Journal of Modern Physics, 2013, 4, 26-41
http://dx.doi.org/10.4236/jmp.2013.47A1004 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Application of the Non-Local Physics in the Theory of
Gravitational Waves and Big Bang
Boris V. Alexeev
Physics Department, Moscow Lomonosov State University of Fine Chemical Technologies, Moscow, Russia
Email: Boris.Vlad.Alexeev@gmail.com
Received April 7, 2013; revised May 10, 2013; accepted June 15, 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 gravitational waves in the frame of non-local quantum hydrodynamics (NLQH) is considered. From cal-
culations follow that NLQH equations for “empty” space have the traveling wave solutions belonging in particular to
the soliton class. The possible influence and reaction of the background microwave radiation is taken into account.
These results lead to the principal correction of the inflation theory and serve as the explanation for the recent discovery
of the universe’s cosmic microwave background anomalies. The simple analytical particular cases and numerical calcu-
lations are delivered. Proposal for astronomers—to find in the center domain of the hefty cold spot the smallest hot spot
as the origin of the initial burst—Big Bang.
Keywords: Foundations of the Theory of Transport Processes; The Theory of Solitons; Generalized Hydrodynamic
Equations; Foundations of Quantum Mechanics; Traveling Wave Solutions; Big Bang
1. Introduction
More than ten years ago, the accelerated cosmological
expansion was discovered in direct astronomical obser-
vations at distances of a few billion light years, almost at
the edge of the observable Universe. This acceleration
should be explained because mutual attraction of cosmic
bodies is only capable of decelerating their scattering. It
means that we reach the revolutionary situation not only
in physics but also in the natural philosophy on the whole.
As result, new idea was introduced in physics about ex-
isting of a force with the opposite sign, which is called
universal antigravitation. Its physical source is called as
dark energy that manifests itself only because of postu-
lated property of providing antigravitation.
It was postulated that the source of antigravitation is
“dark matter” which inferred to exist from gravitational
effects on visible matter. However, from the other side
dark matter is undetectable by emitted or scattered elec-
tromagnetic radiation. It means that new essences—dark
matter, dark energy—were introduced in physics only
with the aim to account for discrepancies between meas-
urements of the mass of galaxies, clusters of galaxies and
the entire universe made through dynamical and general
relativistic means, measurements based on the mass of
the visible “luminous” matter. It could be reasonable if
we are speaking about small corrections to the system of
knowledge achieved by mankind to the time we are liv-
ing. But mentioned above discrepancies lead to affirma-
tion, that dark matter constitutes 80% of the matter in the
universe, while ordinary matter makes up only 20%.
Practically we are in front of the new challenge since
Newton’s Mathematical Principles of Natural Philoso-
phy was published.
Dark matter was postulated by Swiss astrophysicist
Fritz Zwicky [1,2] of the California Institute of Tech-
nology in 1933. He applied the virial theorem to the
Coma cluster of galaxies and obtained evidence of un-
seen mass. Zwicky estimated the cluster’s total mass
based on the motions of galaxies near its edge and com-
pared that estimate to one based on the number of galax-
ies and total brightness of the cluster. He found that there
was about 400 times more estimated mass than was visu-
ally observable. The gravity of the visible galaxies in the
cluster would be far too small for such fast orbits, so
something extra was required. This is known as the
“missing mass problem”. Based on these conclusions,
Zwicky inferred that there must be some non-visible
form of matter, which would provide enough of the mass,
and gravity to hold the cluster together.
Observations have indicated the presence of dark mat-
ter in the universe, including the rotational speeds of
galaxies, gravitational lensing of background objects by
C
opyright © 2013 SciRes. JMP
B. V. ALEXEEV 27
galaxy clusters such as the Bullet Cluster, and the tem-
perature distribution of hot gas in galaxies and clusters of
galaxies.
The work by Vera Rubin (see for example [3,4]) re-
vealed distant galaxies rotating so fast that they should
fly apart. Outer stars rotated at essentially the same rate
as inner ones (~254 km/s). This is in marked contrast to
the solar system where planets orbit the sun with veloci-
ties that decrease as their distance from the centre in-
creases. By the early 1970s, flat rotation curves were
routinely detected. It was not until the late 1970s, however,
that the community was convinced of the need for dark
matter halos around spiral galaxies. The mathematical
modeling (based on Newtonian mechanics and local phy-
sics) of the rotation curves of spiral galaxies was realized
for the various visible components of a galaxy (the bulge,
thin disk, and thick disk). These models were unable to
predict the flatness of the observed rotation curve beyond
the stellar disk. The inescapable conclusion, assuming
that Newton’s law of gravity (and the local physics de-
scription) holds on cosmological scales, that the visible
galaxy was embedded in a much larger dark matter (DM)
halo, which contributes roughly 50% - 90% of the total
mass of a galaxy. As result other models of gravitation
were involved in consideration—from “improved” New-
tonian laws (such as modified Newtonian dynamics and
tensor-vector-scalar gravity [5]) to the Einstein’s theory
based on the cosmological constant [6]. Einstein intro-
duced this term as a mechanism to obtain a stable solu-
tion of the gravitational field equation that would lead to
a static universe.
Computer simulations with taking into account the
hypothetical DM in the local hydrodynamic description
include usual moment equations plus Poisson equation
with different approximations for the density of DM
DM containing several free parameters. Computer si-
mulations of cold dark matter (CDM) predict that CDM
particles ought to coalesce to peak densities in galactic
cores. However, the observational evidence of star dy-
namics at inner galactic radii of many galaxies, including
our own Milky Way, indicates that these galactic cores
are entirely devoid of CDM. No valid mechanism has
been demonstrated to account for how galactic cores are
swept clean of CDM. This is known as the “cuspy halo
problem”. As result, the restricted area of CDM influence
introduced in the theory. As we can see that the concept
of DM leads to many additional problems.

I do not intend to review the different speculations
based on the principles of local physics. I see another
problem. It is the problem of Oversimplification—but
not “trivial” simplification of the important problem. The
situation is much more serious—total Oversimplification
based on principles of local physics, and obvious crisis,
we see in astrophysics, simply reflects the general short-
enings of the local kinetic transport theory. In other words
returning to the previous problems—is it possible using
only Newtonian gravitation law and non-local statistical
description to forecast the Hubble expansion and flat
gravitational curve of a typical spiral galaxy? Both ques-
tions have the positive answers [7].
Another extremely important class of cosmological
problems is connected with gravitational waves, universe
inflation and anomalies in the distribution of the uni-
verse’s cosmic microwave background (CMB). This class
of problems leads also to antigravitation but in absolutely
another sense. In other words what is the origin of the
Big Bang and evolution in the burst regime when “usual”
matter and “dark” matter do not exist on principle?
Let us investigate the possibilities which deliver the
unified generalized quantum hydrodynamics [8-12] for
investigation of these problems. I deliver here some main
ideas and deductions of the generalized Boltzmann physi-
cal kinetics and non-local physics. For simplicity, the
fundamental methodic aspects are considered from the
qualitative standpoint of view avoiding excessively cum-
bersome formulas. A rigorous description can be found,
for example, in the monographs [8-11], see also [12-17].
Let us consider the transport processes in open dissi-
pative systems and ideas of following transformation of
generalized hydrodynamic description in quantum hy-
drodynamics which can be applied to the individual par-
ticle.
The kinetic description is inevitably related to the sys-
tem diagnostics. Such an element of diagnostics in the
case of theoretical description in physical kinetics is the
concept of the physically infinitely small volume (PhSV).
The correlation between theoretical description and sys-
tem diagnostics is well-known in physics. Suffice it to
recall the part played by test charge in electrostatics or by
test circuit in the physics of magnetic phenomena. The
traditional definition of PhSV contains the statement to
the effect that the PhSV contains a sufficient number of
particles for introducing a statistical description; however,
at the same time, the PhSV is much smaller than the
volume V of the physical system under consideration; in
a first approximation, this leads to the local approach in
investigating of the transport processes. It is assumed in
classical hydrodynamics that local thermodynamic equi-
librium is first established within the PhSV, and only
after that the transition occurs to global thermodynamic
equilibrium if it is at all possible for the system under
study.
Let us consider the hydrodynamic description in more
detail from this point of view. Assume that we have two
neighboring physically infinitely small volumes PhSV1
and PhSV2 in a non-equilibrium system. Even the point-
like particles (starting after the last collision near the
boundary between two mentioned volumes) can change
Copyright © 2013 SciRes. JMP
B. V. ALEXEEV
28
the distribution functions in the neighboring volume. The
adjusting of the particles dynamic characteristics for
translational degrees of freedom takes several collisions
in the simplest case. As result, we have in the definite
sense “the Knudsen layer” between these volumes. This
fact unavoidably leads to fluctuations in mass and hence
in other hydrodynamic quantities. Existence of such
“Knudsen layers” is not connected with the choice of
space nets and fully defined by the reduced description
for ensemble of particles of finite diameters in the con-
ceptual frame of open physically small volumes, there-
fore—with the chosen method of measurement. This en-
tire complex of effects defines non-local effects in space
and time.
The physically infinitely small volume (PhSV) is an
open thermodynamic system for any division of macro-
scopic system by a set of PhSVs. But the Boltzmann
equation (BE) [8,18,19]
B
Df DtJ, (1.1)
where
B
J
is the Boltzmann collision integral and DDt
is a substantive derivative, fully ignores non-local effects
and contains only the local collision integral
B
J
. The
foregoing nonlocal effects are insignificant only in equi-
librium systems, where the kinetic approach changes to
methods of statistical mechanics.
This is what the difficulties of classical Boltzmann
physical kinetics arise from. Also a weak point of the
classical Boltzmann kinetic theory is the treatment of the
dynamic properties of interacting particles. On the one
hand, as follows from the so-called “physical” derivation
of BE, Boltzmann particles are regarded as material
points; on the other hand, the collision integral in the BE
leads to the emergence of collision cross sections.
Notice that the application of the above principles also
leads to the modification of the system of Maxwell equa-
tions. While the traditional formulation of this system
does not involve the continuity equation, its derivation
explicitly employs the equation
0
a



j
r
a
a
t
,
(1.2)
where
is the charge per unit volume, and is the
current density, both calculated without accounting for
the fluctuations. As a result, the system of Maxwell
equations is written in the standard notation, namely
a
j
0, ,,
a
 
 
 
a
tt
 
 
 
B
D
Hj
r
,
lafl
jjj
BD E
rr r
(1.3)
contains
af

 . (1.4)
The
eralized Boltzmann equation are given, for example, in
Refs. [9,12,13]. The violation of Bell’s inequalities [20]
is found for local statistical theories, and the transition to
non-local description is inevitable.
f
l
,
f
l
j

fluctuations calculated using the gen-
The rigorous approach to derivation of kinetic equa-
tion relative to one-particle DF f
K
E is based on
employing the hierarchy of Bogoliubov equations. Gen-
erally speaking, the structure of
K
f
E is as follows:
B
nl
Df
J
J
Dt 
nl
, (1.5)
where
J
is the non-local integral term. An approxi-
mation for the second collision integral is suggested by
me in generalized Boltzmann physical kinetics,
nl DDf
JDt Dt



. (1.6)
Here,
is non-local relaxation parameter, in the
simplest case—the mean time between collisions of parti-
cles, which is related in a hydrodynamic approximation
with dynamical viscosity
and pressure p,
p
, (1.7)
where the factor
is defined by the model of collision
of particles: for neutral hard-sphere gas, [21,
22]. All of the known methods of the kinetic equation
derivation relative to one-particle DF lead to approxima-
tion (1.6), including the method of many scales, the
method of correlation functions, and the iteration method.
0.8
In the general case, the parameter
is the non-lo-
cality parameter; in quantum hydrodynamics, its magni-
tude is correlated with the “time-energy” uncertainty re-
lation [9,10,14,15]. Now we can turn our attention to the
quantum hydrodynamic description of individual parti-
cles. The abstract of the classical Madelung’s paper [23]
contains only one phrase: “It is shown that the Schröd-
inger equation for one-electron problems can be trans-
formed into the form of hydrodynamic equations”. The
following conclusion of principal significance can be
done from the previous consideration [14,15]:
1) Madelung’s quantum hydrodynamics is equivalent
to the Schrödinger equation (SE) and leads to the de-
scription of the quantum particle evolution in the form of
Euler equation and continuity equation. Quantum Euler
equation contains additional potential of non-local origin
which can be written for example in the Bohm form. SE
is consequence of the Liouville equation as result of the
local approximation of non-local equations.
2) Generalized Boltzmann physical kinetics leads to
the strict approximation of non-local effects in space and
time and in the local limit leads to parameter
, which
on the quantum level corresponds to the uncertainty
principle “time-energy”.
3) Generalized hydrodynamic equations (GHE) lead to
Copyright © 2013 SciRes. JMP
B. V. ALEXEEV
Copyright © 2013 SciRes. JMP
29
SE as a deep particular case of the generalized Boltz-
mann physical kinetics and therefore of non-local hy-
drodynamics.
In principle GHE needn’t in using of the “time-en-
ergy” uncertainty relation for estimation of the value of
the non-locality parameter
. Moreover the “time-en-
ergy” uncertainty relation does not lead to the exact rela-
tions and from position of non-local physics is only the
simplest estimation of the non-local effects. Really, let us
consider two neighboring physically infinitely small
volumes PhSV1 and PhSV2 in a non-equilibrium system.
Obviously the time
should tend to diminish with in-
creasing of the velocities of particles invading in the
nearest neighboring physically infinitely small volume
(PhSV1 or PhSV2):
u
n
H
u
. (1.8)
But the value
cannot depend on the velocity direc-
tion and naturally to tie
with the particle kinetic en-
ergy, then
2
H
mu
, (1.9)
where
is a coefficient of proportionality, which re-
flects the state of physical system. In the simplest case
is equal to Plank constant and relation (1.9) be-
comes compatible with the Heisenberg relation. Possible
approximations of
—parameter in details in the mono-
graphs [8-10] are considered. But some remarks of the
principal significance should be done.
It is known that Ehrenfest adiabatic theorem is one of
the most important and widely studied theorems in
Schrödinger quantum mechanics. It states that if we have
a slowly changing Hamiltonian that depends on time, and
the system is prepared in one of the instantaneous eigen-
states of the Hamiltonian then the state of the system at
any time is given by an the instantaneous eigenfunction
of the Hamiltonian up to multiplicative phase factors.
The adiabatic theory can be naturally incorporated in
generalized quantum hydrodynamics based on local ap-
proximations of non-local terms. In the simplest case if
is the elementary heat quantity delivered for a sys-
tem executing the transfer from one state (the corre-
Q
in
t
e
t
sponding time moment is ) to the next one (the time
moment ) then

12QT
, (1.10)

tt
where ein
and T is the average kinetic energy.
For adiabatic case Ehrenfest supposes that
12
2,,T
 
,,
(1.11)
where 12
are adiabatic invariants. Obviously for
Plank’s oscillator (compare with (1.9))
2Tnh
. (1.12)
Then the adiabatic theorem and consequences of this
theory deliver the general quantization conditions for
non-local quantum hydrodynamics.
2. Generalized Quantum Hydrodynamic
Equations
Strict consideration leads to the following system of the
generalized hydrodynamic equations (GHE) [9,10,24]
written in the generalized Euler form: continuity equation
for species

 

0
0000
1
0
I
,
tt
p
t
qR
m
 
 
 
 
 

 









 
 
 
v
r
vvvv
rrr
FvB
(2.1)
and continuity equation for mixture

 

0
0000
1
0
I
0.
tt
p
t
q
m

 

 
 

 









 

 
v
r
vvvv
rrr
FvB
(2.2)
Momentum equation for species

 



11
0000 00
1
0000 0
00
I
pq
ttm t
qpq
mt m
pp
t
 
 

 

 
 



 
 






 


00



 

 


 

vvvvFvBF v
rr r
vvvvFvBB
rr
vv vv
r

 



 
1
00 0000
1,,
00000
I2II
dd.
st elst inel
pp
qq mJ mJ
mm
 

  


 


 



 

vv vvvFv
rrr
vFvBvvvBvvvv

(2.3)
B. V. ALEXEEV
30
Generalized moment equation for mixture

 



11
00000
1
0000 0
00 I
pq
tt m
qpq
mt m
pt

 
 

 
 

 
 









0
t







v
r




 



 


vvvvFvBF
rr
vvvvFvBB
rr
vv v
r

 

 
 
0000 000
11
000000
I2I
0
pp
qq
mm
 

 

 


 



 

vvvvv
rrr
F vvFvBvvvB
I
p
v
(2.4)
Energy equation for component

22
2
00
00 0 0
22
00 0 0000 0
2
000 00
3315
22222 2
15 15
22 22
17
22
vv
pnpnv pn
tt
vpn vpn
t
vp

 



 




 


1
0
 






 



 
vvv
r
vvv vvv
r
vv vv
r
Fv
 
 
 
 


2
2
000 0
2
11
2(1)
0
0000
11 1
0000 0
15
22
13 5
222 2
pp
pv np
m
vq q q
vpp n n
mmm
pqn
t



 
 

 


 
 
 
 
  
 
vvF vvF
F FvBvBvBF
Fv FvvvFv
rr
 

11
0



22
,,
dd.
22
st elst inel
mv mv
JJ
 
 

 
 
 
 

B
vv
(2.5)
and after summation the generalized energy equation for mixture

22
2
00
00 0 0
22
000 00000
2
000
3315
222 222
151 5
222 2
17
22
vv
pnpnvp n
tt
vp nvpn
t
vp
 

 



 




 


1
0
 










 


vvv
r
vvvv vv
r
vv v
r
Fv
 
 
  




2
11
2
00 00000
2
11
20
0000
11
000
15
22
13 5
222 2
pp
pv np
m
vqqq
vpp nn
mmm
pqn
t



 
 

 


 
 
 
 
 
 
vvvFv
FFvBvBvB F
FvvvFv
rr
 

00.




B
11
0





vF
v F

1
(2.6)
Here
F
BIq
are the forces of the non-magnetic origin,
magnetic induction,
unit tensor,
charge of
the
component particle, p
static pressure for
component,
internal energy for the particles of
component, 0hydrodynamic velocity for mixture.
For calculations in the self-consistent electro-magnetic
field the system of non-local Maxwell equations should
be added (see (1.3)).
v
It is well known that basic Schrödinger Equation (SE)
of quantum mechanics firstly was introduced as a quan-
tum mechanical postulate. The obvious next step should
be done and was realized by E. Madelung in 1927—the
derivation of special hydrodynamic form of SE after in-
troduction wave function
as
Copyright © 2013 SciRes. JMP
B. V. ALEXEEV 31

,,, ,


i,,,
,, e
x
yzt
zt
xyzt xy
 . (2.7)
Using (2.7) and separating the real and imagine parts
of SE one obtains
22
0
tm




rr


 , (2.8)
and Equation (2.8) immediately transforms in continuity
equation if the identifications in the Madelung’s nota-
tions for density
and velocity v
2


, (2.9)

m
r
,k
pE
v (2.10)
introduce in Equation (2.8). Identification for velocity
(2.10) is obvious because for 1D flow with constant val-
ues



px v
mx
11
k
vmE tpx
xmx







v
,
(2.11)
where
is phase velocity. The existence of the condi-
tion (2.10) means that the corresponding flow has poten-
tial
m
 . (2.12)
As result two effective quantum hydrodynamic equa-
tions take place:

0


 v
r
t
, (2.13)
2
22
m
2
11
vU
tm



 

 
v
rr . (2.14)
But
2
22
1
2
2






r
 , (2.15)
and the relation (2.15) transforms (2.14) in particular
case of the Euler motion equation
1U
tm
rr






vvv , (2.16)
where introduced the efficient potential
2
1
42








r
2
UUm

. (2.17)
Additive quantum part of potential can be written in
the so called Bohm form
2
1
42








r
22
2m
m


 . (2.18)
Then
2
22
1
42
2
qu
UUU
UU
m
m





 





r

2
p
. (2.19)
Some remarks:
1) SE transforms in hydrodynamic form without addi-
tional assumptions. But numerical methods of hydrody-
namics are very good developed. As result at the end of
seventieth of the last century we realized the systematic
calculations of quantum problems using quantum hydro-
dynamics (see for example [8]);
2) SE reduces to the system of continuity equation and
the particular case of the Euler equation with the addi-
tional potential proportional to . The physical sense
and the origin of the Bohm potential are established later
in [14,15];
3) SE (obtained in the frame of the theory of classical
complex variables) cannot contain the energy equation
on principle. As result in many cases the palliative ap-
proach is used when for solution of dissipative quantum
problems the classical hydrodynamics is used with the
insertion of the additional Bohm potential in the system
of hydrodynamic equations;
4) The system of the generalized quantum hydrody-
namic equations contains energy equation written for un-
known dependent value which can be specified as quan-
tum pressure
of non-local origin;
5) Generalized hydrodynamic equations (GHE) (2.1)-
(2.6) can be written in the spherical coordinate system
[11,25].
3. Propagation of Plane Gravitational Waves
in Vacuum with Cosmic Microwave
Background (CMB)
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
weak-field, 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 conserva-
tion 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 ab-
erration 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 Equa-
tions (2.1)-(2.6) for investigation of the gravitational
wave propagation in vacuum using non-stationary 1D
Copyright © 2013 SciRes. JMP
B. V. ALEXEEV
Copyright © 2013 SciRes. JMP
32
(momentum equation, 1D case)
Cartesian description.
Call attention to the fact that Equations (2.1)-(2.6)
contain two forces of gravitational origin, F—the force
acting on the unit volume of the space and g—the force
acting on the unit mass. As result we have from Equations
(2.1)-(2.6):



(continuity equation)

 
0
0000
tt
 
 
 







 
 

v
r
vvvv
rrr
I0,
p 

 


F
r
(3.1)
(continuity equation, 1D case)



0
2
00
v
tt
x
vvv
xtx
 
 
 







 
 

00,
pF
x
 


(3.2)
(momentum equation)






0000
0
00 00
00
II
2I I
p
tt
t
pp
t
pp
 
 

 




 





 
 



 



vvvv
rr
gv
r
vv vv
rr
vv
rr


00 0
00
0
 
F
vv v
FvvF
,
(3.3)

2
000
0
223
0000
0
3
20,
p
vvvF
ttxt
gv
tx
vpvpvpv
xtx
Fv
 
 











 






 


(3.4)
(energy equation) (please see Equation (3.5) below)
(energy equation, 1D case) (please see Equation (3.6)
below)
Nonlinear evolution Equations (3.1)-(3.6) contain forces
F, g acting on space and masses including cross-term
(see for example the last line in Equation (3.6)). The re-
lation
F
g comes into being only after the mass
appearance as a result of the Big Bang.
The cosmic microwave background (CMB) radiation
is an emission of black body thermal energy coming
from all parts of the sky. CMB saves the character traces
of the initial burst evolution. If the quantum density
tends to zero the first term in the third line of the energy
Equation (3.6) can be used for estimation of the initial
fluctuation and for the investigation of the influence of
this initial fluctuation on the following evolution.
2
0
5pp F
FLv
xx








x




, (3.7)
where L is the character length parameter reflecting the
fluctuation influence on the initial physical system. Then
22
00
22
000 000
3 3
22 22
171
222
vv
p p
tt
vp








vv vv
r
22
2
00 00000000
2
2
00 0
151515
2222 22
5 13
2 22
vpvp vp
t
p
pv pv
 


 

 
 


 
 
 








vvFvvv vv
rr
F vvgFg
 

00
t


 

Fv gvv
00 0,
p
p


 

 






v F
rr
(3.5)




223
00000
334222
000000 00
2
2
000
3352
5582
52 0,
vpvpvpv Fv
ttx
vpvvpvvpvFvvF
xtx
pp p
FFvgvvF
xx txx
 



 







 













 
 
 





(3.6)
B. V. ALEXEEV 33




22
00
0
33
00 00
2
0
2
00 0
33
2
55
3
2
0.
vpvp v
tt
Fv
vpv vpv
xt
Fv
F
LvFv gvv
xt



 


 


 


3
00
42
00
0
5
8
pv
x
vpv
x
p
F
xx
 



(3.8)
The system of Equations (3.2), (3.4) and (3.8) can be
transformed as follows (u—velocity in the
x
—direc-
tion):
(continuity equation, 1D case)
0,
pF
xx
 






 (3.9)
(momentum equation, 1D case)
2
pp
30,
pu
FF
x
tx
pp
up
xtx

 




 


 


x x
u
x
 
 


 
 





(3.10)
(energy equation, 1D case)
 
33 52
33 5
55
60
ppup
upu
txx x
ppu
up
ttxx
pu upu
xt x
puF
Fu Lu
xxx
 
 
 
 

2
11
,
F
p
uF
x
u
up x



 




 

 


 




 
(3.11)
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 equation
leading to Newton gravitational description. If non-lo-
cality parameter
is equal to zero the mentioned sys-
tem becomes unclosed.
Let us introduce the length scale 0
, quantum pres-
sure scale 0, the force scale 0
p
, the velocity scale
and approximation for non-local parameter
0
u
0
0
H
p
F
u
. (3.12)
The length scale is taken as 000
pF
, then
is
dimensionless parameter. The principles of the
—ap-
proximation are discussed in [7-10], here I remark only
that the approximation (3.12) is compatible with the
Heisenberg relation (see also (1.8), (1.9)).
Let us introduce the coordinate system moving along
the positive direction of
x
-axis in 1D space with veloc-
ity 0
Cu
equal to phase velocity of considering object
x
Ct . (3.13)
Taking into account the De Broglie relation we should
wait that the group velocity
g
u2u is equal 0. In mov-
ing coordinate system all dependent hydrodynamic val-
ues are function of
,t
. We investigate the possibility
of the object formation of the soliton type. For this solu-
tion there is no explicit dependence on time for coordi-
nate system moving with the phase velocity 0. Write
down the system of Equations (3.9)-(3.11) in the moving
coordinate system:
u
(continuity equation, 1D case)
0,
pF










(3.14)
(momentum equation, 1D case)
230,
pup u
Fp

 

 
 


 

(3.15)
(energy equation, 1D case)
2
254
611 0,
pupu
uFp Fu
uuF
upp Lu


 
 

 
 

 
 
 
(3.16)

 

 
Let us write down these equations in the dimensionless
form, where dimensionless symbols are marked by tildes,
using the introduced scales and approximation (3.12) for
the non-local parameter. The mentioned equations take
the form
2
10,
pF
u













(3.17)
22
230,
puHp Hu
Fp
uu
 


 

 


 


 
 

(3.18)
2
2
22
0
254
611 0,
puHpu
uFpuF
u
HuH uLF
upp u
uu
 
 
 
 
 
 
 
 
 
 


 
  
 

 


 


 

(3.19)
4. Results of Mathematical Modeling
Now we are ready to display the results of the mathe-
Copyright © 2013 SciRes. JMP
B. V. ALEXEEV
34
matical modeling realized with the help of Maple (the
versions Maple 9 or more can be used).
First of all from Equations (3.17)-(3.19) it is possible
to make analytical estimates using the condition
0,
pF

(4.1)
In this case Equation (3.17) is satisfied identically and
Equations (3.18), (3.19) can be written as follows
2
30,
u
p





pH
u




(4.2)
2
22
11
Hu
p
u
0
56
0,
uHu
pu p
u
LF
u


 









 




2u
(4.3)
Multiplying Equation (4.2) by and adding to
(4.3) one obtains
2
2
5211
upHu
pup
u


 
 
 

 
 

0
0,
LF
u
 
(4.4)
or
2
2
ln
52 11
upHu
uu





 

 
 
0
0,
LuF
p

(4.5)
Omitting the derivative of the logarithmic function and
the last term in Equation (4.5) one obtains the equation
2
2.2 ,
uu
H








u
(4.6)
which non-trivial solution is

0,0 exp2.2
x
ut
uux t
H





 (4.7)
for waves propagating in the positive direction of
x
axis. It can be shown that for
x
ut


exists the solu-
tion

0,0exp2.2
x
ut
uux t
H





0,0 0t
const

. (4.8)
It means that qualitative consideration leads to travel-
ing waves with exponential evolution and no surprise
that the solution of full system (3.17)-(3.19) defines soli-
tons.
I emphasize that:
1) Relations (4.7), (4.8) reflecting the exponential law
of the perturbation evolution, are the particular case of
the generalized H—theorem proved by me (see [9,26];
2) If , then (this follows from
Equations (4.1) and (4.2))
uux

p
, ;
3) The physical system is at rest until the appearance
of external perturbations.
The system of Equations (3.17)-(3.19) has the great
possibilities of mathematical modeling as result of chang-
ing of two parameters
and 0
LL
and six Cauchy
conditions describing the character features of initial
perturbations which lead to the soliton formation. Maple
program contains Maple’s notations—for example the
expression
00Du
means in the usual notations
0F

00u

t
, independent variable responds to
.
We begin with investigation of the problem of princi-
ple significance—is it possible after a perturbation (de-
fined by Cauchy conditions) to obtain the object of the
soliton’s kind as result of the self-organization? With this
aim let us consider the initial perturbations:

 
01, 01, 01, 0 0,
00, 00.
upf Du
Dp Df
 

u
p
The following Maple notations on figures are used:
u—velocity , p—pressure , and f—the self consis-
tent force
F

. Explanations placed under all following
figures. In the soliton regime the solution exists only in
the restricted domain of the 1D space and the obtained
object in the moving coordinate system
x
t

1u has
the constant velocity
for all parts of the object. In
this case the domain of the solution existence defines the
character soliton size. The following numerical results
reflect two principally different regime of the physical
system evolution. The distinctive features of evolution
are defined by the sign plus or minus in front of the term
D in the energy equation,
0
LF F
Du Lu
 




L
L
L
1,H
. (4.9)
The term D reflects the interaction of physical system
with the surrounding media and defines the value of per-
turbation. The Maple inscription L on figures corre-
sponds to the dimensionless value including the sign
in front of .
Therefore, Regime I is characterized by the negative
sign in front of . The mentioned calculations are dis-
played in Figures 1-12.
One can see that the first regime is characterized by
the force directed basically (in front of the wave, Figures
2, 5, 8 and 11) against the direction of the wave propaga-
tion. This fact leads to the effect of attraction. Domains
of the solution existence in regime I:
1L
1) For
1,H. (0.7298; 1.1618);
0.1L
2) For
1,H. (0.1571; 0.6347);
. (0.0265; 0.5327); 0.01L
1,H
3) For
4) For
. (0.003801; 0.5186). 0.001L
Regime II is characterized by the positive sign in front
of L. The mentioned calculations are displayed in Fig-
ures 13-22.
Copyright © 2013 SciRes. JMP
B. V. ALEXEEV 35
Figure 1. uvelocity , . u
H1,L
1
Figure 2. fthe self-consistent force , F
H1, L
1.
The second regime is characterized by the force di-
rected along the direction of the wave propagation. This
fact leads to the effect of anti-attraction during the Big
Bang, (Figures 13, 17, 19 and 21). The term “antigravi-
tation” is deeply embedded in the physical literature but
this term unlikely applicable for the vacuum explosion.
As follow from Figures 15 and 16 the fist regime of trav-
eling waves corresponds only to the early life of evolu-
tion and later gives way to regime of the very intensive
explosion which details should be investigated using
non-stationary 3D models on the basement of Equations
Figure 3. ppressure , .
p
H1, L
1
Figure 4. uvelocity , . u
H1,L
0.1
34.582.t

(2.1)-(2.6). For example domain of the solution existence
for the case is shown in Figures 15 and 16:
lim
Important to notice that Hubble expansion can be ex-
plained as result of the matter self-catching in the frame
of the Newtonian law of gravitation [7].
As you see during all investigations 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-
Copyright © 2013 SciRes. JMP
B. V. ALEXEEV
36
Figure 5. fth e se l f - c o n s i s tent force ; F
H1, L
0.1 .
Figure 6. ppressure ; .
p
H1, L
0.1
pF
cal description for the physical system under considera-
tion.
If the matter is absent, the gravitational evolution of
the system in space and time is containing in EE only so
to speak on the “genetic level”; it means the origin of the
EE derivation in the macroscopic case for massive sys-
tem.
Better to speak about evolution of “originating vac-
uum” (OV) which description in time and 3D space on
the level of quantum hydrodynamics demands only quan-
tum pressure , the self-consistent force (acting on
Figure 7. uvelocity , . u
H1, L
0.01
Figure 8. fthe self-consistent force ; . F
H1,L
0.01
v
L
unit of the space volume) and velocity 0. The perturba-
tions of OV lead to two different processes—travelling
waves including the soliton formation (regime I) and the
explosion of the system (regime II, the Big Bang regime).
Both regimes can be incorporated in one scenario.
As follow from calculations (see Figures 13-22) the
most intensive explosion effect achieves for the smallest
perturbations with the positive sign in front of . From
the mathematical point of view we have the typical Ha-
damard instability leading to the Big Bang. Moreover
two regimes differ from one another by the directions of
Copyright © 2013 SciRes. JMP
B. V. ALEXEEV 37
Figure 9. p—pressure ; .
p
H1, L
0.01
Figure 10. u—velocity , . u
H1,L
0.001
forces
F
.
After the Big Bang and interaction of OV with the cre-
ated matter the microwave background radiation should
contain the traces of the travelling waves evolution real-
ized as regime I. Let us look at the last measurements
realized in the frame of the Planck programme. The
temperature variations don’t appear to behave the same
on large scales as they do on small scales, and there are
some particularly large features, such as a hefty cold spot,
that were not predicted by basic inflation models.
From the position of the developed theory it is no sur-
Figure 11. f—the self-consistent force ; H = 1, . F
L
0.001
Figure 12. p—pressure
p
H1, L
0.001
; .
prise. Really, look at the Planck space observatory’s map
(Figure 23) of the universe’s cosmic microwave back-
ground. This map is in open Internet access (see for ex-
ample SPACE.com Staff. Date: 21 March 2013 Time:
11:15 AM ET).
It was reported that CMB is a snapshot of the oldest
light in our Universe, imprinted on the sky when the
Universe was just 380,000 years old. It shows tiny tem-
perature fluctuations that correspond to regions of slightly
different densities, representing the seeds of all future
structure: the stars and galaxies of today.
Copyright © 2013 SciRes. JMP
B. V. ALEXEEV
38
Figure 13. f—the self-consistent force ,
F
H1, L
1.
Figure 14. p—pressure ; . p
H1, L
1
From the position of the developed theory Planck’s
all-sky map contains the regular traces of traveling waves
as the alternation of the “hot” (red) and “cold” (blue)
strips. In Figure 23 the Planck space observatory staff
shows the “mysterious” hefty cold spot as the blue small
area bounded by the white circle.
From the position of the developed theory it is the
area reflecting the initial explosion of OV. In this ca s e
the center domain of the mentioned hefty cold spot should
contain the smallest hot spot as the origin of the initial
Figure 15. u—velocity , . u
H1,L
0.1
Figure 16. u—velocity , . u
H1,L
0.1
burst.
I hope this fact will be established by astronomers af-
ter following more precise observations.
5. Conclusions
During all investigations we needn’t to use the theory
Newtonian gravitation for solution of nonlinear non-local
evolution equations. In contrast with the local physics
this approach in the frame of quantum non-local hydro-
dynamics leads to the closed mathematical description
Copyright © 2013 SciRes. JMP
B. V. ALEXEEV 39
Figure 17. f—the self-consistent force ; . F
H1, L
0.1
Figure 18. p—pressure ; . p
H1,L
0.1
0
v
for the physical system under consideration. If the matter
is absent, non-local evolution equations have neverthe-
less non-trivial solutions corresponding evolution of
“originating vacuum” (OV) which description in time
and 3D space on the level of quantum hydrodynamics
demands only quantum pressure p, the self-consistent
force F (acting on unit of the space volume) and velocity
.
The perturbations of OV lead to two different proc-
esses—travelling waves including the soliton formation
Figure 19. f—the self-consistent force , H = 1, . F
L
0.01
Figure 20. p—pressure
p
H1,L
0.01
F
r
; .
(regime I) and the explosion of the system (regime II, the
Big Bang regime). Both regimes can be incorporated in
one scenario. From the mathematical point of view we
have the typical Hadamard instability (the smaller is an
initial perturbation the greater is the burst intensity) lead-
ing to the Big Bang. Two regimes differ from one an-
other by the directions of forces .
Finally some words concerning the following investi-
gations. Numerical calculations, realized in the spherical
coordinate system for the dependent variables (—radius,
Copyright © 2013 SciRes. JMP
B. V. ALEXEEV
40
t
Figure 21. f—the self-consistent force , H = 1, . F
L
0.001
Figure 22. p—pressure ; . p
H1,L
0.001
Figure 23. Planck space observatory’s map of the universe’s
cosmic microw ave background.
—time) cannot change principal results of the shown
calculations in the Cartesian coordinate system. Increas-
ing of the character distances between “cold” and “hot”
zones (see Figure 23) is obliged to the burst configura-
tion closed to the spherical form. But some other effects
obviously need in 3D non-stationary calculations. This
remark relates first of all to so called “dark flow” de-
scribing a possible non-random component of the pecu-
liar velocity of galaxy clusters.
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