Journal of Modern Physics, 2013, 4, 42-49
http://dx.doi.org/10.4236/jmp.2013.47A1005 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Application of the Non-Local Physics in the Theory of the
Matter Movement in Black Hole
Boris V. Alexeev
Physics Department, Moscow Lomonosov State University of Fine Chemical Technologies, Moscow, Russia
Received April 15, 2013; revised May 17, 2013; accepted June 23, 2013
Copyright © 2013 Boris V. Alexeev. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The theory of the matter movement in a black hole in the frame of non-local quan tum hydrod ynamics (NLQHD ) is con-
sidered. The theory corresponds to the limit case when the matter density tends to in finity. 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 calcu-
lations are delivered.
Keywords: The Theory of Traveling Waves; Generalized Hydrodynamic Equations; Foundations of Quantum
Mechanics; Matter Movement in Black Hole
1. 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 Al-
bert 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 hori-
zon. 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 tem-
perature (and therefore the internal energy) and emit
Hawking radiation through slow dissipation by anti-
protons.
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 an d die. In
1967, John Wheeler gave black holes the name “black
hole” for the first time. Astronomers have identified nu-
merous 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 Pen-
rose proved that black holes must exist.
Let us investigate the possibilities delivered by the
unified generalized quantum hydrodynamics [1-4] for
investigation of these problems. From position of non-
local quantum hydrodynamics (NLQHD) the mentioned
theory has two limit cases con nected with the dens ity

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 int er pre ta-
tion 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 men-
tioned theory are considered in [5]. A rigorous descrip-
tion can be found, for example, in the monographs [3,4,
6], see also [7-11].
Strict consideration leads to the following system of
the generalized hydrodynamic equations (GHE) [3,4,10]
written in the generalized Euler form:

continuity equation for species
C
opyright © 2013 SciRes. JMP

B. V. ALEXEEV 43
  

1
00000
pq
tt t m

0
I,R

 


 
  

 
 

 
 
 

vvvvv F
rrr r









vB
(1.1)
and continuity equation for mixture
 

1
00 000
pq
tt tr

 
 

 
  

 
 

 
 
 


vvvvv F
rr r

0
I0.
m









vB
(1.2)
Momentum equation for species

 




11
0000 0
1
0000 0
00 00
I
pq
tt m
qpq
mt m
pt

 


 
 


 
 


 
 

 

0
t







 








v
r


 
 

 


 



 

 


vvvvFvBF
rr
vvvvFvBB
rr
vv vv
r


 

 
 
00 000
11 ,,
0000 00
I2I
st elst inel
pp
qq mJ mJ
mm
 

  


 


 






 



vv vvv
rr
F vvFvBvvvBvvv

I
dd.
p



 r
v

(1.3)
Generalized moment equation for mixture

 



11
0000 0
1
0000 0
00 0
I
pq
tt m
qpq
mt m
pt


 


 
 



 

 



 
 



0
t








 








v
r



 

 







 





vvvvFvBF
rr
vvvvFvBB
rr
vv vv
r


 

 
 
00000
11
0000 00
I2I
0
pp
qq
mm
 

 


 


 






 




vv vvv
rrr
F vvFvBvvvB
0
Ip




(1.4)
Energy equation for component

22 1
2 0000 0 00
222
00 0 0000000000
3315
22222 2
15151 7
22222 2
vv
pnpnv pn
tt
vpnvpnv p
t

 
 

 

 
 
 
  
 


 
 




 






vvvFv
r
vvvvvv vvvv
rr

 
 

 




2 2
11 11
20
0000 00
11 11
00000
1
2
513
222
pp vq
npvp
mm
q
nn p
mt
 
   


  


 






 



 



 



 

vvFvv FFFvB
vBFFv FvvvF
rr

2
0
0
0
5
22
pv
q
p
m
qn








vB
vB
22
,,
dd.
22
st elst inel
mv mv
JJ
 
 

 
 
 
 

vv




(1.5)

B. V. ALEXEEV 44
and after summation the generalized energy equation for
mixture (please see Equation (1.6) below).

Here

1

F
BI
are the forces of the non-magnetic origin,
—magnetic induction, —unit tensor, qα—charge of
the α—component particle, qα—static pressure for α—
component,


—internal energy for the particles of α—
component, 0—hydrodynamic velocity for mixture. For
calculat ions in the self- cons istent e lectro -mag netic f ield th e
system of non-local Maxwell equations should be added.
v
2. Propagation of Plane Traveling Waves in
Black Hole
Newtonian gravity propagates with the infinite speed.
This conclusion is co nnected only with the description in
the frame of local physics. Usual affirmation-general rela -
tivity (GR) reduces to Newtonian gravity in the weak-
field, low-velocity limit. In literature you can find criti-
cism of this affirmation because the conservation of an-
gular 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 Equ a-
tions (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 Equa-
tions (1.1 ) -(1.6):
F
g
(continuity equation)
 
0
0000
I0
,
tt
p
t

 
 
 













 



v
r
vvvvF
rrr
(2.1)
(continuity equation , 1D case)



0
2
000 0,
v
ttx
p
vvvF
xtxx

 
 
 









(2.2)







(momentum equation)






0000
0
000000 0
0000
II
2I I0
p
tt
t
pp
t
pp
 

 

 







 

 
 



 

 





 




 

 
vvvvF
rr
gv
r
vvvvvvv
rr
vvFvvF
rr


.
(2.3)
(momentum equation, 1D case)






2
000
0
22
00
3
00 0
32 0,
p
vvvF
ttxx
gv
tx
vp vp
xt
vpv Fv
x
 

 



 











 

 
 



 

 










(2.4)

22 1
2
00
00 0 00
22
000 00000
3315
222 222
151 5
222 2
vv
pnpnvp n
tt
vp nvpn
t



 




 
  
 
  

2
000
17
22
vp





 
 




 








vvvFv
r
vvvvvv
r

 
vv v
r
 
 
 






211
2
00 00000
2
11 11
20
00000
11
0
15
22
13 5
222 2
pp
pv np
m
vq q q
vpp n n
mmm
pqn



 
 




















 


vvvFvvF
F FvBvBvBFvF
v Fv
rr
 


00.




B
00
t
  



 

Fvv


(1.6)

B. V. ALEXEEV 45
(energy equation)
22 22
00 00 0000 0
2
22 2
00000 0000
33 1515
2222 2222
1715 13
2222 22
vv
pp vpvp
tt
p
vppvpv

 



 
 
 
 
  
 
 
 

  




 



vvFv vv
rr
vvvvFvvgFg
r
 
2
00 0
15
22
vp
t








vv

0000 0,
p
p
t







 

 



 



Fv gvvvF
rr

(2.5)
(energy equation, 1D case)





22333
00000000
2
22 2
00000
33525 5
25 20,
vpvpvpvFvvpvvpv
ttxxt
pp p
Fvv FFFvgvvF
xx txx
 




 







 

 
   
 

  
 
42
000
8vpv
x
 

(2.6)
Nonlinear evolution Equati ons (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 re-
lation

Fg comes into being only after the mass
appearance as result of the Big Bang.
Let us introduce now the main mentioned before as-
sumption 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 Ep

 and should be consid-
ered as a dependent variable by

.
As result we have the following system of equations:
20,
uuu uE
ug
tx xxtxx

 
 


 

 
 

(2.7)


223
21
32
0.
uuE u
uugg
ttxx x
uEuEu Eugu
xtx


 




 





 

 











(2.8)
 
22333422
33525 583
220,
uEuEuEuguuEuuEuuEu gu
ttxx tx
EuuE
g ug
txx

 
 
  
 
 

 
 
 
 

 

 
52
2Eg gu
xx

  







(2.9)
x
where is the velocity component along the u direc-
tion. 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
Cu
x
Ct

 . (2.10)
Taking into account the De Broglie relation we should
wait that the group velocity
g
u2u

,t
is equal 0. In mov-
ing coordinate system all dependent hydrodynamic val-
ues 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 sys-
tem of Equations (2.7)-(2.9) in the moving coordinate
system using the relation
u
x
ut :

(continuity equation, 1D case)
2
0,
uu E
g 


  



 

 

(2.11)
(momentum equation, 1D case)
35 30
EuuEu
gg E  

  
,

 



 
(2.12)
(energy equation, 1D case)

B. V. ALEXEEV 46

2
2510
11 105
20,
EuuE
ugE u
uE
EE
E
gg
 

6
6
u
uE
u
Eggu




 

 
 



 





 
 





 






,, uEg
(2.13)
Non-local equations are closed system of three differ-
ential equations with three dependent variables .
In this case no needs to use the additional Poisson equa-
tion leading to the Newton gravitation a l 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 u000
x
u

, and scales for the
gravitation acceleration 2
0000 0
g
uux  
2
00
Eu and for the internal energy of the mass unit . Using these
scales one obtains
2
uu


 






 0,
E
g














 (2.14)
35 3
EuuE
gg E 



 




 
 
0,
u


 







(2.15)

2
2510
11 105
20,
EuuE
ugE uu
uE
EE
E
gg



6
6
u
E
u
Eggu








 




 
 



 









 






 



















(2.16)
We need also an approximation for the non-local pa-
rameter

. Take this approximation in the fo rm
2
H
u  

, (2.17)
where H is dimensionless value. In the dimension form
00 2
H
ux u  . (2.18)
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 rela-
tion “time-energy”. Remark now that (as follow from the
numerical calculations) the choice of the non-local pa-
rameter in this case has the small influence on the results
of modeling.
3. Results of Mathematical Modeling
Now we are ready to display the results of the mathe-
matical modeling realized with the help of Maple (the
versions Maple 9 or hi g her can be used).
The system of Equations (2.14)-(2.16) has the great
possibilities of mathematical modeling as result of chang-
ing the parameter
H
and five Cauchy conditions de-
scribing the character features of initial perturbations
which lead to the traveling wave formation. Maple pro-
gram contains Maple’s notations—for example the ex-
pression




00Du

means in the usual notations



00u 



t, independent variable responds to

.
We begin with investigation of the problem of pr in ci ple
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 con-
sider the initial perturbations:











0 1, 0 1, 0 1, 00,
01.
uEg Du
DE

 

u
E

(3.1)
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;1000H

. Figure 1 reflects the evolution of the
dependent values in the area of the event horizon in de-
tails.
Figure 1. u—velocity (dotted line), H = 1, E—energy u

E
 (solid line), and g—acceleration
g
 (dashed line), area
of event horizon.

B. V. ALEXEEV 47
Figure 2. u—velocity (dotted line), H = 1, E—energy u

E
 (solid line), and g—acceleration
g
 (dashed line).
Figure 3. u—velocity (dotted line), H = 1000, E—energy u

E
 (solid line), and g—acceleration
g

0.5  

(dashed line).
In all calculations the boundary of the transition area
of events is limited by the condition (obtained as the
self-consistent result of calculations) .
lim
As follow from calculations (see Figures 1-4) the
variation of
H
-parameter has the weak influence on the
numerical results. Let us show also the results obtained
for (see Figure 5) and the corresponding
numerical results near singularity ; namely:
0.0001H0.5
 
li m
Figure 4. u—velocity (dotted line), H = 0.001, E—energy u

E
 (solid line), and g—acceleration
g
 (dashed line).
Figure 5. u—velocity (dotted line), H = 0.0001, E—en-
ergy
u

E
 (solid line), and g—acceleration
g

0.4999999  

3
0.382 10E 
2615.014g

1u
(dashed line).
H = 0.0001; . We have the following
results of calculations ; ;


1.u
.
As we see the self-consistent solutions lead with the
high accuracy to the relation
(3.2)


Let us use this condition for analytical transfor mations
of the Equations (2.1 4)-(2.1 6 ) . We have corresponding l y

B. V. ALEXEEV 48
0,
Eg
















 (3.3)
0,
Eg





 (3.4)

20 E g 





 

E
E 









 . (3.5)
From (3.4), (3.5 ) follow
0
E
E 











, (3.6)
const
E
E






  (3.7)
and for chosen

 approximation
221
EC C




, CE
 , (3.8)

2
22
0
0
EE







E






.
(3.9)
It means that for large



0
E

20EE












 (3.10)
or in the dimensional form

0
E

20EE








 (3.11)
where
x
ut



const00EgE g



. Taking into account (3.4), (3.6) one
obtains
(3.12)



22
2
00
00
ln
01
Eg
EE
E

0gg














 




(3.13)
and for large




0
01 E
E

0
2
gg












0.5

 . (3.14)
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


0.5
-distance and, on the contrary, the in-
ternal energy of the mass unit increases as

.
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
 

2
3/2
2
02
0
0
d0 d
0
E
E
gg
EE












 














.(3.15)
g
Relation (3.15) reflects the change 
t
in the tidal
force acting at the time moment across the body ele-
ment
x

. This change tends to infinity if the point of
singularity
1
2
0
ln
s
E
















0.5
s
 

(3.16)
which corresponds to the frontier barrier. For example
for Cauchy condition s (3.1) ,
32
1
21
g


 





. (3.17)

g
In this case the change

in the tidal force acting at
the time moment across the body element t
x

turns
into infinity by 0.5 


. In the following if

22
0
00
EE













 (3.18)
g
the

change of the tidal force acting at the time mo-
ment across the body element
x
t has not the catas-
trophic character.
4. Discussion and Conclusion
As one can see during all investig ation we needn’t to use
the theory Newtonian gravitation for solution of nonlin-
ear 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 mathemati-
cal description for the physical system under considera-
tion.
If the density tends to infinity the matter evolution in-
side of “the black channel” (1D Cartesian model) is or-
ganizing in the form of the traveling waves.
Numerical modeling leads to appearance of the singu-
larity on the left side of domain where the gravitational
acceleration turns into infinity. This singularity corre-
sponds 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-

B. V. ALEXEEV
opyright © 2013 SciRe JMP
49

Cs.
turbations the analogical wave p ropagates in the oppo site
directi on

 after the sign change
x
x uu
180
E

, .
In the theory of Black Hole (BH) with the spherical
symmetry it leads near the event horizon to the appear-
ance 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 consid-
eration.
Really, after rotation the right half-plane picture by
two domains (see Figures 1 and 2) create the jo i n ed
domain with the width and minimums for
and 1
 

g

r
t
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 solu tion of the an alogical
problem in the Schrödinger quantum mechanics leads to
the discrete energetic levels. Quantum calculations of
oscillators in the arbitrary potential fields can b e found in
[4].
Finally some words concern the following investiga-
tions. Numerical calculations, realized in the spherical
coordinate system for the dependent variables (—ra-
dius, —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. REFERENCES
[1] B. V. Alexeev, Journal of Modern Physics, Vol. 3, 2012,
pp. 1103-1122. doi:10.4236/jmp.2012.329145
[2] B. V. Alexeev, “Mathematical Kinetic s of Reacting Gases,”
Nauka, Moscow, 1982.
[3] B. V. Alexeev, “Generalized Boltzmann Physical Kinet-
ics,” Elsevier, Amsterdam, 2004.
[4] B. V. Alexeev, “Non-Local Physics. Non-Relativistic The-
ory,” Lambert Academic Press, Saarbrücken, 2011.
[5] B. V. Alexeev, Journal of Modern Physics, Vol. 4, 2013,
pp. 26-41.
[6] B. V. Alexeev and I. V. Ovchinnikova, “Non-Local Phys-
ics. Relativistic Theory,” Lambert Academic Press, Saar-
bracken, 2011.
[7] B. V. Alekseev, Physics-Uspekhi, Vol. 43, 2000, pp. 601-
629. doi:10.1070/PU2000v043n06ABEH000694
[8] B. V. Alekseev, Physics-Uspekhi, Vol. 46, No. 2, 2003,
pp. 139-167. doi:10.1070/PU2003v046n02ABEH001221
[9] B. V. Alexeev, Journal of Nanoelectronics and Optoelec-
tronics, Vol. 3, 2008, pp. 143-158.
doi:10.1166/jno.2008.207
[10] B. V. Alexeev, Journal of Nanoelectronics and Optoelec-
tronics, Vol. 3, 2008, pp. 316-328.
doi:10.1166/jno.2008.311
[11] B. V. Alexeev, Philosophical Transactions of the Royal
Society A, Vol. 349, 1994, pp. 417-443.
doi:10.1098/rsta.1994.0140