Black Holes — Information Models

Estimation of the volume of information in black holes is necessary for generation of restrictions for their formation, development and interconversion. Information is an integral part of the Universe. By its physical essence information is heterogeneity of matter and energy. The universal measure of physical heterogeneity of information is the Shannon information entropy. It is important to note that the Neumann entropy cannot be applied as the universal measure of heterogeneity because it is equal to zero for structured pure state. Therefore information is inseparably connected with matter and energy. The informatics laws of nature are: the basic law of Zeilinger’s quantum mechanics postulates that the elementary physical system (in particular, fundamental particles: quarks, leptons,...) bears one bit of information, the law of simplicity of complex systems, the law of uncertainty (information) conservation, the law of finiteness of complex systems characteristics, the law of necessary variety by W. Ashby, and the theorem of K. Gödel. The law of finiteness of complex systems characteristics and the principle of necessary variety by W. Ashby impose restrictions on the topology and symmetry of the universe. The author’s works testify about the practicality of information laws simultaneously with physical rules for cognition of the Universe. The results presented in this paper show the effectiveness of informational approach to studying the black holes. The article discusses the following questions: The volume of information in the black hole, Emission and absorption of usual substance by a black hole, Formation and development (changing) of black holes, Black hole merger. Black hole is called optimal if information content is minimal at the University region. Optimal black holes can exist when at least the two types of substance are available in the Universe: with non-linear and linear correspondence between information content and mass. Information content of optimal black hole is proportional to squared coefficient correlating information content with mass in usual substance and in inverse proportion to coefficient correlating information content with black hole mass. Concentration of mass in optimal black hole minimizes information content in the system “usual substance—black holes”. Minimal information content of the Universe consisting of optimal black holes only is twice as less as information content available of the Universe of the same mass filled with usual substance only. An information approach along with a physical one allows obtaining new, sometimes more general data in relation to data obtained on the ground of physical rules only.


Introduction
Estimation of the volume of information in cosmological objects, including stars of the Sun type, neutron stars, white dwarfs and black holes, is necessary for generation of restrictions for their formation, development and interconversion.Information is an integral part of the Universe.The basic law of Zeilinger's quantum mechanics postulates that the elementary physical system (in particular, fundamental particles: quark, electron, photon) bears one bit of information.By its physical essence information is heterogeneity of matter and energy.Therefore information is inseparably connected with matter and energy.An information approach along with a physical one allows obtaining new, sometimes more general data in relation to data obtained on the ground of physical rules only.Information is inextricably linked with matter and energy.The universal measure of physical heterogeneity of information is the Shannon information entropy.It is important to note that the Neumann entropy cannot be applied as the universal measure of heterogeneity because it is equal to zero for structured pure state.Use of the information approach, along with the physical, provides a new and sometimes more general information in relation to the information obtained on the basis of only physical laws.The author's works, testify about the practicality of information laws simultaneously with physical rules for cognition of the Universe.The estimates cited below are based on the foundational principle of Zeilinger quantum mechanics i.e. "an elementary system carries one bit of information" and prove it.The elementary systems are fundamental particles (quarks, leptons, photons).

The Informatics Laws of Nature
The informatics laws of nature are [1][2][3]: The main principle of quantum mechanics by A. Zeilinger: Elemental physical systems contain (carry) one bit of information.
The law of simplicity of complex systems.Such variant of complex system is realized, survives which possesses the minimum complexity.
The law of simplicity of complex systems is realized by nature in a number of constructive principles: "Occam Razor"; hierarchical modular construction of complex systems; symmetry; simmorfoz, stability; field interaction (interaction through the carrier or interactions through space-time status, for example, curvature of spacetime); extreme uncertainty (functions of characteristics distribution have extreme uncertainty).
The law of conservation of uncertainty (information).Uncertainty (information) of the isolated (closed) systems is saved at physically realized transformations and only at physically realized transformations.
The law of finiteness of information characteristics of complex systems.All kinds of interaction between systems, their parts and elements have final speed of distribution.Speed of change of system states of elements is limited also.
In any system of coordinates information on event is always final.Duration of signal is always more than zero .Information on coordinates of physical systems in our Univerce is limited by 333 bits.
The law of necessary variety by W. Ashby.For effective functioning of system a variety of operating body should no be less then variety of management object.
Uncertainty (information) is the basic characteristic of a variety of systems.The law of necessary variety by W. Ashby is also realized in a number of concrete principles: Shannon theorems, Kotelnikov theorem, Kholevo theorem, Brillouin theorem, theorem of Margolis-Levitin.
Gödel theorem of incompleteness.In enough rich theories (containing on arithmetic) there are always the unprovable true assertions.
The law of systems complexity growth.During systems evolution its uncertainty (systems information) grows.
Le Chatelier Principle.External influence discompos-ing system, calls in it the processes, aspiring to weaken results of this influence.

The Volume of Information in the Black Hole
The basic law of Zeilinger's quantum mechanics postulates that the elementary physical system (in particular, fundamental particles: quark, electron, photon) bears one bit of information [1].The volume of information in physical systems is estimated the information entropy [4,5].The information volume of a black hole of mass [6,7].The information volume contained in the black hole is proportional to its squared mass.How to explain it?Let us assume that a black hole contains pairwise interacting particles (q-bits).Then the quadratic dependence of the volume of information in the black hole on its mass can be explained by the fact that each interaction forms 1 bit of information.The black hole is described with the wave function As mentioned earlier the volume of information in the system described with the present given function is equal to   Let us compare the estimates of squared mass of a black hole: (squared mass of the particles composing the black hole).We have Eventually we obtain the estimate of mass of particles being part of the black holes: , that agree by its Therefore, a black hole is the aggregate of particles (let us call them black particles) each having a mass equal to 0.23th of Planck mass) and interacting with all other black particles that form a black hole.Characteristics and models used in the paper are taken from [9][10][11][12][13][14][15][16] and [13][14][15][16][17][18][19][20][21][22][23][24].

Emission and Absorption of Usual
Substance by a Black Hole

M m   m
. As far as each black hole has the sub-Planck mass 0 , then the change in energy of a black hole in such a case is equal to .In the event of the loss of emission of one black particle the mass of a black hole becomes equal to In this case the volume of information remaining in the black hole is Change (loss) of information in the black hole comes to bit.n For further estimates we implement the law of conservation of uncertainty (information) [2,3] and energy conservation principle.According to the law of conservation of uncertainty (information) a change in the system "a black hole with the mass 0 n M n m   n -external environment" on emission of one black particle must be balanced by the occurrence of particles containing 1 bit each:   m c Identical dependencies are true for cases when photons are absorbed by black holes.In virtue of the law of conservation of uncertainty (information), the changes in the system "a black hole with the mass -external environment", when the mass of a black hole is increased by the mass of one black particle, must be determined by absorption of n particles containing 1 bit each:  It is known that during supernova explosion the radiation energy is on the order of 10 50 erg, the maximum 10 54 erg.Whereas the average radiation frequency is 10 18 Hz, then the supernova explosion generates

 
ing that ≈100% radiation quanta is used for black hole formation, then the mass of a black hole would be equal to about 1 bit 2 for its formation are needed n n  particles of usual substance each of t (radiation quanta), hem containing 1 bit.For formation of a black hole of the mass equal to Thus, for formation of a black hole with the mass of t anta.
he Sun equal to

. (Columns 1, 2). Information characteristics of bl holes of different masses; (Columns 3, 4). Information (a)
The number of absorbed radiation quanta The number of sub Planck particles in the newly formed black hole .The Table 1 (columns 1 -4) present information characteristics of black holes of different masses.
From the above chart it follows that:  The mass of a black hole formed during the supernova explosion is close to the mass of an optimal black hole in the system "a black hole-radiation".One can expect that during the supernova explosions the black holes be formed having the masses under which the information volume in the adjacent space is close to minimum. For formation of the black holes with the mass equal to million masses of the Sun the volume of information exceeding the volume of information of the Universe (10 90 bit) is required [2,3,25].Significant volumes of information are needed for formation of the black holes with the same mass as of the Sun's.In ack characteristics of black holes of different masses.
(the volume of absorbed information)

Black Hole Merger
onsisting of the two black required.It means that locally (in the zone of black hole formation) there must take place the intensive physical processes of radiation formation.For instance, the supernova explosions and accelerated motion of relativistic particles.
Let us consider a system c holes.Suppose that the black holes have the mass of The mass of the sys to tem is equal ble that a blac Let us consider the following example.Suppose that the mass of the black holes is and they contain the volume of informat , the volume of information in the system is 1 2 I  Is it possible that a black hole is formed as f merging the two black holes of inimal mass and only the two given black hole al We give a system of equations for estimation of the mass of a black hole under formation when the two black holes are me fore merging possess the mass of We believe that the main energy of black holes and usual substance is concentrated in the mass.Then m the energy conservation principle it follows: 1) Thus we have a two-equations system or at (restriction on the mass of the black 2) Next we have The following assertions are valid.

Assert
, without using any additional usual substance, the mass of a newly formed black hole is less than .

During the merger of two black holes having the same masses
, the mass of a newly formed black hole is less than . The mass of a black rmed as a result of the merger of 2 black holes of the sa hole that was fo me masses without using any additional usual substances is 2 times rge less than the sum of masses of merging black holes.The remaining mass is dissipated in the space.
Assertion 5.During the me r of k black holes having the mass of M n m  without using any additional usual substance, the mass of a new formed black ho ly le is less than During the merger of black holes of the same masses n n taining

Optimal Black Holes
By using informatics approach o onstrate the existence of bl information content of arbitra and place lower bound limitations on information content of the Universe.The origin and cause of optimal black holes existence is the occurrence of substance of two different types: with square-law and linear-law dependence of information content on mass.In the presence of substance of only one type, optimal black holes do not exist.Following tasks are solved.The direct task: to discover an information minimum in system "usual substance-black a hole" at the given mass of usual substance and a black hole.The dual task: to discover a mass maximum in system "usual substance-black a hole" at the given information in usual substance and a black hole.We examine optimal black hole characteristics in the systems "radiation (photons)-black holes", "hydrogen (protons)-black holes" and in the system "several types of usual substance-black holes" [16][17][18][19][20][21][22][23][24].

Let us estimate the information content of region of mass
nd one


Let us find a condition for the minimum: .
We will discover an optimality requirement:  at the solution of a direct task (a minimi ion of information content in system "usual substance-a black hole" at the give .Information contents and the masses gained sat n mass of system-assertions 8, 9) and a dual task (a maximisation of mass of system "usual substance-a black hole" at the given information content) assertions 8.a, 9.a), coincide.Thereby concept of an optimum black hole is identical and all subsequent assertions and relationships also are identical.
As far as the black hole mass under which information content is minimal at the Universe region consisting of black hole and common substance does not depend neither on aggregated mass of the Universe (the Universe), nor on usual substance mass in the region under study, then the minimal information content of the Universe region (the Universe) can be reached if the Universe region (the Universe) consists of optimal black holes only.The maximum number of optimal black holes of the Universe region (the Universe) is equal to The minimal information level of the Universe region (the Universe) consisting of black holes on is equal to ly 2 2 4 2

Un bh
Opt bh Opt bh Un

Note 7.
Hereafter for brevity sake we'll speak about the Universe, though the assertions and expressio e also true for arbitrary regions of the Universe.ns ar Assertion 10.Minimal information content of the Universe consisting of optimal black holes only is twice as less as information content available of the Universe of u the same mass filled with sual substance only: Assertions 8 -10 are true for any kind of linear dependence of information volume on usual substance mass.The important agents of usual substanc radiation and hy e are drogen.Let's consider next the optimal black holes characteristics in the systems "radiation (photons)black holes", "hydrogen (protons)-black holes", "several types of usual substance-black holes".

Universe Filled with Radiation
Let's consider the Universe filled with usual substance (radiation).The energy required for transfer, retrieval, can record of one bit under the temperature T than the value ln 2 E kT  .In accor not be less dance with the min Einstein equation, the mass required for transfer, retrieval, record of one bit under the temperature T can not be less than the value , and for radiation in content of op ed in the system "black hole + radiation" is equal to formation timal black hole form , in th of optimal black hole minimizes informati the system "photons-black holes".
We note that the mass of optimal erged in the system "radiation-blac inverse proportion to radiation temperature.
The total number of optimal bla 9.09 10 g 9.09 10 kg. 4 The mass of black hole under which the information minimum is gained in the Universe of mass M c Un onsisting of radiation and black holes does not depend on the gross mass of the Universe and is equal to 9.09 × 10 22  timal black holes that emerged in the systems "radiation -black hole" is equal to the mass of optimal s that emerge "h black hole d in the systems ydrogen (protons)-black ho en atoms do not exist under high temperatures, then in such case the calculations have been done with respe y From informatics point of view, various types of usual le".By virtue of the fact that stable hydrog ct to protons.Note 9.In the period of transition from the Universe with predominant radiation to the Universe with predominant substance (10 4 > T > 10 3 ), the mass of optimal black hole in the system "radiation-black hole" changes from 2.45E + 19 kg to 2.45E + 20 kg.

The Systems Consisting of Black Holes and Several Types of Usual Substance
Let us consider the s stems consisting of black holes and several types of usual substance, for instance, of various kinds of particles.The mass of optimal black hole in the system "sereval type of usual substnce-black holes" under which information content of the system under consideration is getting minimized, is defined by the minimal factor of proportiona ity min The optimal black hole corresponds to the system "usual substance of type 0 i -black holes".Assertion 22.The mass of black hole under which information content is minimal, meaning information under which information content is minimal in the system "several type of usual substance-black holes", is equal to Assertion 23.Information content of optimal black hole in the system "several types of usual substanceblack holes" is proportional to squared minimal coefficient correlating information content with mass in different types of usual substance and inversely proportional to coefficient correlating in tion content with m forma ass in black hole: As the black hole mass under which information content is minimal in the system "several types of usual substance-black holes", does not depend on neither on the total mass of the system  M , M, nor on the mass of usual substance, then the minimum information content of the Universe is ga co ined if the system nsists ofoptimal black holes only.The maximum number of optimal black ho les in the system "several types of usual substanceblack holes" The minimum information content in the system "several types of usual substance-black holes", consisting of black holes only is equal to

Conclusions
in the black hole is square of the mass black hole, and n characteristics of black holes of di Assertion les rious types of atoms of usual substance or a mix of various types of atoms of usual substance masses of optimum black holes and volumes of the information in them are approximately identical 1) A black hole is the aggregate of particles (black particles) each having a mass equal to 0.23th of Planck mass) and interacting with all other black particles that form a black hole.
2) The volume of information proportional to the presents informatio fferent masses.
3) The temperature of the radiation of a black hole coincides up to a factor 2ln 2 0.51  with an estimate of 2.7

Note 4 .
In the general case, there must be absorbed photons having the frequencies In the general case, there must be absorbed particles with aggregated information equal to n bit and aggregated energy .
usual substance (radiati ntioned earlier, such a black hole contains

1 1 2
   .a result o the m s of the minim mass?The mass of a new black hole must be equal to the sum of masses of the original black holes of information in a new black hole must be equal to in a new black hole does not coincide with the volum aggregated volume o rged.Let us suppose that the black holes be the f information in the original black holes.


Un rgn I, it is proportional to a square of coefficient linking an inform ontent with mass in usual substance and inversely proportional to coefficient linking an inform content with mass in a black hole: Assertion 9.a.The mass of a black hole at which the mass of field of Universe containing the given information content maximum,

Assertion 11 .
Concentration of mass

2 ,
the minimal possible information content of the Universe of the Universe of mass Thus Un M consisting of radiation and black holes is proportional to the mass of the Universe, the speed of light squared, inversely proportional to Boltsman constant and the temperature of the Universe.Assertion 13.The Universe of mass Un M , consisting of radiation and black holes, containing of radiation and black holes is proportional to the energy of the Universe, inversely proportional to Boltsman constant the tempe te and rature of the Universe.It is notable that the minimal information con nt in the Universe does not depend neither on gravitation constant nor Plank constant 3 25 22

substance differ in coefficient i  standing for the use of per 1
the given type of usual substance.Assertion 21.

24 .
If optimum black ho are formed of va .

4 )
Black hole cannot be created by means of merging the two black holes and only two black holes.Merging of Open Access OJM

system consisting of the two black holes usual substance.
Let us consider a system consisting of the two black holes and usual substance.The bl Let us consider the Universe filled with usual substance (hydrogen).Wave function of proton with upward-di-It is important to note that the minimal information content of the Universe consisting of hydrogen atoms and black holes does not depend on gravity constant, the speed of light and Plank constant.