a given entropy S H can be located.
evaporation of black holes
We still have one last contradiction , preventing the thermodynamics of black holes from being consistent: a black hole can absorb anything, but it cannot "in return" radiate anything. Therefore, a black hole cannot be in thermodynamic equilibrium with anything, its "temperature" should be absolute zero according to classical physics.
This missing link in the chain of black hole thermodynamics was supplemented by S. Hawking, who in 1974 expressed the hypothesis of quantum radiation of black holes ,  , according to which each black hole spontaneously emits radiation exactly as if it were ordinary (absolutely ) by a black body heated to a temperature of T H = k.h/k proportional to surface gravity k on the horizon.
A certain harbinger of Hawking's effect of quantum evaporation was the quantum analysis of superradiation in the ergosphere of a rotating black hole (mentioned in §4.4) by Zeldovich and Starobinsky , . From the point of view of quantum physics, the amplification of waves is a kind of "induced" radiation, so the existence of spontaneous radiation can also be expected .
The Hawking effect of quantum evaporation is often discussed in connection with thermodynymics, but it has its fundamentally independent significance , even if it is not related to thermodynamics. Its mechanism (discussed below) is important. It can serve as a connecting bridge between general theory of relativity as the physics of gravity and quantum physics - a "window" to quantum gravity.
Mechanism of quantum evaporation
The basic idea of quantum field theory in curved spacetime (semiquant approach) leading to spontaneous formation of radiation particles by quantum effects in an inhomogeneous gravitational field can be clearly summarized as follows. According to quantum field theory (see eg §1.1 " Atoms and atomic nuclei ", passage " Quantum field theory " in the monograph " Nuclear physics and physics of ionizing radiation ") , the vacuum is filled by fluctuating pairs of particles and antiparticles, which are constantly formed and then annihilated. (recombine). The most common such pairs are electron-positron pairs and photon pairs (a photon is its own antiparticle) . According to quantum uncertainity relations d E d t ³ H , a pair of particles with a mass m and energy mc2 is virtual, if their recombination lifetime does not exceed the Compton time ~ h / mc 2 .
In a strongly inhomogeneous gravitational field (eg black holes), the gravitational forces at the particle and antiparticle locations can vary considerably (large gravitational gradients - tidal forces) , so that due to the deviation of geodesics the two originally virtual particles can move significantly apart exceeding h/mc. In this case, recombination may no longer occur and the originally virtual pair becomes a true particle-antiparticle pair, with the corresponding energy being supplied by the gravitational field.
Let's go back for a moment to the effect of superradiation ( §4.4., Part " Penrose process ", passage " Superradiation "). If the quantum process described above releases an originally virtual pair in the ergosphere of a rotating black hole and one of the particles enters an orbit with negative energy with respect to infinity and is captured by the black hole, its partner may have sufficient energy to he could fly out of the ergosphere like a real particle and move away from a black hole. By this mechanism, the rotating black hole will spontaneously emit radiation, gradually carrying away its rotational energy. Surface horizon of a black hole with the same time in accordance with the classical 2.law dynamics of black holes increases . As we said in §4.4, rotating black holes are non-static and "alive", so the possibility of their quantum radiation is not so surprising. Hawking's discovery (so far it is a hypothesis) that even static Schwarzschild black hole emits radiation that leads to a gradual reduction ( "evaporation") of the mass of the black holes, however but was completely unexpected!
This is in clear contradiction with the second law of black hole mechanics (§4.6 " Laws of black hole dynamics ", sentence 4.7) , according to which the horizon of a black hole cannot be reduced. However, for the validity of this law (see §4.6) it is necessary to assume that the intrinsic energy density T oo cannot be negative. However, in the scales corresponding to quantum fluctuations, this assumption may no longer be met *) and thus the 2nd law of Black Hole Dynamics may be violated. Second law of dynamics black holes should therefore be generalized to apply to quantum evaporation; below we show (" Unification of thermodynamics and quantum evaporation of black holes ") that such a generalization also leads to its unification with the second law of thermodynamics.
*) In addition to the positive density of local energy for each observer, a "strong" energy condition also requires an energy flow of only a temporal nature (§2.6). In creating a virtual pair, two particles "materialize" with a spatial interval, so that they are locally a stream of spatial-type energy; this disrupts the energy condition necessary for the 2nd Law of Black Hole Dynamics.
The laws of quantum physics applied in curved spacetime around the horizon of a black hole provide three basic ways to imagine the mechanism of black hole radiation :
1. Absorption of one of the members of a virtual particle-antiparticle pair below the horizon and emission of the other member as real particles.
2. Quantum tunneling phenomenon allowing virtual particles from below the horizon to penetrate into outer space and become real.
3. Quantum theory of gravity (not yet developed).
Basic idea of the mechanism of the Hawking effect of quantum radiation (according to point 1.) is schematically shown in Fig. 4.23. In a strong inhomogeneous gravitational field with universal effects, a certain "polarization" of the vacuum of fluctuating virtual pairs of particles and antiparticles occurs around the black hole. In regions close to the horizon, there is a certain probability that one member of a pair of particles and particles will penetrate below the horizon and be absorbed, while the other member escapes into outer space like a real particle . In such a case, the negative energy needed to release the originally virtual pair is actually brought below the horizon . The particle-antiparticle relationship and vice versa are related to the inversion of time, so the opposite of the second law of black hole dynamics will apply to the given process, ie the horizon will shrink .
|Fig.4.23. Illustration of Hawking's mechanism of quantum black hole radiation by absorbing one part of a virtual particle-antiparticle pair and radiating the other part as a real emitted particle. The horizon gradually decreases.|
The result of quantum analysis (we will not provide technical details of quantum calculations here, see eg , , ) is the conclusion that a black hole will thus produce radiation whose total intensity (power) will be proportional to
|W ~ hc 2 / r g 2 .||(4.64)|
For a general
Kerr-Newman black hole with surface gravity k , the angular velocity term, W H and the electric potential
horizon F H
, is the probability of emission of
particles with the energy E (= h. W where
w is the frequency of the wave), the angular
momentum L (at a looking at the axis of rotation
of the black hole) and an electric charge q, equal to
e-2p(E-LWH-qFH)/h.k . G(E,L,q) ,
where G indicates the probability of absorbing a particle with energy E , the momentum L and the charge q through this black hole. The mean number of <N> particles radiated (as a wave of a quantum) with frequency w , momentum L and electric charge q, is then given by
d<N> = ------------------------------- dw.dL.dq ;
e-2p(E-LWH-qFH)/h.k ± 1
the "+" sign applies to fermions, the
"-" sign to bosons. The surface gravity k , the angular
velocity W H and the electric potential F H of the black hole horizon are given by
the relations (4.44-4.47 in §4.4). The absorption coefficient G is a slowly variable function w and depends on the type of particles or
radiation; is of the order of one (especially for wavelengths
significantly exceeding the size of the horizon given by M ).
For Kerr black hole, this radiation will have the
spectrum of thermal radiation *) of black-body with temperature
TH = k.h/k
= h(rg+-rg-)/8pkMrg+ ; for non-rotating uncharged
Schwarzschild black hole is TH = h/4pkrg (= hc3/8pM G @ 10-7M¤/M [°K]. Distant observer while this
spectrum observed through the "filter" formed
potentials of the field around the black hole.
*) By thermal radiation of temperature Tm we will next understand the radiation of an absolutely black body with temperature T m. It is radiation in a broader sense than conventional thermal radiation means; This radiation may comprise not only photons (electromagnetic waves), but also gravitons (gravitation waves), and at a sufficiently high temperature T m will be present also the particles with nonzero rest mass (electrons, protons, and the like.), together with the respective antiparticles.
It can be seen from Equation (4.65) that at the same w , the emission intensity of particles with a positive momentum L is higher than with a negative momentum, so quantum radiation carries away the rotational momentum of a black hole (similar to electrons and positrons they can carry away with an electric charge). If w <L. W H , the (for boson fields) denominator (4.65) will be negative; however, this inequality is a condition of superradiation, so G will also be negative . For black holes of larger masses (ie at 1 m low temperature T H ) then the quantum emission can noticeably occur only for radiation with w <L. W H , with intensity G , which is in accordance with the quantum analysis of superradiation performed by Starobinskı [ 234] and Unruh .
Quantum radiation of black holes consists mainly of photons (electromagnetic waves whose wavelength will be comparable to the size of the horizon, l ~ r g / 2); electrons, positrons, and other particles with a non-zero rest mass can only emit black minidires (see the " Quantum Evolution of a Black Hole " section below) .
The Hawking effect of quantum radiation is the
result of the application of quantum field theory in curved
spacetime. In general, there are problems in defining the states
of vacuum and the presence of a particle in a strong
gravitational field. For example, in the field of full extension
of Schwarzschild geometry (Kruskal's diagram in Fig.3.19,
describes an "eternal" black hole), the final result of
the flow of quantum radiation at infinity will be influenced by
the choice of initial vacuum and particle states on the horizon
of the past , ; by appropriate selection of these
states, it can be achieved that there will be no radiation.
Problems of this kind do not arise for a black hole created by
gravitational collapse in asymptotically planar spacetime,
because all particles here begin (in the past) and end (in the
future) their motion in areas where the influence of the field
can be neglected. For such asymptotically free particles, a
quantity of asymptotic energy can be used and only asymptotic
field shapes can be used in quantization. In this way Hawking
arrived at his result ; the final result is completely
independent of the details of the course of the collapse, so it
seems that the result will be valid even without the assumption
that the black hole was created by collapse (the
"eternal" black hole). Some other
methods of deriving quantum black hole radiation ,  do
show this without the assumption of gravitational collapse.
In connection with this, it is interesting to ask the question, where they actually arise particles forming that make up the quantum radiation of a black hole? There are basically two diametrically opposed possibilities. According to the first of them, the formation of particles is caused by a variable component of the gravitational field during a gravitational collapse , , ie they are formed by collapsing matter before the formation of the horizon. The collapsing mass remains out of the horizon for the outside observer, with zradiation takes its kinetic energy. The surface of the collapsing object (the area of the "horizon") is thus reduced. The resulting particles are then held by the gravitational field for various lengths of time near the horizon and gradually escape (to infinity). However, a serious problem arises here: the particles would have to be formed with ever-higher self-energy in order to compensate for the growing redshift on their way to infinity. According to this mechanism, quantum radiation would only occur in black holes caused by collapse, while "eternal" black holes would not emit quantum.
The second possibility on which Hawking was based (and which we also used above in the explanation) is that particles are constantly formed in pairs from a polarized vacuum in an inhomogeneous (even static) field outside the horizon. The energy going to infinity is compensated by the flow of negative energy of particles falling below the horizon. All physical arguments testify to the correctness of this second possibility , , from which follows directly the independence of radiation on individual properties and the course of previous collapse, as wel as the thermal shape of the spectrum. According to this, the "eternal" black hole will radiate quantum in the same way as a hole created by collapse; quantum effects thus mean that a truly "eternal" black hole cannot exist.
accelerating quantum radiation
From the point of view of quantum-gravitational analysis in GTR, quantum radiation is a more general dynamic effect , taking place in a vacuum not only in the presence of real gravity, but also inertial forces in non-inertial systems . In 1973-76 S. Fulling, P. Davies and especially W. Unruh  showed, that radiation with a thermal spectrum (such as radiation of an absolutely black body) will be generated in an accelerating moving reference frame in a vacuum . In an inertial system in a vacuum, no such radiation is manifested, while a accelerated-moving observer will register radiation in a vacuum with a thermal spectrum corresponding to the temperature
T a = h .a / (4 p c.k) @ 10 -19 .a [° K] ,
where a is the instantaneous acceleration, k is the Boltzman constant, h is Plank's constant (reduced) . It is analogous to the Hawking radiation of temperature T H = h .g / (4 p c.k) of a black hole, on the horizon of radius r g is the gravitational acceleration g .
In terms of the mechanism of quantum radiation, Hawking evaporation is due to the existence of the gravitational horizon of the black hole , while Unruh's radiation is related to the existence of the kinematic so-called Rindler's horizon in the accelerating frame of reference. This analogy fully corresponds to the principle of equivalence between gravity and the acceleration of a non-inertial system. The concept of Unruh radiation also shows that, from the point of view of the quantum concept, a vacuum filled with virtual fluctuating quantum fields appears in different states for different reference frames - depending on the state of motion of the observer.
Like Hawking's radiation of quantum evaporation of a black hole, Unruh's acceleration radiation is still completely hypothetical , without the possibility of experimental verification *) .
*) During high-energy collisions of particles in accelerators, very high values of acceleration occur, at which several quants of Unruh radiation could arise. However, it is quite impossible to distinguish this faint radiation from the huge background of very strong radiation (arising from particle interactions) ! It remains only as a theoretical interest ...
Nonlocality of quantum
Is important to reflect fundamental spatial nonlocality process quantum evaporation: the characteristic wavelength of the quantum radiation is comparable to the size of the black hole. Each quantum of radiation is formed in a spatial region of a size comparable to a black hole, so it is not possible to locate more precisely in space the location of this quantum. An observer falling into a black hole would not observe any radiation locally at the horizon...
The black hole shrinks during quantum evaporation not so much by the escape of positive energy particles from below the horizon (the horizon is impenetrable outwards), but as a result of bombardment and absorption of negative energy particles from surrounding area. Hawking's quantum mechanism evaporation is actually a process of "external liquidation" of a black hole by a polarized vacuum.
*) In quantum physics (in connection with quantum relations of uncertainty), however, the processes of " tunneling " particles from places below the horizon to places outside the horizon are possible - a particle that is just below the horizon has a non-zero probability of occurrence outside the horizon (for the nature of the tunneling phenomenon, see §1.1, passage " quantum tunneling phenomenon " in the monograph " Nuclear physics and physics of ionizing radiation ") . The originally virtual particle can then become a real particle above the horizon, carrying the appropriate energy from the black hole. These different interpretations lead to the same observable consequences for the quantum evaporation of black holes.
Stochastic character of quantum radiation
The thermal shape of the spectrum, ie that the quantum radiation from a black hole is completely random and uncorrelated, is related to the " black hole has no hair " theorem . Quantum radiation comes from an region, about which the outside observer has no other information than knowledge of the total mass, charge, and angular momentum. Therefore, all configurations of emitted particles that have the same energy, charge, and momentum are equally likely. With regard to such beyond the horizons of "hidden" areas (hypersurfaces), about which the external observer has only limited available information, S. Hawking expressed the so-called "principle of ignorance" : All configurations in "hidden" areas, consistent with the limited principle observable information, are equally probable. It leads to a violation of the predictability of the evolution of these systems (discussed below the passage "Quantum evaporation: the return of materials and information from black holes ? ") .
The black hole will therefore radiate with equal probability *) any configuration of the particles, compatible with the conservation of energy, momentum and charge. In principle, therefore, it is possible for a black hole to emit a material formation of any structure. Hawking states  that a black hole could quantumly emit, for example, even a television or even humans, but all such complicated and "exotic" modes of radiation have very neglible chance (and will never materialize). By far the most likely is the emission of particles with a thermal spectrum .
*) However, not every such particle or set of particles may come to infinity with the same probability, because when interacting with the field of a black hole, there is a certain potential barrier for the emitted particles, on which some particles can "bounce" back into the black hole. Few of the particles released at the horizon or tunneling below the horizon have enough energy to overcome the gravity of the black hole and escape to infinity (escape velocity directed vertically upwards). Most of these particles are then captured by the gravitational field and re-absorbed by the black hole. We can therefore imagine that a black hole has a kind of " quantum-particle atmosphere " just around its horizon (this could only be registered by a hypothetical observer "descending" slowly to the horizon according to Fig. 4.22; a free-falling observer would not observe it due to the principle of equivalence) . However, in the case of macroscopic black holes, such an "atmosphere" is in any case negligibly sparse, it can only occur significantly in the case of black micro-holes (see below " Quantum explosion of a black micro-hole ") .
Some possibilities of quantum evaporation of black holes are further discussed below in the section "Quantum evaporation: mass return and information from a black hole?".
Unification of thermodynamics and
quantum evaporation of black holes
We said that at in quantum radiation is violated the second law of dynamics of black holes. Conversely, in the presence of a black hole, the second thermodynamic theorem is violated - a black hole (for which entropy is not defined in the classical case) absorbs matter and thus draws entropy from the surroundings. However, the absorption of matter is always accompanied by an increase in the surface of the black hole and quantum radiation in turn by a decrease in the surface of the horizon. Offers therefore "save" both the 2nd theorem of thermodynamics and the 2nd law of dynamics of black holes: to combine the entropy of matter outside the black hole Sm with the size of the surface A of the black hole and create from them a quantity S, which could never decrease with time. Thermodynamic analysis (performed by Bekenstein before the discovery of the Hawking effect) shows that such a quantity should be the sum of entropy outside the horizon and a certain multiple of the area of the horizon: S = S + a.A, where a is a constant. Since we consider k / 2p as a temperature, the 1st law of black hole mechanics can be considered as the 1st theorem thermodynamic for a black hole, it can be integrated, which gives a .A = S H = (k / 8p h) .A + const., where however const. = 0, because we require that the entropy of a black hole with a mass approaching zero also go to zero. We can therefore make a statement that generalizes the second law of black hole dynamics and at the same time the second thermodynamic theorem :
|Theorem 4.10 ("generalized 2nd law of black hole dynamics")|
|The sum (k / 8 p h ) -times of the horizon area A of the black hole and the entropy Sm of the mass outside the black hole cannot decrease with time.|
(in the system of units, where h = k = 1 is a = 1/4). Bekenstein  postulated this generalized law even before Hawking's discovery of quantum evaporation. However, without quantum radiation, generalized Second Law 4.10 could be violated in certain situations, such as when a black hole is in a field of thermal radiation below the "temperature" of a black hole; then, when the thermal radiation accretes, the entropy of the black hole S H increases by a smaller value than the entropy of the surrounding thermal radiation decreases (and "heat" passes from the colder environment to the warmer black hole). Quantum evaporation is therefore a necessary prerequisite for the validity of the generalized 2nd Law of Black Hole Dynanics. With the existence of quantum radiation, the thermodynamics of black holes becomes completely consistent - a black hole can be in equilibrium with thermal radiation (see the stability of such an equilibrium, however, below). A black hole behaves like any any other body heated to a certain temperature TH and having a certain entropy SH , which, when on interacting with the environment, governed by basic laws of thermodynamics. All four laws of black hole mechanics can be equivalently formulated in thermodynamic terminology and then talk about the laws of black hole thermodynamics :
|0 . Law of thermodynamics of black holes:|
|Temperature T H = k . h / k of stationary black hole is the same in all places of the horizon.|
|1. Law of thermodynamics of black holes:|
|The law of conservation of total
energy ~ matter applies, where two close equilibrium
states of a black hole are connected by the relation
dMH = TH dSH + WH dJ + FH dQ .
|2nd law of thermodynamics of black holes:|
|The total entropy S = SH + Sm of the black hole SH and the mass outside the black hole Sm cannot decrease with time.|
|3rd law of thermodynamics of black holes:|
|With a finite number of steps, the black hole cannot be cooled to a temperature TH = 0 of absolute zero.|
Let's now look at the
properties of the interaction of a black hole with thermal
radiation. As the weight of the black hole increases, its
temperature decreases. If a black hole absorbs a certain amount
of thermal radiation, its weight will increase, causing its
temperature TH = h /4M to decrease. The black
hole therefore has a negative heat
(dM / dTH = -4 M2 / h) - by supplying energy to
the black hole, its temperature is reduced. Thermodynamically
speaking: by supplying heat (~ energy, ie heating), a black hole does not
heat up, but cools down! The more energy (heat) a black hole
absorbs, the colder it becomes, and the more energy it radiates,
the more it heats up.
Let us have a Schwarzschild black hole *) surrounded by radiation of an absolutely black body with temperature T m the same as the temperature of the black hole T m = T H . Due to quantum evaporation, there will be a thermal equilibrium between the absorbed and emitted radiation, and this balance will, of course, show statistical fluctuations.
*) Because, to be precise, only a non-rotating and electrically neutral black hole can be in equilibrium with thermal radiation. The charged and rotating black hole preferentially emits particles with an electric charge and a momentum of the same "sign" as the black hole, so that this radiation will not have exactly the Planck spectrum of thermal radiation.
Let the black hole absorb at a certain moment due to statistical fluctuation and slightly more heat radiation than it radiated. Because a black hole has a negative heat capacity, its temperature decreases, thereby reducing the intensity of quantum radiation; on the contrary, the intensity of radiation absorption increases somewhat. If the temperature of thermal radiation is kept constant (this will be the case if the whole universe is filled with this thermal radiation, or if the black hole is located inside a thermostat powered by a sufficiently large energy storage - Fig.4.24a), it will lead to increasingly significant uimbalance between thermal radiation accretion and quantum evaporation: the emission of a black hole will constantly slow down (the black hole will constantly cool) and the absorption of thermal radiation will constantly accelerate. The weight of the black hole will increase indefinitely. If, on the other hand, as a result of a statistical fluctuation, a black hole radiates a little more energy for a while than it has absorbed, its temperature (due to its negative heat capacity) will rise, thus radiating even more intensely and absorbing less; this imbalance process will increase avalanche. Finally, the black hole radiates out completely and disappears.
Fig.4.24. To the thermodynamic equilibrium between the quantum evaporation of a black hole and the surrounding thermal radiation.
a) Black Hole enclosed in a cabinet with heat radiation, the temperature is thermostatically maintained at a constant value T m = T H . If the energy storage is able to provide and receive an unlimited amount of energy (which models a situation where the whole universe is filled with thermal radiation of temperature Tm ), the thermodynamic balance between the black hole and the thermal radiation cannot be stable.
b) A stable thermal balance between radiation and a black hole is established only when the black hole interacts with an environment that is able to release and receive only a limited amount of energy (such as the thermally insulated box in the figure).
Thus, the negative heat capacity of a black hole means that the black hole cannot be in stable thermodynamic equilibrium with an environment that can emit or receive an unlimited amount of heat radiation; such a thermal balance is only unstable. A stable thermal equilibrium between a black hole and thermal radiation is possible only in the situation shown in Fig. 4.24b, where the black hole is enclosed in a thermally insulated box of final volume. In this case, a thermal equilibrium is always established, regardless of possibly the difference between the original temperature of the black hole and the thermal radiation in the cabinet. If, under this equilibrium, as a result of statistical fluctuations, the black hole warms up by emitting more energy than it absorbs, the temperature of the thermal radiation in the box increases, accretion prevails and the black hole cools down again. By analogy, by randomly absorbing more radiation, the black hole cools down, but the heat radiation in the box decreases and the temperature drops, quantum evaporation prevails, and the black hole warms up again.
Quantum evolution of a black hole
The effect on the radiant emission of black holes, as a result, somewhat changes the image we created in §4.2-4.6 - that the gravitational collapse creates a black hole, which is the absolute end product of the evolution of sufficiently massive stars. Quantum effects mean that even in empty space no black hole can be exactly stationary, but only quasi-stationary, because mass, momentum and charge gradually decrease as a result of quantum evaporation. Fig. 4.25 schematically shown the basic features thus supplemented idea of the final stages of evolution of a sufficiently massive star in empty asymptotically plane space-time. The gravitational collapse creates a black hole, which at first begins to evaporate extremely slowly, but with ever-increasing speed. The thickness of the barrier that the particles have to cross over the tunnel effect during quantum evaporation is proportional to the gravitational radius of the black hole. In the case of large black holes of stellar masses (with dimensions of the order of kilometers), quantum evaporation is negligible and in practice it is many times covered by the accretion of gases, particles and radiation from the surrounding space into a black hole *); here, however, we assume that the black hole is in a vacuum. Black hole temperature
T H = h / 4 p kM @ 10 -7 M ¤ / M [° K]
will be very close to
absolute zero for stellar mass collapsars. There may be a slight
emission of very weak electromagnetic waves of large wavelengths
*) The situation would be the opposite for small primordial black holes with dimensions of the order of 10 -13 cm, which will appear very "hot" and whose quantum radiation will already be very intense (of the order of 10 9 W), see §4.8 " Astrophysical significance of black holes ", the passage" Primordial Black Holes? " . In the case of an open expanding universe, the temperature of the relic radiation would far below the temperature of the black holes of stellar masses after a sufficiently long time, so that eventually all black holes could evaporate quantum (see §5.6 " The Future of the Universe. The Arrow of Time. Hidden Matter. ") .
Quantum explosion of a black microhole
With gradual evaporation (decreasing r g ) the intensity of radiation and energy of emitted particles is constantly increasing - a smaller black hole glows more - and not only photons but also electrons (+ positrons) and later heavy particles (protons, neutrons, ...) and antiparticles will be emitted. After reduction to microscopic dimensions, quantum evaporation eventually acquires the avalanche character of a kind of quantum black hole explosion . The black hole thus ends its existence with a massive explosion, during which energy of the order of 1023 J is released in the last about 0.1 second. The speed of the final phase of quantum evaporation (i.e. power of quantum explosion) depends on how many different types of elementary particles exists, because in the final stage the temperature of the black hole increases indefinitely and is capable of emitting all types of elementary particles. The above value corresponds to the spectrum of elementary particles as we know them now (particles composed of 6 types of quarks); if there were other elementary particles of higher masses, the final quantum explosion could be several orders of magnitude stronger.
Fig.4.25. Schematic representation of the space-time diagram of the collapse with the formation of a black hole, the gradual quantum evaporation of this black hole up to its quantum explosion and complete radiation - exhaustion and disappearance.
Basically, a black hole
has only two behaviors - depending on its size and the content of
the environment. Either it is an "all-destructive
cannibal" - that is, when
a black hole is sufficiently and its strong gravitational field
literally "sucks" all matter from
space and irreversibly absorbs every body that gets near her. Or, conversely, when
the black hole is very small, it will be a "self-destructive
dealer" - it will glow brightly until it radiates all its
mass and finally ceases
to exist .
Quantum evaporation: mass and information return from a black hole ?
The effect of quantum evaporation could lead to the idea that matter (and related information) absorbed by a black hole during collapse and accretion is only temporarily "trapped" in a black hole and eventually returns to its original space..?.. This straightforward view would be however misleading. In quantum evaporation, only the value of the gravitational mass - energy equivalent to the substance that formed the black hole is "returned" . The substance itself, absorbed by the black hole, irreversibly "melted" in the field - including all its atoms and particles, all its structure disappeared . What is emitted from the black hole during quantum evaporation is not the original absorbed particles, but completely different "new" particles and radiation, creating from the gravitational field gradient. Even by more careful analysis of particles emitted by a black hole during quantum evaporation, therefore, nothing can be found about objects previously absorbed by the black hole - it is not possible to " reconstruct " their composition and structure! Even if we "collected" all the photons, electrons and other particles and antiparticles emitted during the whole evaporation process and measured their energies and other characteristics in the imaginary experiment, it would not tell us anything about the matter from which the black hole was previously formed.
Thus, it cannot be expected that the quantum evaporation of a black hole is a reverse process to the formation of a black hole by collapse *), and that perhaps some individual characteristics of the object before collapse could reappear. All characteristics except weight, charge and rotational momentum, the black hole is irreversibly and permanently "forgoten" already when the horizon is created. For example, the probability of the emission of particles and antiparticles does not depend on whether the black hole originated from ordinary matter or antimatter.
*) That Hawking's process of quantum evaporation would be a time-reversed collapse cannot be expected anymore because completely different physical theories are responsible for both processes: classical GTR for collapse and quantum field theory (against a given background of curved spacetime) for quantum evaporation.
The process of collapse and subsequent quantum evaporation also completely violates the law of conservation of the byryon number (which is considered unshakable in nuclear physics!) . For most of the time, a black hole (which was formed mainly from baryons by its mass) emits almost exclusively photons and other particles with zero rest mass, thus radiating the vast majority of its mass. Electrons, positrons and other leptons will be emited by the black hole until it shrinks enough (and thus "warms up") , and only at the very end , will it begins to emit also baryons. However, the weight of the black hole is already very small at this time, so in principle it could emit only a very small percentage of baryons compared to the number of baryons from which it formed during collapse. And even then, the black hole will emit on average the same number of baryons and antibaryons...
The paradox of information loss ?
In §4.5 " Black hole has no hair " it was shown that when a black hole absorbs a matter-substance below its horizon, it absorbs and irreversibly bury all information about it (except only the total mass, charge and rotational momentum) . This individual information about the particles and structures of the absorbed matter is definitely " forgotten " for the outside world , it can no longer be ascertained in any way. It is not recognizable whether the black hole was formed by the gravitational collapse of hydrogen, helium or iron, ordinary matter or antimatter, such as the representation of electrons, protons, neutrons, various types of fields. In this context, it is sometimes referred to as the " paradox of information loss - from the point of view of the general theory of relativity was discussed in §4.5, passage" Loss of determinism? - An information paradox? Here we will briefly continue this discussion from the point of view of quantum physics (and event. the possibility of combining gravity with quantum physics) :
Determinism in quantum physics is applied at the level of wave functions whose future state is determined by quantum operators that are time inverse; even the past state of wave functions is retrospectively determined - wave functions are unambiguous , unitary , it is assumed here that information should be preserved. This " law of information conservation " in quantum physics (its Copenhagen interpretation) , however, is in fact fulfilled only at the most basic model level: when we have the wave function of a particle without any interaction with the "surrounding world". Then we actually have a particle with a permanently described wave function y (t, x, y, z), whose quadrate I y I 2 indicates the probability density of finding a particle at a given place (x, y, z) at time t . The probability of finding a particle in a certain volume V , is given by the integral of VnIyI2dxdzdy through the volume V . And when we add (integrate) valuesI y I 2 across the whole space, we should get the number " 1 ", ie the certainty that the particle "is somewhere". That is the principle of unitarity ; no information is lost during the existence of the particle. However, in reality, during the formation and extinction of particles, information is constantly lost countless times...
However, when an interaction occurs, or when the position or momentum of a particle is measured, the wave function "collapses". And in quantum field theory, the formation of particles and their extinction commonly occurs, ie the formation and extinction of the respective wave functions. So "information" about the existence or non-existence of particles is constantly appearing and losing ......
Physics of black holes in GTR shows loss of information about all structural details of matter collapsed into a black hole, except weight, el. charge and rotational momentum (§4.5 " Black hole has no hair ") . Some quantum physicists see this as a fundamental contradiction (violation of the principle of unitarity of wave functions) .
One of the proposed solutions could be the hypothesis of quantum perturbations-fluctuations of the horizon : any particle that is absorbed by a black hole would leave a certain quantum "trace" on the horizon ... So a kind of micro-deformed " wrinkled-hairy"horizon ..? .. Hawking radiation of quantum evaporation of a black hole could then carry information from these quantum deformations of the horizon ..? .. But the problem is the instability of these hypothetical quantum micro-deformations of the horizon ..? .. Both at the quantum level and also from the point of view of the general theory of relativity, where all gravitational and electromagnetic asymmetries radiate in the form of gravitational and electromagnetic waves - and thus smooth out ...
From a factual scientific point of view, this " information paradox " is perhaps only an insignificant detail (its astrophysical significance is probably zero...). However, it may be one of the theoretical obstacles in some approaches to unifying gravity with quantum physics..?..
Objective analysis thus shows that in strongly curved spacetime with event horizons the law of conservation of information generally does not apply (similar to the above-discussed law of conservation of baryon number) ! This seemingly padadox loss of information is not, in fact, a contradiction , but a physical reality: the law of space-time "rules" even informations !
of determinism ?
The space-time event of complete evaporation and disappearance of a black hole (quantum explosion) is actually a naked singularity that appears for a moment, so that quantum evaporation leads to a violation of the principle of cosmic censorship discussed in §3.9. In fact, there is no need to wait for the complete evaporation of a black hole, because, as Hawking  has shown, the principle of cosmic censorship is in some sense violated during even in "peaceful" quantum radiation. This is because the emitted particles had an external observer can be considered as particles originating from a singularity, that tunnel across the horizon along the trajectory of the spatial type. In any case, we are affected by quantum radiation from areas, about which we have no available informations (except mass, charge and momentum). The effect of quantum radiation thus reduce "shielding ability" of the horizon agains the singularity and thus leads to a fundamental violation of the determinism of the evolution of spacetime . In §4.5, we have shown that the " black hole has no hair " theorem limits our ability to reconstruct the past, because a black hole irretrievably "buries" all information about the structure of matter from which it originated. Quantum radiation deprives us but also the possibility of predicting the future, because quantum radiation emanating from a black holes (and affecting physical events in the future), we are only able to determine the probabilities of individual radiated configurations. The uncertainty and "randomness" emanating from quantum mechanics is combined here with the classical uncertainty caused by causal relationships in strongly curved spacetime - the horizon of a black hole and its property " not to have hair ".
Albert Einstein, who never internally accepted quantum mechanics (despite his significant contribution to its construction), cited the metaphorical statement " I don't believe God plays dice with the world " * as the main argument against its indeterminism. S.Hawking, on the other hand, characterized quantum processes in black holes with the words: "God not only plays dice with the world, but sometimes throws them even where they are not visible" .
*) The game of dice here symbolizes complete chanceand the unpredictability of what number will fall in each litter. Einstein was convinced of the causal and deterministic nature of the laws of physics, whose "creator and designer" is symbolically God. He did not believe that the behavior of individual microparticles is in principle random and "chaotic" and only with a large number of interactions will a certain exact regularity "break out" from different probabilities. Hawking, on the other hand, believed in the principled nature of stochastic quantum laws (the "dice game") and, as a quantum astrophysicist, emphasized the fact that some particles came below the horizon of a black hole where they were "not visible" from outer space. And the effect of quantum evaporation of a black hole by this "hidden stochasticity" can uncontrollably affect the surrounding universe..!? ..
possible to return information from a black hole ?
The paradox of information loss in the black hole discussed above arouses some displeasure among some experts, especially in the field of quantum physics; all information should be retained ! Recently, even its author S. Hawking himself got used to it and came up with a controversial questioning, which received wide publicity - there were articles with sensational headlines such as "Hawking returns from a black hole".!. :
Allegorical sci-fi depiction of the (non-)possibility of returning information during quantum radiation of a black hole.
Imagine that S.Hawhing decides to "jump" into a black hole. During his fall, his body is "spaghetti" near the black hole and then atomized, all structures are destroyed. In this state, it is absorbed by a black hole (part of its mass is radiated by gravitational waves) .
The question arises whether S.Hawking, resp. information about it, they find in that black hole (perhaps in the form of quantum fluctuations of the horizon) ? And whether, based on the analysis of all quantum particles of Hawking radiation during the process of quantum evaporation and black hole explosion (in an unimaginably long period of about 1080 years) would S.Hawking perhaps be "reconstructed" - a kind of his "return from the black hole"..?..
What is it about from a sober physical point of view? It is an idea (or a wish?), that all information (except mass, charge, and momentum - which is retained) about matter that collapses and accretes to create a black hole may not be irretrievably lost , but some may be re-radiated back in the Hawking's process of quantum evaporation..?.. But by what mechanism could this information escape from a black hole? Two variants in particular were discussed :
S.Hawking recently pointed out, that quantum fluctuations could prevent
the creation of a well-defined event horizon, the "ability
to shield" against the information inside the black hole
would not be as perfect as foreseen in the framework of general
The whole thing the possibility of radiating information from a black hole (other than those related to the "black hole has no hair" theorem, ie mass, charge and momentum) is still completely unclear, it is only a hypothesis on which experts do not have a unified opinion. In light of the above discussion in the passage " Quantum Evaporation: The Return of Mass and Information from a Black Hole? ", this hypothesis is not very likely. Quantum supporters "Law of conservation of information" , even if they "stand on their ears", do not extract any such information from below the horizon of events..!..
Arguing " law of information conservation " is probably out of place here, because a complete gravitational collapse is so universal and everything affecting event, that even the usual notion of information (used in classical physics and standard quantum mechanics) loses its meaning here, becomes illusory... (here everything can be "different "- cf. the above-discussed violation of the law of conservation of the number of baryons, in nuclear physics unacceptable..!..) ...
With a bit of exaggeration, we can say that: The properties of spacetime boss also informations !
|Gravity, black holes and space-time physics :|
|Gravity in physics||General theory of relativity||Geometry and topology|
|Black holes||Relativistic cosmology||Unitary field theory|
|Anthropic principle or cosmic God|
|Nuclear physics and physics of ionizing radiation|
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