Імітаційне моделювання обертання чорної діри та антигравітації
In this article, we show that rotation of a black hole can create antigravity and anti-gravitational waves, given that there is a strong gravity in the black hole, which distorts time and space. At first, we derived the curvature tensors upon Einstein’s field equation, using spherical polar coordina...
Gespeichert in:
| Datum: | 2020 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2020
|
| Schlagworte: | |
| Online Zugang: | https://journal.iasa.kpi.ua/article/view/221372 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | System research and information technologies |
| Завантажити файл: |  |
Institution
System research and information technologies| _version_ | 1867334409457238016 |
|---|---|
| author | Matsuki, Yoshio Bidyuk, Petro I. |
| author_facet | Matsuki, Yoshio Bidyuk, Petro I. |
| author_institution_txt_mv | [
{
"author": "Yoshio Matsuki",
"institution": "The Laboratory for Econometrics and Forecasting at the World Data Center for Geoinformatics and Sustainable Development, the National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
},
{
"author": "Petro I. Bidyuk",
"institution": "Educational and Scientific Complex \"Institute for Applied System Analysis\" of the National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
}
] |
| author_sort | Matsuki, Yoshio |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2021-01-19T12:18:25Z |
| description | In this article, we show that rotation of a black hole can create antigravity and anti-gravitational waves, given that there is a strong gravity in the black hole, which distorts time and space. At first, we derived the curvature tensors upon Einstein’s field equation, using spherical polar coordinates, and then calculated the coefficients of the curvature tensors to simulate the strength of each component of the tensors. It is assumed that the stress-energy tensor, which is located outside of the black hole, can reflect the strength of the gravitational field and the gravitational waves. As the result, we concluded that, if the time and space are distorted in the black hole, the rotation can create antigravity and the anti-gravitational waves. In addition, the result of the simulation shows that the antigravity positively contributes to the stress-energy tensor, which may expand the size of the Universe. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2020.3.09 |
| first_indexed | 2025-07-17T10:27:00Z |
| format | Article |
| fulltext |
Y. Matsuki Y., P.I. Bidyuk, 2020
124 ISSN 1681–6048 System Research & Information Technologies, 2020, № 3
TIДC
МАТЕМАТИЧНІ МЕТОДИ, МОДЕЛІ,
ПРОБЛЕМИ І ТЕХНОЛОГІЇ ДОСЛІДЖЕННЯ
СКЛАДНИХ СИСТЕМ
UDC 519.004.942
DOI: 10.20535/SRIT.2308-8893.2020.3.09
SIMULATING THE ROTATION OF A BLACK HOLE
AND ANTIGRAVITY
Y. MATSUKI, P.I. BIDYUK
Abstract. In this article we show that rotation of a black hole can create antigravity
and anti-gravitational waves, given that there is a strong gravity in the black hole,
which distorts time and space. At first, we derived the curvature tensors upon Ein-
stein’s field equation, using spherical polar coordinates, and then calculated the co-
efficients of the curvature tensors to simulate the strength of each component of the
tensors. It is assumed that the stress-energy tensor, which is located outside of the
black hole, can reflect the strength of the gravitational field and the gravitational
waves. As the result, we concluded that, if the time and space are distorted in the
black hole, the rotation can create antigravity and the anti-gravitational waves. In
addition, the result of the simulation shows that the antigravity positively contributes
to the stress-energy tensor, which may expand the size of the Universe.
Keywords: antigravity, curvature tensor, stress-energy tensor, Einstein’s field equation.
INTRODUCTION (RESEARCH QUESTION)
In our previous two researches [1, 2], we reported as follows: the negative flow of
gravitational waves (anti-gravitational waves) must be described by the expres-
sion: 0,
gg , while Dirac [3] predicted that, 0,
gg , describes
the gravitational waves. This means that the negative waves move backward from
the direction of the positive flow of the waves. Usually the positive flow and the
negative flow should be balanced; therefore, neither of the positive flow nor nega-
tive flow of gravitational waves is observable. However, when a star moves, the
movement of the mass of the star breaks the balance; and then gravitational waves
of both positive and negative flows appear [1]. Upon this conclusion, we investi-
gated the curvature tensors of gravitational waves that are emitted from a black hole
and found that the tensors of the gravitational waves from a black hole share the same
mathematical forms with the tensors of gravitational field of the black hole [2].
And then, we made the next research to investigate the effect of rotation of
the black hole, assuming that the rotation of the black hole breaks the balance of
positive and negative flows so that anti-gravitational waves would appear. We
also examined, whether or not, the antigravity appears when the black hole rotates
and this antigravity creates the energy that may expand size of the Universe to
larger scale. This article reports the results of these investigations.
Simulating the rotation of a black hole and antigravity
Системні дослідження та інформаційні технології, 2020, № 3 125
CURVATURE TENSORS FOR SIMULATION
Gravitational field
According to Einstein and Dirac [3], the gravitational field is described by the
curvature tensors:
,,R ,
where
)(
2
1
,,,
ggggg . (1)
Here, g , are the fundamental tensors that describe the curvature of the
4-dimensional space in spherical polar coordinates, which is diagonal and sym-
metric as shown below:
33323130
23222120
13121110
03020100
gggg
gggg
gggg
gggg
g
23
4
2
3
4
2
3
2
sin)(000
0)(00
00
)(
2
0
0001
m
.
And it makes the geodesic of the kind:
)(
2
1
,,,
ggggg .
Therefore, the equation (1) becomes like this:
g
)(
2
1
,,,
gggg , where
x
g
g , , and x , is the vector in
-th coordinate.
Then, we derived all the components of, R , and then according to Ein-
stein’s rule (
RR ), summated them to obtain:
200
)(9
11
R ;
21001
)(9
4
RR ,
3/4211
)(18
11
)(3
20
m
R ;
2
2423/10222 cot
sin
4
)(9
140
)(9
28 m
R ;
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2020, № 3 126
2
2
2243
23
10
2
33
sin
cot11
sin
4
sin)(9
140
sin)(9
28 m
R .
Here, and m are constants, where, 3/2)22/3( m . All other, 0R .
The non-diagonal components, 01R and 10R , appear because and are not
independent. However, in this research only the space components of the curva-
ture tensors, 11R , 22R , and 33R , are considered.
Gravitational waves
The curvature tensors of gravitational waves, which penetrate the boundary of
a black hole [2], are:
)
2
1
()
2
1
( ,,,,,,, ggggggg
,,,, 2
1
2
1
gggggggg
,,,,,, 2
1
2
1
gggggggg
,,,, 2
1
2
1
gggggggg
,,,,,, 2
1
2
1
gggggggg .
When 0 :
23/1023/1023/4 sin)(9
112
)(9
112
)(9
4
m
;
When 1 :
3/73/1022 )(9)(81
2
)(3
2
)(9
16
mm
2434323/4 sin)(9
224
)(9
224
)(
2
)(9
mm
m
;
When 2 :
24 sin
8
sin
cos24
, when, 3 :
2
2
cot32
)(9
64
.
Distortion of time and space in strong gravity
Using the curvature tensors, Dirac [3] invented a coordinate system that describes
the gravitational field from the center of strong gravity in a black hole, in which
time and space are distorted by affecting each other. He suggested that if we
travel toward the center of the strong gravity, it takes infinite time to reach the
center. Upon this Dirac’s prescription, we assumed that the time and the dis-
tance between the center of the gravitational field and the edge of the Universe
are as shown in Fig. 1 and Fig. 2.
Simulating the rotation of a black hole and antigravity
Системні дослідження та інформаційні технології, 2020, № 3 127
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 rela-
tive distance, which expands and shrinks depending on the time t , where
)(rgt ; and )(rf , and )(rg are functions of r . For the simulation, we as-
sumed Case 1: rrf log)( , and, rerg )( ; and Case 2:
r
rf
1
)( , and rrg )( .
Note: r is the distance from the center of strong gravity, t is the time to travel on the
distance, f and g are given functions, and )(rft ; and )(rgt .
ALGORITHM
Einstein’s field equation [3] 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 [4]. Then, we propose the following
algorithm to calculate the relative intensities of the components of curvature ten-
sors:
)( 2211 ll XcXcXckTRkTH ,
and
2
l2211
2 )}({ lXcXcXckTH ,
where lcc ,,1 are the coefficients, that create a column vector, c .
And, ][ 21 lXXXX , then H XckT . Then we set the con-
straint, 0HX , then 0)( XckTX , where X is transpose matrix of X .
Then, kTXXcX , kTXXXc 1)( , and 12 )()( XXcV , where
2)( cV is the variance of the c , and )/('2 lnee , where kTMe ,
XXXXIM 1)( ; n is the number of rows of each column of X (in this
Fig. 1. Time and distance from the center
of the gravitational field, Case-1 (non-linear
distortion): rrf log)( and rerg )(
Fig. 2. Time and distance from the center
of the gravitational field, Case-2 (linear
distortion): )4/1()( rf and rrg )(
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2020, № 3 128
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 . By calculating c and
)(cV , we estimated the relative strength of each component of, R , to the stress-
energy tensor in the system of spherical polar coordinates.
Rotation of the object that contains strong gravity
When an object rotates as shown in Fig. 3, its coordinate system will be trans-
formed by the transformation matrix D of the Euler’s angles [4]. For the rotation
around one axis, the tensors of the object’s coordinate system will be multiplied
by:
100
0cossin
0sincos
D .
And then the curvature tensor will be transformed to the following form:
33
22
11
00
00
00
100
0cossin
0sincos
R
R
R
DR
33
2211
2211
00
0cossin
0sincos
R
RR
RR
.
The components of R before
and after the rotation are shown in
Tables 1 and 2.
For the simulation, we used the
components, 11cos R , 22cos R and
33R , which correspond to the coordi-
nates that describe the space coordi-
nates, , and . The components of
33R doesn’t change by the rotation,
under the operation of RD , because
133 D . We selected these three diago-
nal components for calculating the co-
efficients of the curvature tensors with
the algorithm mentioned above, which
simulates the relative strength of each
components of the curvature tensor to
the stress-energy tensor. However, we
didn’t use the non-diagonal components, 11sin R and 22sin R , because these
are perpendicular to the diagonal components, therefore do not contribute to the
stress-energy tensor.
Fig. 3. Rotation of an object
Simulating the rotation of a black hole and antigravity
Системні дослідження та інформаційні технології, 2020, № 3 129
SIMULATION
Input data
Time t is set as shown in Fig. 1 for Case-1, and in Fig. 2 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. 4. For simulating the rotation of the object, we set two cases, assuming 1
(Rotation1) and 2 (Rotation 2) also as shown in Fig.4. With these settings, sin ,
cos , cot , and cos of the gravitational field behave like as shown in Fig. 5.
Note: 1 : for gravitational field, 2 : for gravitational waves, 1 : for the rotation 1,
2 : for the rotation 2
2
1
1
2
r
Degree
Fig. 4. 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. 5. Sin , cos , cot , and cos of the simulated gravitational field
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2020, № 3 130
In addition, for this simulation, we set the stress-energy tensor kT to be 1;
because, the purpose of this simulation is to measure the order of magnitude of
the relative strength of each component of, R , to the stress-energy tensor.
RESULTS
Gravitational field
The results of simulation for the gravitational filed are shown in Table 1 for
Case-1, and Table 2 for Case-2. The both Tables show the calculated coefficients
of the simulation with no rotation, with the rotation 1 and the rotation 2.
T a b l e 1 . Results of the simulation of gravitational field, Case-1
Components
of R
c and )(cV
of R before
the rotation
Components
of RD
c and )(cV
(Rotation 1)
c and )(cV
(Rotation 2)
2)(
1
2,902 210
(1,875 410 )
2)(
cos
-4,255 510
(5,078 i 510 )
1,783 510
(7,302 510 )
3/4)(
1
3,496 310
(2,573 310 )
3/4)(
cos
1,390 410
(9,997 310 )
-6,495 310
(2,272 410 )
3/10)(
1
-1,488 210
(1,064 710 )
3/10)(
cos
-2,768 710
(1,608i 810 )
-2,996 710
(1,004 810 )
4)(
1
-2,623 310
(1,808 i 810 )
4)(
cos
-2,676 810
(8,577 i 810 )
1,351 810
(4,202 810 )
2sin
1
1,000
(6,252 210 )
2sin
cos
0,7787
(0,3866)
0,2913
(0,4413)
2cot
-1,000
(0,5245)
2cotcos
-2,788
(0,7892)
-0,7126
(1,432)
24 sin)(
1
-55,86
(3,877 i 610 ) 24 sin)(
1
* -6,454 610
(2,594 i 710 )
-6,129 510
(2,190 610 )
23/10 sin)(
1
-38,63
(2,688 i 610 )
23/10 sin)(
1
*
-4,436 610
(1,736 i 710 )
-1,686 410
(2,473 510 )
2
2
sin
cot
-2,773 710
(0,2009)
2
2
sin
cot
* 2,057 210
(0,2481 i)
0,2367
(0,1400)
The values in the brackets are )(cV . For example, 1,808 i 810 = 161027.3 .
* This component corresponds to the coordinate of the axis of the rotation, therefore
cos is not multiplied.
Simulating the rotation of a black hole and antigravity
Системні дослідження та інформаційні технології, 2020, № 3 131
In Case-1 (non-linear distortion of the time and space), the coefficient, c, of
2
2
sin
cot
changes its sign from minus to plus after the rotation of 1 (the Rota-
tion 1) and of 2 (the Rotation 2). The gravity must be negative, and it is so to
the stress-energy tensor when it doesn’t rotate, but it becomes positive to the
stress-energy tensor after the rotations. This result means that the antigravity ap-
pears after the rotation.
T a b l e 2 . Results of the simulation of gravitational field, Case-2
Components of
R
c and )(cV of
R before the
rotation
Components of
RD
c and )(cV
(Rotation 1)
c and )(cV
(Rotation 2)
2)(
1
-8,518 310
(1,896 210 )
2)(
cos
1,278 410
(4,437 310 )
5,473 310
(3,900 310 )
3/4)(
1
1,217 310
(2,820 310 )
3/4)(
cos
-2,182 310
(7,573 210 )
-9,831 210
(6,530 210 )
3/10)(
1
0,1086
(0,2162) 3/10)(
cos
-6,724 410
(2,833 410 )
-3,707 410
(2,353 410 )
4)(
1
-0,2701
(0,5121) 4)(
cos
9,317 410
(4,145 410 )
4,968 410
(3,173 410 )
2sin
1
1,000
(1,864 510 )
2sin
cos
16,76
(5,815)
8,595
(4,914)
2cot
-1,000
(2,679 510 )
2cotcos
-33,90
(9,405)
-9,779
(9,921)
24 sin)(
1
-2,229 410
(4,083 410 ) 24 sin)(
1
* -1,052 310
(5,370 210 )
-6,204 210
(4,183 210 )
23/10 sin)(
1
-3,371 310
(5,829 310 )
23/10 sin)(
1
*
-2,856 210
(1,743 210 )
-3,109 210
(2,104 210 )
2
2
sin
cot
-1,707 810
(6,560 810 )
2
2
sin
cot
*
0,1293
(3,106 210 )
8,552 310
(4,746 210 )
The values in the brackets are )(cV . For example, 1,808 i 810 = 161027.3 .
* This component corresponds to the coordinate of the axis of the rotation, therefore
cos is not multiplied.
In addition, the coefficients of
4)(
1
and
2
2
sin
cot
change these signs from
minus to plus, only for the rotation of 2 in Case-1. And in Case-2 (linear distor-
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2020, № 3 132
tion of time and space), only the coefficients of
2)(
1
,
4)(
1
, and
2
2
sin
cot
,
change these signs from minus to plus after the both rotations of, 1 , and 2 .
The summation of each of the positive coefficients and the negative coeffi-
cients are shown in Table 5, and in Fig. 6 for Case-1, and Fig. 7 for Case-2. In
case of non-linear distortion of time and space (Case-1), antigravity appears after
the Rotation 2 in Case-1; while in case of linear distortion of time and space
(Case-2), the antigravity appears after both of the Rotation 1 and 2.
Gravitational waves
The results of simulation for the gravitational waves are shown in Table 3 for
Case-1 and Table 4 for Case-2. For the gravitational waves the coefficients of the
Fig. 6. Gravity and antigravity (Case – 1: non-linear distortion of time and space)
Fig. 7. Gravity and antigravity (Case – 2: linear distortion of time and space)
Simulating the rotation of a black hole and antigravity
Системні дослідження та інформаційні технології, 2020, № 3 133
curvature tensors are positive to the stress-energy tensor; while the negative coef-
ficients represent the anti-gravitational waves.
T a b l e 3 . Results of the simulation of gravitational waves, Case-1
Components of
gravitational
waves before
the rotation
c and )(cV
of gravita-
tional waves
Components of
gravitational waves
after the rotation
c and )(cV
Rotation 1
c and )(cV
Rotation 2
2)(
1
1,200 410
(6,038 810 )
2)(
cos
1,185 610
(3,619 1010 )
-1,210 610
(3,772 i 910 )
3/4)(
1
-84,48
(5,520 610 ) 3/4)(
cos
-1,723 410
(7,669)
9,443 310
(8,600 i 610 )
4)(
1
1,360 710
(9,263 i 1110 )
4)(
cos
1,450 910
(6,839 i 1310 )
9,839 810
(5,481 i 1210 )
2sin
1
1,001
(1,747 210 )
2sin
cos
0,1496
(1,703 310 )
2,997 210
(1,422 210 )
2cot
-1,001
(1,877 210 )
2cot *
8,722 210
(6,337 210 )
8,884 310
(11,21)
24 sin)(
1
-6,752 610
(4,939 i 1110 )
24 sin)(
cos
-7,830 810
(4,291 i 1310 )
-8,148 810
(4,387 i 1210 )
23/10 sin)(
1
1,476 710
(7,651 i 1110 )
23/10 sin)(
cos 1,293 910
(2,133 i 1310 )
-3,424 810
(2,597 i 1210 )
3/10)(
1
-1,544 710
(7,783 i 1110 )
3/10)(
cos
-1,335 910
(1,668 i 1310 )
4,209 810
(2,913 i 1210 )
3/7)(
1
-4,509 410
(1,953 910 )
3/7)(
cos
-3,890 610
(1,437 1110 )
4,601 610
(1,723 i
4sin
cos
2,996 710
(0,1763)
4sin
coscos
-2,889 410
(11,36)
1,859 410
(5,165)
– – 2)(
1
*
1,876 410
(6,619 310 )
3,842 210
(5,644 510 )
The values in the brackets are )(cV . For example, 1,808 i 810 = 161027.3 .
* This component corresponds to the coordinate of the axis of the rotation, therefore
cos is not multiplied.
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2020, № 3 134
T a b l e 4 . Results of the simulation of gravitational waves, Case-2
Components of grav-
itational waves be-
fore the rotation
c and
)(cV of
gravitational
waves
Components of
gravitational waves
after the rotation
c and )(cV
Rotation 1
c and )(cV
Rotation 2
2)(
1
2,335 210
(21,38) 2)(
cos
-2,080 410
(7,251 310 )
3,964 310
(1,428 410 )
3/4)(
1
-1,708 310
(1,511) 3/4)(
cos
1,220 310
(4,052 310 )
-5,052 210
(9,136 210 )
4)(
1
1,217
(1,140 310 ) 4)(
cos
-1,463 610
(6,858 510 )
3,541 610
(2,825 610 )
2sin
1
1,000
(9,834 410 )
2sin
cos
0,8678
(0,3550)
0,8594
(0,7481)
2cot
-1,000
(2,405 310 ) 2cot *
5,531 310
(2,318 310 )
8,384 310
(6,984 310 )
24 sin)(
1
-0,9864
(9,137 210 )
24 sin)(
cos
1,169 610
(5,448 510 )
-2,773 610
(2,218 610 )
23/10 sin)(
1
-0,1682
(1,523 210 )
23/10 sin)(
cos 1,568 510
(6,602 410 )
-3,016 510
(2,612 510 )
3/10)(
1
0,2770
(2,517 210 ) 3/10)(
cos
-2,829 510
(1,292 510 )
6,841 510
(5,614 510 )
3/7)(
1
-2,439 210
(22,20) 3/7)(
cos
2,115 410
(8,154 310 )
2,338 410
(2,391 410 )
4sin
cos
8,679 1010
(1,092 610 )
4sin
coscos
3,683 410
(6,707 410 )
-1,286 210
(8,520 310 )
– – 2)(
1
* 30,60
(8,602)
11,74
(6,283)
The values in the brackets are )(cV . For example, 1,808 i 810 = 161027.3 .
* This component corresponds to the coordinate of the axis of the rotation, therefore
cos is not multiplied.
In Case-1 (non-linear distortion of time and space), the coefficient, c, of,
4sin
cos
, changes its sign from plus to minus after the rotation of 1 (the Rotation 1),
and,
2)(
1
, and
23/10 sin)(
1
, change these signs from plus to minus
after the rotation of 2 (the Rotation 2). And in Case-2 (linear distortion of time
Simulating the rotation of a black hole and antigravity
Системні дослідження та інформаційні технології, 2020, № 3 135
and space), the coefficients of
2)(
1
,
4)(
1
and
3/10)(
1
change these
signs from plus to minus after the rotation of 1 (the Rotation 1), while the
coefficient of,
4sin
cos
, changes its sing from plus to minus after the rotation of 2
(the Rotation 2).
T a b l e 5 . Strengths of gravity and antigravity
Case-1 Case-2
Case
Gravity Antigravity Gravity Antigravity
No rotation -2,867 310 1,033 -1,282 1,110
Rotation 1 -3,066 810 1,390 410 -7,079 410 1,060 510
Rotation 2 -3,060 710 1,353 810 -3,899 410 5,516 410
The summation of each of the positive coefficients and the negative coeffi-
cients are shown in Table 6, and in Fig. 8 for Case-1, and Fig. 9 for Case-2. In
both cases, gravity and antigravity are balanced without the rotation of the black
hole, but the balance is broken after the rotations, then antigravity appears with
the Rotation 2 in Case-1, and with both of the Rotation 1 and Rotation 2 in Case-2.
Also, gravitational waves and anti-gravitational waves are balanced without the
rotation, but they appear when the black hole rotates.
T a b l e 6 . Strengths of gravitational waves and anti-gravitational waves
Case - 1 Case - 2
Case Gravitational
waves
Anti-gravitational
waves
Gravitational
waves
Anti-gravitational
waves
No rotation 2,838 710 -2,224 710 2,517 -2,181
Rotation 1 2,744 910 -2,122 910 1,348 610 -1,766 610
Rotation 2 1,409 910 -1,158 910 4,252 610 -3,075 610
Fig. 8. Gravitational waves and anti-gravitational waves (Case-1: non-linear distortion of
time and space)
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2020, № 3 136
PHYSICAL MEANING OF THE RESULT
Here, kTgRgR )2/1( , is the equation of gravitational field of the
Universe [4], where R are curvature tensors, named Ricci tensors, g are fun-
damental tensors, RR , and, gg , where , T is the stress-energy
tensor and k is a constant, and 0 is the cosmological constant named “dark
energy”, which is a positive contribution to kT . The above result of our simula-
tion shows that the rotation of the black hole makes positive contribution to the
stress-energy tensor, which may expand the size of the Universe, however it is
unknown if the antigravity is related to the dark energy.
Fig. 6 and Fig.7 show that the gravity and the antigravity are balanced with-
out the rotation, but balance is broken when the black hole rotates. Also, Fig. 8
and Fig. 9 show that the gravitational waves and the anti-gravitational waves are
balanced without the rotation, but the balance is broken when the black hole ro-
tates. This finding is consistent with our previous report [1].
CONCLUSIONS AND RECOMMENDATIONS
In this simulation, assuming that the coordinates of time and space can be dis-
torted in the strong gravity in a black hole, we investigated whether a rotation of a
black hole can produce antigravity and anti-gravitational waves, or not, by calcu-
lating the relative strengths of the components of the curvature tensors of the
black hole, which are measured by the stress-energy tensor that is placed outside
of the black hole, upon Einstein’s field equation. In order to simulate the curva-
ture in the strong gravity, we used the system of the spherical polar coordinates so
that we could simulate rotation of the black hole with Euler’s angles.
The results of the simulation show that the rotating black hole can produce
the antigravity and anti-gravitational waves, if time and space are distorted line-
arly and non-linearly. Also, the results suggest a possible explanation about the
expansion of the Universe.
Further investigations are needed about the process of the time-space distor-
tions and of the angular momentum of the rotation.
Fig. 9. Gravitational waves and anti-gravitational waves (Case-2: linear distortion of time
and space)
Simulating the rotation of a black hole and antigravity
Системні дослідження та інформаційні технології, 2020, № 3 137
REFERENCES
1. Y. Matsuki, P.I. Bidyuk, “Analysis of negative flow of gravitational waves (Part 5)”,
System Research & Information Technology, no. 4, pp. 7–18, 2019.
2. Y. Matsuki, P.I. Bidyuk, “Numerical Simulation of Gravitational Waves from
a Black hole, using Curvature Tensors (Part 6)”, System Research & Information
Technology, no.1, pp. 54–67, 2020.
3. P.A.M. Dirac, General Theory of Relativity, Florida University, A Wiley-
Interscience Publication, John Wiley & Sons, New York, 1975, pp. 69 .
4. H. Goldstein, C.P. Poole, J.L. Safko, Classical Mechanics, 3rd Edition, published by
Pearson Education, Inc., 2002, pp. 646; [especially Chapter 4 “The Kinematics of
Rigid Body Motion”, pp.134–183, Chapter 7.11 “Introduction to the general
theory of relativity”, pp. 324–328].
Received 02.07.2020
____________________________
From the Editorial Board: the article corresponds completely to submitted manuscript.
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 Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: matsuki@wdc.org.ua
ІМІТАЦІЙНЕ МОДЕЛЮВАННЯ ОБЕРТАННЯ ЧОРНОЇ ДІРИ ТА АНТИГРАВІТАЦІЇ
/ Й. Мацукі, П.І. Бідюк
Анотація. Показано, що обертання чорної діри може створити антигравітацію
та антигравітаційні хвилі за умови, що у чорній дірі існує сильна гравітація,
яка викривлює час і простір. Отримано тензори кривизни на підставі рівняння
поля Ейнштейна з використанням сферичних полярних координат, розрахова-
но коефіцієнти тензорів для моделювання сили кожного компонента тензорів.
Зроблено припущення, що тензор енергії-імпульсу, розміщений за межами чор-
ної діри, може відображати силу гравітаційного поля і гравітаційних хвиль.
У результаті сформовано такий висновок: якщо час і простір викривляються
у чорній дірі, то обертання може створити антигравітацію та антигравітаційні
хвилі. Результат моделювання показав, що антигравітація робить позитивний
внесок у тензор енергії імпульсу, що може розширити розмір Всесвіту.
Ключові слова: антигравітація, тензор кривизни, тензор енергії напруження,
рівняння Ейнштейна для поля.
ИМИТАЦИОННОЕ МОДЕЛИРОВАНИЕ ВРАЩЕНИЯ ЧЕРНОЙ ДЫРЫ И
АНТИГРАВИТАЦИИ / Й. Мацуки, П.И. Бидюк
Аннотация. Показано, что вращение черной дыры может создать антиграви-
тацию и антигравитационные волны при условии, что в черной дыре сущест-
вует сильная гравитация, которая искажает время и пространство. Получены
тензоры кривизны на основании уравнения поля Эйнштейна с использованием
сферических полярных координат, рассчитаны коэффициенты тензоров для
моделирования силы каждого компонента тензоров. Предполагается, что тен-
зор энергии импульса, расположенный за пределами черной дыры, может от-
ражать силу гравитационного поля и гравитационных волн. В результате
сформулирован вывод: если время и пространство искривляются в черной ды-
ре, вращение может создать антигравитацию и антигравитационные волны.
Результат моделирования показал, что антигравитация делает позитивный
вклад в тензор энергии импульса, что может расширить размер Вселенной.
Ключевые слова: антигравитация, тензор кривизны, тензор энергии напряже-
ния, уравнение Эйнштейна для поля.
|
| id | journaliasakpiua-article-221372 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:27:00Z |
| publishDate | 2020 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/08/5cdb46a9b06f4dc21a8402c00bcdee08.pdf |
| spelling | journaliasakpiua-article-2213722021-01-19T12:18:25Z Simulating the rotation of a black hole and antigravity Имитационное моделирование вращения черной дыры и антигравитации Імітаційне моделювання обертання чорної діри та антигравітації Matsuki, Yoshio Bidyuk, Petro I. antigravity curvature tensor stress-energy tensor Einstein’s field equation антигравитация тензор кривизны тензор энергии напряжения уравнение Эйнштейна для поля антигравітація тензор кривизни тензор енергії напруження рівняння Ейнштейна для поля In this article, we show that rotation of a black hole can create antigravity and anti-gravitational waves, given that there is a strong gravity in the black hole, which distorts time and space. At first, we derived the curvature tensors upon Einstein’s field equation, using spherical polar coordinates, and then calculated the coefficients of the curvature tensors to simulate the strength of each component of the tensors. It is assumed that the stress-energy tensor, which is located outside of the black hole, can reflect the strength of the gravitational field and the gravitational waves. As the result, we concluded that, if the time and space are distorted in the black hole, the rotation can create antigravity and the anti-gravitational waves. In addition, the result of the simulation shows that the antigravity positively contributes to the stress-energy tensor, which may expand the size of the Universe. Показано, что вращение черной дыры может создать антигравитацию и антигравитационные волны при условии, что в черной дыре существует сильная гравитация, которая искажает время и пространство. Получены тензоры кривизны на основании уравнения поля Эйнштейна с использованием сферических полярных координат, рассчитаны коэффициенты тензоров для моделирования силы каждого компонента тензоров. Предполагается, что тензор энергии импульса, расположенный за пределами черной дыры, может отражать силу гравитационного поля и гравитационных волн. В результате сформулирован вывод: если время и пространство искривляются в черной дыре, вращение может создать антигравитацию и антигравитационные волны. Результат моделирования показал, что антигравитация делает позитивный вклад в тензор энергии импульса, что может расширить размер Вселенной. Показано, що обертання чорної діри може створити антигравітацію та антигравітаційні хвилі за умови, що у чорній дірі існує сильна гравітація, яка викривлює час і простір. Отримано тензори кривизни на підставі рівняння поля Ейнштейна з використанням сферичних полярних координат, розраховано коефіцієнти тензорів для моделювання сили кожного компонента тензорів. Зроблено припущення, що тензор енергії-імпульсу, розміщений за межами чорної діри, може відображати силу гравітаційного поля і гравітаційних хвиль. У результаті сформовано такий висновок: якщо час і простір викривляються у чорній дірі, то обертання може створити антигравітацію та антигравітаційні хвилі. Результат моделювання показав, що антигравітація робить позитивний внесок у тензор енергії імпульсу, що може розширити розмір Всесвіту. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2020-12-07 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/221372 10.20535/SRIT.2308-8893.2020.3.09 System research and information technologies; No. 3 (2020); 124-137 Системные исследования и информационные технологии; № 3 (2020); 124-137 Системні дослідження та інформаційні технології; № 3 (2020); 124-137 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/221372/223561 Copyright (c) 2021 System research and information technologies |
| spellingShingle | антигравітація тензор кривизни тензор енергії напруження рівняння Ейнштейна для поля Matsuki, Yoshio Bidyuk, Petro I. Імітаційне моделювання обертання чорної діри та антигравітації |
| title | Імітаційне моделювання обертання чорної діри та антигравітації |
| title_alt | Simulating the rotation of a black hole and antigravity Имитационное моделирование вращения черной дыры и антигравитации |
| title_full | Імітаційне моделювання обертання чорної діри та антигравітації |
| title_fullStr | Імітаційне моделювання обертання чорної діри та антигравітації |
| title_full_unstemmed | Імітаційне моделювання обертання чорної діри та антигравітації |
| title_short | Імітаційне моделювання обертання чорної діри та антигравітації |
| title_sort | імітаційне моделювання обертання чорної діри та антигравітації |
| topic | антигравітація тензор кривизни тензор енергії напруження рівняння Ейнштейна для поля |
| topic_facet | antigravity curvature tensor stress-energy tensor Einstein’s field equation антигравитация тензор кривизны тензор энергии напряжения уравнение Эйнштейна для поля антигравітація тензор кривизни тензор енергії напруження рівняння Ейнштейна для поля |
| url | https://journal.iasa.kpi.ua/article/view/221372 |
| work_keys_str_mv | AT matsukiyoshio simulatingtherotationofablackholeandantigravity AT bidyukpetroi simulatingtherotationofablackholeandantigravity AT matsukiyoshio imitacionnoemodelirovanievraŝeniâčernojdyryiantigravitacii AT bidyukpetroi imitacionnoemodelirovanievraŝeniâčernojdyryiantigravitacii AT matsukiyoshio ímítacíjnemodelûvannâobertannâčornoídíritaantigravítacíí AT bidyukpetroi ímítacíjnemodelûvannâobertannâčornoídíritaantigravítacíí |