Application of the Non-Local Physics in the Theory of the Matter Movement in Black Hole

The theory of the matter movement in a black hole in the frame of non-local quantum hydrodynamics (NLQHD) is considered. The theory corresponds to the limit case when the matter density tends to infinity. From calculations follow that NLQHD equations for the black hole space have the traveling wave solutions. The domain of the solution existence is limited by the event horizon where gravity tends to infinity. The simple analytical particular cases and numerical calculations are delivered.


Introduction
The first ideas about the existence of cosmic objects which gravitation is so big that the escape velocity would be faster than the speed of light, were formulated in 1783 by English geologist named John Mitchell.In 1796, Pierre-Simon Laplace promoted the same idea in his book Exposition du système du Monde.In 1916 Albert Einstein introduced an explanation of gravity called general relativity.According to the general theory of relativity, a black hole is a region of space from which nothing, including light, can escape.It is the result of the denting of spacetime caused by a very compact mass.Around a black hole there is an undetectable surface which marks the point of no return, called an event horizon.It is called "black" because it absorbs all the light that hits it, reflecting nothing, just like a perfect black body in thermodynamics.Black holes possess a temperature (and therefore the internal energy) and emit Hawking radiation through slow dissipation by antiprotons.
In 1930, Subrahmanyan Chandrasekhar predicted that stars heavier than the sun could collapse when they ran out of hydrogen or other nuclear fuels to burn and die.In 1967, John Wheeler gave black holes the name "black hole" for the first time.Astronomers have identified numerous stellar black hole candidates, and have also found evidence of supermassive black holes at the center of every galaxy.In 1970, Stephen Hawking and Roger Penrose proved that black holes must exist.
Let us investigate the possibilities delivered by the unified generalized quantum hydrodynamics [1][2][3][4] for investigation of these problems.From position of nonlocal quantum hydrodynamics (NLQHD) the mentioned theory has two limit cases connected with the density  evolution: 1) The density 0   .From the physical point of view this case corresponds to the motion in the Big Bang regime.This regime is considered in my previous paper published in this issue [5]; 2) The density    .From the physical point of view this case corresponds to the matter motion in the Black Hole regime.
Here we intend to consider the second limit case on the basement of non-local physics which particular interpretation is the generalized Boltzmann physical kinetics.We need not to deliver here main ideas and deductions of the generalized Boltzmann physical kinetics and non-local physics.The fundamental methodic aspects of the mentioned theory are considered in [5].A rigorous description can be found, for example, in the monographs [3,4,6], see also [7][8][9][10][11].
Strict consideration leads to the following system of the generalized hydrodynamic equations (GHE) [3,4,10] written in the generalized Euler form: and continuity equation for mixture Energy equation for component Copyright © 2013 SciRes.JMP and after summation the generalized energy equation for mixture (please see Equation (1.6) below).
  are the forces of the non-magnetic origin, -magnetic induction,  -unit tensor, q α -charge of the α-component particle, q α -static pressure for αcomponent,   -internal energy for the particles of αcomponent, 0 -hydrodynamic velocity for mixture.For calculations in the self-consistent electro-magnetic field the system of non-local Maxwell equations should be added.v

Propagation of Plane Traveling Waves in Black Hole
Newtonian gravity propagates with the infinite speed.This conclusion is connected only with the description in the frame of local physics.Usual affirmation-general relativity (GR) reduces to Newtonian gravity in the weakfield, low-velocity limit.In literature you can find criticism of this affirmation because the conservation of angular momentum is implicit in the assumptions on which GR rests.Finite propagation speeds and conservation of angular momentum are incompatible in GR.Therefore, GR was forced to claim that gravity is not a force that propagates in any classical sense, and that aberration does not apply.But here I do not intend to join to this widely discussed topic using only unified non-local model.Let us apply generalized quantum hydrodynamic Equations (1.1)-(1.6)for investigation of the traveling wave propagation inside the black hole using non-stationary 1D Cartesian description.It means that consideration corresponds so to speak to "the black channel".
Call attention to the fact that Equations (1.1)-(1.6)contain two forces of gravitational origin, -the force acting on the unit volume of the space and -the force acting on the unit mass.As result we have from Equations (1.1)-(1.6): Nonlinear evolution Equations (2.1)-(2.6)contain forces F, g acting on space and masses including cross-term (see for example the last line in Equation (2.6)).The relation   F g comes into being only after the mass appearance as result of the Big Bang.
Let us introduce now the main mentioned before assumption leading to the theory of motion inside the black holes: the density    .Derivating the basic system of equation we should take into account two facts: 1) The density can tend to infinity by the arbitrary law; 2) The ratio of pressure to density defines the internal energy of the mass unit E p   and should be consid-ered as a dependent variable by    .
As result we have the following system of equations: x where is the velocity component along the u direction.Let us introduce the coordinate system moving along the positive direction of x -axis in 1D space with velocity equal to phase velocity of considering object 0

C u 
x Ct    . (2.10) Taking into account the De Broglie relation we should wait that the group velocity g u 2u

 
,t is equal 0 .In moving coordinate system all dependent hydrodynamic values are function of  .We investigate the possibility of the traveling wave formation.For this solution there is no explicit dependence on time for coordinate system moving with the phase velocity 0 .Write down the system of Equations (2.7)-(2.9) in the moving coordinate system using the relation Non-local equations are closed system of three differential equations with three dependent variables .In this case no needs to use the additional Poisson equation leading to the Newton gravitational description.
If the non-locality parameter  is equal to zero the mentioned system becomes unclosed.
Let us introduce the length scale 0  , the velocity scale 0 , time scale u , and scales for the gravitation acceleration and for the internal energy of the mass unit . Using these scales one obtains We need also an approximation for the non-local parameter  .Take this approximation in the form where H is dimensionless value.In the dimension form

.18)
It means that the nonlocal parameter is proportional to the kinematic velocity and inversely with square of the velocity.Relation (2.18) resembles the Heisenberg relation "time-energy".Remark now that (as follow from the numerical calculations) the choice of the non-local parameter in this case has the small influence on the results of modeling.

Results of Mathematical Modeling
Now we are ready to display the results of the mathematical modeling realized with the help of Maple (the versions Maple 9 or higher can be used).
The system of Equations (2.14)-(2.16)has the great possibilities of mathematical modeling as result of changing the parameter H and five Cauchy conditions describing the character features of initial perturbations which lead to the traveling wave formation.Maple program contains Maple's notations-for example the expression responds to   .We begin with investigation of the problem of principle significance-is it possible after a perturbation (defined by Cauchy conditions) to obtain the traveling wave as result of the self-organization?With this aim let us consider the initial perturbations: The following Maple notations in figures are used: u-velocity , E-energy , and g-acceleration g  .Explanations are placed under all following figures.The mentioned calculations are displayed in Figures 1-4.
All calculations are realized using the conditions (3.1) but by the different value of the H parameter, namely 0.001; 1; 1000 H  . Figure 1 reflects the evolution of the dependent values in the area of the event horizon in details.In all calculations the boundary of the transition area of events is limited by the condition (obtained as the self-consistent result of calculations) .
From (3.4), (3.5) follow and for chosen  approximation It means that for large or in the dimensional form where x ut . Taking into account (3.4), (3.6) one obtains (3.12)The influence of the tidal force on the object in the black channel can be calculated using (3.13), (3.14).From (3.13) follows which corresponds to the frontier barrier.For example for Cauchy conditions (3.1) ,

Discussion and Conclusion
As one can see during all investigation we needn't to use the theory Newtonian gravitation for solution of nonlinear non-local evolution equations (EE).In contrast with the local physics this approach in the frame of quantum non-local hydrodynamics leads to the closed mathematical description for the physical system under consideration.
If the density tends to infinity the matter evolution inside of "the black channel" (1D Cartesian model) is organizing in the form of the traveling waves.
Numerical modeling leads to appearance of the singularity on the left side of domain where the gravitational acceleration turns into infinity.This singularity corresponds to event horizon and the whole neighboring area of the strong gravitational variation can be named as the transition area of events, (see Figure 1).
x ut All calculations are realized for the case    , corresponding to the wave traveling along the positive direction of the x -axis.Obviously after the initial per- , .In the theory of Black Hole (BH) with the spherical symmetry it leads near the event horizon to the appearance of black body radiation which was predicted by Stephen Hawking.Hawking radiation reduces the mass and the energy of the black hole and is therefore also known as black hole evaporation.The structure of this radiation significantly depends on the topological features of BH.
Usually the appearance of the analogical picture in the left hand half-plane does not lead to information of the principal significance, but not for the case under consideration.
Really, after rotation the right half-plane picture by two domains (see Figures 1 and 2) create the joined domain with the width and minimums for and in the centre of the infinite square well.On the whole the configuration reminds the known quantum mechanical problem of the particle evolution in a box with the infinite potential barriers of the gravitational origin.It is well known that the solution of the analogical problem in the Schrödinger quantum mechanics leads to the discrete energetic levels.Quantum calculations of oscillators in the arbitrary potential fields can be found in [4].
Finally some words concern the following investigations.Numerical calculations, realized in the spherical coordinate system for the dependent variables ( -radius, -time) cannot change principal results of the shown calculations in the Cartesian coordinate system.
But some other effects (where the real form of the black hole is significant) obviously need in a 3D non-stationary calculation.

5 -
.14) After the penetration through the frontier barrier the external matter is moving in the black channel in the form of the traveling wave.In this 1D Cartesian model the gravitational acceleration decreases as    with the rise of the   0.distance and, on the contrary, the in-ternal energy of the mass unit increases as   .
15) g Relation (3.15) reflects the change  t in the tidal force acting at the time moment across the body element x  .This change tends to infinity if the point .18) g the  change of the tidal force acting at the time moment across the body element x t  has not the catastrophic character.