Розрахунок щільності енергії та щільності спіну гравітаційних хвиль Місяця у прямолінійних координатах
In this research the energy density was calculated and the spin momentum density of Moon’s gravitational waves in the rectilinear coordinates’ system of Moon’s gravity and Earth’s global temperature. At first, we assumed an action principle that combines the gravitational field and gravitational wav...
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System research and information technologies| _version_ | 1866302618264928256 |
|---|---|
| author | Matsuki, Yoshio Bidyuk, Petro I. |
| author_facet | Matsuki, Yoshio Bidyuk, Petro I. |
| author_sort | Matsuki, Yoshio |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2019-12-13T15:15:18Z |
| description | In this research the energy density was calculated and the spin momentum density of Moon’s gravitational waves in the rectilinear coordinates’ system of Moon’s gravity and Earth’s global temperature. At first, we assumed an action principle that combines the gravitational field and gravitational waves, which formulate a closed system, together with Earth’s global temperature. And, then, we calculated the energy densities of those energy field and waves, which are calculated as their variances in the rectilinear coordinates, also to calculate their coefficients and standard errors of the calculated coefficients. The calculated results are consistent with the findings of our previous research [1], which shows the negative contribution of gravitational waves to Earth’s global temperature, while the gravitational field positively contributes to the global temperature. We also calculated spin momentum of Moon’s gravitational waves in the system of rectilinear coordinates. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2019.3.01 |
| first_indexed | 2025-07-17T10:26:28Z |
| format | Article |
| fulltext |
Y. Matsuki, P.I. Bidyuk, 2019
Системні дослідження та інформаційні технології, 2019, № 3 7
TIДC
ПРОГРЕСИВНІ ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ,
ВИСОКОПРОДУКТИВНІ КОМП’ЮТЕРНІ
СИСТЕМИ
UDC 519.004.942
DOI: 10.20535/SRIT.2308-8893.2019.3.01
CALCULATING ENERGY DENSITY AND SPIN MOMENTUM
DENSITY OF MOON’S GRAVITATIONAL WAVES
IN RECTILINEAR COORDINATES
Y. MATSUKI, P.I. BIDYUK
Abstract. In this research the energy density was calculated and the spin momentum
density of Moon’s gravitational waves in the rectilinear coordinates’ system of
Moon’s gravity and Earth’s global temperature. At first, we assumed an action prin-
ciple that combines the gravitational field and gravitational waves, which formulate
a closed system, together with Earth’s global temperature. And, then, we calculated
the energy densities of those energy field and waves, which are calculated as their
variances in the rectilinear coordinates, also to calculate their coefficients and stan-
dard errors of the calculated coefficients. The calculated results are consistent with
the findings of our previous research [1], which shows the negative contribution of
gravitational waves to Earth’s global temperature, while the gravitational field posi-
tively contributes to the global temperature. We also calculated spin momentum of
Moon’s gravitational waves in the system of rectilinear coordinates.
Key words: Moon, Earth, global temperature, gravitational field, gravitational
waves, rectilinear coordinates, energy density, spin momentum density.
INTRODUCTION
Our previous research [1] showed a relation between Moon’s gravitational field
and gravitational waves. After that report we continued investigating the charac-
ters of gravitational waves, using the same data set, but this time with the theories
of relativity and quantum mechanics.
Our mathematical method starts from an action principle, which assumes
that there is an action integral that describes the motion of the waves, which must
be stationary to be consistent with the law of conservation of energy within the
boundary of a closed space. We also assume the rectilinear coordinates of the
flat-space for time and space by tensors that represent the energy field as well as
the pseudo-tensors that represent the flow of energy.
Dirac [2, 3] predicted that the pseudo-tensor can be built in the coordinates
of the tensors’ field only when the motion of the gravitational waves, which is
expressed by the pseudo-tensors, occurs in one direction. In addition, he also cre-
ated a basic equation that describes the gravitational field and the other energy
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2019, № 3 8
flows, such as motion of the gravitational waves, and they can be added together
linearly, which agrees with the theory of the special relativity [2].
And then, upon the theory [2, 3], we set up an equation to describe the gravi-
tational waves in the gravitational field. Then we solved the equation to calculate
the energy densities and coefficients of the gravitational field, and the gravita-
tional waves toward Earth’s global temperature in the space between Moon and
Earth. After that we also investigate the character of the gravitational waves by
calculating the momentum density of the spin of the gravitational waves.
DATA
Table 1 shows the descriptive statistics of the data, from 1987 till 2009, of the
global temperature (increased degree Celsius since 1978) [4], the distance be-
tween Moon and Earth ( r : kilometers, km) [5], and calculated
2
1
r
((kilometers)--22,,
kkmm--22)), which we use for our calculations.
T a b l e 1 . Descriptive statistics
Variable
Global temperature
(oC) *
Distance between
Moon and Earth
(r : km)
2
1
r
((km)--22))
Mean 0,29130 3,62618 105 7,60509 10-12
Standard deviation 0,12125 5,98411 102 2,51097 10-14
Minimum 0,10000 3,61583 105 7,56999 10-12
Maximum 0,43000 3,63483 105 7,64865 10-12
Skewness -0,21063 -0,15249 0,15787
Kurtosis 1,29401 1,67498 1,67879
Valid number
of observations
23 23 23
* Increased degree Celsius since 1978
CALCULATIONS
Gravitational waves in gravitational field
Dirac [2] created the basic equation for a quantum theory of the Born-Infield elec-
tro-dynamics in the rectilinear coordinates, which agrees with the special relativ-
ity theory. It defines the action integral of the motions of particles in the electro-
magnetic field: xdFgI 4)det( , where, )det( FgI is the
energy density of the electro-magnetic field that provides the action principle of
a particle in this energy field, g are fundamental tensors,
1000
0100
0010
0001
g .
Calculating energy density and spin momentum density of Moon’s gravitational waves …
Системні дослідження та інформаційні технології, 2019, № 3 9
F are tensors that describe the field quantities of the electro-magnetic
field, which are made by two vectors, x and x , while 3,2,1,0 and
3,2,1,0 , where each suffix represents each of 4 coordinates of the flat-space.
We use this mathematical formula in order to calculate the energy density of the
gravitational waves in the gravitational field, and have made
2
30 1
r
cbxaxYI , where a , b , and c are coefficients that are to be
constants; Y is Earth’s global temperature as we assume that it is influenced by
the motion of gravitational waves; 0x is the metric that refers the time-coordinate
of the flat-space, which is made of a time vector (=1), where it is constant. (Be-
cause in the theory of the special relativity, nothing can exceed the speed of light;
therefore, nothing moves and it is 1 .) And 3x is the metric in the flat-space,
which we consider as only one direction with a distance r between Moon and
Earth.
For calculating the energy density of the gravitational waves, we assume that
it is
2
1
r
c ; because, we assume that the solutions of 0,
gg are the gravita-
tional waves traveling with the velocity of light, which satisfies d’Alambert equa-
tion [3]. Here,
1000
0100
0010
0001
g is made of contravariant vectors, and
g are made of their covariant vectors, while
xx
g
g
2
, , where xx ,
are the contravariant vectors that are described in the rectilinear coordinates of the
flat-space. Here, , , , are the suffixes that indicate the coordinates of time
and space. Because of
xx
g
g
2
, , the gravitational waves must have di-
mension of
2
1
r
. Now we set
2
2
2 1
r
cbraYH .
Now calculate the coefficients a , b and c , with the constraints:
0
)( 2
a
HE
, 0
)( 2
b
HE
, and 0
)( 2
c
HE
, And then the equations are trans-
formed to: 0)(2)( 2
HEHE
a
, with 0)( HE ; 0)(2)( 2
HrEHE
b
,
with 0)( HrE ; 0
1
2
r
HE
c
, then 0
1
2
r
HE , where )( 2HE is the ex-
pected value of 2H and where
2
1
r
cbraL is Lagrangean. (Note: here in
this Lagrangean, the minus-sign of the space coordinate appears as the minus-sign
of the coefficient b, because in the special theory of relativity, the geodesic of the
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2019, № 3 10
time and space is described by the expression: 23222120 )()()()( dxdxdxdx ,
where 0dx is a displacement of the time-vector, and 1dx , 2dx and 3dx are dis-
placements of the space-vectors). And then we can calculate these coefficients:
a , b and c , algebraically as follows: at first, we make a matrix },,{ 321 xxxX ,
where 11 x (time-coordinate), rx 2 (space-coordinate, which is a distance be-
tween Moon and Earth), and
23
1
r
x . Here, the above mentioned constraints,
0)( HE , 0)( HrE and 0
1
2
r
HE are generally described by the matrices
0HX , then:
21-5-10-
5-126
-106
101,33027106,34278 101,74917
106,34278103,02432108,34022e
101,74917108,34022 23,00000
XXQ ,
where 'X is the transposed matrix of X ; '1XQA , where 1Q is the inverse
matrix (reciprocal matrix) of Q ; AY , and where is the vector of three coef-
ficients, a , b and c ; XAN ; NIM , where I is a unity matrix, in which
all the diagonal elements are 1, and non-diagonal elements are 0; MYe ;
)/(' 1 kneQe , where is the matrix that contains variances and covari-
ances of the variables; 'e is the transposed vector of e ; n is the number of data
(in this analysis 23); and k is the degree of freedom (number of variables, in this
analysis 3k ).
The results of the calculations are as follows:
14
2-
3
102,50844
101,05217
105,72334
c
b
a
;
291318
134-2
1828
102,42964 101,01952 105,54473
101,01952 104,27814 102,32669
105,54473 102,32669 101,26538
.
Standard errors of the coefficient vector
14
2-
4
29
4-
8
104,92914
102,06837
101,12489
102,42964
104,27814
101,26538
b .
Spin momentum density of gravitational waves
From the above calculations, we found that the energy of gravitational waves has
negative coefficient, 14102,50844 c to Earth’s global temperature, while the
coefficient of the gravitational field has positive coefficient, -2101,05217 b .
Calculating energy density and spin momentum density of Moon’s gravitational waves …
Системні дослідження та інформаційні технології, 2019, № 3 11
It means that the flow of the gravitational waves doesn’t increase the potential
energy of this system between Moon and Earth. Then we investigated the charac-
ter of the gravitational waves, assuming that the vectors of the coordinates formu-
late the motion of the waves, with the theory made by Dirac [3] upon the equation
of the motion of the gravitational waves, 0,
gg .
At first, lug , , while
x
g
g , , where x are the contravariant
vectors that describe the coordinates of time and space, and , , are the suf-
fixes that indicate those coordinates; while we analyze only one direction of the
space, , , = 0, or 3, where 0 is for time, and 3 is for one direction of the
space. Also, we put uuuu
, where u are contravariant two-vector
tensors and u are covariant two-vector tensors, and uu ; here l are con-
stants, which satisfy 0
llg . Then it was assumed that the gravitational
waves are traveling in the empty space where only the gravitational field exists,
and then this condition leads to
ulluglug
2
1
2
1
, and then we get
ullu
2
1
.
Now, when 0 , we have:
3
0
0
03
3
02
2
01
1
00
0
0
3
0
0 00 uululululuulu
uululuuuuugug
2
1
2
1
2
1
)1( 00300030003
33
00
00 ,
where 10 l , 01 l , 02 l and 13 l .
When 1 , then
3
1
0
13
3
12
2
11
1
10
0
1
3
0
1 00 uululululuulu
0
2
1
)1( 11310131013
33
10
00 uluuuuugug .
When 2 , we have:
3
2
0
23
3
22
2
21
1
20
0
2
3
0
2 00 uululululuulu
0
2
1
)1( 22320232023
33
20
00 uluuuuugug .
When 3 , then we have
3
3
0
33
3
32
2
31
1
30
0
3
3
0
3 00 uululululuulu
uuluuuuugug
2
1
2
1
)1( 33330333033
33
30
00 .
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2019, № 3 12
Thus,
uuuuuuuuuuuuu
2
1
2
1
)()( 33003330030033300300 ,
where 3003 uu . Also, 01
1
11111 lugu , and 02
2
22222 lugu , therefore
332211003300 uuuuuuu ; and 02211 uu .
Also, )(2)(22 330033000033000003 uuuuuuuuuu , because
uuu
2
1
0300 .
Here, 100 g , 1332211 ggg , and
0323130232120131210030201 gggggggggggg .
On the other hand the general formula of the action integral for the waves
moving in one direction is:
32104 ))((
4
1
dxdxdxdxlulululululugxLd
,
where L is Lagrangean that describes the motion of the waves [3]. With the con-
straint, 0L , the general solution of the pseudo-tensor
t that represents the
spin momentum densities of the gravitational waves are:
16
lluuut )
2
1
(
2
1 2 ,
where l is one direction, in which the waves are moving in. Here,
02
02
01
01
33
33
22
22
11
11
00
00
2 22
2
1
uuuuuuuuuuuuuuu
231
31
23
23
12
12
03
03 2
1
2222 uuuuuuuuu
01
1100
0133
3333
3322
2222
2211
1111
1100
0000
00 2 ugguugguugguugguuggu
23
3322
2312
2211
1203
3300
0302
2200
02 2222 ugguugguugguuggu
2
02
2
01
2
33
2
22
2
11
2
00
2
31
1133
31 2)1(2)1(
2
1
2 uuuuuuuuggu
2
12
2
22
2
11
2
3300
2
31
2
23
2
12
2
03 2)(
2
1
2222)1( uuuuuuuuu
2
12
2
2211 2)(
2
1
uuu ;
and here
uggu , 100 g , 111 g , 122 g , 133 g ,
)2(
2
1
2 2
333300
2
00
2
03 uuuuu , )2(
2
1
)(
2
1 2
333300
2
00
2
2300 uuuuuu ,
Calculating energy density and spin momentum density of Moon’s gravitational waves …
Системні дослідження та інформаційні технології, 2019, № 3 13
2
31
2
13
2
01
2
10 uuuu because 01310 uu , and, 2
23
2
02
2
20 uuu because
02320 uu , and from 02211 uu , 1122 uu , and then
2
11
2
11
2
11
2
22
2
11 2)( uuuuu , and finally: 2
1111
2
2211 ))(()( uuuu ,
2
11)2( u so 2
2211
2
11
2
11
2
22
2
11 )(
2
1
)2(
2
1
2 uuuuuu .
And then the spin momentum density of the gravitational waves becomes of
the form:
16 }){(
4
1 2
12
2
2211
0
0 uuut , with 0
0
3
0 tt .
Now assume an infinitesimal rotation operator, R , in the plane of con-
travariant vectors 21xx . If it is applied to any vector, 1A , 2A , it has the effect:
21 ARA , 12 ARA , and 11
2 AAR , so iR must have the eigenvalues
1 when applied to the vector [3]; here, 1iR . So, the operator R makes anti-
symmetric change of the vectors. When we apply this infinitesimal rotation opera-
tor, R , to AAu , the rotations will occur as follows:
121221211211111111 2)(()()( uuuAAAAARAARAAARRu ,
where 1221 uu ;
1122112221212112 )()()()( uuAAAARAAARAAARRu ;
122112122122222222 2)()()()( uuuAAAARAAARAAARRu ;
)()()()()()( 2222111122112211 RAAARARAAARAAAAARuuR
0122112112 AAAAAAAA ;
12211221211222112211 44)()( uAAAAAAAAAAAAAARuuR ;
))(()( 22112211
2 uuRRuuR
121212122211 22)2(2()( RRuuuRRuRuR
)(4)(4)(2)(2 2211112211221122 uuuuuuuu .
Thus, 2211 uu is invariant (constant), while iR has the eigenvalues 2
when applied to 2211 uu or 12u . Therefore, the components of u that contrib-
ute to the momentum density of gravitational waves correspond to spin 2 [3].
Upon the above theory, it was calculated the spin momentum of the gravita-
tional waves by assuming
xl as coordinates of 4 dimensional flat-space (recti-
linear coordinates), 4310 ,,, xxxx , and then, we examined the parity of the u ,
where each element of vector 10 x (time-coordinate), 01 x , 02 x and
rx 3 , and each of those are 123 vector.
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2019, № 3 14
Now },,,{ 3210 xxxxX , and then we calculated the u as shown below:
126
6
103,0243200108,34022
0000
0000
108,34022000.23
XXu .
We see that: 2300 u , 222113121120100201 uuuuuuuuu
0323123 uuu , 6
0330 108,34022 uu , and 12
33 103,02432 u .
And then the rotation operator was applied, R , to u , to see the following:
12122111 200 uuuRu , 0112212 uuRu , 12211222 2uuuRu .
So, 0)( 2211 uuR , 122211 4)( uuuR , and ).(4)( 22112211
2 uuuuR
This result shows that the calculated u , shown above, are consistent with the
report by Dirac [3] about the infinitesimal rotational operator and the spin mo-
mentum of the gravitational waves.
THEORETICAL JUSTIFICATION OF OUR CALCULATIONS
Our equation for calculating the energy densities of gravitational energy
field and gravitational waves is:
2
1
r
cbraYH , where
2
1
r
cbraL , Y is the global temperature. And then we calculated the co-
efficients a , b and c of
2
1
r
cbraL , after giving the constraints:
0
)( 2
a
HE
, 0
)( 2
b
HE
, and 0
)( 2
c
HE
, where )( 2HE is the expected
value of 2H .
Below we show a theoretical justification of this our calculation. In general,
if ),( nn pqff and ),( nn pqgg are arbitrary functions, and
if
nnnn q
g
p
f
p
g
q
f
gf
],[ , and then, for example, ],[],[ mm gUHgg ,
where Mm ,....,1 , which distinguishes independent functions ),( pqm . And if
0],[ Hm , it gives the constraint to find the solutions of the problem. Here now
we assume that, m ’s are the functions that describe the gravitational waves,
where H is named as Hamiltonian, where nq are coordinates, and Nn ,......1 ,
while N is the number of degrees of freedom. Also, LqpH nn , where
),( qqLL is Lagrangean,
t
q
q n
n
, t is time-coordinate,
nn q
L
q
L
dt
d
, and
n
n q
L
p
are momenta. In the theory of special relativity, N is finite; but, in the
Calculating energy density and spin momentum density of Moon’s gravitational waves …
Системні дослідження та інформаційні технології, 2019, № 3 15
theory of general relativity of 4 dimensional curved space, N is infinite; and then,
xx qpL , where the coefficient of xq in the integrand in L is defined to be
momenta xp [2].
The action integral of Born-Infield electro-dynamics is
xdFgI 4)det( , where F give the electromagnetic field. Here, the
coordinates of electrodynamics is rA , where 3,2,1,0r ; and, the related mo-
menta rD are the components of electric induction [2]. rA and rD satisfy
)'(]',[ xxgDA s
r
s
r , where )'( xx are the changes of the coordinates from
r to s , where s
rg is the Kronecker delta function and )'( xx is the delta func-
tion of 'xx . Now, only A remains as Hamiltonian, H . And then,
st
rst
st
rstr AFB ,2
1
, where stA , are differentials of tA , differentiated by the
coordinate vector s , and 1rst when )3,2,1()( rst . Here there are only 3 co-
ordinates, because in electromagnetic dynamics the time-coordinate 0r doesn’t
have meaning. And now, )'(,]',[ xxDB t
rstsr . Then the momentum density
is s
rsr DFK . Also the energy density is:
2
1
2 })({ ut
surt
rssrsr
rs DDFFBBDDK ,
where rs is the metric in three-dimensional surface and rs det2 .
In these calculations Earth’s global temperature Y is assumed to be, nnqp ;
and the coefficients, a , b , and c are translated as np of the Lagrangean, L . Also
here, H is the only Hamiltonian; and now, the energy densities of the gravita-
tional waves and the gravitational field are calculated with
2
1
r
cbraYH .
Here, rba is generalized as rs det2 ; and,
2
1
r
c is generalized as
ut
surt
rssrsr
rs DDFFBBDD )( . Also, 0
)( 2
a
HE
, 0
)( 2
b
HE
, and
0
)( 2
c
HE
are the constraints that we used in our calculations. Here the gener-
alized expression of our constraint 0HX is 0],[ HX , and X represents the
energy densities that includes the gravitational waves ),( pq . Here we have to
note that the number of order of freedom of coordinates is finite, 4N , as we
assumed only 4 vectors, 0x , 1x , 2x , and 3x , of the rectilinear coordinates in
our calculation.
Similarly, in our calculation about the spin momentum of the Gravitational
waves, the generalized form of the pseudo tensor is:
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2019, № 3 16
Lgq
q
L
gt n
n
,
,
)det( is ],[ nqL ,
where, ))((
4
1
lulululululgL , is the La-
grangean that describes the motion of the gravitational waves. And then,
0],[ nqL , gives the constraint to calculate the spin momentum of the gravita-
tional waves, which is followed by calculations we showed above. In addition,
)'(,]',[ xxDB t
rstsr , in the theory of the electromagnetic field is the original
idea of the infinitesimal rotation operator, R , in our analysis shown above. It
changes the variables, but it doesn’t change the physical system, which is the con-
sistent argument that agrees with the theory of relativity [2].
Here we have to report one more aspect of the gravitational waves: the gen-
eral solution of the momentum density of the gravitational waves,
16
lluuut )
2
1
(
2
1 2 , involves both contravariant vectors, x ’s, and
covariant vectors, x ’s. Dirac [3] predicted that the gravitational waves appear
only in one direction. Then, the momentum density of the gravitational wave be-
comes 16 }){(
4
1 2
12
2
2211
0
0 uuut , where 0
0
3
0 tt . These contravariant vec-
tors, x ’s, and covariant vectors, x ’s, are exchanged each other through funda-
mental tensors, g ’s, as we showed as
uggu , and this operation
changes the sign ( ) of the vectors. And the momentum density of gravitational
waves is calculated as the scalar-products of those two different coordinates’ sys-
tems. However, contravariant vectors and covariant vectors are in different coor-
dinates’ systems, and the momentum density can be calculated when two different
coordinates’ systems meet, although the contravariant vectors are not yet observ-
able in the real physical system. This issue may be further investigated for ex-
plaining the negative contribution of the gravitational waves to the gravitational
energy field.
CONCLUSIONS AND RECOMMENDATION
In our previous research, [1], we compared the influences of
r
1
(as a surrogate for
Newton’s gravitational field) and
2
1
r
(as the surrogate for the gravitational
waves’ movement) to Earth’s global temperature, assuming as if they are inde-
pendent variables for the Least Squares Estimation of Classical Regression Mod-
el. Instead in this report, we have calculated the energy density of gravitational
waves in the rectilinear coordinates of time and space (the empty space where
only gravitational field exists). For these calculations we set an action integral in a
rectilinear coordinate system, which linearly combines the gravitational field, the
gravitational waves and Earth’s global temperature, where each of them describes
the field of energy. Then we calculated the coefficients of those energy fields
Calculating energy density and spin momentum density of Moon’s gravitational waves …
Системні дослідження та інформаційні технології, 2019, № 3 17
from the energy densities algebraically with a constraint, in which the derivatives
of the energy density are zero, and as a result we found that the gravitational field
has more effect on Earth’s global temperature, while the energy of Moon’s gravi-
tational waves has a negative contribution to it.
In order to investigate the nature of the negative contribution of gravitational
waves to the Earth’s global temperature, we also examined the spin momentum of
the assumed gravitational waves, in the rectilinear coordinate system. Although
the spin momentum is very small and it doesn’t raise the potential energy in the
theory of quantum mechanics, it must exist on theory [6]. The result or our calcu-
lation indicated that the gravitational waves in our coordinate system had the spin
2. On the other hand in this analysis, we calculated the scalar products of con-
travariant vectors and covariant vectors, while contravariant vectors are not ob-
servable in the real physical field, which leaves the issue for the further inves-
tigation.
REFERENCES
1. Matsuki Y., Empirical Investigation on Influence of Moon’s Gravitational-Field to
Earth’s global temperature / Y. Matsuki, P.I. Bidyuk // System Research & In-
formation Technology, 1/2019, 2019. — N 2. — P. 107–118.
2. Dirac P.A.M. Lectures on Quantum Mechanics. — New York: Belfer Graduate
School of Science, Yeshiva University, 1964. — 87 p.
3. Dirac P.A.M., General Theory of Relativity. Florida State University. — New York:
John Wiley & Sons, 1975. — 69 p.
4. UK Department of Energy and Climate Change (DECC). — Available at:
http://en.openei.org/datasets/dataset/b52057cc-5d38-4630-8395-b5948509f764/
resource/f42998a9-071e-4f96-be52-7d2a3e5ecef3/download/england.surface.
temp1772.2009.xls
5. Moon Distance Calculator – How Close is Moon to Earth? Available at:
https://www.timeanddate.com/astronomy/moon/distance.html?year=1987&n=367
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Clarendon Press, 1958. — 312 p.
Receive 06.02.2019
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From the Editorial Board: the article corresponds completely to submitted manuscript.
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| id | journaliasakpiua-article-183719 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:26:28Z |
| publishDate | 2019 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/ed/d0c73f37921d715249187d16624ccced.pdf |
| spelling | journaliasakpiua-article-1837192019-12-13T15:15:18Z Calculating energy density and spin momentum density of Moon’s gravitational waves in rectilinear coordinates Расчет плотности энергии и плотности спина гравитационных волн Луны в прямолинейных координатах Розрахунок щільності енергії та щільності спіну гравітаційних хвиль Місяця у прямолінійних координатах Matsuki, Yoshio Bidyuk, Petro I. Moon Earth global temperature gravitational field gravitational waves rectilinear coordinates energy density spin momentum density Луна Земля глобальная температура гравитационное поле гравитационные волны прямолинейные координаты плотность энергии плотность спина Місяць Земля глобальна температура гравітаційне поле гравітаційні хвилі прямолінійні координати щільність енергії щільність спина In this research the energy density was calculated and the spin momentum density of Moon’s gravitational waves in the rectilinear coordinates’ system of Moon’s gravity and Earth’s global temperature. At first, we assumed an action principle that combines the gravitational field and gravitational waves, which formulate a closed system, together with Earth’s global temperature. And, then, we calculated the energy densities of those energy field and waves, which are calculated as their variances in the rectilinear coordinates, also to calculate their coefficients and standard errors of the calculated coefficients. The calculated results are consistent with the findings of our previous research [1], which shows the negative contribution of gravitational waves to Earth’s global temperature, while the gravitational field positively contributes to the global temperature. We also calculated spin momentum of Moon’s gravitational waves in the system of rectilinear coordinates. Рассчитано плотность энергии и плотность спина гравитационных волн Луны в прямолинейной системе координат притяжения на поверхности Луны и глобальной температуры Земли. Рассмотрен принцип действия, соединяющий гравитационное поле и гравитационные волны, образующие замкнутую систему, вместе с глобальной температурой Земли. Рассчитано значение плотности энергии этих энергетических полей и волн, которые рассчитывались как дисперсии в прямолинейных координатах, а также рассчитаны их коэффициенты и стандартные погрешности рассчитанных коэффициентов. Результаты расчетов согласовываются с результатами предыдущего исследования, которое показывает негативное влияние гравитационных волн на глобальную температуру Земли, в то время как гравитационное поле влияет на нее положительно. Рассчитан также спин гравитационных волн Луны в прямолинейной системе координат. Розраховано щільність енергії та щільність спіну гравітаційних хвиль Місяця у прямолінійній системі координат тяжіння на поверхні Місяця та глобальної температури Землі. Розглянуто принцип дії, що поєднує гравітаційне поле та гравітаційні хвилі, які утворюють замкнену систему разом із глобальною температурою Землі. Розраховано значення щільності енергії розрахованих енергетичних полів та хвиль як дисперсію у прямолінійних координатах, а також розраховано їхні коефіцієнти та стандартні похибки розрахованих коефіцієнтів. Результати розрахунків узгоджуються з результатами попереднього дослідження, яке демонструє негативний вплив гравітаційних хвиль на глобальну температуру Землі, у той час як гравітаційне поле впливає на неї позитивно. Розраховано також спін гравітаційних хвиль Місяця у прямолінійній системі координат. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2019-10-07 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/183719 10.20535/SRIT.2308-8893.2019.3.01 System research and information technologies; No. 3 (2019); 7-17 Системные исследования и информационные технологии; № 3 (2019); 7-17 Системні дослідження та інформаційні технології; № 3 (2019); 7-17 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/183719/183556 Copyright (c) 2021 System research and information technologies |
| spellingShingle | Місяць Земля глобальна температура гравітаційне поле гравітаційні хвилі прямолінійні координати щільність енергії щільність спина Matsuki, Yoshio Bidyuk, Petro I. Розрахунок щільності енергії та щільності спіну гравітаційних хвиль Місяця у прямолінійних координатах |
| title | Розрахунок щільності енергії та щільності спіну гравітаційних хвиль Місяця у прямолінійних координатах |
| title_alt | Calculating energy density and spin momentum density of Moon’s gravitational waves in rectilinear coordinates Расчет плотности энергии и плотности спина гравитационных волн Луны в прямолинейных координатах |
| title_full | Розрахунок щільності енергії та щільності спіну гравітаційних хвиль Місяця у прямолінійних координатах |
| title_fullStr | Розрахунок щільності енергії та щільності спіну гравітаційних хвиль Місяця у прямолінійних координатах |
| title_full_unstemmed | Розрахунок щільності енергії та щільності спіну гравітаційних хвиль Місяця у прямолінійних координатах |
| title_short | Розрахунок щільності енергії та щільності спіну гравітаційних хвиль Місяця у прямолінійних координатах |
| title_sort | розрахунок щільності енергії та щільності спіну гравітаційних хвиль місяця у прямолінійних координатах |
| topic | Місяць Земля глобальна температура гравітаційне поле гравітаційні хвилі прямолінійні координати щільність енергії щільність спина |
| topic_facet | Moon Earth global temperature gravitational field gravitational waves rectilinear coordinates energy density spin momentum density Луна Земля глобальная температура гравитационное поле гравитационные волны прямолинейные координаты плотность энергии плотность спина Місяць Земля глобальна температура гравітаційне поле гравітаційні хвилі прямолінійні координати щільність енергії щільність спина |
| url | https://journal.iasa.kpi.ua/article/view/183719 |
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