Числове моделювання гравітаційних хвиль чорної діри за допомогою тензорів викривлення
In this research we formulated the curvature tensors with the system of spherical polar coordinates, which describe the gravitational field and gravitational waves of a black hole; and then we calculated eigenvalues of the curvature tensors to estimate the relative strengths of their components to t...
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| Дата: | 2020 |
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| Формат: | Стаття |
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2020
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Репозитарії
System research and information technologies| _version_ | 1867334405145493504 |
|---|---|
| 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 | 2020-08-11T08:50:57Z |
| description | In this research we formulated the curvature tensors with the system of spherical polar coordinates, which describe the gravitational field and gravitational waves of a black hole; and then we calculated eigenvalues of the curvature tensors to estimate the relative strengths of their components to the stress-energy tensor in Einstein’s field equation. For this simulation, we assumed that the time and the distance interact with each other if we travel from Earth to the inside of the black hole, and then the result of the simulation showed that the gravitational waves carry the same components of the gravitational field of the black hole. On the other hand, when we assumed that the time and the distance are independent, which resembles the situation outside of the boundary of the black hole toward Earth, the curvature tensors are different between those of the gravitational field and the gravitational waves. Upon the results of the simulation we conclude that the gravitational waves that come from the inside of the black hole carry the information of the gravitational field inside of the black hole, if we assume that time and space are dependent each other. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2020.1.05 |
| first_indexed | 2025-07-17T10:26:43Z |
| format | Article |
| fulltext |
Y. Matsuki, P.I. Bidyuk, 2020
54 ISSN 1681–6048 System Research & Information Technologies, 2020, № 1
UDC 519.004.942
DOI: 10.20535/SRIT.2308-8893.2020.1.05
NUMERICAL SIMULATION OF GRAVITATIONAL WAVES
FROM A BLACK HOLE, USING CURVATURE TENSORS
Y. MATSUKI, P.I. BIDYUK
Abstract. In this research we formulated the curvature tensors with the system of
spherical polar coordinates, which describe the gravitational field and gravitational
waves of a black hole; and then we calculated eigenvalues of the curvature tensors to
estimate the relative strengths of their components to the stress-energy tensor in Ein-
stein’s field equation. For this simulation, we assumed that the time and the distance
interact with each other if we travel from Earth to the inside of the black hole, and
then the result of the simulation showed that the gravitational waves carry the same
components of the gravitational field of the black hole. On the other hand, when we
assumed that the time and the distance are independent, which resembles the situa-
tion outside of the boundary of the black hole toward Earth, the curvature tensors are
different between those of the gravitational field and the gravitational waves. Upon
the results of the simulation we conclude that the gravitational waves that come
from the inside of the black hole carry the information of the gravitational field in-
side of the black hole, if we assume that time and space are dependent each other.
Keywords: Gravitational field, gravitational waves, curvature tensor, black hole,
spherical polar coordinates.
INTRODUCTION (Theory)
Curvature tensors of gravitational field and gravitational waves of a black hole
According to Einstein and Dirac [1], the gravitational field of a black hole is de-
scribed by the curvature tensors:
,,R ,
where )(
2
1
,,,
ggggg , and g are the funda-
mental tensors that describe the curvature of the 4-dimensional space.
Outside of black hole
Gravitational field outside of a black hole. According to Dirac [1], the funda-
mental tensors, g , of gravitational field outside of a black hole in spherical po-
lar coordinates are as follows:
22
2
1
sin000
000
00
2
10
000
2
1
r
r
r
m
r
m
g
Numerical simulation of gravitational waves from a black hole, using curvature tensors
Системні дослідження та інформаційні технології, 2020, № 1 55
22
2
2
2
sin000
000
000
000
r
r
e
e
,
where m2 is a constant, and m is assumed to be the mass of a black hole; and
g are functions of a network of curvilinear coordinates, which provide a geo-
desic, 2332222112002 dgdgdrgdtgds , outside of the black hole. And
then, the curvature tensors R of gravitational field outside of a black hole in
spherical polar coordinates are described as follows:
222
00
2
e
r
R ,
r
R
22
11 ,
1)1( 2
22 errR ; 22
33 sin}1)1{( errR , (1)
while the rest of R are all zero. Here, and are functions of r , and
dr
d
,
2
2
dr
d
, and
dr
d
.
Gravitational waves outside of a black hole. The gravitational waves in cur-
vature tensors are described by the equations:
)
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 .
Note. In Dirac’s original article [1], only the first term ,
,g , is dis-
cussed, but here in our article we also consider those terms from the second to the
12th, which were neglected in [1].
If ,
,,,, gggggggg ; 1
ggg if
; 0
ggg if ; 0,
gg if ; otherwise,
0,
gg ; and 0,
gg if ; otherwise, 0,
gg . Therefore,
many terms become zero, and only the following terms remain:
2
2
,
1
11,00
00 2
1
1
2
2
1
2
1
2
1
r
m
r
m
r
m
r
m
r
m
gg
rr
=
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2020, № 1 56
3
1
2
2
42
1
22
12
r
m
r
m
r
m
r
m
;
)2(
42
1
2
1
2
,
1
11,11
11
mrr
m
r
m
r
m
gg
rr
;
2
42
,
32
,
22
11,22
22 6
6)2())((
r
rrrrrrgg rrr ;
rrrrgg ,
2222
11,33
33 )sin)(sin(
2
422
,
322 6
6sin)2(sin
r
rrrr r ;
,
3222
,
2222
22,33
33 )cossin2)(sin()sin)(sin( rrrrgg
)1cos3(sin2 2 ;
22341,111,11
11
1,1,111,11
11
1,
44
4
)
2
1
()
2
1
(
mrmrr
m
gggggg
;
3
2
1,111,1111
11
1,111,1111
11 8
2
1
2
1
r
m
gggggggg
;
3
2
1,111,1111
11
1,1,111,1111
11
1,
42
2
1
2
1
r
mmr
gggggggg
;
)2(
8
2
1
2
1
5
3
1,111,111,11
11
1,111,111,11
11
mrr
m
gggggggg
, (2)
where
rr
(*)
(*), and
2
2
,
(*)
(*)
r
rr
; and, (*) is any given function.
Penetrating the boundary of a black hole
As shown in the equation (1), there is a singularity at mr 2 in one of the curva-
ture tensors, 11R , which means the presence of the boundary of a black hole, and
we cannot see the inside, where mr 2 , from the outside, where mr 2 . In order
to look inside of the black hole, Dirac [1] invented different coordinate system,
assuming that the time and space are dependent, by )(rft ; and,
)(rgt , where t is time, and r is distance. The steps given below show how
Dirac [1] described the system of coordinates that extends to the inside of a black
hole.
At first, the geodesic in the spherical polar coordinates is defined as follows:
2332222112002 dgdgdrgdtgds ; but, if t and r are dependent on each
other, the first two terms, about time and distance, change to the following:
Numerical simulation of gravitational waves from a black hole, using curvature tensors
Системні дослідження та інформаційні технології, 2020, № 1 57
22222
1
2 )(
2
)(
22
1
2
1 drgdt
r
m
drfdtd
r
m
ddr
r
m
dt
r
m
)2(
2
2 222222 drgdtdrgdt
r
m
drfdtdrfdt
2222 22
2
2
1 drg
r
m
fdtdrg
r
m
fdt
r
m
,
where
r
f
f
and
r
g
g
.
The necessary conditions for satisfying
2
1
2 2
1
2
1 dr
r
m
dt
r
m
22 2
d
r
m
d are:
0
2
2
dtdrg
r
m
f , g
r
m
f
2
and
1
22 2
1
2
r
m
fg
r
m
.
And then,
1
2
2
2 2
1
2
1
222
r
m
r
m
r
m
gg
r
m
g
r
m
; then
2
2 2
1
2
r
m
m
r
g , and then
1
2
1
2
1
2
r
m
m
r
g .
Now, assume that 2yr and, 22 am . If mr 2 , 22 ay , then ay :
a
y
ay
y
a
y
ay
y
y
ay
a
y
y
a
a
y
g
dr
dg
22
2
2
2
22
21
2
222
1
2
21
2
22
1
2
2
1
;
ry 2 , so,
22
4
22
3 12
)(
1
22
aya
y
aya
y
y
dr
dg
y
dy
dr
dr
dg
dy
dg
.
Then,
dyygdy
dy
dr
gdy
ayaya
y
g
2
))((
12 4
;
ay
ay
aayy
a
g log2
3
2 23 ;
2
1
1
2
1
2
2
1
2
1
2
2
1
2
'
m
r
r
m
r
m
m
r
r
m
gg
r
m
gfg ;
fgr
m
)2/3(
2
1
3
2
.
Thus, 3
2
)( r , where
3
2
2
2
3
m , and 3
2
3
2
)(2
2
3
mr ,
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2020, № 1 58
2
3
2)(2/3 mr , and
3/2
)(2
2
3
mr .
Therefore,
2332222112002 dgdgdrgdtgds
drdrdr
r
m
dt
r
m 22222
1
2 sin
2
1
2
1
223/4224/322
3/2
2 sin)()(
)(
2
ddd
m
d .
Therefore, the fundamental tensors g of gravitational field, which pene-
trate the boundary of a black hole in spherical polar coordinates, are:
33323130
23222120
13121110
03020100
gggg
gggg
gggg
gggg
g
23/42
3/42
3/2
sin)(000
0)(00
00
)(
2
0
0001
m
,
where g provide functions of a geodesic, 2222112002 dgdrgdtgds
233 dg , which penetrates the boundary of the black hole.
THE CURVATURE TENSORS FOR NUMERICAL SIMULATION
Table 1 and Table 2 show the curvature tensors of gravitational field and gravita-
tional waves, which we made for our numerical simulation. These tensors don’t
have the mathematical singularity; therefore we can simulate the inside of the
black hole.
The curvature tensors of gravitational field are:
,,R .
In our simulation, we included the component of,
, which
Dirac [1] neglected.
For example, for 0 , 2 and 1 we have:
1
02
2
01
1
21
2
00
2
2,00
2
0,0200R
Numerical simulation of gravitational waves from a black hole, using curvature tensors
Системні дослідження та інформаційні технології, 2020, № 1 59
102
11
201
22
121
11
200
22
2,200
22
0,202
22 gggggg
2
,,
)3/7(3/42
)(3
2
)(3
2
))(1(
3
4
})({
2
1
,
where
{*}
{*}, and
2
2
,
{*}
{*}
, and {*} is any given function.
T a b l e 1 . Curvature tensors of gravitational field, which extends beyond the
boundary of a black hole
Para-
meter 00R 1001 RR 11R 22R 33R
0 ,
0 0 0 3/4)(6
m 3/102 )(9
14
23/102 sin)(9
14
1 0 0 3/4)(6
m 3/102 )(3
4
23/102 sin)(3
4
2 0 0 3/4)(18
5
m 3/102 )(9
22
23/102 sin)(
2
3 0 0 3/4)(18
5
m 3/102 )(
2
23/102 sin)(9
14
1 ,
0 2)(3
1
0 3/4)(18
m 42 )(
4
m
243 sin)(
4m
1 2)(9
2
0 0 42 )(9
32
m
243 sin)(9
32m
2 2)(3
1
2)(9
2
2)(9
2
42 )(
4
m
243 sin)(9
44m
3 2)(3
1
2)(9
2
2)(9
2
42 )(9
44
m
243 sin)(
4m
2 ,
0 2)(3
2
2)(3
2
2)(3
2
3/102 )(9
4
2
2
sin
cot31
1 2)(3
2
2)(3
2
2)(3
2
42 )(9
8
m
2
2
sin
cot31
2 2)(9
10
2)(9
10
2)(9
10
0
2
2
sin
cot31
3 2)(3
2
2)(3
2
2)(3
2
0
2
2
sin
cot31
3 ,
0 2)(3
2
2)(3
2
2)(3
2
2sin
1
23/102 sin)(9
4
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2020, № 1 60
Continued Tabl. 1
Para-
meter 00R 1001 RR 11R 22R 33R
1 2)(3
2
2)(3
2
2)(3
2
2sin
1
243 sin)(9
8m
2 2)(3
2
2)(3
2
2)(3
2
2sin
1
2
2
sin
cot
3 2)(9
10
2)(9
10
2)(9
10
2
2
cot
sin
1
0
For 2002 RR , only 3
23
2
02
)(3
cot2
( 2 , 3 ) and 3
23
3
03
)(3
cot2
( 3 , 3 ) remain: therefore 2002 RR 3
23
2
02 03
23
3
03 ;
for 2112 RR , only 3
23
2
12
)(3
cot2
( 2 , 3 ), and
)(3
cot23
23
3
13
( 3 , 3 ) remain. Therefore 2112 RR 3
23
2
12 03
23
3
13 ; and all other
components such as 13R , 31R , 23R and 32R are 0. Then, according to Einstein’s
rule (
RR ) we summate all the components:
2
242
3
42
cot
sin
8
)(9
140
)(18
11
)(9
56 m
m
R
23
10
2
2
2
243
sin)(9
28
sin
cot35
sin)(9
140m
.
The curvature tensors of gravitational waves, which penetrate the boundary
of a black hole, 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 .
Numerical simulation of gravitational waves from a black hole, using curvature tensors
Системні дослідження та інформаційні технології, 2020, № 1 61
If ,
,,,, gggggggg ; 1
ggg if ;
0
ggg if ; 0,
gg if , otherwise 0,
gg ; and,
0,
gg if , otherwise 0,
gg .
T a b l e 2 . Curvature tensors of gravitational waves, which penetrates the
boundary of a black hole
0 ,
3,2,1
3,2,1,0
,
3/2
00,11
00 )(
2m
gg
,
)3/1()(
3
2
)1(
2m
3/4)(9
m
,
3/10200,22
00
)(9
28
gg ,
23/10200,33
00
sin)(9
28
gg
1 ,
0 ,
3,2,1,0
20,11
11
0,
)(9
4
gg
1 ,
1 ,
0
,
)(9
4
)(9
2
2
1
2
1
221,111,11
11
1,1,111,11
11
1,11,11
11
gggggggg
3/1001,110,1100
11
)(81
2
2
1
m
gggg ,
3/110,110,1100
11
1,
)(272
1
m
gggg ,
3/70,110,111,00
11
)(272
1
m
gggg , 0
2
1
0,110,111,00
11 gggg ,
3/701,110,1100
11
)(272
1
m
gggg ,
3/70,1101,1100
11
)(108
4
2
1
m
gggg ,
3/70,110,1100
11
1,
)(27
2
2
1
m
gggg ,
3/40,110,111,00
11
)(92
1
m
gggg
1
221,111,11
11
1,1,111,11
11
1,11,11
11
)(9
4
)(9
2
)
2
1
()
2
1
(
gggggggg ,
3/52
2
11,111,1111
11
)(542
1
m
gggg ,
3/52
2
1,1111,1111
11
)(542
1
m
gggg ,
3/52
2
1,111,1111
11
1,
)(272
1
m
gggg ,
3/52
2
1,1111,111,11
11
)(272
1
m
gggg ,
3/52
2
11,111,1111
11
)(542
1
m
gggg ,
3/52
2
1,1111,1111
11
)(542
1
m
gggg ,
3/52
2
1,111,1111
11
1,
)(272
1
m
gggg ,
3/52
2
1,1111,111,11
11
)(272
1
m
gggg
2
1,111,11
11
1,1,111,11
11
1,11,11
11
2
1
2
1
gggggggg
22 )(9
4
)(9
2
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2020, № 1 62
Continued Tabl. 2
3
1,111,11
11
1,1,111,11
11
1,11,11
11
2
1
2
1
gggggggg
22 )(9
4
)(9
2
1 ,
3,2
3,2,1,0
4311,22
11
)(9
56
m
gg ,
24311,33
11
sin)(9
56m
gg
2 ,
3,1,0 ,
3,2,1,0
20,22
22
1,
)(9
16
gg ,
21,22
22
1,
)(9
16
gg ,
2422,33
22
sin
2
sin
cos6
gg
3 ,
,2,1,0
3,2,1,0
2
2
2
0,33
33
0, cot4
sin
cos
4gg ,
21,33
33
1,
)(9
16
gg ,
2
2,33
33
2, cot4gg
Note. All other components are zero.
Now, we summate all the components:
2223/1023/1023/4 )(3
2
)(9
16
sin)(9
112
)(9
112
)(9
4
m
4323/43/73/10 )(9
224
)(
2
)(9)(9)(81
2 m
mmm
2
224243
cot32
)(9
64
sin
8
sin
cos24
sin)(9
224m
.
NUMERICAL SIMULATION
Algorithm
Einstein’s field equation [1] that rules the motion of particles in the gravitational
field is: 0
2
1
,
RgR , where R is a scalar tensor,
RgRR ,
which are 00
000
0 RgRR , 11
111
1 RgRR , 22
222
2 RgRR , 3
3RR
33
33Rg . Then, for example, 00000000
00
0000 2
1
2
1
2
1
RRRRggR .
Now, we use the relation: kTRgR 2
1
, where T is the stress-energy
tensor and k is a constant [2]. Then, we set the following algorithm to simulate
the relative intensities of the components of curvature tensors.
For example, when 0 , 00000000
00
0000 2
1
2
1
2
1
RRRRggR . In
our simulation, we omit the factor of, 2/1 , because we only use the selected vec-
Numerical simulation of gravitational waves from a black hole, using curvature tensors
Системні дослідження та інформаційні технології, 2020, № 1 63
tors of the coordinates, which are taken from the obtained curvature tensors in
Table 1 for gravitational field and in Table 2 for gravitational waves.
Then, from Table 1, for example when 1 and 0 ,
200
)(3
1
R
and when 3 and 0 ,
222
sin
1
R . Then, for example, if we only summate
these two components of 00R (when 1 and 0 ) and 22R (when 3 and
0 ), and then, our algorithm is shown as follows:
}{
sin
1
)(3
1
)( 21222200 bXaXkTkTRRkTH
,
2
21
2
2200
2 }]{[)}({ bXaXkTRRkTH ,
where a and b are constants.
Then, we assume: 0)(22
)(
1
22
HXE
a
H
HE
a
H
E
a
HE
, and
0)(22
)(
2
22
HXE
b
H
HE
b
H
E
b
HE
, where )( 1HXE and
)( 2HXE are operators to calculate averages of HX1 and, HX 2 .
0)( 1 HXE and 0)( 2 HXE are equivalent to solving the problem of
0),( 1 HXC and 0),( 2 HXC , where ),( 1 HXC is the covariance of 1X and
H ; and, ),( 2 HXC is the covariance of 2X and, H .
Now,
b
a
c , where a and b are the eigenvalues of 2200 RR , and
][ 21 XXX ; then )( 2200 RRkTH XckT , and 0HX , then
0)(' XckTX , where X is transpose matrix of X . Then, kTXXcX ,
kTXQkTXXXc 11)( , and 1212 )()( XXQcV , where
2)( cV is the variance of c , and )/('2 lnee , where kTMe ,
XXXXIM 1)( , n is a number of rows of each column of X (in this
simulation 23n ), l is a number of columns of X , I is a 2323 unit matrix,
1)( XX is the inverse matrix of XX , and 'e is the transpose vector of e .
By calculating c and, )(cV , we can estimate the strength of each compo-
nent of )( 2200 RR to the stress-energy tensor, in the system of spherical polar
coordinates.
We can also expand the size of matrix not only, 2l , but also to 2l such
as ],,,[* 21 kcccc , and ][ 21 kXXXX , where *c is a transposed vector
of c , so that we were able to calculate not only )( 2200 RR , but more general
R in our numerical simulation.
Input data
At first, our time t on Earth is set as shown in Fig. 1 and Fig. 2, with which its
slope to the distance r from the center of a black hole toward outside is a con-
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2020, № 1 64
stant, which is consistent with Einstein’s theory that the speed of light is a con-
stant. Then, 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 , )(tg are conjugate functions of t and r . (For the simulation, we as-
sumed case-1: rrf log)( and retg )( ; and, case-2: rrf )( and
)4/1()( tg ). According to Dirac [1], the collapse of a star into a black hole
would take an infinite time at our clocks on Earth, but it takes only a finite time
relatively to the collapsing matter on the star itself. From this Dirac’s statement,
we assumed that is larger when the relative distance, , from the center of the
black hole is smaller. Also, in this simulation we set the stress-energy tensor kT
to be 1; because, the purpose of this simulation is to measure the order of magni-
tude of the relative strength of each component of R to the stress-energy tensor.
For the simulation of the gravitational waves we assumed the angles, , as
shown in Fig. 3, as if it becomes smaller in far distance from a black hole; on the
τ
ρ
t
r
Fig. 1. Input data for simulation Case 1:
rrf log)( and retg )(
r
ρ
t
τ
Fig. 2. Input data for simulation Case 2:
rrf )( and 4/1)( tg
Fig. 3. Angles for simulating gravitational
field and gravitational waves
θ
θ
r
θ
θ
r
θ
sin θ,cos θ cot θ
Fig. 4. sin , cos and cot for simulation
of gravitational waves
Numerical simulation of gravitational waves from a black hole, using curvature tensors
Системні дослідження та інформаційні технології, 2020, № 1 65
other hand, for the simulation of the spatial expansion of the gravitational field of
a black hole, we assumed as if becomes larger in far distance. With this as-
sumption, the gravitational waves of sign cosign and cotangent curves behave like
as shown in Fig. 4.
Results
Table 3 shows the calculated eigenvalues c of R and )(cV for the gravita-
tional field, and Fig. 5, 6 and Fig. 7 show selected functions that have the negative
coefficients, which mean the gravity, and of which each element of )(cV is
smaller than each element of c . Table 3 also shows the calculated eigenvalues of
the gravitational waves, and Fig. 8 shows the function of
2sin
1
that have the
positive coefficients, and of which )(cV is smaller than its value of c . We se-
lected only,
2sin
1
, because it has the positive coefficient, assuming that the
gravitational waves should give positive impact to the stress-energy tensor, kT .
T a b l e 3 . Results of the simulation of gravitational field and gravitational waves
c and )(cV
of Gravitational field
c and )(cV
of Gravitational waves Components
Case-1 Case-2 Case-1 Case-2
2)(
1
-1,850 210
(6,298)
4,406 410
(0,0002776)
5,950 810
(-1,059 1610 )
-3,190 310
(0,7698)
3/4)(
1
-5,750 210
(2,060 )10 17
5,118 810
(3,211 810 )
-2,088 1210
(0,03356)
-5,308 810
(1,511 510 )
4)(
1
-24,58
(6546,0)
-6,932 310
(0,002527)
6,591 1210
(2,672 1910 )
-6,957 210
(33,92)
2sin
1
1,000
(6,550 510 )
1,000
(2,146 610 )
0,1870
(-1,753 810 )
1,000
(9,408 510 )
2cot
-1,000
(2,430 410 )
-1,000
(6,034 710 )
-1,180
210
(-1,373 910 )
-1,000
(3,128 410 )
24 sin)(
1 -0,4142
(111,2)
-1,714 410
(4,359 510 )
6,311 1210
(1,833 1910 )
5,784 210
(26,55)
23/10 sin)(
1 -0,2985
(79,94)
-2,705 410
(8,060 510 )
8,995 1210
(-1,442 1910 )
-2,258 210
(9,089)
2
2
sin
cot 9,974 810
(4,800 510 )
1,732 810
(1,324 810 )
- -
3/10)(
1
– – -1,630 1210
(-1,618 1910 )
2,608 210
(14,21)
3/7)(
1
– – -1,065 1010
(-5,221 1610 )
6,675 310
(1,866)
4sin
cos – – -3,160 410
(6,635 610 )
5,047 1010
(1,469 710 )
Note. The numeric values in the brackets are, )(cV , the standard errors of c .
Y. Matsuki, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2020, № 1 66
Note. (Left axis) means that the scale of the function is shown in the left
axis; and, (Right axis) in the right axis.
CONCLUSION AND RECOMMENDATION
In this research, we investigated the structure of gravitational field inside of
a black hole, assuming that time and space interact each other with )(rgt ,
)(rft , ( t is the time, r is the distance from the center of the black hole,
and f and, g , are functions of r ). And we also investigated the functions of
gravitational waves that are emitted from the inside of the black hole. As the re-
sult we found that some of the functions of gravitational waves carry the compo-
nents of the curvature tensor of gravitational field. It means that we are able to
investigate the structure of a black hole by the information carried by the gravita-
tional waves.
Fig. 6. Components of gravitational
field, case 2 (1)
2cos
23/10 sin)(
1 (Left axis)
4)(
1
24 sin)(
1
r
Fig. 5. Components of gravitational field,
case 1
2cot
r
2)(
1
2cot
r
Fig. 7. Components of gravitational field,
case 2 (2)
2sin
r
Fig. 8. Components of gravitational field,
case 2
Numerical simulation of gravitational waves from a black hole, using curvature tensors
Системні дослідження та інформаційні технології, 2020, № 1 67
Inside of the black hole the functions shown in Table 1 for the gravitational
field and in Table 2 for the gravitational waves, show the same functions:
4)( , 2cot and 2sin . The function of 4)( in Fig. 6 suggests the
presence of a boundary of the black hole, where the gravitational waves are dis-
turbed once before they are moving toward the outside.
On the other hand, if we don’t assume the dependence of time and space, the
gravitational waves are to be made outside of the black hole, and they do not
carry the same functions of the gravitational field. The equations (1) about the
gravitational field are 1
00
rR , 1
11
rR , rR 22 and rR 33 ; while the
equations (2) about the gravitational waves are 2r , 3r , 4r and 6r .
In this research, we used the spherical polar coordinate system to describe
the curvature of gravitational field, and this system helped us to make numerical
simulation possible. However, further mathematical investigations are needed
about the curvature tensors.
REFERENCE
1. Dirac P.A.M. General Theory of Relativity / P.A.M. Dirac. — New York: Florida
University, A Wiley Interscience Publication, John Wiley & Sons, 1975. — 69 p.
2. Goldstein H. Classical Mechanics / H. Goldstein, C.P. Poole, J.L. Safko. — 3rd Edi-
tion. — Pearson Education, Inc., 2002. — P. 646 (especially Chapter 7.11 “In-
troduction to the general theory of relativity”), P. 324–328.
Received 06.01.2020
____________________________
From the Editorial Board: the article corresponds completely to submitted manu-
script.
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| id | journaliasakpiua-article-209134 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:26:43Z |
| publishDate | 2020 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/04/4bc17906568b77784b291670ce015c04.pdf |
| spelling | journaliasakpiua-article-2091342020-08-11T08:50:57Z Numerical simulation of gravitational waves from a black hole, using curvature tensors Численное моделирование гравитационных волн черной дыры с помощью тензоров искривления Числове моделювання гравітаційних хвиль чорної діри за допомогою тензорів викривлення Matsuki, Yoshio Bidyuk, Petro I. гравітаційне поле гравітаційні хвилі тензор кривизни чорна діра сферичні полярні координати гравитационное поле гравитационные волны тензор кривизны черная дыра сферические полярные координаты Gravitational field gravitational waves curvature tensor black hole spherical polar coordinates In this research we formulated the curvature tensors with the system of spherical polar coordinates, which describe the gravitational field and gravitational waves of a black hole; and then we calculated eigenvalues of the curvature tensors to estimate the relative strengths of their components to the stress-energy tensor in Einstein’s field equation. For this simulation, we assumed that the time and the distance interact with each other if we travel from Earth to the inside of the black hole, and then the result of the simulation showed that the gravitational waves carry the same components of the gravitational field of the black hole. On the other hand, when we assumed that the time and the distance are independent, which resembles the situation outside of the boundary of the black hole toward Earth, the curvature tensors are different between those of the gravitational field and the gravitational waves. Upon the results of the simulation we conclude that the gravitational waves that come from the inside of the black hole carry the information of the gravitational field inside of the black hole, if we assume that time and space are dependent each other. Сформулированы тензоры искривления по системе полярно-сферических координат, которые описывают гравитационное поле и гравитационные волны черной дыры; вычислены собственные значения тензоров искривления с целью оценки относительных сил их компонентов по сравнению со значением тензора напряжения-энергии в уравнении поля Эйнштейна. В рамках моделирования сделано предположение, что время и расстояние взаимосвязаны при условии, если двигаться от Земли до внутренней части черной дыры; результат моделирования показал, что гравитационные волны имеют одинаковые компоненты гравитационного поля черной дыры. При предположении, что время и расстояние независимы, что напоминает ситуацию за пределами черной дыры по отношению к Земле, тензоры искривления отличались от тензоров гравитационного поля и гравитационных волн. По результатам моделирования сделан вывод, что гравитационные волны, поступающие изнутри черной дыры, несут информацию о гравитационном поле внутри черной дыры при условии, если предполагать, что время и пространство взаимосвязаны. Cформулювано тензори викривлення за системою полярно-сферичних координат, які описують гравітаційне поле та гравітаційні хвилі чорної діри; обчислено власні значення тензорів викривлення з метою оцінення відносних сил їх компонентів порівняно зі значенням тензора напруги-енергії в рівнянні поля Ейнштейна. У межах моделювання зроблено припущення, що час і відстань взаємопов’язані за умови, якщо рухатися від Землі до внутрішньої частини чорної діри; результат моделювання показав, що гравітаційні хвилі мають однакові компоненти гравітаційного поля чорної діри. За припущення, що час і відстань є незалежними, що нагадує ситуацію поза межами чорної діри відносно Землі, тензори викривлення відрізнялися від тензорів гравітаційного поля і гравітаційних хвиль.За результатами моделювання зроблено висновок, що гравітаційні хвилі, які надходять із середини чорної діри, містять інформацію про гравітаційне поле всередині чорної діри за умови, якщовважати, що час та простір взаємопов’язані. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2020-06-23 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/209134 10.20535/SRIT.2308-8893.2020.1.05 System research and information technologies; No. 1 (2020); 54-67 Системные исследования и информационные технологии; № 1 (2020); 54-67 Системні дослідження та інформаційні технології; № 1 (2020); 54-67 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/209134/209531 Copyright (c) 2021 System research and information technologies |
| spellingShingle | гравітаційне поле гравітаційні хвилі тензор кривизни чорна діра сферичні полярні координати Matsuki, Yoshio Bidyuk, Petro I. Числове моделювання гравітаційних хвиль чорної діри за допомогою тензорів викривлення |
| title | Числове моделювання гравітаційних хвиль чорної діри за допомогою тензорів викривлення |
| title_alt | Numerical simulation of gravitational waves from a black hole, using curvature tensors Численное моделирование гравитационных волн черной дыры с помощью тензоров искривления |
| title_full | Числове моделювання гравітаційних хвиль чорної діри за допомогою тензорів викривлення |
| title_fullStr | Числове моделювання гравітаційних хвиль чорної діри за допомогою тензорів викривлення |
| title_full_unstemmed | Числове моделювання гравітаційних хвиль чорної діри за допомогою тензорів викривлення |
| title_short | Числове моделювання гравітаційних хвиль чорної діри за допомогою тензорів викривлення |
| title_sort | числове моделювання гравітаційних хвиль чорної діри за допомогою тензорів викривлення |
| topic | гравітаційне поле гравітаційні хвилі тензор кривизни чорна діра сферичні полярні координати |
| topic_facet | гравітаційне поле гравітаційні хвилі тензор кривизни чорна діра сферичні полярні координати гравитационное поле гравитационные волны тензор кривизны черная дыра сферические полярные координаты Gravitational field gravitational waves curvature tensor black hole spherical polar coordinates |
| url | https://journal.iasa.kpi.ua/article/view/209134 |
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