Регресійний аналіз спостережень LIGO-Virgo WG200115 із використанням тензорів кривизни з рівнянь Ейнштейна та гравітаційних хвиль Дірака
This research presents the mathematical derivation of curvature tensors from the Schwarzschild solution of the gravitational field, along with the derivation of gravitational waves as the second derivative of the field. The elimination of the event horizon is the core contribution of this paper, bui...
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{
"author": "Yoshio Matsuki",
"institution": "National University of Kyiv Mohyla Academy, Kyiv"
}
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| description | This research presents the mathematical derivation of curvature tensors from the Schwarzschild solution of the gravitational field, along with the derivation of gravitational waves as the second derivative of the field. The elimination of the event horizon is the core contribution of this paper, building on Paul Dirac’s approach in his 1975 book, General Theory of Relativity. This paper introduceslinear and non-linear models for the spacetime distortion function. The fit of the tensor equation for gravitational waves is tested using geometrical information from the LIGO-Virgo observation of WG200115, employing econometric techniques with indicators such as R-squared for the robustness of the given data and the fit of the model to the data distribution, as well as the Durbin–Watson statistic for time-series predictability. The results show a good fit of the derived mathematical model to the observed geometry. Different models for simulating spacetime distortion exhibit varying degrees of fit to the observed geometry, with a less distorted model providing a better explanation for the observed gravitational waves. The model’s fit decreases when considering the rotation of the source object, whether it be a neutron star or a black hole. Further modeling effort is needed to accurately represent the NSBH merger process and its role in the formation of gravitational waves. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2026.2.07 |
| first_indexed | 2026-07-01T01:00:13Z |
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Y. Matsuki, 2026
102 ISSN 1681–6048 System Research & Information Technologies, 2026, № 2
TIÄC
МАТЕМАТИЧНІ МЕТОДИ, МОДЕЛІ,
ПРОБЛЕМИ І ТЕХНОЛОГІЇ ДОСЛІДЖЕННЯ
СКЛАДНИХ СИСТЕМ
UDC 519.004.942
DOI: 10.20535/SRIT.2308-8893.2026.2.07
REGRESSION ANALYSIS OF LIGO-VIRGO OBSERVATIONS
WG200115 USING CURVATURE TENSORS FROM EINSTEIN’S
EQUATIONS AND DIRAC’S GRAVITATIONAL WAVES
Y. MATSUKI
Abstract. This research presents the mathematical derivation of curvature tensors
from the Schwarzschild solution of the gravitational field, along with the derivation
of gravitational waves as the second derivative of the field. The elimination of the
event horizon is the core contribution of this paper, building on Paul Dirac’s
approach in his 1975 book, General Theory of Relativity. This paper introduces
linear and non-linear models for the spacetime distortion function. The fit of the tensor
equation for gravitational waves is tested using geometrical information from the
LIGO-Virgo observation of WG200115, employing econometric techniques with
indicators such as R-squared for the robustness of the given data and the fit of the
model to the data distribution, as well as the Durbin–Watson statistic for time-series
predictability. The results show a good fit of the derived mathematical model to the
observed geometry. Different models for simulating spacetime distortion exhibit
varying degrees of fit to the observed geometry, with a less distorted model providing
a better explanation for the observed gravitational waves. The model’s fit decreases
when considering the rotation of the source object, whether it be a neutron star or
a black hole. Further modeling effort is needed to accurately represent the NSBH mer-
ger process and its role in the formation of gravitational waves.
Keywords: curvature tensors, Schwarzschild solution, gravitational waves, event
horizon, LIGO-Virgo observation.
INTRODUCTION
The Schwarzschild solution is a fundamental solution to Einstein’s equations,
which describes the gravitational field outside a spherical, non-rotating mass such
as a black hole. However, this solution contains a singularity, or a point of infinite
curvature, at a distance r = 2m from the center of the black hole, known as the event
horizon. The Schwarzschild metric is given by:
ds2 = g00dt2 – g11dr2 – g22dθ2 – g33dϕ2=
= (1 – 2m/r)dt2 – (1 – 2m/r)-1dr2 – r2dθ2 – r2sin2θdϕ2. (1)
Here, ds2 represents the spacetime interval; gμν is the fundamental tensor of
the metric; m is the mass of the black hole; and r is the distance from the center of
the black hole in the radial coordinate. The diagonal elements of gμν are g00 =
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= (1 – 2m/r), g11 = (1 – 2m/r)-1, g22 = r2, and g33 = r2sin2θ, respectively in the spherical
polar coordinates. Equation (1) is known as the Schwarzschild solution. It is held
outside the surface of a star or black hole. [1]
To address the issue of the event horizon and extend the solution to regions
where r < 2m, Paul Dirac proposed a transformation to a non-static coordinate sys-
tem. This involves introducing new time and radial coordinates, τ and ρ, which vary
with the original coordinates t and r as follows:
τ = t + f(r), (2)
ρ = t + g(r), (3)
where f(r) and g(r) are functions of r. This transforms the Schwarzschild solution
into a new form:
ds2 = dτ2 – 2m/{μ(ρ – τ)2/3}dρ2 – μ2(ρ – τ)4/3(dθ2 + sin2θdϕ2). (4)
In his 1975 book “General Theory of Relativity,” Dirac also derived an equa-
tion for gravitational waves in a region of empty space, where the gravitational field
is weak and the metric tensor gμν is approximately constant [1, p. 64]. He started
with the following equation, which is applicable in such a weak gravitational field:
According to [1, p. 25, p. 27–28, p. 64], in the empty space where only the
gravitational field exists, the Ricci curvature tensor Rμν is zero:
. (5)
Therefore,
, (6)
where
. (7)
Rμν is called as the Ricci tensor; Γαμν is the Christoffel symbol; and Γαμν,σ is the
first derivative of the Christoffel symbol by the vector σ; gαμ,ν is the first derivative
of the fundamental tensor gαμ by the vector ν; and gαα is inverse of the fundamental
tensor gαα. Equation (6) shows Einstein’s assumption that in empty space, the Ricci
tensor is zero [1].
If the gravitational field is weak, the curvature is also weak and gμν is approx-
imately constant. Therefore,
, (8)
then
. (9)
On the other hand,
. (10)
Equation (10) is another expression of (6), but with the Riemann–Christoffel
tensor (the curvature tensor) Rμνρσ, which is another variation of the Ricci tensor.
The second term of (10) disappears in the weak gravitational field. But, in this re-
search, it is included in the mathematical transformation to make the model of grav-
itational waves in the strong gravitational field, such as in the space near a black
hole or a dense star.
By interchanging ρ and μ, and neglecting ΓβμσΓβνρ – ΓβμρΓβνσ , we get
Y. Matsuki
ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 104
= 0. (11)
Then
= 0. (12)
On the other hand, the moving particle in a scalar field of the potential energy
V follows d’Alembert equation,
. (13)
In order to describe a particle moving in the 4-dimensional time-space, V is
replaced by vectors xλ of the 4-dimensional coordinates in which xλ,α = gλα , then
d’Alembert equation becomes
. (14)
Then
. (15)
Meanwhile,
. (16)
So,
, (17)
where
, (18)
where μ and ν are in symmetrical relation in the equation, therefore they are
exchangeable; and
, (19)
where μ = 0, 1, 2, 3, while xμ, a vector in μ-th dimension, is located in the
N-dimensional physical space of yn, n = 1, 2, .., N.
Therefore,
. (20)
Then, in order to describe the waves moving in the gravitational field, it is
differentiated by xσ.
. (21)
In the equation (17),
. (22)
because gρμ is approximately constant in the weak gravitational field. Therefore,
. (23)
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By interchanging ρ and σ of (21),
. (24)
By adding (12), (21) and (22), we get
. (25)
This satisfies the d’Alembert equation, describing waves propagating at the
speed of light in empty space with a weak gravitational field. It is noted that gravi-
tational waves are mathematically expressed as the second derivatives of the grav-
itational field tensor. [1]
HIGHLIGHTS OF THIS RESEARCH PAPER
While Dirac’s work derived gravitational waves in the weak gravitational field, the nov-
elty of this research lies in deriving gravitational waves in a strong gravitational field,
where the metric tensor is not constant, as opposed to a weak one. Econometric techniques
[2] are then applied to assess how well these tensors fit the observational data from the
LIGO-Virgo Collaboration, a global scientific collaboration dedicated to detecting grav-
itational waves, with a particular focus on the WG200115 event [3] This approach incor-
porates both linear and non-linear models of spacetime distortion to evaluate their effec-
tiveness in explaining the observed gravitational waves. GW200115 was identified as a
compact binary coalescence candidate by multiple low-latency matched-filtering pipelines
in LIGO Hanford and LIGO Livingston. The event was detected on January 15, 2020, at
04:23:10 UTC and was significant enough to be reported within six minutes. The initial
sky map indicated a 90 % credible region of 900 square degrees, providing a spatial local-
ization of the event in the sky. This rich data set, including the precise detection time and
well-defined sky map, offers a reliable basis for evaluating the fit of the derived
mathematical models to the observed geometry of gravitational waves.
NOVEL TENSOR EQUATION OF GRAVITATIONAL WAVES IN
STRONG CURVATURE
The following derivations extend beyond what Dirac covered in his publication.
The equation (25) describes waves moving in a weak gravitational field that
approximates a flat 4-dimensional spacetime. However, to accurately predict the
behavior of these waves, it is essential to describe them in curvilinear coordinates
within stronger gravitational fields at their source, as the waves are formed from
the movement of compact objects with significant mass.
By ensuring our model transitions smoothly from the strong gravitational field
at the source to the nearly flat field during propagation to detectors such as LIGO
and Virgo, we provide a comprehensive representation of gravitational waves’ be-
havior. This approach allows us to accurately simulate both the emission of the
waves in a strong gravitational field and their travel through the universe, where
the spacetime curvature is much weaker. This comprehensive model
negates the need for separate simulations for different curvature regimes, provided
it can be validated against known weak-field results.
To simulate the flow of waves coming from a stronger gravitational field, the
following terms are necessary [6–8], which were neglected by Dirac [1, pp. 27–28,
p. 64]:
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ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 106
(26)
for the equation (8),
(27)
for the neglected last two terms in the equation (10), and
(28)
for the equation (22).
The equations below from (29) to (33) show the same process from (20) to
(25) that derive the equation of the waves in the approximation of the curvilinear
coordinates, but the equations below include those neglected terms in order to
derive them in curvilinear coordinates. By including the neglected terms into the
equation (20), we get:
. (29)
Interchanging ρ, μ, σ and ν, we get
. (30)
By including the neglected terms into the equation (30), we get
. (31)
Interchanging ρ and σ in (31), we get
. (32)
Adding (30), (31) and (32), we get
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. (33)
Numbers, 0, 1, 2 and 3 are assigned to each of μ, ν, ρ, σ and β of (33); then by
the rule of tensor, all terms are summed up. The actual functions of the curvature
tensors are derived from the diagonal elements, g00, g11, g22, and g33, of equation
(4). Also,
If β = ρ, gρμgββgρμ,βgνμ,βσ = gρμgρρgρν,ρgνμ,ρσ.
Then if μ = ρ, gρμgρρ = gμρ =1.
And if μ ≠ ρ, gρμgρρ = gμρ = 0.
If μ = ρ = β, gρμ,σ∙gββ ≠ 0.
But if not μ = ρ = β, gρμ,σ∙gββ = 0.
And also, if μ = ρ = β, gρμgββ,σ ≠ 0.
But if not μ = ρ = β, gρμgββ,σ = 0.
All the derived elements of gravitational waves are shown in Appendix 1 Cur-
vature tensors of gravitational waves. If the simulation of gravitational waves is
conducted, focusing their spatial distribution, not on the time distribution in the 4-
dimensional time-space, the element for μ = ν = 0 is excluded from the simulation,
and then only the following diagonal elements of a 3 by 3 matrix are considered:
For μ = ν = 1,
. (34)
For μ = ν = 2,
. (35)
For μ = ν = 3,
. (36)
HAMILTONIAN LIKE APPROACH FOR REGRESSION ANALYSIS
The next step is the derivation of gravitational waves emitted from a rotating black
hole. First, the following 3 by 3 matrix from (34), (35) and (36) are formulated:
(37)
The stress energy tensor, T, reflects the energy intensities of the waves, where
k is a constant.
(38)
Then each of the coefficients of (37) relates to the elements of the waves such
as 1/(ρ–τ)-2, 1/(ρ–τ)-3/11, and so on. This is not Hamiltonian, but form of the equation
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as well as calculations of the coefficients resonate the Hamiltonian equation of mo-
tion in the perturbation theory in quantum mechanics.
GRAVITATIONAL WAVES FROM A ROTATING BLACK HOLE
For derive the tensor equation, Euler angles (39) [4, pp.150–154] is multiplied to (38).
(39)
This operation results (40).
(40)
With the next equation (41), the coefficients, c1, c2, ∙∙∙ and c6 are calculated for
the energy intensity of the gravitational waves emitted from a black hole that
rotates.
(41)
where T is a stress-energy tensor, which is explained below; and k is a constant
value, which is 1 in this research. The equation (41) is non-linear, however in this
simulation, all variables are assigned by the inputdata, which have been calculated
in advance as shown in the following section. Therefore, each term has a coefficient
and variable, then the coefficient is calculated by linear-regression analysis.
GEOMETRIC DATA FROM WG200115 OBSERVATION FOR NUMERIC
SIMULATION
Event GW200115, detected on January 15, 2020, at 04:23:10 UTC, was observed
at a luminosity distance of 300 Mpc from Earth. This distance is a measure of how
far the event is, based on the brightness of the gravitational wave signal. Addition-
ally, the event exhibited a spin magnitude with a negative spin projection, mean-
ing the spin of the primary object (likely the black hole) was oriented opposite to
the orbital angular momentum of the binary system.
These values are integrated at appropriate points in Table 1, while maintaining
the overall structure across 25 grid points. The table reflects the luminosity distance
(r) ranging from 0 to 300 Mpc to encompass the observed values, with θ (theta) and
ϕ (phi) values derived from typical ranges for these angles in spherical coordinates.
For simplicity, θ ranges from 0 to π, and ϕ ranges from 0 to 2π.
In the context of simulating gravitational wave detections from sources like
GW200115, observed by LIGO and Virgo, it’s essential to align the theoretical
model’s timescale with the practical detection window. The timescale depicted in
Fig. 1 and Table 1 spans from 0 to approximately 3.15×107 seconds, corresponding
to the light travel time over a cosmological distance of 300 Mpc (megaparsecs).
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Given that 1 Mpc equals 3.086×1022 meters, the total distance of 300 Mpc is
9.258×1024 meters. Using the speed of light (3×108 meters per second), the calcu-
lated light travel time is approximately 3.086×1024÷3×108=1.029×1016 seconds, or
about 326 million years.
The reported distance to the source is the distance from the Earth to the region
in the sky where the source is likely located. This distance comes with an uncer-
tainty range. The detected area in the sky (with 90% probability) indicates the lo-
cation but not the precise point of the spinning neutron star-black hole (NSBH)
system or other source.
For the calculations, the time and distance values over 25 grid points are con-
verted into logarithm, in order to use the statistical software tool, SST.
Fig. 1. Time and distance taken from geometric data from WG200115
The polar angle (θ) can be interpreted from the sky localization data provided
by the gravitational wave observations. This data is typically available in the form
of sky maps with credible regions indicating the probable location of the source.
However, the goal of this simulation is to perform a general simulation to explore
the overall behavior of gravitational waves in a broad sense expressed by the cur-
vature tensor equation, by investigating the fitting of the equation in a possible
geometry between observable gravitational waves and the ground observatories
such as LIGO-Virgo observatories. Therefore, the input data set is chosen as:
r ranging from 0 to 300 Mpc (to include the observed values), and θ and ϕ values
derived from typical ranges for these angles in spherical coordinates. For simplic-
ity, it was assumed θ ranges from 0 to π.
The azimuthal angle (ϕ) is related to the rotational orientation of the system.
This can often be cyclic, meaning it repeats over a 360-degree range. For this study,
a distribution was assumed, based on the characteristics of the system. For a rotat-
ing black hole in an NSBH system, the azimuthal angle could be inferred from the
angular momentum vector orientation or other related parameters. However, the
curvature tensor equation (41) doesn’t model the binary system such as NSBH, in
which there are individual object’s spin as well as the orbit rotation. Therefore, we
assumed as if one object rotates, to examine how rotation may generally influence
the fitting of the derived curvature tensor equation. And it was assumed ϕ ranges
from 0 to 2π, which is a single cycle of the rotation during the time period of the
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light travels from the observed gravitational waves to the LIGO-Virgo observa-
tories.
Fig. 2 shows the values of θ and ϕ over the 23 discrete r values for a single
cycle.
Fig. 2. Values of θ, ϕ, cos θ and cos ϕ
CONSTRUCTION OF THE DEPENDENT VARIABLE FOR THE
REGRESSION
The dependent variable for the regression to evaluate the fitting of the model in the
data taken from the observation is constructed by the stress energy tensor of the
Einstein’s field equation [4]. The stress energy tensor in 4-dimensional space
time is:
, (42)
where ρ is energy density; J is angular momentum of the rotation of the emission
source; r is distance; pr, pθ, pϕ are pressures of the waves (which might be zero or
negligible in vacuum) [5]. In this simulation, only spacial elements are considered
as shown in (43):
. (43)
In regions far from the black hole where the gravitational waves are detected,
the pressures might be very small but not exactly zero. As typically in a vacuum
solution (black hole exterior), these elements, pr, pθ, pϕ, are negligible, but in order
to calculate the coefficients of the regress-able equations, very small values are
used as shown in Fig. 3 and Table 1 are considered as the effects of surrounding
field.
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However, because this analysis focuses on the space coordinates, r, θ and ϕ,
excluding the time coordinates, the dependent variable is the sum of the sum the
three pressure values pr, pθ, and pϕ to create a single dependent variable Y for the
regression model. This approach simplifies the tensor/matrix form into a scalar
equation that can be handled using standard regression techniques. Evaluation of
the fitting of the model is made by assessing the significance of each coefficient
(e.g., using p-values) and the overall fit of the model (e.g., using R2).
POSSIBLE VALUES FOR PRESSURES
For the simulation small, non-zero values for the pressures are assumed for the sake
of your regression analysis.
Fig. 3. Values of Stress Energy Tensor
T a b l e 1 . Input data (1/2)
r (Mpc) t (sec) τ, linear τ, non-linear ρ, linear ρ, non-linear
0 0 0 0 0 0
12 2.90e+07 2.90e+07 2.90e+07 2.90e+07 2.92e+07
24 2.77e+07 2.77e+07 2.77e+07 2.77e+07 2.65e+10
36 2.64e+07 2.64e+07 2.64e+07 2.64e+07 4.31e+15
48 2.52e+07 2.52e+07 2.52e+07 2.52e+07 7.02e+20
... ... ... ... ... ...
276 1.26e+06 1.26e+06 1.26e+06 1.26e+06 7.33e+119
T a b l e 1 . Input data (2/2)
r (Mpc) Θ
(radians)
ϕ
(radians) pr pθ pϕ
0 0 0 0 0 0
12 0.13 0.26 0.001 0.001 0.001
24 0.26 0.52 0.002 0.002 0.002
36 0.39 0.78 0.003 0.003 0.003
48 0.52 1.04 0.004 0.004 0.004
... ... ... ... ... ...
276 3.00 6.01 0.025 0.025 0.025
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MODELS FOR SPACE TIME DISTORTION
For spacetime distortion models, each of f(r) and g(r) shown in (2) and (3) was
given by linear distortion model and non-linear distortion model as shown in
Table 2. The values of distorted spacetime, τ and ρ, are shown in Table 2.
T a b l e 2 . Two models for simulating the distorted time-space
Case-1
(Non-linear model)
Case-2
(Linear model)
f (r) log∙r 1/r
g (r) er r
RESULT
The coefficients of equations (38) and (41), as well as the R-squared values and
Durbin–Watson statistics, were calculated using SST, a statistical software devel-
oped by Jeffrey Dubin and Douglas Rivers at Caltech in 1986 with the assistance
of several Caltech students. The coefficients represent the strength of each term in
the equations. These values are accompanied by the standard errors of the estimated
coefficients and the corresponding t-statistics. t-statistics are calculated by dividing
the coefficient value by its standard error; higher t-statistics indicate greater statis-
tical significance of the coefficient. After establishing statistical significance, the
magnitude of the coefficient itself reflects the comparative strength of each term in
the equation. If one or more statistically significant coefficients are larger than oth-
ers, these terms have a greater influence on the stress-energy tensor, which reflects
the energy intensity on the right-hand side of the equation and consists of the terms,
i.e., independent variables, in regression analysis.
The R-squared value, as calculated using equation (44), indicates how well the
model explains the observed data related to the geometry of gravitational waves.
It ranges from 0 to 1.0, with a value of 1.0 signifying a perfect fit to the observed
data, while values between 0 and 1.0 represent varying degrees of fit. The objective
of this report is to determine which terms in equations (38) and (41) are most influ-
ential on the stress-energy tensor and to evaluate which model—linear or nonlinear
for spacetime distortion, and whether including or excluding rotation of the emis-
sion source—best explains the observed gravitational wave phenomena.
, (44)
where is the i-th value of the dependent variable; is called the expected value,
which is the calculated i-th value of the dependent variable upon the obtained for-
mula (37), where y = kT of (38); and is the average value of the dependent vari-
able .
The Durbin–Watson statistic, as defined by equation (45), measures the degree
to which the current value of a variable can be predicted based on its previous value.
A Durbin–Watson statistic value close to 2.0 suggests that the immediate future
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values can be predicted well by the immediate past values, indicating a well-be-
haved model. Conversely, values significantly less than 2.0 suggest that past values
may not effectively predict future values. This indicates that the model might not
fully account for the time-dependency or spatial relationship in the data, potentially
affecting the accuracy of the predictions.
, (45)
where
, (46)
, (47)
. (48)
All estimated coefficients are shown in Appendices from 2–1 to 2–4. Table 3
presents the tensor terms that hold statistically significant and relatively large esti-
mated coefficients, along with R-squared values and Durbin–Watson statistics for
cases involving linear spacetime distortion with and without the source’s rotation,
as well as non-linear spacetime distortion without rotation. For non-linear distortion
with rotation, SST detected correlations between the independent variables, result-
ing in the termination of the calculation process. Consequently, for this case, the
coefficients’ values, the R-squared value and Durbin–Watson statistics were calcu-
lated manually using SST’s manual mode.
Table 4 displays the independent variables with statistically and relatively sig-
nificant estimated coefficients, standard errors, and t-statistics for gravitational
waves in the linear spacetime distortion model without assuming source rotation.
Table 5 presents data for the linear spacetime distortion model with source rotation,
while Table 6 shows results for the non-linear spacetime distortion model without
source rotation. For the non-linear spacetime distortion model with source rotation,
standard errors and t-statistics were not obtained using SST, and manual calcula-
tions were not performed as this case appeared to be irrelevant to this report. How-
ever, the manually calculated coefficient values, R-squared value, and Durbin–Wat-
son statistics for this case are shown in Appendix 2–4.
In the linear spacetime distortion model, the R-squared value is higher without
rotation, indicating a better fit to the geometric information of the observed gravi-
tational waves when the source’s rotation is not assumed. However, the Durbin–
Watson statistic is closer to the ideal value of 2.0 when rotation is assumed, sug-
gesting that the gravitational waves exhibit smoother behavior over spacetime dis-
tribution with rotation. For the non-linear spacetime distortion model, only the no-
rotation case provides meaningful results with the R-squared value comparable to
the linear model, but the Durbin–Watson statistic is less informative, at 0.75492.
When rotation is assumed, the analysis yields no meaningful results, as the
R-squared value becomes excessively large and the Durbin–Watson statistic too
small as shown in Table 3. These findings suggest that the spacetime between the
observed gravitational waves and the Earth observatory is better represented by the
linear spacetime distortion model than the non-linear model.
Tables 7 and 8 compare the correlations of the independent variables in the
linear and non-linear models, both with the assumption of source rotation. The non-
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linear model shows more correlations among the independent variables than the
linear model, which led to SST terminating its operation for the non-linear case due
to these correlations.
Figs. 4 and 5 compare the stress-energy tensor as the input data for the de-
pendent variable with the predicted dependent variable after calculating the coeffi-
cient values in both the linear and non-linear models. The non-linear case exhibits
anomalies in the predicted dependent variable near the location of the gravitational
waves, resulting in an extremely poor R-squared value and a low Durbin–Watson
statistic (Table 3).
Regarding the terms in the tensor equations, trigonometric functions are influ-
ential in the linear model, while spacetime functions become influential in the non-
linear model when source rotation is not assumed.
T a b l e 3 . Significant independent variables, R-Squared and Durbin–Watson
statistics
Tensor terms with statistically
significant and larger
coefficients
R-squared
Durbin–
Watson statis-
tics
No rotation,
linear space
time distortion
0.95030 1.18063
Rotation,
linear space
time distortion
0.78633 1.87168
No rotation,
non-linear
spacetime
distortion
, ,
0.95736 0.75429
Non-Linear –
Rotation
SST stopped the analysis, due to
the correlations between
the independent variables
6.05462e+002
(Manually
calculated)
0.32756
(Manually
calculated)
T a b l e 4 . No rotation, linear space time distortion
Selected Independent
Variable
Estimated
Coefficient Standard Error t-statistic
0.10526 3.59601e-003 29.27186
T a b l e 5 . Rotation, linear space time distortion
Selected Independent
Variable
Estimated
Coefficient Standard Error t-statistic
-5.09843e-002 5.09539e-003 -10.00596
-9.87054e-003 8.15690e-003 -1.21008
* 5.17654e-002 5.10418e-003 10.14176
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T a b l e 6 . No rotation, non-linear spacetime distortion
Selected Independent
Variable
Estimated
Coefficient Standard Error t-statistic
1.22587 0.32131 3.81529
-0.68259 4.58014e-002 -14.90334
0.26481 1.38169e-002 19.16566
0.13871 8.71783e-003 15.91077
T a b l e 7 . Correlations of the independent variables
(Linear spacetime distortion with rotation of the emission source)
cosϕ/(ρ-τ)2 cosϕ/(ρ-τ)11/3 –cosϕ/(ρ–τ)13/3 cosϕsin2θ cosϕ (cos2θ–sin2θ) 1/(ρ-τ)2
cosϕ/(ρ-τ)11/3 0.99499
–cosϕ/(ρ–τ)13/3 -3.80095e-002 -3.80696e-002
cosϕsin2θ 7.10911e-002 6.82249e-002 0.73498
cosϕ (cos2θ–sin2θ) 0.11066 0.13808 0.28929 0.43686
1/(ρ-τ)2 0.98018 0.96950 -6.85755e-002 1.51910e-002 2.84118e-002
cot2θ 7.14286e-002 6.85684e-002 0.73603 1.00000 0.43675 1.56263e-002
T a b l e 8 . Correlations of the independent variables
(Non-linear spacetime distortion with rotation of the emission source)
cosϕ/(ρ-τ)2 cosϕ/(ρ-τ)11/3 –cosϕ/(ρ–τ)13/3 cosϕsin2θ cosϕ (cos2θ–sin2θ) 1/(ρ-τ)2
cosϕ/(ρ-τ)11/3 1.00000
–cosϕ/(ρ–τ)13/3 -0.14510 -0.14510
cosϕsin2θ 0.71015 0.71015 0.49518
cosϕ
(cos2θ–sin2θ) 0.29430 0.29430 -4.80880e-002 0.42042
1/(ρ-τ)2 1.00000 1.00000 -0.14510 0.71015 0.29430
cot2θ 0.67187 0.67187 0.52691 0.99714 0.41251 0.67187
Fig. 4. Stress Energy Tensor and Predicted Dependent Variable (Linear, Rotation)
Y. Matsuki
ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 116
Fig. 5. Stress Energy Tensor and Predicted Dependent Variable (Non-linear, Rotation)
DISCUSSION INTERPRETATION OF THE NEGATIVE COEFFICIENTS
Negative coefficients are shown in Appendices from 2.1 to 2.4. The reasons are not
identified yet, however possible reasons are:
1. Reflection or Scattering of Waves: A negative coefficient could indicate
that some portion of the gravitational waves is being reflected or scattered back,
rather than all the energy being absorbed or transmitted. This reflection could result
in a negative pressure effect in the stress-energy tensor as perceived from the
original propagation direction.
2. Wave’s Destructive Interference: Gravitational waves can interfere
constructively or destructively. The negative coefficient might represent regions
where destructive interference occurs, reducing the net amplitude of the wave and
thus appearing as a negative contribution in the stress-energy tensor.
3. Energy Flux and Rotation (Frame-Dragging Effects): In the presence of
rotation, frame-dragging effects (also known as the Lense-Thirring effect) can alter
the energy distribution and propagation of gravitational waves. This could lead to
regions where the effective stress-energy contribution is negative due to the com-
plex interactions between the wave and the rotating spacetime.
4. Centrifugal Effects: The rotation of the source or the spacetime might
introduce centrifugal effects that could lead to a redistribution of energy in such a
way that some components of the stress-energy tensor become negative.
SIGNIFICANCE OF θ IN THE MODEL
The statistical significance of the trigonometric terms involving θ (the angle in spherical
coordinates) in both the non-rotating and rotating cases suggests that the angular depend-
ence of gravitational waves is crucial. This aligns with the theoretical understanding of
gravitational waves, which exhibit angular dependence due to the quadrupole nature of
the source. In particular, NSBH systems emit gravitational waves with strong quadrupole
moments, meaning the angular orientation of the source significantly influences the
waveform. This explains why the trigonometric terms involving θ are statistically signif-
icant in the calculations presented in this report, as they capture the effect of the source’s
angular position on the gravitational wave signal.
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CONCLUSIONS AND RECOMMENDATIONS
The tensor equation for gravitational waves presented in this study applies to strong
curvature, whereas Dirac originally described it in the context of a weak gravita-
tional field with a constant metric tensor. This study’s novelty lies in its ability to
trace the source of gravitational waves—such as merging binary compact objects –
and simulate the potential spinning motion of the emission source. To model the
spinning source, the singularity is eliminated by assuming spacetime distortion. The
gravitational waves emitted from the spinning source are included in the simulation,
taking into account the geometric configuration between GW200115 and the
LIGO-Virgo observatories.
The simulation results indicate that the fitting of the gravitational waves’
tensor deteriorates when the spin of the source is included. This suggests that the
proposed tensor equation may not fully explain this phenomenon. Additionally,
there are uncertainties in the observed data regarding the angle of the spinning axis
and the orbital rotation cycle of the source objects, which this analysis has not
accounted for.
However, the actual observed detection period of the gravitational wave event
GW200115 spans merely 6 minutes (360 seconds) on January 15, 2020. To adapt
the theoretical timescale for practical numerical simulation, we may re-scale the
extensive light travel timescale to fit the brief detection window. This adjustment
ensures the model accounts for the actual duration over which the gravitational
waves were observed, rather than the total travel time from the source, which this
report took into account. By focusing on the observation period, we may be able to
simulate the gravitational wave’s impact within the time frame during which it in-
teracted with the detector, providing a pragmatic and relevant representation of the
event.
It is important to note that the mathematical model employed in this study
only accommodates the rotation of a single compact strong mass, such as a neutron
star or a black hole. It is not designed to simulate the more complex dynamics of
a neutron star-black hole (NSBH) system, which involves both orbital rotation and
individual spin components from each object. Therefore, while the model success-
fully captures the rotational characteristics of individual massive objects, it does
not fully account for the intricate behavior of a binary NSBH system like the one
for GW200115.
Despite this limitation, this study provides a crucial step in the exploration of
gravitational waves emitted by a rotating compact object. The findings offer a clear
and useful mathematical framework for understanding the emission of gravitational
waves from single compact masses, which is applicable to many other scenarios.
This research offers a model for the general behavior of gravitational wave emis-
sions and the impact of spin and rotation in strong gravitational fields.
As part of future work, the current model can be extended to better simulate
binary systems, incorporating both the spin and orbital dynamics of NSBH mergers.
Further refinement of the mathematical framework will involve expanding the ten-
sor equations to capture two-body interactions and gravitational interactions be-
tween the neutron star and the black hole. Addressing these aspects will signifi-
cantly enhance the ability to model complex gravitational wave sources and could
provide a more comprehensive understanding of events like GW200115.
Y. Matsuki
ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 118
REFERENCE
1. P.A.M. Dirac, General Theory of Relativity. Florida University, New York: A Wiley-
Interscience Publication, John Wiley & Sons, 1975, 69 p.
2. A.S. Goldberger, A Course in Econometrics. Harvard University Press,1991, 405 p.
3. “Observation of gravitational waves from two neutron star–black hole coalescences,
The LIGO Scientific Collaboration, the Virgo Collaboration, and the KAGRA Collab-
oration,” LIGO Caltech, 2021. Accessed on 25 June 2024. Available:
https://www.ligo.caltech.edu/page/ligo-data
4. H. Goldstein, C.P. Poole, J.L. Safko, Classical Mechanics; 3rd Edition. Pearson Educa-
tion, Inc., 2002, 646 p.
5. Christopher M. Hirata, “Lecture IV: Stress-energy tensor and conservation of energy
and momentum,” Caltech (Tapir), October 7, 2011. Available: http://www.tapir.cal-
tech.edu/~chirata/ph236/lec04.pdf
6. Y. Matsuki, 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. doi: https://doi.org/10.20535/SRIT.2308-8893.2020.1.05
7. Y. Matsuki, P.I. Bidyuk, “Simulating angular momentum of gravitational field of
a rotating black hole and spin momentum of gravitational waves,” System Research
& Information Technology, no. 1, pp. 7–20, 2021. doi:
https://doi.org/10.20535/SRIT.2308-8893.2021.1.01
8. Y. Matsuki, P.I. Bidyuk, A Course in Black Hole Simulation. LAP LAMBERT Aca-
demic Publishing, 2021, 84 p.
Received 27.12.2024
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Appendix 1. Curvature tensors of gravitational waves
μ = ν = 0
ρ = σ = 1
β = 0, 1, 2, 3
,
,
μ = ν = 1
ρ = σ = 0
β = 0, 1, 2, 3
μ = ν = 1
ρ = σ = 1
β = 0
,
, ,
, ,
, ,
μ = ν = 1
ρ = σ = 1
β = 1
, ,
, ,
, ,
, ,
,
μ = ν = 1
ρ = σ = 1
β = 2
μ = ν = 1
ρ = σ = 1
β = 3
Y. Matsuki
ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 120
μ = ν = 1
ρ = σ = 2, 3
β = 0, 1, 2, 3 ,
μ = ν = 2
ρ = σ = 0, 1, 3
β = 0, 1, 2, 3 , ,
μ = ν = 3
ρ = σ = 0
β = 0, 1, 2, 3
μ = ν = 3
ρ = σ = 1
β = 0, 1, 2, 3
μ = ν = 3
ρ = σ = 2
β = 0, 1, 2, 3
Note: All other components are 0.
Appendix 2. Estimated Coefficients, Standard Errors, t-statistics
2.1. No rotation, linear space time distortion (for equation (34) with the input data Table 2
Two models for simulating the distorted time-space - Case 2)
Dependent Variable: stress energy tensor
Independent
Variable
Estimated
Coefficient
Standard
Error t-statistic Estimated Coefficient
by SST manual mode
-6.02704e-014 5.20397e-015 -11.58162 -6.02704e-014
1.37710e-024 1.46926e-025 9.37274 1.37710e-024
-3.97632e-031 4.84817e-032 -8.20170 -3.97632e-031
0.10526 3.59601e-003 29.27186 0.10526
6.82777e-002 3.71969e-003 18.35576 6.82777e-002
2.21982e-005 1.38632e-004 0.16012 2.21982e-005
Number of Observations 23
R-squared 0.95030
Corrected R-squared 0.93568
Sum of Squared Residuals 4.52692e-004
Standard Error of the Regression 5.16032e-003
Durbin–Watson Statistic 1.18063
Mean of Dependent Variable 3.60000e-002
R-squared by SST manual mode 0.95029
Durbin–Watson Statistic by SST manual mode 1.18067
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2.2. Rotation, linear space time distortion (for equation (41) with the input data
Table 2 Two models for simulating the distorted time-space - Case 2)
Dependent Variable: stress energy tensor
Independent
Variable
Estimated
Coefficient
Standard
Error t-statistic Estimated Coefficient
by SST manual mode
3.82906e-014 3.10774e-014 1.23211 3.82906e-014
2.56705e-025 7.38201e-025 0.34774 2.56705e-025
-5.28814e-031 1.04920e-031 -5.04016 -5.28814e-031
-5.09843e-002 5.09539e-003 -10.00596 -5.09843e-002
-9.87054e-003 8.15690e-003 -1.21008 -9.87054e-003
*
-4.75441e-014 1.08187e-014 -4.39461 -4.75441e-014
* 5.17654e-002 5.10418e-003 10.14176 5.17654e-002
* These components are on the axis of the rotation and belong to W33; therefore, cosф is not
multiplied.
Number of Observations 23
R-squared 0.78633
Corrected R-squared 0.70621
Sum of Squared Residuals 1.94607e-003
Standard Error of the Regression 1.10286e-002
Durbin–Watson Statistic 1.87168
Mean of Dependent Variable 3.60000e-002
R-squared calculated by SST manual mode 0.80481
Durbin–Watson Statistic by SST manual mode 1.88005
2.3. No rotation, non-linear spacetime distortion (for equation (34) with the input data
Table 2 Two models for simulating the distorted time-space - Case 1)
Dependent Variable: stress energy tensor
Independent
Variable
Estimated
Coefficient Standard Error t-statistic Estimated Coefficient
by SST manual mode
1.22587 0.32131 3.81529 1.20947
-5.37506e-005 1.40882e-005 -3.81530 -5.30312e-005
-0.68259 4.58014e-002 -14.90334 -0.68105
0.26481 1.38169e-002 19.16566 0.26428
0.13871 8.71783e-003 15.91077 0.13834
1.07687e-003 5.27743e-004 2.04052 1.10100e-003
Number of Observations 23
R-squared 0.95736
Corrected R-squared 0.94481
Sum of Squared Residuals 3.88397e-004
Standard Error of the Regression 4.77984e-003
Y. Matsuki
ISSN 1681–6048 System Research & Information Technologies, 2026, № 2 122
Durbin–Watson Statistic 0.75429
Mean of Dependent Variable 3.60000e-002
R-squared calculated by SST manual mode 0.95777
Durbin–Watson Statistic calculated by SST manual mode 0.76027
2.4. Rotation, non-linear spacetime distortion (for equation (41) with the input data Table 2
Two models for simulating the distorted time-space - Case 1)
Dependent Variable: stress energy tensor
Independent Variable Estimated Coefficient
by SST manual mode
-7.95519
1.09155e-004
-1.77473e-002
2.32505e-003
4.55818e-002
*
5.28200
* 5.48484e-004
R-squared calculated by SST manual mode 6.05462e+002
Durbin–Watson Statistic calculated by SST manual mode 0.32756
INFORMATION ON THE ARTICLE
Yoshio Matsuki, ORCID: 0000-0002-5917-8263, National University of Kyiv Mohyla
Academy, Ukraine, e-mail: matsuki@wdc.org.ua
РЕГРЕСІЙНИЙ АНАЛІЗ СПОСТЕРЕЖЕНЬ LIGO-VIRGO WG200115
ІЗ ВИКОРИСТАННЯМ ТЕНЗОРІВ КРИВИЗНИ З РІВНЯНЬ ЕЙНШТЕЙНА
ТА ГРАВІТАЦІЙНИХ ХВИЛЬ ДІРАКА / Й. Мацукі
Анотація. Подано математичне виведення тензорів кривизни з розв’язку
Шварцшильда для гравітаційного поля разом із виведенням гравітаційних хвиль
як другої похідної поля. Усунення горизонту подій є основним внеском цієї
статті, яка ґрунтується на підході Пола Дірака в його книзі 1975 року «Загальна
теорія відносності». Представлено лінійні та нелінійні моделі для функції
викривлення простору-часу. Відповідність тензорного рівняння для гравітацій-
них хвиль перевіряється з використанням геометричної інформації зі спостере-
жень LIGO-Virgo WG200115 із застосуванням економетричних методів із
такими індикаторами, як R-квадрат надійності даних і відповідність моделі
розподілу даних, а також статистики Дарбіна–Ватсона для передбачуваності
часових рядів. Результати показують хорошу відповідність отриманої матема-
тичної моделі спостережуваній геометрії. Різні моделі для моделювання викри-
влення простору-часу демонструють різний ступінь відповідності спостережу-
ваній геометрії, при цьому менш спотворена модель забезпечує краще
пояснення спостережуваних гравітаційних хвиль. Узгодженість моделі зменшу-
ється, якщо враховувати обертання об’єкта-джерела, хоч нейтронної зірки, хоч
чорної діри. Необхідні подальші зусилля з моделювання, щоб точно предста-
вити процес злиття чорних дір (NSBH) і його роль у формуванні гравітаційних
хвиль.
Ключові слова: тензори кривизни, рішення Шварцшильда, гравітаційні хвилі,
горизонт подій, спостереження LIGO-Virgo.
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| id | journaliasakpiua-article-365261 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-01T01:00:13Z |
| publishDate | 2026 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/bd/90bcd504eafce0bcb57451511f78f5bd.pdf |
| spelling | journaliasakpiua-article-3652612026-06-30T06:14:59Z Regression analysis of LIGO-Virgo observations WG200115 using curvature tensors from Einstein’s equations and Dirac’s gravitational waves Регресійний аналіз спостережень LIGO-Virgo WG200115 із використанням тензорів кривизни з рівнянь Ейнштейна та гравітаційних хвиль Дірака Matsuki, Yoshio тензори кривизни рішення Шварцшильда гравітаційні хвилі горизонт подій спостереження LIGO-Virgo curvature tensors Schwarzschild solution gravitational waves event horizon LIGO-Virgo observation This research presents the mathematical derivation of curvature tensors from the Schwarzschild solution of the gravitational field, along with the derivation of gravitational waves as the second derivative of the field. The elimination of the event horizon is the core contribution of this paper, building on Paul Dirac’s approach in his 1975 book, General Theory of Relativity. This paper introduceslinear and non-linear models for the spacetime distortion function. The fit of the tensor equation for gravitational waves is tested using geometrical information from the LIGO-Virgo observation of WG200115, employing econometric techniques with indicators such as R-squared for the robustness of the given data and the fit of the model to the data distribution, as well as the Durbin–Watson statistic for time-series predictability. The results show a good fit of the derived mathematical model to the observed geometry. Different models for simulating spacetime distortion exhibit varying degrees of fit to the observed geometry, with a less distorted model providing a better explanation for the observed gravitational waves. The model’s fit decreases when considering the rotation of the source object, whether it be a neutron star or a black hole. Further modeling effort is needed to accurately represent the NSBH merger process and its role in the formation of gravitational waves. Подано математичне виведення тензорів кривизни з розв’язку Шварцшильда для гравітаційного поля разом із виведенням гравітаційних хвиль як другої похідної поля. Усунення горизонту подій є основним внеском цієї статті, яка ґрунтується на підході Пола Дірака в його книзі 1975 року «Загальна теорія відносності». Представлено лінійні та нелінійні моделі для функції викривлення простору-часу. Відповідність тензорного рівняння для гравітаційних хвиль перевіряється з використанням геометричної інформації зі спостережень LIGO-Virgo WG200115 із застосуванням економетричних методів із такими індикаторами, як R-квадрат надійності даних і відповідність моделі розподілу даних, а також статистики Дарбіна–Ватсона для передбачуваності часових рядів. Результати показують хорошу відповідність отриманої математичної моделі спостережуваній геометрії. Різні моделі для моделювання викривлення простору-часу демонструють різний ступінь відповідності спостережуваній геометрії, при цьому менш спотворена модель забезпечує краще пояснення спостережуваних гравітаційних хвиль. Узгодженість моделі зменшується, якщо враховувати обертання об’єкта-джерела, хоч нейтронної зірки, хоч чорної діри. Необхідні подальші зусилля з моделювання, щоб точно представити процес злиття чорних дір (NSBH) і його роль у формуванні гравітаційних хвиль. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2026-06-30 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/365261 10.20535/SRIT.2308-8893.2026.2.07 System research and information technologies; No. 2 (2026); 102-122 Системные исследования и информационные технологии; № 2 (2026); 102-122 Системні дослідження та інформаційні технології; № 2 (2026); 102-122 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/365261/350710 |
| spellingShingle | тензори кривизни рішення Шварцшильда гравітаційні хвилі горизонт подій спостереження LIGO-Virgo Matsuki, Yoshio Регресійний аналіз спостережень LIGO-Virgo WG200115 із використанням тензорів кривизни з рівнянь Ейнштейна та гравітаційних хвиль Дірака |
| title | Регресійний аналіз спостережень LIGO-Virgo WG200115 із використанням тензорів кривизни з рівнянь Ейнштейна та гравітаційних хвиль Дірака |
| title_alt | Regression analysis of LIGO-Virgo observations WG200115 using curvature tensors from Einstein’s equations and Dirac’s gravitational waves |
| title_full | Регресійний аналіз спостережень LIGO-Virgo WG200115 із використанням тензорів кривизни з рівнянь Ейнштейна та гравітаційних хвиль Дірака |
| title_fullStr | Регресійний аналіз спостережень LIGO-Virgo WG200115 із використанням тензорів кривизни з рівнянь Ейнштейна та гравітаційних хвиль Дірака |
| title_full_unstemmed | Регресійний аналіз спостережень LIGO-Virgo WG200115 із використанням тензорів кривизни з рівнянь Ейнштейна та гравітаційних хвиль Дірака |
| title_short | Регресійний аналіз спостережень LIGO-Virgo WG200115 із використанням тензорів кривизни з рівнянь Ейнштейна та гравітаційних хвиль Дірака |
| title_sort | регресійний аналіз спостережень ligo-virgo wg200115 із використанням тензорів кривизни з рівнянь ейнштейна та гравітаційних хвиль дірака |
| topic | тензори кривизни рішення Шварцшильда гравітаційні хвилі горизонт подій спостереження LIGO-Virgo |
| topic_facet | тензори кривизни рішення Шварцшильда гравітаційні хвилі горизонт подій спостереження LIGO-Virgo curvature tensors Schwarzschild solution gravitational waves event horizon LIGO-Virgo observation |
| url | https://journal.iasa.kpi.ua/article/view/365261 |
| work_keys_str_mv | AT matsukiyoshio regressionanalysisofligovirgoobservationswg200115usingcurvaturetensorsfromeinsteinsequationsanddiracsgravitationalwaves AT matsukiyoshio regresíjnijanalízspostereženʹligovirgowg200115ízvikoristannâmtenzorívkriviznizrívnânʹejnštejnatagravítacíjnihhvilʹdíraka |