Политематический сетевой электронный научный журнал Кубанского государственного аграрного университета, 2012, №84

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1

УДК 531.9+539.12.01 
UDC 531.9+539.12.01 
 
 
МОДЕЛИРОВАНИЕ МЕТРИКИ АДРОНОВ НА 
ОСНОВЕ УРАВНЕНИЙ ЯНГА-МИЛЛСА  
HADRONS METRICS SIMULATION ON THE 
YANG-MILLS EQUATIONS   
 
 
Трунев Александр Петрович 
к.ф.-м.н., Ph.D. 
Alexander Trunev 
Cand.Phys.-Math.Sci., Ph.D. 
Директор, A&E Trounev IT Consulting, Торонто, 
Канада 
Director, A&E Trounev IT Consulting, Toronto, 
Canada  
 
В работе рассмотрена система уравнений ЯнгаМиллса в связи с уравнениями Эйнштейна и 
Максвелла. Сформулирована модель метрики, 
удовлетворяющая основным требованиям физики 
элементарных частиц и космологии 
 

In this article we consider the Yang-Mills theory in 
connection with the Einstein and Maxwell equations. 
The model of a metric satisfying the basic 
requirements of particle physics and cosmology is 
proposed 

Ключевые слова: АДРОНЫ, МЕТРИКА, 
ПРОТОН, УРАВНЕНИЯ МАКСВЕЛЛА, 
УРАВНЕНИЯ ЭЙНШТЕЙНА, УРАВНЕНИЯ 
ЯНГА-МИЛЛСА 

Keywords: EINSTEIN EQUATIONS, HADRONS, 
MAXWELL EQUATIONS, METRICS, PROTON, 
YANG-MILLS EQUATIONS.  

 

 

Introduction 

The hypotheses on the global structure of space-time have been formulated in 

the famous Einstein's paper [1]. Einstein assumed that the universe is stationary, 

and the average density of matter and the total mass of the universe does not 

change over time. Friedman [2] has shown that the universe is expanding, it was 

confirmed by the astronomical data, and also served as a further recognition of 

general relativity. However, Einstein's theory of gravity was incompatible with 

Maxwell's equations, so there were numerous attempts to create a unified field 

theory [3] in the space of five dimensions [4-6]. At present, interest in the 

classical field theory models waned, and Einstein's equations themselves are 

primarily used for the solution of cosmological problems. 

Due to the development of quantum theory, is a very relevant question of the 

structure of space-time scale of the proton, as modern lattice models of quantum 

chromodynamics (QCD) has been successfully used to predict the properties of 

hadrons [7]. The answer to the question of the structure of space-time can be 

obtained from the Yang-Mills theory [8-9], which is widely used in particle 

physics. Connection of the Yang-Mills equations to the Maxwell equations and 

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2

Einstein equations as well was established in [10-12]. It has been shown that on 

4-manifolds with conformal connection [13], the system of the Yang-Mills 

equations split into Einstein's equations, Maxwell's equations and the equations 

of motion of matter. So the necessary prerequisites for a unified field theory 

have been creating. In this paper we formulated a model of hadrons metric 

satisfying the basic requirements of particle physics and cosmology. 

The basic equations of the model of the cosmological scale 

Consider the example of a purely temporary solution of the Yang-Mills 

equations in the space of torsion-free [11]. We define a metric space as 

2
3
2
2
2
2
2
1
2
2
)
)(
(
)
)(
(
)
)(
(
dx
t
c
dx
t
dx
t
a
dt
j
i
ij
+
+
+
−
=
=
β
ω
ω
η
ψ
         (1) 

Here 

ij
ij
η
η =
is the metric tensor of the Minkowski space of signature (- + + +), 

3
4
2
3
1
2
1
)
(
,
)
(
,
)
(
,
dx
t
c
dx
t
dx
t
a
dt
=
=
=
=
ω
β
ω
ω
ω
 . 

Yang-Mills equations can be reduced in this case to Einstein's equations, 

Maxwell's equations, and equations of motion of matter accordingly, we have 

 








+
+
−








+
+
=
c
c
ac
c
a
a
a
c
c
a
a
b
β
β
β
β
β
β
&&
&
&
&
&
&&
&&
&&
6
1
3
1
11
                            (2) 









+
+
+








+
+
−
=
c
c
c
c
ac
c
a
a
a
a
a
b
β
β
β
β
β
β
&&
&&
&&
&
&
&
&
&&
6
1
3
1
22
 







+
+
+








+
+
−
=
ac
c
a
c
c
a
a
c
c
a
a
b
&
&
&&
&&
&&
&&
&
&
6
1
3
1
33
β
β
β
β
β
β
 









+
+
+








+
+
−
=
β
β
β
β
β
β
a
a
a
a
c
c
c
c
ac
c
a
b
&
&
&&
&&
&&
&&
&
&
6
1
3
1
44
 

N
c
b
M
c
b
=
=
β
β
34
12
,
 

2
22
11
22
)
(
S
a
a
b
b
b
=
+
+
&
&
 

3
33
11
33
)
(
S
b
b
b
=
+
+
β
β&
&
 

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3

4
44
11
44
)
(
S
c
c
b
b
b
=
+
+
&
&
 







+
+
+








+
+
+






−
+
−
+

=








+
+

44
22
44
33
22
33
11
22
11
2
34
2
12

2
2

2
2
b
b
ac
c
a
b
b
b
a
a
b
b
b
a
a
b
b
b

c
c
S
S

&
&
&
&
&&

&
&
&

β
β

β
β

 









+
+
+








+
+
+








−
+
−
−
−

=






+
+

44
33
44
33
22
22
11
33
11
2
34
2
12

3
3

2
2
b
b
c
c
b
b
b
a
a
b
b
b
b
b
b

c
c
a
a
S
S

β
β
β
β
β
β
&&
&
&
&&

&
&
&

 









+
+
+






+
+
+






−
+
−
−
−

=








+
+

44
33
33
44
22
22
11
44
11
2
34
2
12

4
4

2
2
b
b
c
c
b
b
b
ac
c
a
b
b
b
c
c
b
b
b

a
a
S
S

β
β

β
β

&&
&
&
&&

&
&
&

 

Here 
ij
ij
ij
ij
ji
ij
T
b
b
b
=
−
+
η
η
)
(
2
is the energy-momentum tensor of matter; 

N
M ,
- the parameters characterizing the electromagnetic field. Note that the 

Einstein equations in this notation have the form: 

ij
ij
ji
ij
R
b
b
b
=
+
+
η
                                                  (3) 

 
ij
ij
ij
R
b
b
;
η
=
is Ricci tensor. Einstein equations (3) can be reduced to the first 

four equations (2) in the case of the metric (1).  In contrast to the standard 

Einstein equations, they do not contain dimensional parameters characterizing 

the interaction of the gravitational field with the distribution of matter. This is 

due to the fact that the quantities in equations (2) - (3) and described by the 

classical Yang-Mills equations are geometric quantities, like the Ricci tensor in 

the right-hand side of equation (3). 

  

Some exact solutions and numerical model 

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4

In [11] found a particular solution of equations (2) in elementary functions 

 

2
0
44
33
22
2
0
11

4
2
2
0

0

)
(
6
1
;
)
(
6
5

;
12
1
;
2
;
2

t
t
b
b
b
t
t
b

Q
N
M
Q
t
t
c
t
t
PQ
a

+
=
=
=
+
=

=
+
+
=
=
+
=
β

                       (4) 

Here
0
,
,
t
Q
P
are the arbitrary constants. 

The solution (4) describes a singular time-a process in which the energy density 

increases at 
0t
t
−
→
. Among such processes in our universe, we can specify a 

hypothetical explosion – Big Bang. In this case, the characteristic density of 

baryonic matter decreases with time, whereas the density of electromagnetic 

energy remains constant. Note that in the model (2), these densities are diagonal 

and off-diagonal components of the tensor 
ijb , respectively. 

We point out one particular solution of the system (2) 

6
;
3

;
12
;
,
),
exp(

2

44
33

2

22
11

4
2
2
2
2
0

λ
λ

λ
β
λ

=
=
=
−
=

=
+
=
=
=

b
b
b
b

C
B
N
M
C
c
B
t
a
a

                     (5) 

         

2
3
2
2
2
2
2
1
2
2
0
2
)
(
)
(
)
(
dx
C
dx
B
dx
e
a
dt
t
+
+
+
−
=
λ
ψ
. 

Here
λ
,
,
,
0
C
B
a
 are the arbitrary constants. The solution (5) gives us an 

example of a space of constant negative curvature 

2
2λ
η
−
=
=
ij
ijR
R
. In this 

case, the density of baryonic matter and electromagnetic energy density remains 

constant over time. 

Solutions (4-5) were used to adjust the numerical model for the system (2). 

Explore different modes of transition of the solution (4) to a solution of the type 

(5) - Fig. 1. We found that if in the initial time we set
const
b
=
11
, then in the 

next moment, all the diagonal tensor components 
ijb  tend to constant values, 

and the components of the metric tensor are increasing exponentially - Fig. 1. 

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5

 
Fig. 1. The exact solution (4) (top) and the transition from the solution (4) to a 

solution in which all diagonal components 
ijb tends to a constant (bottom): on 

the left components of the metric tensor, on the right the accuracy of the solution 

are shown. 

 

However, only on the solution of the type (5), the density of electromagnetic 

energy remains constant over time. In all other cases, the density varies 

according to the equations: 
N
c
b
M
c
b
=
=
β
β
34
12
,
.  

This disappearance of the electromagnetic field at a large scale is against the 

astronomical observations, so let's mark solutions (5), as the hypothetical 

scenario that describes the universe at a large scale. 

An alternative scenario is the type (4), but it contradicts the experiments in high
energy physics. Indeed, a decrease in the density of baryonic matter means, 

among other things, that the proton is unstable. This follows from the fact that 

the density of the model of baryonic matter and proton density are linked by the 

fact that the Einstein equation (3) contains only the geometric parameters that 

describe the density distribution at any scale. However, experimental evaluation 

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6

of the proton lifetime show that the half-life of a proton is more than 1033 years, 

i.e. greatly exceeds the lifetime of the universe (about 1010 years). 

For this reason, we should discard the other cosmological models with 

decreasing density of matter. In this sense, Einstein's hypothesis [1] on the 

stationary distribution of the density of matter in the universe is correct, 

however, this hypothesis does not imply that the metric is also stationary. We 

obtained the solution (5) combines the properties of Einstein's model [1], and 

Friedman’s model [2] as well, describes the universe as a time-dependent 

metric, and with a constant density distribution of baryonic matter and 

electromagnetic field. 

Model of the proton scale 

Note that the Einstein equations in the form (3) are universal, i.e. describe the 

metric in any scale, because their solution depends only on the initial conditions. 

Choosing these conditions across the proton, we create model that describes the 

metric of hadrons. Let us consider the metric of the proton, and other elementary 

particles. In [12] obtained all the solutions of the Yang-Mills equations in the 

case of a centrally symmetric metric. A particular case of the centrally 

symmetric metric is 

 

κσ
θ
σ

ϕ
θ
σ
θ
ω
ω
η
ψ
ν

−
=

+
+
+
−
=
=

2

2

2
2
2
2
2
2
)
(

d
d

d
d
dr
e
dt
j
i
ij
                   (6) 

Here 
const
=
κ
is 
the 
Gaussian 
curvature 
of 
the 
quadratic 
form 

2
2
2
)
(
ϕ
θ
σ
θ
d
d
+
, function 
)
,
(
t
r
ν
ν =
 is determined by solution of the Yang
Mills equations. As we address the fundamental geometric structure of the 

observable universe, then we are interested, first of all, the periodic solutions, 

which form a lattice. This requirement stems from the obvious fact that all 

points of space should be equal to each other, but each observer can reproduce 

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7

all the observed variety of phenomena. This is only possible if the basis of the 

space is a periodic structure. 

Among all the solutions of the Yang-Mills equations, obtained in [12] in the 

case of the metric (6), there is one, which is expressed in terms of Weierstrass 

elliptic function. In this case, the Yang-Mills equations simplified to the form: 

  

.0
,
3
6
1
,
6
3
1

),
,
;
12
/
(
12

,
),
(
2
1

21
12
44
33
22
11

3
2
3
3

0
2
2

=
=
−
=
=
−
=
−
=

℘
=

+
±
=
=
−
=

b
b
A
b
b
A
b
b

g
g
A

r
t
A
e
A
A

κ
κ

τ

τ
τ
κ
τ
ν
ττ

                (7) 

Here 
3
2, g
g
are the invariants of the Weierstrass function, and
3
2
2
12
κ
=
g
, 

0
τ is a free parameter related to the choice of origin. Metric corresponding to 

this solution, apparently, can describe the fundamental structure of the universe 

and, in particular, the structure of elementary particles. The solution of (7) and 

the corresponding metrics are characterized by two periods, which, obviously, 

should be related to the parameters of elementary particles.  

Note that the sum of the diagonal components of the energy-momentum tensor 

in this case is 
A
Tii
2
=
. Therefore, setting the characteristic density of matter 

and space-time scale, we can obtain the general solution, which describes the 

metric at any scale. We show that the metric (5) described by equations (7) as 

well. To do this, we consider the solution of the first equation (7) near constant 

level of density of matter, given by Eq. 
2
2
κ
=
A
. 

Introduce a new function and variables according to the formulas 

r
t
A
f
+
=
−
=
τ
κ,
 

 Suppose that 

2
2
κ
<<
f
, then the equations of the model (7) and the solution 

are of the form 

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8

τ
κ
τ
τ
τ
κ
ν

τ
ν
ττ

κ
κ
κ

τ
τ
κ

κ

e
f
e
f
e
f
e

k
f
f
f

f
e
f
f

k
0
1
0

1
0
),
exp(
)
exp(

,

→
−
=

−
+
=

=
=

∞
→
−
                                 (8) 

In general, as it is known, the corresponding metric is reduced to [12, 14] 

 

2
0
2
2
2
0
2
2
)
(
cos
)
(
ϕ
θ
θ
κ
θ
κ
ψ
d
d
dr
t
t
ch
dt
+
+
+
+
+
−
=
           (9) 

Consider the universe at the moment of its existence in a small neighborhood of 

the coordinate system fixed to the solar system, where
t
≈
τ
. It is known that the 

observed average density of matter in the universe is a small quantity compared 

to the baryon density, so
1
<<
κ
. Obviously, under these conditions, to match 

the metrics (5) and (9) it will be enough to put 
2
λ
κ =
. 

We thus proved that the metric of the observable universe is associated with a 

metric of the periodic lattice, given by the Weierstrass function (7). Note that 

although the metric (5) and (9) are similar, but the tensors 
ijb  are not similar. 

Indeed, the metric (5) is compatible with zero electromagnetic field that 

describes the components of the tensor 
c
N
b
c
M
b
β
β
/
,
/
34
12
=
=
. Whereas 

in the metric (7) electromagnetic field is zero, since 
0
12 =
b
. 

Therefore, it is necessary to explain the mechanism of the electromagnetic field 

in the initial lattice (7), which does not contain the electromagnetic field. For 

this we note that the Yang-Mills field in the linear case is divided into a set of 

independent electromagnetic fields [15]. Consequently, the electromagnetic field 

generated at a low energy density, starting from the atomic nuclei and atoms. 

For its occurrence does not require any additional sources other than the Yang
Mills field and the lattes. In this case the electromagnetic fields have a 

wavelength multiple of the lattes scale. These waves travel at a constant speed 

(light) from any source along the lattes in accordance with Maxwell's equations. 

Thus, the quantization of the electromagnetic waves is a consequence of the 

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9

presence of the lattice in the metric (7), the period of which is evident in all of 

the observed phenomena. 

Metric model of elementary particles 

Suppose that
1
,
12
3
3
2
=
=
g
g
, then the half-periods of the Weierstrass 

function defined as
i
61260
.1
66501
.0
,
33003
.1
2
1
+
=
=
ω
ω
 . Calculation of 

half-periods and the construction of appropriate 3D images of the Weierstrass 

function and its first derivative module carried out using the Wolfram 

Mathematica 9.0 [16]. Consider the metric lattice is formed at the specified 

parameters - Fig. 2. 

 

Figure 2: Metric parameters 
τ
ν
A
e =
 and 
A
Tii
=
2
/
 in the case 

1
,
12
3
3
2
=
=
g
g
. 

 

As follows from Fig. 2 data, the peaks of the Weierstrass function fuse to form 

solid 
walls, 
stretching 
along 
the 
lines 
const
r
t
=
±
of 
the 

period
09
.6
12
2
3
1
≈
ω
. These characteristics describe convergent (sign +) or 

diverging (minus sign) spherical waves. It is easy to see that the solutions of the 

first equation (7) are symmetric with respect to the change 
t
t
−
→
, however, 

the metric (5) is asymmetric with respect to time reversal. Therefore, although 

the microscopic events are reversible in time, macroscopic events are known to 

be irreversible, which is due, in particular, increasing of entropy. It can be 

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10

assumed that the emission of spherical waves is the main process in the 

expanding universe, and then the determination of the metric lattes should 

assume that
0
τ
τ
+
−
=
r
t
. 

From classical electrodynamics it is known that in the process of 

electromagnetic wave generation an electrical charge acts as source. In this 

model, it is most natural to assume that the electromagnetic waves are produced 

in the interaction of the charge with the lattice. According to present science a 

charge of hadrons occurs as the sum of fractional charges of quarks. But the 

origin of the electric charge of the quarks themselves also requires explanation. 

In the lattice model can be defined lattice defect type bubble. Lattice with a 

single bubble is described by the model (7), and the corresponding metric has 

the form (6). In the bubble we put
2
2
κ
=
A
, while in the outer region the 

solution given in the form (7), therefore we have,  

 

0
2
1
3
3

0
2
2

,
),
,
,
12
/
(
12

,0
,

τ
τ
τ

τ
τ
κ

τ
ν

ν

>
=
℘
=

<
=
=

A
e
g
g
A

e
A
                                (10) 

On the borders of the bubble are continuous function A  and its first derivative, 

  
0
2
1
3
0
3
,0
),
,
,
12
/
(
12
τ
τ
τ
κ
τ
=
=
℘
=
A
g
g
                             (11) 

In the particular case of the lattice in Fig. 2, with the invariants given in the 

form
1
,
12
3
3
2
=
=
g
g
, we find the first zero and the corresponding parameter 

of the metric 
1038034
.2
,
0449983
.3
0
=
=
κ
τ
. The corresponding lattice is 

shown in Fig. 3.  

Similarly, the solution is constructed for the other roots of the second equation 

(10). All of these roots only effect on the size of the bubble, whereas the value 
κ does not change. Bubble moves at the speed of light along the characteristics 

const
r
t
=
−
- Fig. 3. To the outside observer, the bubble is still a homogeneous 

particle with a finite length. 

Any bubble can be turned inside out, just reversed inequalities in (10) - Fig. 4.