Evaporation of the black holes

See also: Black hole (homonymy)

The radiation of Hawking is the phenomenon according to which an observer looking at a Black hole can detect negligible a emanating Rayonnement of black body of the surface of this one. This phenomenon is also called, for obvious reasons, evaporation of the black holes . He was predicted by Stephen Hawking in 1975, and is regarded as one of his more important achievements.

The theoretical discovery of this phenomenon gave a justification to a branch of the study of the black holes called Thermodynamique of the black holes, developed little before the discovery of Hawking, and which suggested that it was to be possible to associate a temperature with a black hole. However, at the traditional level, it was shown that a black hole could not emit radiation (it is even, to some extent, the definition of a black hole). This paradox was solved by Stephen Hawking, which showed that effects of quantum origin was at the origin of such a phenomenon.

The radiation of Hawking proves extraordinarily weak for the black holes resulting from the stellar evolution and even weaker for the other black holes indirectly detected in the universe (intermediate black holes and black holes supermassifs), also its description is it impossible at present. It could be made possible by the existence of black holes of small size (microscopic). Such objects could be produced at the time of the Big Bang (one speaks about paramount black holes), to even be produced in particle accelerator within the framework of certain theories beyond the standard Modèle of the Physique of the particles.

There exists a kinematic analog with the phenomenon of radiation of Hawking, the Effet Unruh, name of the Canadian Physicien William Unruh, which predicted it in 1976. This one predicted that an observer which looked at a Miroir animated of a movement accelerated would have the impression that this one emits a thermal radiation, whose temperature is proportional to the acceleration of the mirror. In a a little different context, the Schwinger effect, which describes the creation of particles charged in a Electric field, can be seen like an electrostatic analog of the radiation of Hawking.

On the other hand, various processes resulting from traditional mechanics allowing to extract from the energy of a black hole, the Process of Penrose, or its undulatory analog, the Superradiance, are not to confuse with the Hawking effect. In the same way, the phenomena of matter ejection by the intermédaire of jet S resulting from a Disque of accretion surrounding the black hole does not have strictly anything to see with the phenomenon of evaporation of the black holes.

The radiation of Hawking

The Quantum theory of the fields (i.e. laws of the quantum Mechanical applied within the framework of the restricted Relativity) explains the existence of the fluctuations of the vacuum: pairs particle - Antiparticule is permanently generated by the vacuum. Effects of these fluctuations of the vacuum can be highlighted by varied phenomenon like the Effet Casimir in Physique of the particles or Déplacement of Lamb in the spectrum of the energy levels of a electron in a Atome of Hydrogène.

In a general way, these pairs of particle-antiparticles are destroyed at once, except if a physical phenomenon makes it possible to separate them from/to each other in a time lower than the typical lifespan of the pair. In the case of the Hawking effect, with the horizon of a black hole, the forces of tide generated by the gravitational Champ of the black hole are so intense that they can move away the particle from its antiparticle, before which they do not destroy. One is absorbed by the Black hole, while the other (the particle emitted ) moves away from there in an opposite direction. In a heuristic way, the energy of the pair particle antiparticle, measured by an observer located far from the black hole is negative, owing to the fact that the two particles are trapped in the Puits of potential of the black hole. In a diagrammatic way, it is possible that the distribution of energy within the pair particle antiparticle gives to the one of both an energy which would be regarded as positive by a distant observer, i.e. enabling him to escape from its gravitational field. In such a case, the absorption of the other particle can be seen like the absorption of a particle of negative energy, producing a reduction in its mass. The energy of the emitted particles increases with the temperature of the black hole. In lower part of a temperature limits, the emission is done only with particles of null mass like the photons or the Graviton S (and possibly the Neutrino S. With the top, the emission of all types of particles is possible, though this mode relates to only the any end of the evolution of the black holes. During largest with their existence, those radiate particles without mass (see below). of the radiation of Hawking ==

The quantum theory says that the particles come from an “empty” space located just outside the black hole and not inside this one. That implies that space “empties” would not be if empty that because, if it were it, all the fields - with knowing electromagnetic or gravitational - would be null. However, according to the Principle of uncertainty of Heisenberg, one cannot precisely know the value and the rate of variation in the time of a field (the more one knows some about one, the less one knows some about the other.) Then there should be a quantum fluctuation (minimal quantity of uncertainty) in the value of this field. One can compare these variations to pairs of particles (of light and gravity) which can be destroyed. These particles are known as virtual . Always according to the principle of uncertainty, in this “vacuum” could also exist pairs of particles/virtual matter antiparticles (electron S, Quark S) which would be destroyed too.

It is also said that this “empty” space is a minimum state of energy. However energy must be produced by something. As the pair particle/antiparticle must be cancelled, they must have energies of contrary signs. A particle which will have lost its partner (fallen in the black hole and become real particle or antiparticle) will be able either to fall into the black hole, or to flee the black hole if it has a positive energy, in which case, for an observer external with the black hole, a particle will have been emitted by this one.

The issue rate and the apparent temperature of a black hole will be all the more intense as the distance necessary, so that a virtual particle of negative energy becomes positive, will be short. The produced radiation, which has a positive energy, will be compensated inside a black hole by a current of particle of negative energy. However according to Einstein, energy is proportional to the mass (relation $\backslash \; Epsilon\; =\; m\; c^2$). Thus there is loss of mass well. --> Simplified interpretation but more exact

It is based on the implementation of a particular frame of reference, the coordinates of Rindler, which describe an observer in constant acceleration in a flat Espace-temps. These coordinates represent, with the first order, the space time existing in the vicinity of the horizon.

If one carries out an evaluation of the quantum states of the field of vacuum (thanks to the formalism of functional spaces of Fock ) to express them in the metric of Rindler, one realizes that those differ according to whether one moves in the direction of acceleration or the opposite direction. Contrary to what occurs in the Espace de Minkowski, where the fields corresponding to frequencies positive and negative and thus their integrals are compensated, in the case of metric of Rissner, it is isotropic appears a shift, an excess of positive frequency in the direction opposed to that of acceleration. This phenomenon is called radiation of Unruh , of the name of the physicist who has it for the first time highlighted in 1976.

In the case of a black hole, the bottom line, for an observer located ad infinitum, represents an emanating current of particles of the accelerated area (there is reciprocity of the phenomenon under the terms of the principle of Relativity). The Black hole rayon thus, according to a law similar to that of a black Body. Conversely, towards the black hole, there exists current preferential of frequency, therefore of energy, negative, which corresponds to a loss of mass.

As this shift of the vacuum is proportional to the intensity of acceleration on the horizon, and that the latter is in its turn inversely proportional to the mass of the black hole, it results from it that the more massive one black hole is, the less it radiates, and vice versa. The phenomenon of evaporation is thus cumulative. TO REMAKE -->

Formulas and orders of magnitude

A famous calculation having given rise to so that one calls the Thermodynamique black holes makes it possible to show that one can express the Masse M of a black hole according to his size (makes the Surface of it has of sound horizon), and other macroscopic parameters characterizing it, namely, for a black hole of the astrophysical type, his electric Charge Q and its kinetic Moment L . There thus exists a function of the form

$M\; =\; M\; (has,\; Q,\; L)$.

The quantity $\backslash \; partial\; M/\backslash \; partial\; A$ can be written in the form

$\backslash \; frac\; \{\backslash \; partial\; M\}\; \{\backslash \; partial\; has\}\; =\; \backslash \; frac\; \{1\}\; \{8\; \backslash \; pi\}\; \backslash \; frac\; \{1\}\; \{G\}\; \backslash \; kappa$,

where G is the Constante of Newton and κ is a quantity called Gravité of surface of the black hole, and who determines at which speed the gravitational Champ of a black hole diverges as one approaches the horizon. Calculations of Hawking on the evaporation of the black holes indicate that the Température T which can be to them associated is given by

$T\; =\; \backslash \; frac\; \{1\}\; \{k\_\; \{\backslash \; mathrm\; \{B\}\}\}\; \backslash \; frac\; \{\backslash \; hbar\; \backslash \; kappa\}\; \{2\; \backslash \; pi\; C\}$,

where K is the Boltzmann constant, C is the Speed of light and $\backslash \; hbar$ the reduced Constante of Planck. This justifies a priori the whole of calculations on the thermodynamics of the black holes: indeed, the differential of the mass according to the surface of other quantities is identified with the formula of the First principle of thermodynamics,

$\{\backslash \; rm\; D\}\; U\; =\; T\; \{\backslash \; rm\; D\}\; S\; +\dots$,

where the energy interns U is replaced in the case of the black holes by their mass (representing their total energy), and the Entropie S is, according to calculations of the thermodynamics of the black holes, proportional to surface has of their horizon. To return the whole of the coherent thermodynamics of the black holes, it had to be shown that the black holes could have a temperature proportional to the gravity of surface, which was carried out by Hawking.

Case of a black hole of Schwarzschild

In the case of a Black hole of Schwarzschild, one has a simple relation between the mass and the surface of the black hole: the Rayon of Schwarzschild R is written

$R\; =\; \backslash \; frac\; \{2\; G\; M\}\; \{c^2\}$,

and its surface

$A\; =\; 16\; \backslash \; pi\; \backslash \; frac\; \{G^2\; M^2\}\; \{c^4\}$.

The gravity of surface east thus

$\backslash \; kappa\; =\; \backslash \; frac\; \{1\}\; \{4\}\; \backslash \; frac\; \{c^4\}\; \{G\; M\}$,

and the temperature

$T\; =\; \backslash \; frac\; \{1\}\; \{8\; \backslash \; pi\; k\_\; \{\backslash \; mathrm\; \{B\}\}\}\; \backslash \; frac\; \{\backslash \; hbar\; c^3\}\; \{G\; M\}$.

The temperature is thus inversely proportional to the mass. This result is hardly surprising: the only scale length of the problem is the ray of Schwarzschild, proportional to the mass. The temperature of a radiation is determined by the typical energy of the particles (in fact of the photons), of frequency ν , according to a formula of the type $k\_\; \{\backslash \; mathrm\; \{B\}\}\; T\; \backslash \; simeq\; H\; \backslash \; naked\; \backslash \; simeq\; H\; C/\backslash \; lambda$. It is natural that the wavelength of the photons is determined by the scale length available, proportional to the mass, also the temperature is it naturally inversely proportional to the latter.

The preceding formula can be rewritten in units of Planck according to

$\backslash \; frac\; \{T\}\; \{T\_\; \{\backslash \; mathrm\; \{Pl\}\}\}\; =\; \backslash \; frac\; \{1\}\; \{8\; \backslash \; pi\}\; \backslash \; frac\; \{M\_\; \{\backslash \; mathrm\; \{Pl\}\}\}\; \{M\}$.

In a more interesting way, one can rewrite it in solar unit of Masse, which gives:

$\backslash \; frac\; \{T\}\; \{1\; \backslash ;\; \{\backslash \; mathrm\; \{K\}\}\}\; =\; 6,\; \backslash !\; 15\; \backslash \; times\; 10^\; \{-\; 8\}\; \backslash \; frac\; \{M\_\; \backslash \; odot\}\; \{M\}$.

Thus, a black hole of a solar Masse has a temperature of about the ten millionth of Kelvin. This temperature increases as the mass of the black hole decreases. There thus exists an effect of racing: the more the black hole radiates, the more it loses energy (its mass will decrease, see below), and the more it is hot. This is not without pointing out the stellar evolution, where a star is the seat of nuclear reactions increasingly fast as its evolution continues.

It should be noted that if this order of magnitude is correct in general, it is significantly erroneous in several cases. In particular, for a extreme Black hole, i.e. having a maximum value of the electric charge or kinetic moment, then one shows that the temperature of the black hole is strictly null. Such a result is the analog in thermodynamics of the black holes of the Third principle of the thermodynamics, which indicates that it is not possible to reach a state of null temperature (or minimal Entropie) by any physical process.

Evolution of a isolated black hole

Another consequence of the opposite dependence between temperature and mass is that a black hole cannot be in balance with a thermal bath: if the temperature of the bath is higher than that of the black hole, the black hole more will absorb radiation which he will not emit, and thus to increase his mass and will decrease his temperature. The variation in temperature between the black hole and the thermal bath thus will increase. It is the same if this time the temperature of the bath is lower than that of the black hole. This time, it is the temperature of the black hole who will increase and differ more and more from that of the bath.

At present, the whole universe bathes in a thermal radiation, the cosmological diffuse Fond. This radiation is at a temperature of 2,7 Kelvin S, which is thus higher than that of a stellar black hole. Such a black hole, even if it is completely insulated (not of Accrétion of matter of the interstellar Milieu or a companion), thus will absorb radiation. This phase will last until the temperature of the cosmological diffuse bottom sufficiently lowered because of Expansion of the universe. The duration of this phase can be calculated in an approximate way by using the parameters resulting from the standard Modèle of cosmology. At present, one attends a Accélération of the expansion of the universe, which results in the fact that the expansion seems to tend towards an exponential law, where the distance are multiplied by 2,7 in a time about the Temps of Hubble, that is to say 13,5 billion years. The temperature of the cosmological diffuse bottom decrease like the reverse of the dilation lengths. Thus, so that the temperature of the cosmological diffuse bottom reaches the value of 6,15× 10 -8   K, it is necessary to wait approximately 18 times of Hubble. Black holes less massive than of the stellar black holes could, them, being evaporating today. For that, one needs that their mass is lower than

$M\_m\; =\; \backslash \; frac\; \{6,\; \backslash !\; 15\; \backslash \; times\; 10^\; \{-\; 8\}\}\; \{2,\; \backslash !\; 7\}\; M\_\; \backslash \; odot\; \backslash \; simeq\; 4,\; \backslash !\; 5\; \backslash \; times\; 10^\; \{22\}\; \{\backslash \; mathrm\; \{kg\}\}$,

that is to say a mass ranging between those of Mercury and Pluto.

No known astrophysical process makes it possible to carry out black holes of also small mass, but it is possible that such objects were formed in the paramount Univers. Such paramount black holes could exist, and reveal their signature by the phenomenon of evaporation. Indeed, at the end of the evaporation, whereas the black hole only reaches a mass of a few billion tons, one black hole radiates at a temperature of about 10 11   K, is in the field of the Gamma rays, and could leave an observational signature in this field wavelength.

Time of evaporation

It is possible to estimate (with uncertainties, to see below) the time of evaporation of a black hole. Usually, the energy radiated by a spherical body whose temperature of surface east T and ray are R writes

$L\; =\; 4\; \backslash \; pi\; R^2\; \backslash \; sigma\; T^4$,

where σ is the Constante of Stefan-Boltzmann. The loss of energy of mass of a black hole is thus written in theory

$\backslash \; frac\; =\; -\; 4\; \backslash \; pi\; R^2\; \backslash \; T^4$ sigma.

For a black hole of Schwarzschild, this is written

$\backslash \; frac\; =\; -\; 4\; \backslash \; pi\; \backslash \; left\; (\backslash \; frac\; \{2\; G\; M\}\; \{c^2\}\; \backslash \; right)\; ^2\; \backslash \; sigma\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; hbar\; c^3\}\; \{8\; \backslash \; pi\; G\; M\}\; \backslash \; right)\; ^4$,

maybe, by replacing the constant of Stefan-Boltzmann by his value,

$\backslash \; dowry\; M\; =\; -\; \backslash \; frac\; \{1\}\; \{15360\; \backslash \; pi\}\; \backslash \; frac\; \{c^4\; \backslash \; hbar\}\; \{G^2\}\; \backslash \; frac\; \{1\}\; \{M^2\}$.

The time of evaporation of a black hole of mass M is thus written

$t\_e\; =\; 5120\; \backslash \; pi\; \backslash \; frac\; \{G^2\}\; \{c^4\; \backslash \; hbar\}\; M^3$.

In solar units of masses, one obtains

$\backslash \; frac\; \{t\_e\}\; \{1\; \backslash ;\; \{\backslash \; mathrm\; \{S\}\}\}\; =\; 6,\; \backslash !\; 6\; \backslash \; times\; 10^\; \{74\}\; \backslash \; left\; (\backslash \; frac\; \{M\}\; \{M\_\; \backslash \; odot\}\; \backslash \; right)\; ^3$.

Such a duration is approximately 10 57 time larger than the age of the universe, illustrating the fact that the evaporation of stellar black holes is completely negligible. On the other hand, of the objects of mass 10 19 time weaker than a solar mass, is about a billion tons, have a time of evaporation lower than the age of the universe. So of such objects were produced at the time of Big Bang, then their evaporation takes place today.

Limits of the calculation of the time of evaporation

If derivation above is overall correct, it comprises a certain number of approximations which make it inaccurate. In particular, the first equation (which gives the loss of energy according to the surface of emission) proves to be incorrect, because the quantity which should intervene is not surface, but the cross Section of a black hole. Usually, these two quantities are identical except for a factor 4 (surface is worth 4  π   R 2 and the cross section π   R 2 ). However, in the case of a black hole, the angular Taille of a black hole is taller of a factor $3\; \backslash \; sqrt\; \{3\}/2$ than than a naive calculation would give. Thus, for the surface of the black hole in the formula above, it would be necessary to add a factor 27/4.

Moreover, this type of calculation is made in the approximation of the geometrical Optique, where it is supposed that the Photon S can be comparable with specific particles, or in any case of very small size always in front of other dimensions of the problem. However in the case of the evaporation of the black holes, the Wavelength of the photons is of the same order as the physical size of the black hole. It would thus be advisable to be placed in calculations within the framework of physical optics, where would be taken into account the exact form of the wave front in the gravitational field of the black hole.

Lastly, it was supposed here that the black hole emits only Photon S. In practice, the phenomenon of evaporation relates to all the existing particles, in any case all those whose energy of mass is lower than the typical energy of the particles of the radiation. In practice, the black hole radiates also Graviton S, even of the Neutrino S (if their mass is sufficiently low) in addition to the photons. Towards the end of its life, when its temperature reaches the field of the Gigaélectronvolt, it can radiate Quark S, Muon S even of other particles for the time being unknown. However, these last stages do not relate to the any end of the evolution of the black hole. The first detailed calculations of the rates of evaporation one carried out by Gift Page in 1976.

With final, these effects are not judicious to affect the general result, but could correct it of a numerical factor which could be extremely different from 1, but it appears not very probable that it moves away in a disproportionate way of 1. Also, the fact that a stellar black hole of mass cannot evaporate in a time lower than the age of the Universe is an extremely robust result.

Black holes and information

See also: Theorem of baldness

The theorem No to hate (baldness) which states that only three macroscopic parameters define the state of a black hole poses a problem with the eyes of the quantum theory. If one sends in a black hole a unit known as pure of particles, i.e. a coherent beam (for example, a laser beam, a pair of Cooper), the return of this coherent energy is done in the form of an incoherent energy, a thermal radiation, a unit known as mixed . However, the functions of waves which describe these two types of units are different: in the case of the pure unit the functions of waves are added vectoriellement, in the case of a mixed unit , they are the squares of the modules of the functions of waves which are added. The transformation of a unit into another is not possible with the quantum direction, since it is not a question of a unit transformation (which preserves the standard of the function of wave).

From where problem. Stephen Hawking announced to have solved this paradox, but the details of its solution are not yet (02/2005) known.

See too

• Thermodynamic Entropy of the black holes

• of the black holes
• irreducible Mass
• black Body
• Effect Unruh
• Process of Penrose
• Superradiance

References

• , pages 412 to 414.

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