Моделювання кутового моменту гравітаційного поля обертової чорної діри і спін-моменту гравітаційних хвиль
In this research, we simulated the angular momentum of gravitational field of a rotating black hole and the spin momentum of gravitational waves emitted from the black hole. At first, we calculated energy densities of the rotating gravitational field and spinning gravitational waves as the vectors,...
Збережено в:
| Дата: | 2021 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2021
|
| Теми: | |
| Онлайн доступ: | https://journal.iasa.kpi.ua/article/view/236334 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | System research and information technologies |
| Завантажити файл: |  |
Репозитарії
System research and information technologies| _version_ | 1866302736847339520 |
|---|---|
| author | Matsuki, Yoshio Bidyuk, Petro |
| author_facet | Matsuki, Yoshio Bidyuk, Petro |
| author_sort | Matsuki, Yoshio |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2021-07-13T11:01:37Z |
| description | In this research, we simulated the angular momentum of gravitational field of a rotating black hole and the spin momentum of gravitational waves emitted from the black hole. At first, we calculated energy densities of the rotating gravitational field and spinning gravitational waves as the vectors, which were projected on the spherical curved surface of the gravitational field and of the gravitational waves. Then we calculated the angular momentum and the spin momentum as the vectors perpendicular to the curved surface. The earlier research by Paul Dirac, published in 1964, did not select the curved surface to calculate the motion of quantum particles; but, instead, he chose the flat surface to develop the theory of quantum mechanics. However, we pursued the simulation of the gravitational waves in spherical polar coordinates that form the spherical curved surface of the gravitational waves. As a result, we found that a set of anti-symmetric vectors described the vectors that were perpendicular to the spherical curved surface, and with these vectors we simulated the angular momentum of the rotating black hole’s gravitational field and the spin momentum of gravitational waves. The obtained results describe the characteristics of the rotation of a black hole and of spinning gravitational waves. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2021.1.01 |
| first_indexed | 2025-07-17T10:27:13Z |
| format | Article |
| fulltext |
Y. Matsuki, P.I. Bidyuk, 2021
Системні дослідження та інформаційні технології, 2021, № 1 7
TIДC
ПРОГРЕСИВНІ ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ,
ВИСОКОПРОДУКТИВНІ КОМП’ЮТЕРНІ
СИСТЕМИ
UDC 519.004.942
DOI: 10.20535/SRIT.2308-8893.2021.1.01
SIMULATING ANGULAR MOMENTUM OF GRAVITATIONAL
FIELD OF A ROTATING BLACK HOLE AND SPIN MOMENTUM
OF GRAVITATIONAL WAVES
Y. MATSUKI, P.I. BIDYUK
Abstract. In this research, we simulated the angular momentum of gravitational
field of a rotating black hole and the spin momentum of gravitational waves emitted
from the black hole. At first, we calculated energy densities of the rotating gravita-
tional field and spinning gravitational waves as the vectors, which were projected on
the spherical curved surface of the gravitational field and of the gravitational waves.
Then we calculated the angular momentum and the spin momentum as the vectors
perpendicular to the curved surface. The earlier research by Paul Dirac, published in
1964, did not select the curved surface to calculate the motion of quantum particles;
but, instead, he chose the flat surface to develop the theory of quantum mechanics.
However, we pursued the simulation of the gravitational waves in spherical polar
coordinates that form the spherical curved surface of the gravitational waves. As a
result, we found that a set of anti-symmetric vectors described the vectors that were
perpendicular to the spherical curved surface, and with these vectors we simulated
the angular momentum of the rotating black hole’s gravitational field and the spin
momentum of gravitational waves. The obtained results describe the characteristics
of the rotation of a black hole and of spinning gravitational waves.
Keywords: gravitational waves, angular momentum, curvature tensor, stress-energy
tensor, black hole.introduction
INTRODUCTION
Research question
In our previous two researches [1, 2, 3], we reported that the antigravity and anti-
gravitational waves appear when a black hole rotates, but we also reported that
further research is needed to identify the vectors, which are pependicular to the
rotating axis of the gravitational field and the gravitational waves. Then, in this
new research, in order to further investigate this problem, we assumed as if the
“ sin component” to the rotational axis of represents the perpendicular direc-
tion of the curved surface described by spherical polar coordinates.
Theory of movement of the curved surface in the time-space coordinates
Dirac [4] explained two types of coordinate systems that describe time and space:
one is in the flat space-like surface (Fig. 1), and another is the curved space-like
surface (Fig. 2). In each figure three-dimensional surfaces ( 1321 ,,, SSSS in Fig. 1
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2021, № 1 8
and S in Fig. 2) are placed in four-dimensional time-space, where 0x is for time
and 321 ,, xxx are for the space. (Einstein’s special theory of relativity is explained
in Fig.1, while the general theory of relativity [5] is explained in Fig. 2) Dirac [4]
described that Fig. 2 represents a three-dimensional curved surface in a four-
dimensional space-time, which has the property of being everywhere space-like,
and the normal (perpendicular) vector to the suruface must be in the light-cone in
the Fig. 2. Dirac predicted that this perpendicular movement to the curved space
must be physically meaningful.
In our research, we simulated the gravitational field and gravitational waves
in the curved space-like surface using the spherical polar coordinate system so
that we could still use the orthogonal transformation for modelling the rotation of
a black hole. In our previous research [6], we simulated the energy density of the
gravitational field and of the gravitational waves, also with the spin angular mo-
mentum of the gravitational waves on the flat surface; but, not the movements of
the vectors perpendicular to the curved surface. In this article we report the result
of our next research about the simulation of perpendicular component of the
movement of the curvature tensors.
Curvature tensors
In this research, we used the same curvature tensors that we derived for our previ-
ous researches [2, 3], but we reorganized the components in the following for-
mula for the gravitational field:
33
22
11
00
00
00
R
R
R
R ,
where
3/4211 )(18
11
)(3
20
m
R ;
2
2423/10222 cot
sin
4
)(9
140
)(9
28 m
R ,
and
2
2
224323/10233 sin
cot11
sin
4
sin)(9
140
sin)(9
28 m
R ,
Fig. 1. Flat space-like surface
(adapted from Reference [4])
x0
x1x2x3
1S
S3
S2
S1
Fig. 2. Curved space like surface
x0
x1x2x3
S
Simulating angular momentum of gravitational field of a rotating black hole and spin
Системні дослідження та інформаційні технології, 2021, № 1 9
then we formulated the body vector of the black hole: T332211 RRRR .
And for the simulation of the gravitational waves, we used the following
formula, as three diagonal components of a 33 matrix:
For 1 :
3/73/1022 )(9)(81
2
)(3
2
)(9
16
mm
2434323/|4 sin)(9
224
)(9
224
)(
2
)(9
mm
m
.
For 2 :
24 sin
8
sin
cos24
, and for 3 :
2
2
cot32
)(9
64
and similarly we formulated the wave vector: T321 gggg .
The curvature tensors of gravitational waves, which penetrate the boundary
of a black hole [2], are:
+ggg+ggg+g ))2/1(())2/1(( σςη,ησς,
ςη
ρ,ρςη,ηρς,
ςη
σ,
ςηρσ,ςη
βης,gggg+gggggggg βρς,ββ
ρς
σ,βης,βσρς,ββ
ρς
βσης,βρς,ββ
ρς )2/1()2/1()2/1(
βςη,βρσς,ββ
σς
βρςη, )2/1()2/1()2/1( gggg+gggg+gggg βσς,ββ
σς
βης,βρς,σββ,
ρς
βης,βσς,ρββ,
σς
βης,βσς,ββ
σς
ρ, )2/1()2/1( gggg+gggg+ .
Distortion of time and space in strong gravity
We used the same assumption of our previus research [3] for simulating the dis-
tortion of time and space, as shown in Fig. 3 and Fig. 4. In these figures, is a
relative time in the coordinate system, which expands and shrinks depending on
the distance r , where )(rft ; and is the relative distance, which expands
and shrinks depending on the time t , where )(rgt ; and )(rf and )(rg are
functions of r . For the simulation we assumed Case-1: rrf log)( ; and
rerg )( (non-linear); and Case-2: rrf /1)( , and rrg )( (linear).
Fig. 4. Time and distance from the center
of the gravitational field, Case-2 (linear
distortion): )4/1()( rf and rrg )(
Fig. 3. Time and distance from the center
of the gravitational field, Case-1 (non-linear
distortion): rrf log)( and rerg )(
t t
t
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2021, № 1 10
ALGORITHM
We used the same algorithm that we used for our previous research [3] to simu-
late the relative strengths (intensities) of the curvature tensors, which are reflected
by the stress-energy tensor that is placed at the end of the distance r in Fig. 3 and
Fig. 4.
Einstein’s field equation [5] that rules the motion of particles in the gravitational
field is as follows: 0))2/1(( ,
RgR . Then kTRgR )2/1( , where
T is the stress-energy tensor and k is a constant [7]. By calculating, c and
)(cV , as shown below, we estimated the relative strength of each component of
R to the stress-energy tensor in the system of spherical polar coordinates:
)( 332211 XcXcXckTRkTH ,
and 2
3 32211
2 )}({ XcXcXckTH ,
where 1c , 2c , and 3c are the coefficients, which make a column vector c . And
321 XXXX , then H XckT . Then we set the constraint 0HX ,
then 0)( XckTX , where 'X is transpose matrix of X .
Then kTXXcX , kTXXXc 1)( and 12 )(ˆ)( XXcV , where
2)( cV is the variance of the c and )/(ˆ 2 lnee , where MkTe ,
XXXXIM 1)( , n is the number of rows of each column of X (in this
simulation 23n ), l is the number of columns of X , I is a 2323 unit matrix
that holds 1 on all diagonal elements and 0 for the other elements, 1)( XX is the
inverse matrix of XX , and e is the transpose vector of e .
Rotation of the black hole (an object), which contains strong gravity that
distorts time and space
When an object rotates as shown
in Fig. 5, its coordinate system
will be transformed by the trans-
formation matrix D of the Eu-
ler’s angles [7]. For the rotation
around one axis of the ten-
sors of the object’s coordinate
system will be multiplied by the
matrix
100
0cossin
0sincos
D .
And then the curvature ten-
sor R will be transformed to:
Fig. 5. Rotation of an object
r
r
dr
d
z
x
y
Simulating angular momentum of gravitational field of a rotating black hole and spin
Системні дослідження та інформаційні технології, 2021, № 1 11
33
2211
2211
33
22
11
00
0cossin
0sincos
00
00
00
100
0cossin
0sincos
R
RR
RR
R
R
R
RD .
Here the components 22sin R and 11sin R are anti-symmetrical, which
are perpendicular to the rotational axis 3xz for of Fig. 5.
Given the above transformed curvature tensor after the rotation, at first we
calculated the relative strength of the principal moment of the rotation by the
diagonal components of DR , which are
33
22
11
33
22
11
00
0cos0
00cos
00
00
00
R
R
R
R
dR
dR
to formulate )( 333222111 RcdRcdRckTH , then the algorithm follows as
explained above.
And then we also calculated the anti-symmetrical components of DR ,
which are
000
00
00
000
00sin
0sin0
311
322
11
22
dR
dR
R
R
to calculate
0
311
322
33
22
11
dR
dR
dR
dR
dR
to formulate 3221( dRckTH
)3112 dRc , then the same algorithm follows as explained above.
Here,
000
00
00
3
3
d
d
is an infinitesimal rotation operator; while in general;
0
0
0
12
13
23
dd
dd
dd
, according to Reference [7]; but in this our simula-
tion 021 dd , and sin3d . It calculates rotated vector as the cross-
product of R and d ,
122211
311133
233322
3
2
1
33
22
11
33
22
11
dRdR
dRdR
dRdR
d
d
d
R
R
R
dR
dR
dR
dR
.
For the simulation of gravitational waves, simply 11R , 22R and 33R , are re-
placed by the diagonal components of the gravitational waves, and henceforward,
the same algorithm follows.
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2021, № 1 12
SIMULATION
Input data
Time t is set as shown in Fig. 3 for Case-1 and in Fig. 4 for Case-2, with which
its slope to the distance r from the center of the gravitational field is a constant.
For simulating the spatial expansion of the gravitational field we assumed, as if
becomes larger in far distance. On the other hand, for simulating the flow of grav-
itational waves we assumed, that becomes smaller in far distance, as shown in
Fig. 6. For simulating the rotation of the object we set two cases assuming 1 (the
rotation1) and 2 (the rotation 2) also as shown in Fig. 6. With these settings
sin , cos , cot and cos behave like it is shown in Fig. 7.
2
1
1
2
r
Degree
Fig. 6. Angles and for simulating gravitational field and gravitational waves
cot 2
cos 2
sin 1
cos 2
sin 2
cos 1
cos 1
cot 1
cos 1
cos 2
sin 1
sin 2
cos 1
cos 2
cot 1 cot 2
t
Fig. 7. Sin , cos , cot , and cos of the simulated gravitational field
Simulating angular momentum of gravitational field of a rotating black hole and spin
Системні дослідження та інформаційні технології, 2021, № 1 13
In addition, for this simulation we set the stress-energy tensor 1Tk ;
because, the purpose of this simulation is to measure the order of magnitude of
the relative strength of each component of R and the gravitational waves to the
stress-energy tensor.
RESULT
Gravitational field
Fig. 8 (Table 1) shows the relative strengths of the gravitational field of the black
hole, which are the gravitational field energy, projected on the spherical curved
surface, and the angular momentum on the perpendicular vector to the surface, in
Case-1 (non-linear distortion of the time and space) and Case-2 (linear distortion
of time and space). As the rotation becomes more frequent from the rotation 1 to
the rotation 2, the angular momentum (the perpendicular vector) changes from
positive to negative. It means that the direction of the angular momentum chang-
es, depending on the frequency of the rotation of the black hole. On the other
hand, on the curved surface the gravitational field energy is negative (gravity)
before the rotation in Case-1, but it changes to positive (antigravity) in the rota-
tion 1, and then to negative (gravity) again in the rotation 2. It means that the an-
tigravity appears, depending on the frequency of the rotation of the black hole. In
Case-2, the gravitational field energy is positive (but smaller than in Case-1, and
closer to zero) when the black hole doesn’t rotate; while the gravitational field
energy (negative) becomes larger when the black hole rotates faster. The angular
momentum of Case-2 changes as it changes in Case-1.
Fig. 9 (Table 2) and Fig. 10 (Table 3) show the strengths of gravitational
field energy, projected on the spherical curved surface in Case-1 and in Case-2, in
Fig. 8. Gravitational fields on curved surface and on perpendicular direction from the
surface
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2021, № 1 14
3 directions (projected on the spherical curved surface in the coordinates of r , ,
and , which are the generalized coordinates of xx 1 , yx 2 , and zx 3 of
Cartesian coordinate system). In Case-1, only the r direction appears on the sur-
face in all cases (no rotation, the rotation 1 and the rotation 2); while in Case-2, in
addition to the direction of r , the component of appears as the black hole rotates.
Fig. 11 (Table 4) and Fig. 12 (Table 5) show the strengths of the rotation’s
angular momentum in two directions ( r and , which are perpendicular to the ro-
tating axis, ( zx 3 of Fig. 5)). Similar to the gravitational field energy, only the
vector’s component of r appears in Case-1; while the vector’s component of
also appears in Case-2. In both Case-1 and Case-2, as the frequency of the rota-
tion increases from the rotation 1 to the rotation 2, the direction of the momentum
changes from plus to minus. It suggests that the rotation of a black hole reverses
its direction of the momentum when the frequency of the rotation changes.
Fig. 9. Gravitational field energy in 3 directions on the spherical curved surface, Case-1
Fig. 10. Gravitational field energy in 3 directions on the spherical curved surface, Case-2
r
Fig. 11. Rotation momentum of the gravitational field in two directions of r and , Case-1
Fig. 12. Rotation momentum of the gravitational field in two directions of r and , Case-2
Simulating angular momentum of gravitational field of a rotating black hole and spin
Системні дослідження та інформаційні технології, 2021, № 1 15
Gravitational waves
Fig 13 (Table 6) shows the srength of the gravitational waves emitted from the
black hole, which are the energy (projected on the spherical curved surface of the
wave) and the spin momentum of the rotation (projected on the pependicular di-
rection to the surface), in Case-1 and Case-2. On spherical curved surface the en-
ergy of gravitational waves are not affected by the rotation; while on the perpen-
dicular direction the rotational momentum (spin) appears, and it changes its
direction from positive to negative when the frequency of the rotation changes
from the rotation 1 to the rotation 2. It suggests, that the gravitationa waves make
spin as the waves move on the direction of r as the emitter (the black hole) ro-
tates, and it changes its spinning direction when the frequency of the rotation
changes from the rotation 1 to the rotation 2.
Fig. 14 (Table 7) and Fig. 15 (Table 8) show the energy density of gravita-
tional waves, projected on the spherical curved surface in three directions in Case-
1 and Case-2. These figures suggest that the waves have negative energy density
on the direction of r in Case-1; while the negative energy density appears also on
the direction of rotation in Case-2. These figures suggest the appearance of
anti-gravitational waves on the spherical curved surface. (The anti-gravitational
waves must have negative sign [1], while gravitational waves must have positive
sign). This finding is different from our previous report [3], in which either the
gravitational waves or the anti-gravitational waves didn’t appear when the black
hole didn’t rotate. The difference came from the configuration in the algorithm to
formulate )( 332211 XcXcXckTH . In this new research we reorganized
the components of the curvature tensor into three vectors, 1X , 2X and 3X ; while
in our previous report [3] we calculated the relative strength of every component
of the curvature tensor by, )( 2211 nn XcXcXckTH .
Fig. 13. Gravitational waves on curved surface and on perpendicular direction from the
surface
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2021, № 1 16
Fig. 16 (Table 9) and Fig. 17 (Table 10) show the spin momentum of gravi-
tational waves, projected in the directions of r and in Case-1 and Case-2. In
both cases the vector component of appears in positive direction in the rotation
1, and in the negative direction in the rotation 2. These figures suggest that the
gravitational waves changes its direction of spin as the rotation’s speed of the black
hole changes. its dsrection of spin as the rotation’s speed of the black hole changes
PHYSICAL MEANING OF THE RESULTS
The components of cos in this analysis, which are the projections of the gravi-
tational field and the gravitational waves on the spherical curved surface, are
comparable to the vector component on the curved surface shown in Fig. 2 (at the
beginning of this article). And the movements of these vectors are the movement
of the curved surface itself. On the other hand, the vector components of sin are
perpendicular to the curved surface, which is shown in the light-cone of Fig. 2,
and the movement perpendicular to the curved surface must have real physical
meaning. However, the earlier research by Paul Dirac [4] reported that it was
problematic to quantize the movement of a quantum particle (gravitation is one of
r
Fig. 15. Gravitational waves energy density on the curved surface, Case-2
r
Fig. 14. Gravitational waves energy density on the curved surface, Case-1
r
Fig. 17. Spin momentum of gravitational waves in two directions of r and , Case-2
Fig. 16. Spin momentum of gravitational waves in two directions of r and , Case-1
Simulating angular momentum of gravitational field of a rotating black hole and spin
Системні дослідження та інформаційні технології, 2021, № 1 17
them) such as calculating its momentum in the direction of the vectors, perpen-
dicular to the curved surface. Henceforward, the theory of quantum mechanics
was not developed on the curved surface (Fig. 2), but on the flat surface (Fig. 1).
In this research we simulated the gravitational waves on the surface of
spherical polar coordinates as a surrogate of the general curved surface. And we
used the cross product of anti-symmetrical vectors for simulating the components
of sin as the mathematical model of the momentum in the light-cone of the
general curved surface (Fig. 2).
CONCLUSIONS AND RECOMMENDATIONS
In this research we investigated the angular momentum of the gravitational field
and the spin momentum of the gravitational waves by simulating the perpendicu-
lar component that is the “ sin component” to the rotational axis of . We used
the system of the spherical polar coordinates so that we could simulate the rota-
tion with the orthogonal transformation of Euler’s angles.
The result of the simulation shows that the rotating black hole can produce
the antigravity and anti-gravitational waves; and the gravitational waves changes
their spinning direction as the frequency of the black hole’s rotation changes.
These findings are consistent with our previous researches: [1] in which we
reported that the negative flow of gravitational waves must have the clockwise
spin, while the positive flow has the counter-clockwise spin; and [3] in which we
reported that the antigravity and anti-gravitational waves appear when the black
hole rotates.
In this research we used the system of spherical polar coordinates as the
surrogate of the general curved surface; however, in near future, the developed
computer technologies must increase a possibility of using general curved surface
of Einstein’s gravitational field equation also for solving the equation of motion
of quantum particles.
T a b l e 1 . Strengths of gravitational field
Case-1 Case-2
Case On curved
surface
On perpendicular
vector
On curved
surface
On perpendicular
vector
No rotation -78,55 — 1,770 —
Rotation 1 20,00 36,90 -8,178 14,77
Rotation 2 -41,96 -43,00 -21,31 -15,29
T a b l e 2 . Strength of gravitational field on principal axis z , Case-1
Diagonal
Components
of R
C and )(cV
of R before
the rotation
Diagonal
Components
of rotated R
C and )(cV
(Rotation 1)
C and )(cV
(Rotation 2)
11R -78,68 (26,49) 11cos R 20,01 (58,22) -42,05 (52,30)
22R
0,1307
(3,369 210 ) 22cos R
-5,516 310
(8,030 210 )
8,903 210
(6,829 210 )
33R
-6,803 510
(2,557 410 )
33R
-3,290 410
(3,869 410 )
-2,990 410
(5,403 410 )s
The values in the blackets are )(cV
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2021, № 1 18
T a b l e 3 . Strength of gravitational field on principal axis z , Case-2
Diagonal
Components
of R
C and )(cV
of R before
the rotation
Diagonal
Components
of rotated R
C and )(cV
(Rotation 1)
C and )(cV
(Rotation 2)
11R
1,767
(7,364) 11cos R -8,368
(11,85)
-21,79108
(16,36)
22R 2,862 310
(0,1469) 22cos R
0,1924
(0,2427)
0,48490
(0,3369)
33R
1,110 510
(2,224 310 )
33R
-2,854 310
(3,673 310 )
-7,281 310
(5,099 310 )
T a b l e 4 . Strength of the perpendicular vector to the principal axis z of gravi-
tational field, Case-1
Rotation C and )(cV (Rotation 1) C and )(cV (Rotation 2)
3221 dRdx
22sin R
9,077 210
(5,072 210 )
-4,816 210
(5,931 210 )
3112 dRdx
11sin R
36,83
(46,33)
-42,94
(45,44)
T a b l e 5 . Strength of the perpendicular vector of the rotating gravitational field,
Case-2
Rotation C and )(cV (Rotation 1) C and )(cV (Rotation 2)
3221 dRdx
22sin R
0,28213
(0,2621)
-0,22852
(0,2597)
3112 dRdx
11sin R
14,48
(16,65)
-15,06
(16,69)
T a b l e 6 . Strengths of gravitational waves
Case-1 Case-2
Case On curved
surface
On perpendicular
vector
On curved
surface
On perpendicular
vector
No rotation -28,13 — -0,5738 —
Rotation 1 -28,21 1,125 210 -0,5396 1,341
Rotation 2 -31,84 -1,505 210 -0,6123 -1,113
T a b l e 7 . Strength of gravitational waves, Case-1
Components of
gravitational tensor
C and )(cV of R
before the rotation
C and )(cV
(After Rotation 1)
C and )(cV
(After Rotation 2)
1xx component
-28,06
(14,20)
-28,15
(15,52)
-31,84
(19,70)
2xy component
-4,594 410
(2,398 410 )
5,535 410
(2,987 410 )
1,167 310
(1,181 310 )
3xz component
-6,855 210
(3,083 210 )
-5,955 210
(2,711 210 )
-8,950 310
(7,179 310 )
Simulating angular momentum of gravitational field of a rotating black hole and spin
Системні дослідження та інформаційні технології, 2021, № 1 19
T a b l e 8 . Strength of gravitational waves. Case-2
Components of
gravitational tensor
C and )(cV of R
before the rotation
C and )(cV
(After Rotation 1)
C and )(cV
(After Rotation 2)
1xx component
-0,4164
(0,2513)
-0,3967
(0,2734)
-0,4850
(0,3506)
2xy component
-1,116 310
(4,614 410 )
-1,009 310
(4,484 410 )
-8,918 410
(4,123 410 )
3xz component
-0,1562
(6,097 210 )
-0,1419
(5,927 210 )
-0,1264
(5,444 210 )
T a b l e 9 . Strength of the perpendicular vector to the principal axis z of gravi-
tational waves. Case-1
Rotation C and )(cV (Rotation 1) C and )(cV (Rotation 2)
3221 dRdx
22sin R
-9,438 410
(5,800 310 )
6,081 310
(3,813 310 )
3112 dRdx
11sin R
1,125 210
(1,426 210 )
-1,505 210
(1,027 210 )
T a b l e 1 0 . Strength of the perpendicular vector of the rotating gravitational waves.
Case-2
Rotation C and )(cV (Rotation 1) C and )(cV (Rotation 2)
3221 dRdx
22sin R
1,025 210
(5,865 310 )
-5,437 310
(4,966 310 )
3112 dRdx
11sin R
1,33031
(1,053)
-1,10749
(0,8583)
REFERENCES
1. Y. Matsuki and P.I. Bidyuk, “Analysis of negative flow of gravitational waves”, Sys-
tem Research & Information Technology, no. 4, pp. 7–18, 2019.
2. Y. Matsuki and P.I. Bidyuk, “Numerical simulation of gravitational waves from a
black hole, using curvature tensors, System Research & Information Technology,
no. 1, pp. 54–67, 2020.
3. Y. Matsuki and P.I. Bidyuk, “Simulating the rotation of a black hole and antigrav-
ity”, System Research & Information Technology, no. 3, pp. 124–137, 2020.
4. P.A.M. Dirac, Lectures on quantum mechanics; originally published by the Belfer
Graduate School of Science. Yeshiva University, New York, 1964, 90 p.
5. P.A.M. Dirac, General Theory of Relativity. Florida University, A Wiley-
Interscience Publication, John Wiley & Sons, New York, 1975, 70 p.
6. Y. Matsuki and P.I. Bidyuk, “Calculating energy density and spin momentum den-
sity of moon’s gravitational waves in rectilinear coordinates”, System Research &
Information Technology, no. 3, pp. 7–17, 2019.
7. H. Goldstein, C.P. Poole, and J.L. Safko, Classical Mechanics. 3rd edition; published
by Pearson Education Inc., 2002, 690 p.
Received 02.05.2020
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2021, № 1 20
INFORMATION ON THE ARTICLE
Petro I. Bidyuk, ORCID: 0000-0002-7421-3565, Educational and Scientific Complex
“Institute for Applied System Analysis” of the National Technical University of Ukraine
“Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: pbidyuke_00ukr.net
Yoshio Matsuki, ORCID: 0000-0002-5917-8263, National University of Kyiv-Mohyla
Academy, Ukraine, e-mail: matsuki@wdc.org.ua
МОДЕЛЮВАННЯ КУТОВОГО МОМЕНТУ ГРАВІТАЦІЙНОГО ПОЛЯ
ОБЕРТОВОЇ ЧОРНОЇ ДІРИ І СПІН-МОМЕНТУ ГРАВІТАЦІЙНИХ ХВИЛЬ /
Й. Мацукі, П.І. Бідюк
Анотація. Змодельовано момент імпульсу гравітаційного поля обертової
чорної діри і спін-моменту гравітаційних хвиль, що випромінюються з чорної
діри. Спочатку обчислено питому енергію обертового гравітаційного поля і
спін-моменту гравітаційних хвиль як векторів, що проєктуються на сферичну
криволінійну поверхню гравітаційного поля та гравітаційних хвиль.Обчислено
момент імпульсу та спін-момент як вектори, перпендикулярні до кри-
волінійної поверхні. У своєму дослідженні, опублікованому в 1964 р., Поль
Дірак обрав не криволінійну поверхню для обчислення руху квантових части-
нок, а плоску поверхню для розроблення теорії квантової механіки. У цій ро-
боті зроблено спробу змоделювати гравітаційні хвилі у сферичних полярних
координатах, які утворюють сферичну криволінійну поверхню гравітаційних
хвиль. З’ясовано, що множина антисиметричних векторів описує вектори,
перпендикулярні до сферичної криволінійної поверхні; з такими векторами
змодельовано момент імпульсу гравітаційного поля обертової чорної діри і
спін-момент гравітаційних хвиль. Отримані результати описують характери-
стики обертання чорної діри та обертання гравітаційних хвиль.
Ключові слова: гравітаційні хвилі, кутовий момент, тензор кривизни, тензор
енергії напруження, чорна діра.
МОДЕЛИРОВАНИЕ УГЛОВОГО МОМЕНТА ГРАВИТАЦИОННОГО ПОЛЯ
ВРАЩАЮЩЕЙСЯ ЧЕРНОЙ ДЫРЫ И СПИН-МОМЕНТА ГРАВИТАЦИОННЫХ
ВОЛН / Й. Мацуки, П.И. Бидюк
Аннотация. Смоделирован момент импульса гравитационного поля вращаю-
щейся черной дыры и вращающего момента гравитационных волн, излучаю-
щихся из черной дыры. Сначала вычислена удельная энергия вращающегося
гравитационного поля и вращающих гравитационных волн как векторов, про-
ецирующихся на сферическую криволинейную поверхность гравитационного
поля и гравитационных волн. Вычислены момент импульса и вращающий мо-
мент как векторы, перпендикулярные к криволинейной поверхности. В своем
исследовании, опубликованном в 1964 г., Поль Дирак выбрал не криволиней-
ную поверхность для вычисления движения квантовых частиц, а плоскую по-
верхность для разработки теории квантовой механики. В этой работе предпри-
нята попытка смоделировать гравитационные волны в сферических полярных
координатах, которые образуют сферическую криволинейную поверхность
гравитационных волн. В результате выяснено, что множество антисимметрич-
ных векторов описывает векторы, перпендикулярные к сферической криволи-
нейной поверхности; с такими векторами смоделированы момент импульса
гравитационного поля вращающейся черной дыры и спин-момент гравитаци-
онных волн. Полученные результаты описывают характеристики вращения
черной дыры и вращения гравитационных волн.
Ключевые слова: гравитационные волны, угловой момент, тензор кривизны,
тензор энергии напряжения, черная дыра.
|
| id | journaliasakpiua-article-236334 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:27:13Z |
| publishDate | 2021 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/12/a7942fb4ff114d63a0520c40c169dc12.pdf |
| spelling | journaliasakpiua-article-2363342021-07-13T11:01:37Z Simulating angular momentum of gravitational field of a rotating black hole and spin momentum of gravitational waves Моделирование углового момента гравитационного поля вращающейся черной дыры и спин-момента гравитационных волн Моделювання кутового моменту гравітаційного поля обертової чорної діри і спін-моменту гравітаційних хвиль Matsuki, Yoshio Bidyuk, Petro гравитационные волны угловой момент тензор кривизны тензор энергии напряжения черная дыра гравітаційні хвилі кутовий момент тензор кривизни тензор енергії напруження чорна діра gravitational waves angular momentum curvature tensor stress-energy tensor black hole In this research, we simulated the angular momentum of gravitational field of a rotating black hole and the spin momentum of gravitational waves emitted from the black hole. At first, we calculated energy densities of the rotating gravitational field and spinning gravitational waves as the vectors, which were projected on the spherical curved surface of the gravitational field and of the gravitational waves. Then we calculated the angular momentum and the spin momentum as the vectors perpendicular to the curved surface. The earlier research by Paul Dirac, published in 1964, did not select the curved surface to calculate the motion of quantum particles; but, instead, he chose the flat surface to develop the theory of quantum mechanics. However, we pursued the simulation of the gravitational waves in spherical polar coordinates that form the spherical curved surface of the gravitational waves. As a result, we found that a set of anti-symmetric vectors described the vectors that were perpendicular to the spherical curved surface, and with these vectors we simulated the angular momentum of the rotating black hole’s gravitational field and the spin momentum of gravitational waves. The obtained results describe the characteristics of the rotation of a black hole and of spinning gravitational waves. Смоделирован момент импульса гравитационного поля вращающейся черной дыры и вращающего момента гравитационных волн, излучающихся из черной дыры. Сначала вычислена удельная энергия вращающегося гравитационного поля и вращающих гравитационных волн как векторов, проецирующихся на сферическую криволинейную поверхность гравитационного поля и гравитационных волн. Вычислены момент импульса и вращающий момент как векторы, перпендикулярные к криволинейной поверхности. В своем исследовании, опубликованном в 1964 г., Поль Дирак выбрал не криволинейную поверхность для вычисления движения квантовых частиц, а плоскую поверхность для разработки теории квантовой механики. В этой работе предпринята попытка смоделировать гравитационные волны в сферических полярных координатах, которые образуют сферическую криволинейную поверхность гравитационных волн. В результате выяснено, что множество антисимметричных векторов описывает векторы, перпендикулярные к сферической криволинейной поверхности; с такими векторами смоделированы момент импульса гравитационного поля вращающейся черной дыры и спин-момент гравитационных волн. Полученные результаты описывают характеристики вращения черной дыры и вращения гравитационных волн. Змодельовано момент імпульсу гравітаційного поля обертової чорної діри і спін-моменту гравітаційних хвиль, що випромінюються з чорної діри. Спочатку обчислено питому енергію обертового гравітаційного поля і спін-моменту гравітаційних хвиль як векторів, що проєктуються на сферичну криволінійну поверхню гравітаційного поля та гравітаційних хвиль.Обчислено момент імпульсу та спін-момент як вектори, перпендикулярні до криволінійної поверхні. У своєму дослідженні, опублікованому в 1964 р., Поль Дірак обрав не криволінійну поверхню для обчислення руху квантових частинок, а плоску поверхню для розроблення теорії квантової механіки. У цій роботі зроблено спробу змоделювати гравітаційні хвилі у сферичних полярних координатах, які утворюють сферичну криволінійну поверхню гравітаційних хвиль. З’ясовано, що множина антисиметричних векторів описує вектори, перпендикулярні до сферичної криволінійної поверхні; з такими векторами змодельовано момент імпульсу гравітаційного поля обертової чорної діри і спін-момент гравітаційних хвиль. Отримані результати описують характеристики обертання чорної діри та обертання гравітаційних хвиль. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2021-07-13 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/236334 10.20535/SRIT.2308-8893.2021.1.01 System research and information technologies; No. 1 (2021); 7-20 Системные исследования и информационные технологии; № 1 (2021); 7-20 Системні дослідження та інформаційні технології; № 1 (2021); 7-20 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/236334/234842 |
| spellingShingle | гравітаційні хвилі кутовий момент тензор кривизни тензор енергії напруження чорна діра Matsuki, Yoshio Bidyuk, Petro Моделювання кутового моменту гравітаційного поля обертової чорної діри і спін-моменту гравітаційних хвиль |
| title | Моделювання кутового моменту гравітаційного поля обертової чорної діри і спін-моменту гравітаційних хвиль |
| title_alt | Simulating angular momentum of gravitational field of a rotating black hole and spin momentum of gravitational waves Моделирование углового момента гравитационного поля вращающейся черной дыры и спин-момента гравитационных волн |
| title_full | Моделювання кутового моменту гравітаційного поля обертової чорної діри і спін-моменту гравітаційних хвиль |
| title_fullStr | Моделювання кутового моменту гравітаційного поля обертової чорної діри і спін-моменту гравітаційних хвиль |
| title_full_unstemmed | Моделювання кутового моменту гравітаційного поля обертової чорної діри і спін-моменту гравітаційних хвиль |
| title_short | Моделювання кутового моменту гравітаційного поля обертової чорної діри і спін-моменту гравітаційних хвиль |
| title_sort | моделювання кутового моменту гравітаційного поля обертової чорної діри і спін-моменту гравітаційних хвиль |
| topic | гравітаційні хвилі кутовий момент тензор кривизни тензор енергії напруження чорна діра |
| topic_facet | гравитационные волны угловой момент тензор кривизны тензор энергии напряжения черная дыра гравітаційні хвилі кутовий момент тензор кривизни тензор енергії напруження чорна діра gravitational waves angular momentum curvature tensor stress-energy tensor black hole |
| url | https://journal.iasa.kpi.ua/article/view/236334 |
| work_keys_str_mv | AT matsukiyoshio simulatingangularmomentumofgravitationalfieldofarotatingblackholeandspinmomentumofgravitationalwaves AT bidyukpetro simulatingangularmomentumofgravitationalfieldofarotatingblackholeandspinmomentumofgravitationalwaves AT matsukiyoshio modelirovanieuglovogomomentagravitacionnogopolâvraŝaûŝejsâčernojdyryispinmomentagravitacionnyhvoln AT bidyukpetro modelirovanieuglovogomomentagravitacionnogopolâvraŝaûŝejsâčernojdyryispinmomentagravitacionnyhvoln AT matsukiyoshio modelûvannâkutovogomomentugravítacíjnogopolâobertovoíčornoídíriíspínmomentugravítacíjnihhvilʹ AT bidyukpetro modelûvannâkutovogomomentugravítacíjnogopolâobertovoíčornoídíriíspínmomentugravítacíjnihhvilʹ |