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Article
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Physics
Relativistic Equation Failure for LIGO Signals
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
LIGO announced that the detected signal wave with monotonously increasing frequency is the
gravitational wave radiated by the merger of spiral binaries, thus confirming the general relativity
of basing on the assumption that the speed of light is constant. However, the assumption that the
sp eed of light is constant has not been proved to be universally valid, and there are many ground
signals with monotonic frequency increase, which means that the qualitative of LIGO signal is lack of
sufficient basis. Only by finding out the precise law of the characteristic physical quantity of LIGO
signal and comparing it with the precise law of the characteristic physical quantity of spiral binary
gravitational wave, can we judge whether LIGO signal is spiral binary gravitational wave according to
the similarity between the signal and the theoretical mo del. This decisive scientific research process
has b een intentionally or unintentionally ignored by LIGO. Here I propose a signal wave spectrum
analysis metho d to clarify the real conclusions of numerical calculation and image analysis of gw150914
signal wave. Firstly, numerical calculation results of GW150914 signal wave frequency change rate
ob ey the com quantization law which needs to be accurately describ ed by integers, and there is an
irreconcilable difference between the results and the generalized relativistic frequency equation of the
gravitational wave. Secondly, the assignment of the frequency and frequency change rate of GW150914
signal wave to the generalized relativistic frequency equation of gravitational wave constructs a non-
linear equation group for the mass of wave source, and the computer image solution shows that the
equation group has no GW150914 signal wave solution. Thirdly, it is not unique to calculate the chirp
mass of the wave source from the different frequencies and change rates of the numerical relativistic
waveform of the GW150914 signal wave, and the numerical relativistic waveform of the GW150914
signal wave deviates too far from the original waveform actually. Other LIGO signal waveforms do
not have obvious characteristics of gravitational frequency variation of spiral binary stars and lack
precise data, so they cannot be used for numerical analysis and image solution. Therefore, LIGO
signals represented by GW150914 signal do not supp ort the relativistic gravitational wave frequency
equation, and it is not credible to claim that the GW150914 signal wave is the gravitational wave of
a spiral binary star. However, whether gravitational wave signals from spiral binaries that may be
detected in the future follow the same com quantization law? Only the numerical analysis results of
detailed observation data can give an accurate answer.
Keywords: GW150914 signal wave; Lagrange frequency change rate; Blanchet frequency equation;
com quantization.
PACS number(s): 02.60.-x—Numerical approximation and analysis; 04.30.-w—Gravitational
waves; 04.60.Bc—Phenomenology of quantum gravity; 04.70.-s—Physics of black holes; 04.70.Bw—
Classical black holes; 04.80.Nn—Gravitational wave detectors and experiments.
1 Introduction
In 1915, Einstein established the equation of a gravita-
tional field
[1]
and founded general relativity
[2]
. In 1916,
Einstein predicted the existence of gravitational waves
based on general theory of relativity, and published the
first paper on gravitational waves
[3]
. According to the
general theory of relativity, when an object accelerates,
it will generate gravitational radiation and escape from
the gravitational field source and propagate in the vacu-
um to form gravitational waves, which is considered that
accelerated masses stimulate fluctuations in space-time.
In fact, gravitational waves are fluctuating gravitation-
al fields. In 1916, Schwarzschild calculated the static
spherical symmetric solution
[4, 5]
of the Einstein field e-
quation, and the singularity in this particular solution
was interpreted as the radius of the black hole’s horizon.
In February 1918, Einstein published the second paper
on gravitational waves and gave formulas for the calcu-
lation of gravitational radiant energy
[6]
. In 1963, Kerr
found the solution of the rotating black hole
[7]
for the
field equation.
Since Einstein put forward the concept of gravitation-
al wave, the theory and experimental detection principle
Citation: Dongfang, X. D. Relativistic Equation Failure for LIGO Signals. Mathematics & Nature 1, 202103 (2021).
*The first draft of this article is titled “Relativistic Differences of LIGO Signal” and published in arXiv.org (1909.05072). It is hereby
declared that Rui Chen and X. D. Dongfang are different signatures of the same author.
2 X. D. Dongfang Relativistic Equation Failure for LIGO Signals
of gravitational wave
[8-10]
have been continuously devel-
oped within the framework of general relativity, which
resulting in the illusion that gravitational wave cannot
be described without general relativity, even though the
assumption of constant speed of light as the basis of rel-
ativity is not universally established
[11]
. Because of the
complex mathematics of general relativity, gravitational
waves appear more profound and mysterious. In fact,
gravitational waves exist widely, but because the energy
of gravitational waves is very weak, they are generally
difficult to detect. According to theoretical prediction-
s, massive binary black holes or binary compact stars
can generate powerful energy gravitational waves when
merger, so compact binary stars like binary black holes
become an important model for the study of gravita-
tional wave theory and experimental detection princi-
ples. The main vibration frequency and energy of the
gravitational waves generated by the binary star during
the process of inspiral, merger and ringdown first in-
crease and then decrease with time
[12, 13]
. In 1974, Hulse
and Taylor used a radio telescope to find a pulsed bi-
nary neutron star that rotates one revolution every 8
hours
[14]
. The radiation of gravitational waves in the
process of binary satellite spiraling reduces the system
energy and decreases the revolution period. Precise mea-
surement results show that the period of revolution of
the twin neutron star is reduced by 10
−4
second per
year, which is in accordance with the theoretical value.
The discovery of the pulsed binary neutron star is consid-
ered to prove indirectly the existence of the gravitational
wave. In 1995, Blanchet derived the Blanchet frequency
equation
[15]
of gravitational wave from the binary star
wave source, which is one of the important corollaries of
the general relativistic gravitational wave theory.
In February 2016, LIGO’s two detectors at Liv-
ingston and Hanford received the signal wave named
GW150914
[16]
at 6.9 ms intervals on September 14, 2015.
In the signal waveforms, the vibration curve with the s-
train not exceeding about 1.2 × 10
−21
for a period of
time from 0.25 s to 0.45 s is considered as the gravita-
tional wave of the spiral binary black holes, which was
generated by the merger of two black holes whose mass
was 29
+4
−4
M
⊙
and 36
+5
−4
M
⊙
respectively from the solar
system 1.5 billion light years away, 1.3 billion years ago,
and the mass of the merged black hole is 62
+4
−4
M
⊙
[17]
. In
less than two years after that, LIGO successively an-
nounced the detection of four gravitational wave sig-
nals GW151226
[18]
, GW170104
[19]
, GW170814
[20]
, and
GW170817
[21]
from the spiral binary holes or spiral bi-
nary neutron stars. The widely accepted conclusion is
that these signals are gravitational waves generated by
the merging of spiral binary black holes or spiral bi-
nary neutron stars. They constitute the last piece of
Einstein’s general relativity. Not only does the black
hole predicted by general relativity exist, but the bina-
ry black holes also merge frequently, even though space-
time is ancient-far. Why do all the gravitational wave
signals detect by LIGO come from the merger of ancient-
far binary black holes or binary neutron stars? How
is the detailed frequency law of the gravitational wave
from binary stars? Scientific assertions require scientific
argumentation. The vibration curve of signal wave of
GW150914 is clear, and the part where the frequency
changes monotonously accords with the characteristics
of spiral binaries, but this is only a qualitative conclu-
sion. However, the frequency distribution of signals such
as GW151226, GW170104, GW170814, GW170817 does
not have precise rules, and quantitative calculations can-
not be made to arrive at reliable conclusions, and there
is actually a lot of uncertainty. Therefore, up to now,
only GW150914 signal wave can be used to quantitative-
ly explain the degree of LIGO signal conforming to the
generalized relativistic gravitational wave equation.
Here the numerical analysis method of signal wave
frequency is introduced, and the data of GW150914 sig-
nal wave is analyzed in detail and the precise law of fre-
quency distribution and variation of signal wave is found
out, thus the scientific conclusion of the relationship be-
tween GW150914 signal wave and general relativity is
obtained. First, the time of main strain peak is extract-
ed from GW150914 signal wave database, and the time
of wide peak and uncertain peak is corrected within the
error range, and the frequency of main strain peak is
determined. Then, the Lagrange frequency change rate,
which represents the average frequency change rate, is
calculated. It is proved that the frequency change of
GW150914 signal wave presents a generalized quanti-
zation law called com quantum law which needs to be
described by integers, but it does not support the gener-
alized relativistic Blanchet frequency equation of a grav-
itational wave. Thirdly, the broken line of frequency av-
erage change rate of GW150914 signal wave is compared
with the function curve of Blanchet frequency equation
by using the image method. It is further proved that
the generalized relativistic Blanchet frequency equation
of GW150914 signal wave, which is regarded as gravita-
tional wave of spiral binary stars, is invalid. Finally, the
frequency change rate of LIGO’s numerical relativistic
gravitational waveform is calculated, and the result al-
so do es not conform to the general relativistic Blanchet
frequency equation.
2 Frequency distribution and change laws
of GW150914 signal wave
The frequency of the main vibration part of the G-
W150914 signal wave increases monotonously, and the
strain of the main vibration part shows a tendency to
change synchronously with the frequency, but does not
increase strictly monotonously, which is mainly caused
by noise. The detection of gravitational waves usually
uses filtering techniques to shield the noise. It should be
pointed out that the filtering technology can only shield
Mathematics & Nature (2021) Vol. 1 3
the noise of the expected frequency distribution, and the
isolated wave or the noise of unintended frequency dis-
tribution can still reach the detector. The noise with
similar energy mixed in the gravitational wave strength-
ens or weakens the strain at a certain moment, so that
the gravitational waveform is distorted to varying de-
grees. The gravitational wave waveform is also distorted
by the forced vibration of the detection instrument due
to the tremor of the crust without a fixed frequency.
Therefore, the strain corresponding to the gravitational
wave signal may be shifted or masked. Strain distribu-
tion of the GW150914 signal wave is more complex than
its frequency distribution. Here we mainly analyze its
frequency distribution and its variation laws.
As shown in Figure 1, the main positive and nega-
tive strain peak time are extracted from the Hanford
database
[22]
of the GW150914 signal wave and marked
by vertical lines in reverse time sequence. The right-
most vertical line corresponds to the sequence number
1. Correct the time of wide or uncertain peak within
the allowable range of error. The accuracy of record-
ing time in LIGO database reaches 10
−9
s. According
to the monotonic increase of frequency, the correction
time of wide peak or uncertain peak is obtained by us-
ing the characteristic equation correction method, and
the accuracy of 10
−9
s is naturally retained.The correc-
tion time of wide or uncertain peaks are obtained by
using the characteristic equation correction method and
the accuracy of 10
−9
s is naturally retained. The for-
mulas for the period and frequency are T
i
= t
i
− t
i+1
and f
i
= 1/T
i
, respectively. The time of the positive
and negative strains, the corresponding period, and the
frequency calculation result are listed in Table 1 in re-
versed time order. The positive strain peaks have 6 main
frequencies from 36.55320819Hz to 197.3975904Hz; the
negative strain peaks have a total of seven main frequen-
cies from 35.37953557Hz to 230.7600891Hz. The period
and frequency of the strain peaks characterize the period
and frequency of the signal wave.
Signal GW150914 Hanford
0.25
0.30
0.35
0.40
0.45
-1.0
-0.5
0.0
0.5
1.0
Time HsL
Strain H10
-21
L
Figure 1 The positive and negative strain time of the G-
W150914 signal wave
[22]
.
LC of Positive Strain Peaks
LC of Negative Strain Peaks
LC of Positive and Negative Strain Peaks
0.37 0.38 0.39 0.40 0.41 0.42
0
50
100
150
200
Time (s)
frequency,(Hz)
Figure 2 Frequency-time polylines of strain peaks of the G-
W150914 signal wave.
Table 1 Positive and negative strain peak times of the GW150914 signal wave and its frequency distribution
i
Positive strain observation values Negative strain observation values
t
i
/s T
i
/s f
i
/Hz t
i
/s T
i
/s f
i
/Hz
1 0.428222656 0.005065918 197.3975904 0.430297903 0.004333505 230.7600891
2 0.423156738 0.008573092 116.6440307 0.425964398 0.007333624 136.3582345
3 0.414583646 0.012607488 79.31794085 0.418630774 0.010784741 92.72359945
4 0.401976158 0.017110162 58.44479852 0.407846033 0.014636434 68.32265222
5 0.384865996 0.022037889 45.37639637 0.393209599 0.018851727 53.04553744
6 0.362828106 0.027357380 36.55320819 0.374357872 0.023402144 42.73112738
7 0.335470727 0.350955728 0.028264927 35.37953557
8 0.322690801
As shown in Figure 2, according to the positive and
negative strain time t
i
and frequency f
i
listed in Table 1,
the frequency and time curves of the positive and nega-
tive strains of the GW150914 signal wave are plotted re-
spectively. The two polylines have the same monotonous
change trend, and they are very close to each other. A
single polyline that plots the frequency and time of the
positive and negative strain mixtures reflects the fluctu-
4 X. D. Dongfang Relativistic Equation Failure for LIGO Signals
ation characteristics of the frequency distribution.
It is assumed that the signal wave of GW150914 is
the gravitational wave of a helical double black hole. In
order to determine the chirp mass of GW150914 wave
source, LIGO uses the low frequency approximation of
the highly nonlinear Blanchet frequency equation of gen-
eral relativity. In fact, the high and low frequencies are
relative and there is no clear demarcation. To judge
whether the frequency distribution and variation laws of
a signal wave accord with the Blanchet frequency equa-
tion, it is necessary to calculate the time derivative of
the frequency. The frequencies of the GW150914 signal
wave and their change rates are all discrete. The classical
method of calculation is to first modify the graph of fre-
quency change over time into a smooth curve, draw the
tangent of the position of each frequency, and then mea-
sure the slope of these tangents. Therefore, the deriva-
tive of the frequency versus time
˙
f
i
at each frequency
is obtained. However, the correction of a polyline to
a smooth curve has great uncertainty, and the tangent
line also has uncertainty. The calculation of the change
rate is actually uncertain. As we all know, the mathe-
matical significance of the average rate of change of the
discrete variation is consistent with the Lagrange mean
value theorem
[23]
, so the Lagrange mean value theorem
can be used to calculate the time derivative of the fre-
quency of the gravitational wave, that is, the Lagrange
frequency change rate. The integer i in a table is an
inverse time sequence, and the variation rate of discrete
frequency conforming to the meaning of Lagrange mean
value theorem is called Lagrange frequency change rate.
It is defined as,
˙
f
i
=
f
i−1
− f
i+1
T
i
+ T
i−1
(1)
The Lagrange change rate has a definite value for calcu-
lating the average change rate of the discrete frequency,
which is better than the tangent slope measured after the
frequency time polyline is corrected to a smooth curve,
because the correction of the curve and the measurement
results of the slope are uncertain.
Table 2 Relativistic differences of frequency change rule of the GW150914 signal wave
i
Positive strain observation values Negative strain observation values
f
i
˙
f
i
˙
f
i
/
f
2
i
˙
f
i
/
f
11/3
i
f
i
˙
f
i
˙
f
i
/
f
2
i
˙
f
i
/
f
11/3
i
1 197.3975904 230.7600891
2 116.6440307 8657.49422 0.636307692 0.000228505 136.3582345 11831.23042 0.636307692 0.000176143
3 79.31794085 2747.763841 0.436753649 0.000298275 92.72359945 3755.061951 0.436753649 0.000229925
4 58.44479852 1142.134177 0.33436853 0.000379882 68.32265222 1560.827218 0.33436853 0.000292831
5 45.37639637 559.1999942 0.271585859 0.000470461 53.04553744 764.1961764 0.271585859 0.000362654
6 36.55320819 42.73112738 418.0919129 0.228972362 0.000438405
7 35.37953557
One kind of experimental data can often be used to
explain different theories. In the absence of quantitative
conclusions, experimental data are used to qualitatively
illustrate that any viewpoint is unreliable. This means
that the GW150914 signal may not be a spiral double
star gravitational wave, or it may be a spiral double star
gravitational wave, but the wave source may not be a
spiral double black hole, or it may actually be a signal
for the ground locomotive to start, or even the possibility
of using a computer to generate several data superposi-
tion results according to several functions is not ruled
out. However, in theory, there must be a discrimination
method to explain whether the GW150914 signal meets
the relativity, and finally determine the true source of
the signal.
The column in Table 2 that
˙
f
i
is located lists the La-
grange frequency change rates of the main positive and
negative strain peaks of the GW150914 signal wave. Ac-
cording to dimension analysis, the time derivative of the
frequency should be represented by the square of the
frequency. The column in Table 2 where
˙
f
i
˙
f
−2
i
is lo-
cated also lists the ratio of the frequency change rate
of the positive and negative strains of the GW150914
signal wave to their square of the frequency. Numerical
results show that the frequency distribution and changes
in the positive and negative strain peaks are different,
but the ratio
˙
f
i
˙
f
−2
i
has the same distribution with high
accuracy, which is expressed as the following equations,
˙
f
+
2
= 0.636307692f
2
2
˙
f
+
3
= 0.436753649f
2
3
˙
f
+
4
= 0.334368530f
2
4
˙
f
+
5
= 0.271585859f
2
5
.
.
.
⇔
˙
f
−
2
= 0.636307692f
2
2
˙
f
−
3
= 0.436753649f
2
3
˙
f
−
4
= 0.334368530f
2
4
˙
f
−
5
= 0.271585859f
2
5
˙
f
−
6
= 0.228972362f
2
6
(2)
The positive and negative superscripts represent the pos-
itive and negative strains, respectively. Discrete laws of
physical quantities need to be described by integers. Dis-
crete laws are essentially generalized quantization laws
including quantization laws, which are called com quan-
tum laws. Equation (2) shows that the discrete frequen-
cies of the GW150914 signal wave imply a generalized
Mathematics & Nature (2021) Vol. 1 5
quantization law closely related to quantum numb ers in
accordance with the dimensional law. This opens the
prelude to the gravitational com quantum theory, which
systematically describes the quantization law of gravi-
tational systems. It also predicts that the GW150914
gravitational wave does not correspond to the Blanchet
frequency equation of general relativity.
3 Numerical Proof of Relativistic Equation
Failure for GW150914 Signal wave
According to the theory of general relativity, frequen-
cy distribution and variation law of a gravitational wave
of spiral binaries are highly nonlinear Blanchet frequen-
cy equations
[15]
, which can not be solved accurately.
The frequency and strain of the signal wave increase
monotonously, which is only qualitatively characterized
by the merging of spiral binaries. GW150914 signal was
regarded as gravitational wave of spiral double black hole
only because of its qualitative characteristics. In the
literature, the zero order approximate equation of rela-
tivistic Blanchet frequency equation
[16]
at low frequency
˙
f =
96π
8/3
G
5/3
m
1
m
2
5c
5
(m
1
+ m
2
)
1/3
f
11/3
(3)
is used to infer the mass of binary black holes before
merging. Among them, the universal gravitational con-
stant is G = 6.674 ×10
−11
m
3
kg
−1
s
−2
, the speed of light
in a vacuum is c = 2.998 × 10
8
m s
−1
, and m
1
and m
1
is the masses of two black holes in binary black hole’s
gravitational wave source respectively. However, low fre-
quency approximation is not a scientific method. Be-
cause low-frequency and high-frequency are only rela-
tive, there is no clear boundary between them. Theo-
retically, the frequency conversion motion has a low fre-
quency approaching to zero. Low frequency approxima-
tion (3) makes it easy for readers to shift their attention
to the so-called chirp mass, while ignoring the difference
between the frequency distribution of the GW150914 sig-
nal wave and the Blanchet frequency equation.
Note that the approximate theoretical value
˙
f ∝
f
11/3
derived from the formula (3) does not conform
to the observed value (2) clearly. The low-frequency
approximation of the Blanchet frequency equation re-
quires
˙
f
i
˙
f
−11/3
i
to be an approximate constant, but the
˙
f
i
˙
f
−11/3
i
value of the GW150914 signal wave listed in
Table 2 varies with frequency, which is a prominent con-
tradiction, manifesting as that the first significant digits
of the
˙
f
i
˙
f
−11/3
i
values corresp onding to each frequen-
cy of positive and negative strain p eaks are quite dif-
ferent. This difference cannot be eliminated by a nu-
merical method that corrects the strain peak time or
redefines the change rate of the discrete frequency. Al-
though Equation (3) is the zero-order approximation of
the Blanchet’s frequency equation at low frequencies,
the zero-order approximation embodies the main rule
of the Blanchet’s frequency equation. The high-order
approximation of the Blanchet frequency equation has
a small effect on the first significant figure, and it can-
not change the conclusion that the
˙
f
i
˙
f
−11/3
i
values of
the GW150914 signal wave are not constant and it are
inconsistent with the Blanchet equation.
The amplitudes and orbital contraction rates of the
general relativistic quadrupole moments of binary stars
with unequal masses are related to the system’s mass,
which happens to be in the zero-order approximation (3)
of the Blanchet’s frequency equation,
M =
(m
1
m
2
)
3/5
(m
1
+ m
2
)
1/5
=
c
3
G
5
96
π
−8/3
f
−11/3
˙
f
3/5
(4)
According to the report of LIGO, the masses of the two
black holes of the GW150914 signal source are 29
+4
−4
M
⊙
and 36
+5
−4
M
⊙
, respectively, where M
⊙
represents the
mass of the sun. From this, the estimated value of the
chirp mass of the wave is
M
LIGO
= 28.10
+3.89
−3.51
M
⊙
The relevant literature does not explain how to cal-
culate the lowest frequency to reach the ab ove conclu-
sion. If the GW150914 signal wave is the gravitational
wave radiated by a pair of spiral binary stars, the re-
sults of chirp mass calculated by different frequencies
of the gravitational waves should be approximately e-
qual, because the chirp mass of a spiral binary star
is unique. However, according to the lower frequen-
cy f
5
= 45.37639637Hz of the positive strain of the
GW150914 signal wave and its Lagrange change rate
˙
f
5
= 559 .1999942Hzs
−1
listed in Table 2, it is esti-
mated that the chirp mass of the signal wave source is
M = 36.090M
⊙
, and according to the lower frequen-
cy f
6
= 42.73112738Hz of the negative strain of the
GW150914 signal wave and its Lagrange change rate
˙
f
6
= 418.0919129Hzs
−1
, it is estimated that the chirp
mass of the signal wave source is M = 30.873M
⊙
. It
can be seen that the estimation of chirp masses from
different frequencies are very different, and there is no
basis for making trade-offs. This poses a challenge to
the general relativity Blanchet frequency equation. The
low frequency required by the zero order approximation
of Blanchet’s frequency equation is relative, the signal
wave has some lower frequencies which have not been
recorded, and the GW150914 signal certainly has other
different chirp quality estimates. Using the frequencies
and their time derivatives of all main positive and neg-
ative strain peaks of GW150914 signal wave, the results
of estimating the chirp quality of wave source are very
different, as follows,
6 X. D. Dongfang Relativistic Equation Failure for LIGO Signals
M
+
2
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 45.376
−11/3
× 559.200
3/5
M
⊙
1.989 × 10
30
= 55.664M
⊙
M
+
3
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 58.445
−11/3
× 1142.134
3/5
M
⊙
1.989 × 10
30
= 48.960M
⊙
M
+
4
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 79.318
−11/3
× 2747.764
3/5
M
⊙
1.989 × 10
30
= 42.347M
⊙
M
+
5
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 116.644
−11/3
× 8657.494
3/5
M
⊙
1.989 × 10
30
= 36.090M
⊙
M
−
2
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 42.731
−11/3
× 418.092
3/5
M
⊙
1.989 × 10
30
= 53.356M
⊙
M
−
3
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 53.045537
−11/3
× 764.196
3/5
M
⊙
1.989 × 10
30
= 47.616M
⊙
M
−
4
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 68.323
−11/3
× 1560.827
3/5
M
⊙
1.989 × 10
30
= 41.881M
⊙
M
−
5
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 92.724
−11/3
× 3755.062
3/5
M
⊙
1.989 × 10
30
= 36.224M
⊙
M
−
6
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 136.358
−11/3
× 11831.23
3/5
M
⊙
1.989 × 10
30
= 30.873M
⊙
Among them, the positive and negative sup erscript
indicates the calculation of positive and negative strain-
s. The above approximate calculation results show that
the frequency distribution of the positive and negative s-
train of the GW150914 signal wave determines two kinds
of the chirp mass distribution of the wave source, which
vary monotonously with the frequency of the change of
the gravitational wave signal, and these values are very
different. Before the merger, the rest mass of the t-
wo stars of the wave source of the GW150914 signal is
invariable. Results of chirp mass estimation from dif-
ferent frequencies of the GW150914 signal wave are far
from each other, and it is impossible to correct to be
the approximate equivalent results. This shows that G-
W150914 signal wave does not support the general rela-
tivistic gravitational wave frequency equation.
Since the estimation of the chirp mass is derived from
the zero order approximation of the general relativis-
tic Blanchet frequency equation, the large difference of
chirp quality corresponding to the above frequencies may
be explained as the error caused by the zero order ap-
proximation of Blanchet frequency equation. It seems
that there is no such obvious difference between the o-
riginal Blanchet frequency equation and the observed
value, or there will be no obvious difference between the
advanced approximation of the Blanchet frequency e-
quation and the observed value. Is the result calculated
according to the exact Blanchet frequency equation or
its high-order approximation highly consistent with the
GW150914 signal? This problem can be solved by the
consistent conclusion of high precision multi image solu-
tion.
4 Graphical proof of Relativistic Equation
Failure for GW150914 Signal wave
After detecting the gravitational wave signal and con-
firming that the wave source is a binary star gravitation-
al system, the mass of the wave source can be determined
by the frequency and strain distribution of the gravita-
tional wave in theory. The Blanchet frequency equation
is a highly nonlinear equation and cannot be solved ac-
curately. However, in order to calculate the chirp mass,
there is no scientific basis for choosing the zero order
approximation of the equation under the low frequency
condition. Ignoring the spin of the black hole, remove
the spin-spin and spin-orbit interactions of the Blanchet
frequency equation. The high-level approximation of the
Blanchet equation is,
π
˙
f =
96G
5/3
m
1
m
2
(πf)
11/3
5c
5
(m
1
+ m
2
)
1/3
×
1 −
743
336
+
11m
1
m
2
4(m
1
+ m
2
)
2
Gπf (m
1
+ m
2
)
c
3
2/3
+4π
Gπf
c
3
(m
1
+ m
2
)
+
34103
18144
+
13661m
1
m
2
2016(m
1
+ m
2
)
2
+
59(m
1
m
2
)
2
18(m
1
+ m
2
)
4
Gπf
c
3
(m
1
+ m
2
)
4/3
(5)
Mathematics & Nature (2021) Vol. 1 7
Although this highly nonlinear equation cannot be
solved accurately, it does not mean that it is possible
to abandon the calculation and assert that the mass-
es of the two black holes m
1
and m
2
are 29
+4
−4
M
⊙
and
36
+5
−4
M
⊙
respectively and that the combined black holes
have a mass of 62
+4
−4
M
⊙
[?]
. In fact, the computer technol-
ogy having developed to today, for some non-linear and
implicit function equations which can not be solved ac-
curately, approximate solutions with high accuracy can
be obtained by numerical calculation or image solution
to prove the reliability of qualitative conclusions. In this
section, we prove that the relativistic Blanchet frequency
equation of GW150914 signal wave is invalid by means
of image solution.
4.1 Incompatibility between theoretical and ex-
perimental curves of GW150914 signal wave
An image solution method of Blanchet equation (5)
is to give different double black hole masses for the e-
quation to draw various theoretical curves of the fre-
quency change rate and frequency relationship, and then
draw the Lagrange frequency curve (i.e. the experimen-
tal curve) of GW150914signal wave that represents the
frequency change rate and frequency relationship. If the
experimental curve coincides or approaches to coincide
with one of the multiple theoretical curves, or the experi-
mental curve is located between a certain two theoretical
curves, the quality of the wave source is the closest to
the quality assignment corresponding to this specific the-
oretical curve; Otherwise, if the experimental curve can-
not coincide with or tends to coincide with any theoret-
ical curve, especially if the experimental curve cuts mul-
tiple theoretical curves, it is proved that the Blanchet
frequency equation has no GW150914signal wave solu-
tion.
As shown in Figure 3, the broken line is the La-
grange frequency broken line (experimental curve L-
C) of the frequency change rate of GW150914 signal
wave strain peak. In order to describe the relativis-
tic Blanche frequency equation curve (theoretical curve
BC) of GW150914 signal wave, seven groups of double
star masses were selected, m
1
= 29m
⊙
, m
2
= 36m
⊙
;
m
1
= 7m
⊙
, m
2
= 58m
⊙
; m
1
= 15m
⊙
, m
2
= 50m
⊙
;
m
1
= 20m
⊙
, m
2
= 45m
⊙
; m
1
= 10m
⊙
, m
2
= 55m
⊙
;
m
1
= 10m
⊙
, m
2
= 65m
⊙
; m
1
= 30m
⊙
, m
2
= 50m
⊙
,
in which m
⊙
= 1.989 × 10
30
kg is the solar mass. Sub-
stituting the mass assignments of the above groups into
equation (5) respectively, the following seven equations
are obtained,
BCHm
1
=29m
, m
2
=36m
L
BCHm
1
=7m
, m
2
=58m
L
BCHm
1
=15m
, m
2
=50m
L
BCHm
1
=25m
, m
2
=40m
L
BCHm
1
=10m
, m
2
=55m
L
BCHm
1
=10m
, m
2
=65m
L
BCHm
1
=30m
, m
2
=50m
L
LC of GW150914
0
20
40
60
80
100
120
140
0
2000
4000
6000
8000
10000
12000
f HHzL
f
HHz s
-1
L
Figure 3 Lagrange frequency curve (experimental curve LC)
and Blanchet frequency curve (theoretical curve BC) of GW150914
signal wave. Graph lines intuitively reflect that the slope of each
Blanchet frequency curve changes rapidly with increasing frequen-
cy, and the Lagrange frequency polyline cannot coincide with a
certain Blancchet frequency curve, indicating that the GW150914
signal does not support the generally relativistic Blanchet equa-
tion.
π
˙
f =
96
6.674 × 10
−11
5/3
29 × 1.989 × 10
30
36 × 1.989 × 10
30
(π × f)
11/3
5 × (2.998 × 10
8
)
5
(29 × 1.989 × 10
30
+ 36 × 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11 × 29 × 36
4(29 + 36)
2
6.674 × 10
−11
π × f × (29 + 36) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × f × (29 + 36) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661 × 29 × 36
2016(29 + 36)
2
+
59 × (29 × 36)
2
(29 + 36)
4
6.674 × 10
−11
π × f × (29 + 36) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
π
˙
f =
96
6.674 × 10
−11
5/3
7 × 1.989 × 10
30
58 × 1.989 × 10
30
(π × f)
11/3
5 × (2.998 × 10
8
)
5
(7 × 1.989 × 10
30
+ 58 × 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11 × 7 × 58
4(7 + 58)
2
6.674 × 10
−11
π × f × (7 + 58) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
8 X. D. Dongfang Relativistic Equation Failure for LIGO Signals
+4π ×
6.674 × 10
−11
π × f × (7 + 58) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661 × 7 × 58
2016(7 + 58)
2
+
59 × (7 × 58)
2
(7 + 58)
4
6.674 × 10
−11
π × f × (7 + 58) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
π
˙
f =
96
6.674 × 10
−11
5/3
15 × 1.989 × 10
30
50 × 1.989 × 10
30
(π × f)
11/3
5 × (2.998 × 10
8
)
5
(15 × 1.989 × 10
30
+ 50 × 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11 × 15 × 50
4(15 + 50)
2
6.674 × 10
−11
π × f × (15 + 50) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × f × (15 + 50) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661 × 15 × 50
2016(15 + 50)
2
+
59 × (15 × 50)
2
(15 + 50)
4
6.674 × 10
−11
π × f × (15 + 50) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
π
˙
f =
96
6.674 × 10
−11
5/3
25 × 1.989 × 10
30
40 × 1.989 × 10
30
(π × f)
11/3
5 × (2.998 × 10
8
)
5
(25 × 1.989 × 10
30
+ 40 × 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11 × 25 × 40
4(25 + 40)
2
6.674 × 10
−11
π × f × (25 + 40) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × f × (25 + 40) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661 × 25 × 40
2016(25 + 40)
2
+
59 × (25 × 40)
2
(25 + 40)
4
6.674 × 10
−11
π × f × (25 + 40) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
π
˙
f =
96
6.674 × 10
−11
5/3
10 × 1.989 × 10
30
55 × 1.989 × 10
30
(π × f)
11/3
5 × (2.998 × 10
8
)
5
(10 × 1.989 × 10
30
+ 55 × 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11 × 10 × 55
4(10 + 55)
2
6.674 × 10
−11
π × f × (10 + 55) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × f × (10 + 55) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661 × 10 × 55
2016(10 + 55)
2
+
59 × (10 × 55)
2
(10 + 55)
4
6.674 × 10
−11
π × f × (10 + 55) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
π
˙
f =
96
6.674 × 10
−11
5/3
10 × 1.989 × 10
30
65 × 1.989 × 10
30
(π × f)
11/3
5 × (2.998 × 10
8
)
5
(10 × 1.989 × 10
30
+ 65 × 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11 × 10 × 65
4(10 + 65)
2
6.674 × 10
−11
π × f × (10 + 65) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × f × (10 + 65) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661 × 10 × 65
2016(10 + 65)
2
+
59 × (10 × 65)
2
(10 + 65)
4
6.674 × 10
−11
π × f × (10 + 65) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
π
˙
f =
96
6.674 × 10
−11
5/3
30 × 1.989 × 10
30
50 × 1.989 × 10
30
(π × f)
11/3
5 × (2.998 × 10
8
)
5
(30 × 1.989 × 10
30
+ 50 × 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11 × 30 × 50
4(30 + 50)
2
6.674 × 10
−11
π × f × (30 + 50) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
Mathematics & Nature (2021) Vol. 1 9
+4π ×
6.674 × 10
−11
π × f × (30 + 50) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661 × 30 × 50
2016(30 + 50)
2
+
59 × (30 × 50)
2
(30 + 50)
4
6.674 × 10
−11
π × f × (30 + 50) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
These seven Blanchet frequency curves (BC) are also
plotted in Figure 3. Theoretically, the Lagrange fre-
quency polyline of the signal wave should be approx-
imately coincident with the Blanchet frequency curve
of the 36M
⊙
and 29M
⊙
mass combinations. Therefore,
the masses of the two black holes of the wave source
can be found the best approximation from the curve.
However, the fact is that the Lagrange polylines of the
GW150914 signal waves deviate far from the Blanchet
frequency curve of gravitational waves from binary black
holes with masses of 36M
⊙
and 29M
⊙
, which is contra-
dictory! More importantly, the Lagrange polyline of the
GW150914 signal wave cuts the Blanchet curves of dif-
ferent mass combinations. This shows that there is no
mass assignment of any group of binary stars to make
the Blanchet frequency curve coincide with the Lagrange
frequency polyline of the GW150914 signal wave.
Taking the combined values of different binary star
masses m
1
and m
2
into the Blanche frequency equation
can draw a lot of Blancchet frequency curves. When
the total mass is kept constant, the masses of the bi-
nary stars are closer, and the Blanchet frequency curve
is steeper with increasing frequency. The trend of al-
l these Blanchet curves and Lagrange frequency poly-
lines predicts that the Lagrange polyline must cut all
the Blanchet curves. It is also impossible to modify the
vibration curve of the GW150914 signal wave so that its
frequency distribution satisfies the Blanchet frequency
equation.
Therefore, the frequency variation law of GW150914
signal wave does not satisfy the relativistic Blanchet fre-
quency equation. It is lack of the minimum standard of
scientific proof to declare that GW150914 signal wave
comes from the merging process of double black holes
with 36 and 29 solar masses.
4.2 Unsolvability of Blanchet equations for G-
W150914 signal
The Lagrange frequency polyline of the GW150914
signal wave cutting the Blanchet frequency curve of d-
ifferent mass combinations shows that the Blanchet fre-
quency equation does not have any solution that satisfies
the GW150914 signal wave. This conclusion can also be
rigorously proved by another image solution.
Substituting the frequencies of the positive and nega-
tive strain peaks in Table 1 and their Lagrange deriva-
tives into the Blanche frequency equation (5) in order, a
system of nine Blanchet equations is obtained,
π × 559.1999942 =
96
6.674 × 10
−11
5/3
m
1
× 1.989 × 10
30
m
2
× 1.989 × 10
30
(π × 45.37639637)
11/3
5 × (2.998 × 10
8
)
5
(m
1
× 1.989 × 10
30
+ m
2
× 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11m
1
× m
2
4(m
1
+ m
2
)
2
6.674 × 10
−11
π × 45.37639637 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × 45.37639637 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661m
1
m
2
2016(m
1
+ m
2
)
2
+
59(m
1
m
2
)
2
(m
1
+ m
2
)
4
×
6.674 × 10
−11
π × 45.37639637 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
π × 1142.134177 =
96
6.674 × 10
−11
5/3
m
1
× 1.989 × 10
30
m
2
× 1.989 × 10
30
(π × 58.44479852)
11/3
5 × (2.998 × 10
8
)
5
(m
1
× 1.989 × 10
30
+ m
2
× 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11m
1
m
2
4(m
1
+ m
2
)
2
6.674 × 10
−11
π × 58.44479852 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × 58.44479852 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661m
1
m
2
2016(m
1
+ m
2
)
2
+
59(m
1
m
2
)
2
(m
1
+ m
2
)
4
6.674 × 10
−11
π × 58.44479852 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
10 X. D. Dongfang Relativistic Equation Failure for LIGO Signals
π × 2747.763841 =
96
6.674 × 10
−11
5/3
m
1
× 1.989 × 10
30
m
2
× 1.989 × 10
30
(π × 79.31794085)
11/3
5 × (2.998 × 10
8
)
5
(m
1
× 1.989 × 10
30
+ m
2
× 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11m
1
m
2
4(m
1
+ m
2
)
2
6.674 × 10
−11
π × 79.31794085 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × 79.31794085 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661m
1
m
2
2016(m
1
+ m
2
)
2
+
59(m
1
m
2
)
2
(m
1
+ m
2
)
4
6.674 × 10
−11
π × 79.31794085 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
π × 8657.49422 =
96
6.674 × 10
−11
5/3
m
1
× 1.989 × 10
30
m
2
× 1.989 × 10
30
(π × 116.6440307)
11/3
5 × (2.998 × 10
8
)
5
(m
1
× 1.989 × 10
30
+ m
2
× 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11m
1
m
2
4(m
1
+ m
2
)
2
6.674 × 10
−11
π × 116.6440307 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × 116.6440307 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661m
1
m
2
2016(m
1
+ m
2
)
2
+
59(m
1
m
2
)
2
(m
1
+ m
2
)
4
6.674 × 10
−11
π × 116.6440307 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
π × 418.0919129 =
96
6.674 × 10
−11
5/3
m
1
× 1.989 × 10
30
m
2
× 1.989 × 10
30
(π × 42.73112738)
11/3
5 × (2.998 × 10
8
)
5
(m
1
× 1.989 × 10
30
+ m
2
× 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11m
1
m
2
4(m
1
+ m
2
)
2
6.674 × 10
−11
π × 42.73112738 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × 42.73112738 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661m
1
m
2
2016(m
1
+ m
2
)
2
+
59(m
1
m
2
)
2
(m
1
+ m
2
)
4
6.674 × 10
−11
π × 42.73112738 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
π × 764.1961764 =
96
6.674 × 10
−11
5/3
m
1
× 1.989 × 10
30
m
2
× 1.989 × 10
30
(π × 53.04553744)
11/3
5 × (2.998 × 10
8
)
5
(m
1
× 1.989 × 10
30
+ m
2
× 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11m
1
m
2
4(m
1
+ m
2
)
2
6.674 × 10
−11
π × 53.04553744 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × 53.04553744 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661m
1
m
2
2016(m
1
+ m
2
)
2
+
59(m
1
m
2
)
2
(m
1
+ m
2
)
4
6.674 × 10
−11
π × 53.04553744 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
π × 1560.827218 =
96
6.674 × 10
−11
5/3
m
1
× 1.989 × 10
30
m
2
× 1.989 × 10
30
(π × 68.32265222)
11/3
5 × (2.998 × 10
8
)
5
(m
1
× 1.989 × 10
30
+ m
2
× 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11m
1
m
2
4(m
1
+ m
2
)
2
6.674 × 10
−11
π × 68.32265222 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × 68.32265222 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661m
1
m
2
2016(m
1
+ m
2
)
2
+
59(m
1
m
2
)
2
(m
1
+ m
2
)
4
6.674 × 10
−11
π × 68.32265222 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
Mathematics & Nature (2021) Vol. 1 11
π × 3755.061951 =
96
6.674 × 10
−11
5/3
m
1
× 1.989 × 10
30
m
2
× 1.989 × 10
30
(π × 92.72359945)
11/3
5 × (2.998 × 10
8
)
5
(m
1
× 1.989 × 10
30
+ m
2
× 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11m
1
m
2
4(m
1
+ m
2
)
2
6.674 × 10
−11
π × 92.72359945 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × 92.72359945 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661m
1
m
2
2016(m
1
+ m
2
)
2
+
59(m
1
m
2
)
2
(m
1
+ m
2
)
4
6.674 × 10
−11
π × 92.72359945 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
π × 11831.23042 =
96
6.674 × 10
−11
5/3
m
1
× 1.989 × 10
30
m
2
× 1.989 × 10
30
(π × 136.3582345)
11/3
5 × (2.998 × 10
8
)
5
(m
1
× 1.989 × 10
30
+ m
2
× 1.989 × 10
30
)
1/3
×
1 −
743
336
+
11m
1
m
2
4(m
1
+ m
2
)
2
6.674 × 10
−11
π × 136.3582345 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
2/3
+4π ×
6.674 × 10
−11
π × 136.3582345 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
+
34103
18144
+
13661m
1
m
2
2016(m
1
+ m
2
)
2
+
59(m
1
m
2
)
2
(m
1
+ m
2
)
4
6.674 × 10
−11
π × 136.3582345 (m
1
+ m
2
) × 1.989 × 10
30
(2.998 × 10
8
)
3
4/3
PeakH45.376Hz,559.200HzsL
PeakH58.445Hz,1142.134HzsL
PeakH79.318Hz,2747.764HzsL
PeakH116.644Hz,8657.494HzsL
TroughH42.731Hz,418.092HzsL
TroughH53.046Hz,764.196HzsL
TroughH68.323Hz,1560.827Hz
2
L
TroughH92.724Hz,755.062Hz
2
L
TroughH136.358Hz,11831.2Hz
2
L
20
40
60
80
100
120
20
40
60
80
100
120
m
1
Hm
L
m
2
Hm
L
Figure 4 The Blanchet mass curves of the GW150914 signal
wave. The group of curves is similar to the isotherm without two
intersection regions, showing that the Blanchet frequency equation
has no a GW150914 signal wave solution.
Theoretically, the solution of this system of equations
must exist and be unique. Otherwise, the result will
prove that the Blanchet frequency equation does not
have a GW150914 signal wave solution. The unknown-
s of the system of equations have only two masses, m
1
and m
2
, and the positions of the two can b e exchanged.
The curves of these equations are plotted. According to
the image solution, these curves should intersect in two
very small regions within the error range. The coordi-
nates correspond to the mass of the two black holes of
the source. However, as shown in Figure 4, similar to the
isotherm of the ideal gas in the closed container, the nine
Blanchet curves corresponding to the frequency and cor-
responding time derivative of the signal wave strain peak
of the GW150914 signal wave are disjoint, and the re-
sults negate the existence and uniqueness of the solution
to the equation, that is, the GW150914 signal wave dos
not support the Blanchet frequency equation of general
relativity.
The image solution can explain intuitively and con-
cisely whether the specified signal wave solution of the
non-linear Blanchet equation exists or is unique. In fact,
computer graphics can also give high-precision numeri-
cal solutions. Therefore, the image solution is one of the
best methods for dealing with gravitational wave detec-
tion data and solving nonlinear Blancht frequency equa-
tions. Observations of the GW150914 signal wave are
inconsistent with the results of the image solution of the
Blanchet frequency equation, which is a general relativ-
ity inference, and there is a big difference between the
general relativity prediction and the real gravitational
wave signal. This is the reason why a reasonable result
cannot be determined by using the Blanche frequency
equation to estimate the chirp mass of the GW150914
signal source.
In summary, if the GW150914 signal wave comes from
the merger of two spiral black holes with masses of 29
and 36 solar masses respectively, then the Lagrange fre-
quency polyline of the GW150914 signal wave will in-
12 X. D. Dongfang Relativistic Equation Failure for LIGO Signals
evitably coincide with the Blanchet frequency curve.
However, the Lagrange frequency polyline of the G-
W150914 signal wave cuts all the Blanchet frequency
curves of wave source mass combinations including 29
and 36 solar masses, so the Blanchet frequency equation
does not have a GW150914 signal wave solution. How-
ever, the Lagrange frequency polyline of the GW150914
signal wave cuts all the Blanchet frequency curves of
different wave source mass combinations including 29
and 36 solar masses, so the Blanchet frequency equa-
tion does not have a GW150914 signal wave solution.
The Blanchet frequency equation set determined by the
frequency and its time derivative contains only the mass
parameters, but the Blanch curve corresponding to the s-
train peak frequency of the GW150914 signal wave is dis-
crete without intersections, which also indicates that the
Blanchet frequency equation does not have a GW150914
signal wave solution. There are indelible essential differ-
ences between the GW150914 signal wave and the gen-
eral relativistic Blanchet frequency equation.
Among all LIGO signals, only GW150914 signal has
obvious monotonic change characteristics, and accurate
data record can be used for quantitative analysis. It is
concluded that the relativistic equation of LIGO signal is
invalid, or specifically, the Blanchet frequency equation
of LIGO signal is invalid.
5 Uncertainty of chirp mass of numerical
relativistic waveform
There is a difference between the frequency distribu-
tion law of the GW150914 signal wave and the general
relativity Blanket frequency equation. Let us now study
the frequency distribution law of the so-called numeri-
cal relativistic waveform of the GW150914 signal wave
drawn by LIGO to understand the credibility of the nu-
merical relativistic waveform. As shown in Figure 5, the
time of the positive and negative strain peaks of the nu-
merical relativistic waveform of LIGO is first extracted,
and the corresponding periods, frequencies, and frequen-
cy change rates are calculated. All the results are listed
in Table 3. It is shown from the calculation results that
the
˙
f
i
˙
f
−2
i
values of the positive strain peaks and the
negative strain peaks of the numerical relativistic wave-
form do not have the same distribution, which deviates
from the law that the
˙
f
i
˙
f
−2
i
values of the positive s-
train peaks and the negative strain peaks of the original
GW150914 waveforms have the same distribution. On
the other hand, the
˙
f
i
˙
f
−11/3
i
value of the numerical rel-
ativistic waveform is also not a constant approximated
by the zero-order approximation (3) of the Blanchet’s
frequency equation, and the value of
˙
f
i
˙
f
−11/3
i
at high
frequencies is very different in particular. It can be seen
that the drawing of the numerical relativistic wavefor-
m of GW150914 does not meet the requirement of logic
self-consistent.
0.25
0.30
0.35
0.40
0.45
-1.0
-0.5
0.0
0.5
1.0
Time HsL
Strain H10
-21
L
Figure 5 The time of the positive and negative strain peak-
s of the numerical relativistic waveform of the GW150914 signal
wave
[22]
.
Table 3 The positive and negative strain frequencies and their change rates of the numerical relativistic waveform
i
Positive strain of LIGO relativistic waveform Negative strain of LIGO relativistic waveform
t
i
f
i
˙
f
i
˙
f
i
f
2
i
˙
f
i
f
11/3
i
t
i
f
i
˙
f
i
˙
f
i
f
2
i
˙
f
i
f
11/3
i
1 0.2805 0.2644
2 0.3103 33.55705 0.2955
3 0.3377 36.49635 132.99008 0.09984 0.00024864 0.3243 32.15434
4 0.3624 40.48583 216.78090 0.13226 0.00027706 0.3507 34.72222 103.70375 0.08602 0.00023276
5 0.3839 46.51163 399.33703 0.18459 0.00030687 0.3737 37.87879 177.24775 0.12353 0.00028915
6 0.4017 56.17978 901.12942 0.28551 0.00034647 0.3934 43.47826 301.70102 0.15960 0.00029688
7 0.4151 74.62687 3215.89835 0.57745 0.00043654 0.409 50.76142 584.25788 0.22674 0.00032582
8 0.4231 125.0000 9644.08726 0.61722 0.00019751 0.4195 64.10256 1704.08712 0.41471 0.00040391
9 0.4281 200.0000 10880.0774 0.27200 0.00003977 0.4259 95.23810 5452.51100 0.60114 0.00030266
10 0.4325 227.2727 5164.99283 0.09999 0.00001181 0.4306 156.2500 10588.0957 0.43369 0.00009568
11 0.4366 243.9024 0.4346 212.7660 10775.8621 0.23804 0.00003139
Mathematics & Nature (2021) Vol. 1 13
The specific operational procedure of the numerical
relativistic waveform has not been disclosed, and the
real physical meaning has not caused the attention it
deserves. The GW150914 signal wave was identified as
coming from a far-ancient spectacle of the merger of spi-
ral binary black holes with 29M
⊙
and 36 M
⊙
respective-
ly, and mass of the combined black hole is 62M
⊙
. How
to draw such a conclusion, the calculation process of ar-
gument is missing. However, these processes of demon-
stration are one of the key procedures to test whether
LIGO signal wave is the binary’s gravitational wave pre-
dicted by general relativity. Now we use the so-called
numerical relativistic gravitational waveform of the G-
W150914 signal wave to estimate the chirp mass of the
wave source. According to the values in Table 3, the
peaks and troughs of the numerical relativistic wave-
form shown in Figure 5 have eight Lagrange frequency
derivatives, respectively, which are substituted into the
approximate equation (4) to estimate the chirp mass.
The result has 16 different values,
M
+
3
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 36.49635
−11/3
× 132.99008
3/5
M
⊙
1.989 × 10
30
= 37.9661M
⊙
M
+
4
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 40.48583
−11/3
× 216.78090
3/5
M
⊙
1.989 × 10
30
= 43.0746M
⊙
M
+
5
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 46.51163
−11/3
× 399.33703
3/5
M
⊙
1.989 × 10
30
= 43.0746M
⊙
M
+
6
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 56.17978
−11/3
× 901.12942
3/5
M
⊙
1.989 × 10
30
= 46.3286M
⊙
M
+
7
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 74.62687
−11/3
× 3215.89835
3/5
M
⊙
1.989 × 10
30
= 53.2188M
⊙
M
+
8
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 125.00000
−11/3
× 9644.08726
3/5
M
⊙
1.989 × 10
30
= 33.0680M
⊙
M
+
9
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 200.00000
−11/3
× 10880.07737
3/5
M
⊙
1.989 × 10
30
= 12.6406M
⊙
M
+
10
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 227.27270
−11/3
× 5164.99283
3/5
M
⊙
1.989 × 10
30
= 6.1023M
⊙
M
−
3
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 34.72222
−11/3
× 103.70375
3/5
M
⊙
1.989 × 10
30
= 36.4917M
⊙
M
−
4
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 37.87879
−11/3
× 177.24775
3/5
M
⊙
1.989 × 10
30
= 41.5652M
⊙
M
−
5
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 43.47826
−11/3
× 301.70102
3/5
M
⊙
1.989 × 10
30
= 42.2280M
⊙
M
−
6
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 50.76142
−11/3
× 584.25788
3/5
M
⊙
1.989 × 10
30
= 44.6521M
⊙
M
−
7
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 64.10256
−11/3
× 1704.08712
3/5
M
⊙
1.989 × 10
30
= 50.7952M
⊙
M
−
8
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 95.23810
−11/3
× 5452.51100
3/5
M
⊙
1.989 × 10
30
= 42.7197M
⊙
M
−
9
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 156.25000
−11/3
× 10588.09569
3/5
M
⊙
1.989 × 10
30
= 21.4062M
⊙
M
−
10
=
2.998 × 10
8
3
6.674 × 10
−11
5
96
π
−8/3
× 212.76600
−11/3
× 10775.86207
3/5
M
⊙
1.989 × 10
30
= 10.9683M
⊙
The estimate of the chirp mass corresponding
to the numerical relativistic waveform varies non-
monotonously with frequency and cannot be approxi-
mately equal, indicating that the frequency distribution
of the numerical relativistic waveform does not satisfy
the general relativistic Blanche equation. Although the
14 X. D. Dongfang Relativistic Equation Failure for LIGO Signals
concept of numerical relativistic wave has been beauti-
fied and rendered, it has failed to achieve the expect-
ed goal. Moreover, the so-called numerical relativistic
waveform drawing should always have a specific opera-
tion procedure. The procedure of drawing numerical rel-
ativistic waveform should be open, so that readers can
understand why it is unique, and then judge whether it
is reasonable.
LIGO announced the detection of gravitational waves
in several spiral binary black holes and one spiral binary
neutron star system, but avoided the calculation of the
distribution and variation law of frequency, which are the
most basic physical quantity. The drawing procedure of
numerical relativistic gravitational waveform is also lack
of the necessary introduction. All conclusions are on-
ly qualitative inferences. In fact, only after calculating
the frequency distribution and frequency change rate of
the signal wave, can we find out the law of frequency
distribution and change, so as to determine whether the
detected signal wave really tests the general relativity
theory.
6 Classical estimation method for total
mass of spiral binaries
It is possible for different wave sources to produce sig-
nal waves with similar frequency variation, and there is
no one-to-one correspondence between signal waves and
wave sources. The GW150914 signal wave is regarded
as the gravitational wave of the merging spiral binary
black holes, which is just one of the countless possibili-
ties. Even if it is considered that it is the only possibility,
any one of values of the so-called chirps mass of the bina-
ry black holes has an infinite set of double star masses.
The estimation of chirp quality can not accurately ob-
tain the information of a wave source, especially when
there are obvious contradictions in the estimation result-
s. Here we introduce the classical estimation results of
the total mass of the source when GW150914 signal wave
is regarded as the gravitational wave of a spiral double
black hole.
As we all know, the calculation results of the classical
mechanics of the planetary revolution cycle are consis-
tent with the astronomical observations. A precise grav-
itational theory is bound to apply both to the strong
gravitational field and the weak gravitational field. As
an approximate analysis, the estimation of the total
mass of a spiral binary black hole by the classical e-
quation of the orbit period will not be too far from the
predictions of any exact theory, otherwise there would
be an exact dividing point that does not actually ex-
ist between the strong gravitational field and the weak
gravitational field. If the binary black holes of mass m
1
and m
2
are combined into one large black hole, it can be
assumed that a particle of mass m
0
performs a uniform
circular motion on the Schwarzschild horizon
[24]
of the
large black hole. The orbital frequency f
s
of the par-
ticle is the Schwarzschild frequency. According to the
gravitation Laws and circular motion laws are obtained,
G (m
1
+ m
2
) m
0
R
2
s
= m
0
R
s
(2πf
s
)
2
(6)
Among them, the Schwarzschild radius of a large
black hole generated by a binary black hole merger is
R
s
= 2G (m
1
+ m
2
)
c
2
, which is replaced by the above
formula to get
m
1
+ m
2
=
c
3
2
5/2
πGf
s
(7)
The gravitational radiation generated by the micro-
scopic particles around the dense star is very small and
has no observation effect. Therefore, it is usually as-
sumed that two spiral black holes merge to generate
strong gravitational radiation. The Schwarzschild radi-
i of the two black holes before the merger are R
s1
=
2Gm
1
c
2
and R
s2
= 2Gm
2
c
2
, respectively. The two
black holes run around their mass centers, and detoured
Schwarzschild frequency f
s
still satisfies equation (7).
Assuming that the maximum frequency of the G-
W150914 signal wave is the Schwarzschild frequency f
s
of the wave source, then substituting the maximum fre-
quency f
s
= 230.760Hz of the negative strain in Table
1 into equation (7) gives the total mass m
1
+ m
2
=
9.845 × 10
31
kg = 49.497M
⊙
of two black holes or bina-
ry compact stars, where the mass of the sun is M
⊙
=
1.989×10
30
kg. And the other estimate of the total mass
of the two black holes obtained by substituting the max-
imum frequency f
s
= 197.398Hz of the positive strain in
Table 1 into equation (7) is 59.43M
⊙
. There are some
differences between the two estimates. We choose the
maximum value that meets the expectations,
m
1
+ m
2
= 59.43M
⊙
(8)
The chirp mass of the source estimated by Blanchet fre-
quency equation is very uncertain, so the uncertainty of
the total mass is also very large. But the uncertainty
of estimating the total mass of the binary black hole ac-
cording to the Schwarzschild orbit frequency is relatively
small. Of course this is just an estimate. Accurate cal-
culation belongs to the content of com quantum theory,
which systematically studies the laws of com quantiza-
tion in nature, what is given therein is the unique value
of the statistical average.
7 Conclusions and comments
The conclusion of physical deduction and experimen-
tal analysis must conform to the unitary principle
[11, 25]
.
Otherwise its logic must be unreliable. Often an ex-
perimental phenomenon can be explained by different
theories
[26]
. Choosing only a certain qualitative judg-
ment as the final conclusion obviously violates the u-
nitary principle. The important feature of GW150914
Mathematics & Nature (2021) Vol. 1 15
signal is the monotonic increase in frequency. We s-
tudied in detail the relationship between the frequency
distribution of GW150914 signal wave and the gener-
alized relativistic Blanchet frequency equation. It was
pointed out that the similarity between GW150914 sig-
nal wave and the wave predicted by general relativity
is only qualitative. However, frequency distribution and
variation law of GW150914 signal do not support the
non-linear Blanchet frequency equation, and the differ-
ence between them is far beyond the error range. On
the other hand, the numerical relativistic waveform de-
viates too far from the original GW150914 signal wave-
form. The other LIGO signals do not have the obvious
characteristic of monotonous increase in frequency, so
they can’t be used for accurate spectrum analysis. In
short, the LIGO signal does not support the relativistic
Blanchet frequency equation of spiral binaries merging
gravitational waves.
There is no precise demarcation point b etween the
strong gravitational field and the weak gravitational
field. The classical theory of gravitation is based on
a large number of astronomical observations. There is
no principle difference between the inferences of classi-
cal theory describing the gravitational system of black
holes and the correct inference beyond classical theory.
Therefore, if GW150914 signal wave belongs to gravi-
tational wave of spiral binary stars, the Schwarzschild
orbital frequency can be combined with classical theo-
ry to estimate the total mass of the wave source. The
problem also restores its simple and easy-to-understand
nature. The maximum frequency of positive strain of G-
W150914 signal wave is used to estimate the total mass
of the wave source. The result is in line with expec-
tation, but the maximum frequency of negative strain
of GW150914 is used to estimate the total mass of the
wave source, the result is not in line with expectation.
Is there any scientific basis for making a unique choice
between the two estimation of the total mass of the wave
source? What kind of exact equation does the frequency
of the GW150914 signal wave satisfy? How to accurately
calculate the gravitational wave source mass and deter-
mine the exact position of the gravitational wave source?
How to accurately distinguish the different gravitational
wave signals from the binary black hole, the dense bi-
nary star, the multi black hole or the dense multi star
gravitational system? What are the necessary and suffi-
cient conditions leading to the formation and merging of
spiral binary black holes? All these are urgent problems
to be solved by gravitational theory.
Although the frequency distribution of the GW150914
signal wave accords with the motion law of the classi-
cal process of spiral binary star, the numerical calcula-
tion results of the discrete frequency and rate of change
of the positive and negative strain show that only use
the quantum number can accurately describe the law
of gravitational wave. This is the com quantum theory
which is different from the traditional quantum theory.
An accurate theory of gravitational waves is bound to be
highly consistent with the exact results of experimental
observations. The signal wave with monotonic frequency
change detected by LIGO is not necessarily the gravita-
tional wave of spiral binary stars, and the GW150914
signal wave is more likely to be a ground signal. It is
unscientific to qualitatively judge that the signal wave
detected by the laser interference gravitational wave de-
tector belongs to the gravitational wave combined by
the spiral binary black hole or spiral binary neutron star
based on the monotonic increase of the frequency and
strain of the signal wave. The strict proof of the conclu-
sion requires that the high-precision numerical analysis
results of the observation data conform to the theoret-
ical equation. Identification of frequency varying signal
waves is a new technology to be developed. The gener-
alized quantization characteristics of GW150914 signal
contain new scientific theories. Perhaps gravitational
waves of spiral binaries that will be discovered in the
future have the same generalized quantization laws, and
the theory of gravity will also be further developed and
perfected.
[1] Einstein, A. Die Feldgleichungun der Gravitation. Sitzungs-
ber. K. Preuss. Akad. Wiss., 844-847 (1915).
[2] Einstein, A. The foundation of the general theory of relativ-
ity. Annalen Phys. 14, 769-822 (1916).
[3] Einstein, A. N¨aherungsweise Integration der Feldgleichungen
der Gravitation. Sitzungsber. K. Preuss. Akad. Wiss., 688-
696 (1916).
[4] Schwarzschild, K.
¨
Uber das Gravitationsfeld eines Massen-
punktes nach der Einsteinschen Theorie. Sitzungsber. K.
Preuss. Akad. Wiss., 189-196 (1916).
[5] Schwarzschild, K.
¨
Uber das Gravitationsfeld eines Massen-
punktes nach der Einsteinschen Theorie. Sitzungsber. K.
Preuss. Akad. Wiss., 424-434 (1916).
[6] Einstein, A.
¨
Uber Gravitationswellen. Sitzungsber. K.
Preuss. Akad. Wiss., 154-167 (1918).
[7] Kerr, R. P. Gravitational field of a spinning mass as an exam-
ple of algebraically special metrics. Physical review letters
11, 237-238 (1963).
[8] Weber, J. & Mcvittie, G. C. General Relativity and Gravi-
tational Waves. (Interscience Publishers, Inc, 1961).
[9] Abramovici, A. et al. LIGO: The laser interferometer
gravitational-wave observatory. Science 256, 325-333 (1992).
[10] Harry, G. M. & Collaboration, L. S. Advanced LIGO: the
next generation of gravitational wave detectors. Classical
and Quantum Gravity 27, 084006 (2010).
[11] Dongfang, X. D. On the relativity of the speed of light.
Mathematics & Nature 1, 202101 (2021).
[12] Baker, J., Br¨ugmann, B., Campanelli, M., Lousto, C. O.
& Takahashi, R. Plunge waveforms from inspiralling binary
black holes. Physical review letters 87, 121103 (2001).
[13] Damour, T., Nagar, A., Hannam, M., Husa, S. & Br¨ugmann,
B. Accurate effective-one-body waveforms of inspiralling and
coalescing black-hole binaries. Physical Review D 78, 044039
(2008).
16 X. D. Dongfang Relativistic Equation Failure for LIGO Signals
[14] Hulse, R. & Taylor, J. A high-sensitivity pulsar survey. The
Astrophysical Journal 191, L59 (1974).
[15] Blanchet, L., Damour, T., Iyer, B. R., Will, C. M. & Wise-
man, A. G. Gravitational-radiation damping of compact bi-
nary systems to second post-Newtonian order. Physical Re-
view Letters 74, 3515 (1995).
[16] Abbott, B. P. et al. Observation of gravitational waves from
a binary black hole merger. Physical review letters 116,
061102 (2016).
[17] Abbott, B. et al. Localization and broadband follow-up of
the gravitational-wave transient GW150914. The Astrophys-
ical journal letters 826, L13 (2016).
[18] Abbott, B. et al. GW151226: Observation of gravitation-
al waves from a 22-solar-mass binary black hole coalescence.
Physical Review Letters 116, 241103 (2016).
[19] Scientific, L. et al. GW170104: observation of a 50-solar-
mass binary black hole coalescence at redshift 0.2. Physical
Review Letters 118, 221101 (2017).
[20] Abbott, B. P. et al. GW170814: a three-detector observation
of gravitational waves from a binary black hole coalescence.
Physical review letters 119, 141101 (2017).
[21] Abbott, B. P. et al. GW170817: observation of gravitational
waves from a binary neutron star inspiral. Physical Review
Letters 119, 161101 (2017).
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tional Wave Strain Data.
[23] Sahoo, P. & Riedel, T. Mean value theorems and functional
equations. (World Scientific, 1998).
[24] Hawking, S. W. Black holes in general relativity. Communi-
cations in Mathematical Physics 25, 152-166 (1972).
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bers. Mathematics & Nature 1, 202102 (2021).
[26] Wang, R., Zheng, Y. & Yao, A. Generalized Sagnac effect.
Physical Review Letters 93, 143901 (2004).
Appendix A Mathematica Code of Figure 3
Show
ContourPlot
π
˙
f ==
96
(
6.674×10
−11
)
5/3
(
29×1.989×10
30
)(
36×1.989×10
30
)
(π× f )
11/3
5×(2.998×10
8
)
5
(29×1.989×10
30
+36×1.989×10
30
)
1/3
×
1 −
743
336
+
11×29×36
4(29+36)
2
6.674×10
−11
π ×f ×(29+36)×1.989×10
30
(2.998×10
8
)
3
2/3
+ 4π ×
6.674×10
−11
π ×f ×(29+36)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×29×36
2016(29+36)
2
+
59×(29×36)
2
(29+36)
4
6.674×10
−11
π ×f ×(29+36)×1.989×10
30
(2.998×10
8
)
3
4/3
,
π
˙
f ==
96
(
6.674×10
−11
)
5/3
(
7×1.989×10
30
)(
58×1.989×10
30
)
(π× f )
11/3
5×(2.998×10
8
)
5
(7×1.989×10
30
+58×1.989×10
30
)
1/3
×
1 −
743
336
+
11×7×58
4(7+58)
2
6.674×10
−11
π ×f ×(7+58)×1.989×10
30
(2.998×10
8
)
3
2/3
+ 4π ×
6.674×10
−11
π ×f ×(7+58)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×7×58
2016(7+58)
2
+
59×(7×58)
2
(7+58)
4
6.674×10
−11
π ×f ×(7+58)×1.989×10
30
(2.998×10
8
)
3
4
/
3
,
π
˙
f ==
96
(
6.674×10
−11
)
5/3
(
15×1.989×10
30
)(
50×1.989×10
30
)
(π× f )
11/3
5×(2.998×10
8
)
5
(15×1.989×10
30
+50×1.989×10
30
)
1/3
×
1 −
743
336
+
11×15×50
4(15+50)
2
6.674×10
−11
π ×f ×(15+50)×1.989×10
30
(2.998×10
8
)
3
2/3
+ 4π ×
6.674×10
−11
π ×f ×(15+50)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×15×50
2016(15+50)
2
+
59×(15×50)
2
(15+50)
4
6.674×10
−11
π ×f ×(15+50)×1.989×10
30
(2.998×10
8
)
3
4/3
,
π
˙
f ==
96
(
6.674×10
−11
)
5/3
(
25×1.989×10
30
)(
40×1.989×10
30
)
(π× f )
11/3
5×(2.998×10
8
)
5
(25×1.989×10
30
+40×1.989×10
30
)
1/3
×
1 −
743
336
+
11×25×40
4(25+40)
2
6.674×10
−11
π ×f ×(25+40)×1.989×10
30
(2.998×10
8
)
3
2/3
+ 4π ×
6.674×10
−11
π ×f ×(25+40)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×25×40
2016(25+40)
2
+
59×(25×40)
2
(25+40)
4
6.674×10
−11
π ×f ×(25+40)×1.989×10
30
(2.998×10
8
)
3
4/3
,
π
˙
f ==
96
(
6.674×10
−11
)
5/3
(
10×1.989×10
30
)(
55×1.989×10
30
)
(π× f )
11/3
5×(2.998×10
8
)
5
(10×1.989×10
30
+55×1.989×10
30
)
1/3
×
1 −
743
336
+
11×10×55
4(10+55)
2
6.674×10
−11
π ×f ×(10+55)×1.989×10
30
(2.998×10
8
)
3
2/3
+ 4π ×
6.674×10
−11
π ×f ×(10+55)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×10×55
2016(10+55)
2
+
59×(10×55)
2
(10+55)
4
6.674×10
−11
π ×f ×(10+55)×1.989×10
30
(2.998×10
8
)
3
4/3
,
π
˙
f ==
96
(
6.674×10
−11
)
5/3
(
10×1.989×10
30
)(
65×1.989×10
30
)
(π× f )
11/3
5×(2.998×10
8
)
5
(10×1.989×10
30
+65×1.989×10
30
)
1/3
×
1 −
743
336
+
11×10×65
4(10+65)
2
6.674×10
−11
π ×f ×(10+65)×1.989×10
30
(2.998×10
8
)
3
2/3
+ 4π ×
6.674×10
−11
π ×f ×(10+65)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×10×65
2016(10+65)
2
+
59×(10×65)
2
(10+65)
4
6.674×10
−11
π ×f ×(10+65)×1.989×10
30
(2.998×10
8
)
3
4/3
,
π
˙
f ==
96
(
6.674×10
−11
)
5/3
(
30×1.989×10
30
)(
50×1.989×10
30
)
(π× f )
11/3
5×(2.998×10
8
)
5
(30×1.989×10
30
+50×1.989×10
30
)
1/3
Mathematics & Nature (2021) Vol. 1 17
×
1 −
743
336
+
11×30×50
4(30+50)
2
6.674×10
−11
π ×f ×(30+50)×1.989×10
30
(2.998×10
8
)
3
2/3
+ 4π ×
6.674×10
−11
π ×f ×(30+50)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×30×50
2016(30+50)
2
+
59×(30×50)
2
(30+50)
4
6.674×10
−11
π ×f ×(30+50)×1.989×10
30
(2.998×10
8
)
3
4/3
,
{f, 0, 150},
˙
f, 0, 12000
, Axes → True, GridLinesStyle → Directive[Dashed],
PlotLegends → Placed [{"BC(m
1
=29m
⊙
, m
2
=36m
⊙
)", "BC(m
1
=7m
⊙
, m
2
=58m
⊙
)",
"BC(m
1
=15m
⊙
, m
2
=50m
⊙
)", "BC(m
1
=25m
⊙
, m
2
=40m
⊙
)", "BC(m
1
=10m
⊙
, m
2
=55m
⊙
)",
"BC(m
1
=10m
⊙
, m
2
=65m
⊙
)", "BC(m
1
=30m
⊙
, m
2
=50m
⊙
)"} , {0.250, 0.60}] , Ticks → {Range[−10, 10, 2]},
ContourStyle → {{Hue[0.7], AbsoluteThickness[3]}, {Hue[0.9], AbsoluteThickness[3]},
{Hue[0.2], AbsoluteThickness[3]}, {Hue[0.3], AbsoluteThickness[3]},
{Hue[0.6], AbsoluteThickness[3]}, {Hue[0.5], AbsoluteThickness[3]}, {Hue[0.8], AbsoluteThickness[3]}},
FrameLabel →
“f (Hz)”, "
˙
f (Hz s
−1
)"
,
ListLinePlot[{{136.3582345, 11831.23042}, {116.6440307, 8657.49422},
{92.72359945, 3755.061951}, {79.31794085, 2747.763841}, {68.32265222, 1560.827218},
{58.44479852, 1142.134177}, {53.04553744, 764.1961764}, {45.37639637, 559.1999942},
{42.73112738, 418.0919129}}, PlotStyle → {Thickness[0.010], RGBColor[1, 0, 0]},
PlotLegends → Placed[{“LC of GW150914”}, {0.250, 0.60}]], AspectRatio → 0.86]
Appendix B Mathematica Code of Figure 4
ContourPlot
π × 559.1999942 ==
96
(
6.674×10
−11
)
5/3
(
m
1
×1.989×10
30
)(
m
2
×1.989×10
30
)
(π× 45.37639637)
11/3
5×(2.998×10
8
)
5
(m
1
×1.989×10
30
+m
2
×1.989×10
30
)
1/3
×
1 −
743
336
+
11×m
1
×m
2
4(m
1
+m
2
)
2
6.674×10
−11
π ×45.37639637(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
2/3
+4π ×
6.674×10
−11
π ×45.37639637(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×m
1
×m
2
2016(m
1
+m
2
)
2
+
59×(m
1
×m
2
)
2
(m
1
+m
2
)
4
6.674×10
−11
π ×45.37639637(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
4/3
,
π × 1142.134177 ==
96
(
6.674×10
−11
)
5/3
(
m
1
×1.989×10
30
)(
m
2
×1.989×10
30
)
(π× 58.44479852)
11/3
5×(2.998×10
8
)
5
(m
1
×1.989×10
30
+m
2
×1.989×10
30
)
1/3
×
1 −
743
336
+
11×m
1
×m
2
4(m
1
+m
2
)
2
6.674×10
−11
π ×58.44479852(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
2/3
+4π ×
6.674×10
−11
π ×58.44479852(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×m
1
×m
2
2016(m
1
+m
2
)
2
+
59×(m
1
×m
2
)
2
(m
1
+m
2
)
4
6.674×10
−11
π ×58.44479852(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
4/3
,
π × 2747.763841 ==
96
(
6.674×10
−11
)
5/3
(
m
1
×1.989×10
30
)(
m
2
×1.989×10
30
)
(π× 79.31794085)
11/3
5×(2.998×10
8
)
5
(m
1
×1.989×10
30
+m
2
×1.989×10
30
)
1/3
×
1 −
743
336
+
11×m
1
×m
2
4(m
1
+m
2
)
2
6.674×10
−11
π ×79.31794085(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
2/3
+4π ×
6.674×10
−11
π ×79.31794085(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×m
1
×m
2
2016(m
1
+m
2
)
2
+
59×(m
1
×m
2
)
2
(m
1
+m
2
)
4
6.674×10
−11
π ×79.31794085(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
4/3
,
π × 8657.49422 ==
96
(
6.674×10
−11
)
5/3
(
m
1
×1.989×10
30
)(
m
2
×1.989×10
30
)
(π× 116.6440307)
11/3
5×(2.998×10
8
)
5
(m
1
×1.989×10
30
+m
2
×1.989×10
30
)
1/3
×
1 −
743
336
+
11×m
1
×m
2
4(m
1
+m
2
)
2
6.674×10
−11
π ×116.6440307(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
2/3
+4π ×
6.674×10
−
11
π ×116.6440307(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×m
1
×m
2
2016(m
1
+m
2
)
2
+
59×(m
1
×m
2
)
2
(m
1
+m
2
)
4
6.674×10
−11
π ×116.6440307(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
4/3
,
π × 418.0919129 ==
96
(
6.674×10
−11
)
5/3
(
m
1
×1.989×10
30
)(
m
2
×1.989×10
30
)
(π× 42.73112738)
11/3
5×(2.998×10
8
)
5
(m
1
×1.989×10
30
+m
2
×1.989×10
30
)
1/3
×
1 −
743
336
+
11×m
1
×m
2
4(m
1
+m
2
)
2
6.674×10
−11
π ×42.73112738(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
2/3
+4π ×
6.674×10
−11
π ×42.73112738(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×m
1
×m
2
2016(m
1
+m
2
)
2
+
59×(m
1
×m
2
)
2
(m
1
+m
2
)
4
6.674×10
−11
π ×42.73112738(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
4/3
,
π × 764.1961764 ==
96
(
6.674×10
−11
)
5/3
(
m
1
×1.989×10
30
)(
m
2
×1.989×10
30
)
(π× 53.04553744)
11/3
5×(2.998×10
8
)
5
(m
1
×1.989×10
30
+m
2
×1.989×10
30
)
1/3
18 X. D. Dongfang Relativistic Equation Failure for LIGO Signals
×
1 −
743
336
+
11×m
1
×m
2
4(m
1
+m
2
)
2
6.674×10
−11
π ×53.04553744(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
2/3
+4π ×
6.674×10
−11
π ×53.04553744(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×m
1
×m
2
2016(m
1
+m
2
)
2
+
59×(m
1
×m
2
)
2
(m
1
+m
2
)
4
6.674×10
−11
π ×53.04553744(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
4/3
,
π × 1560.827218 ==
96
(
6.674×10
−11
)
5/3
(
m
1
×1.989×10
30
)(
m
2
×1.989×10
30
)
(π× 68.32265222)
11/3
5×(2.998×10
8
)
5
(m
1
×1.989×10
30
+m
2
×1.989×10
30
)
1/3
×
1 −
743
336
+
11×m
1
×m
2
4(m
1
+m
2
)
2
6.674×10
−11
π ×68.32265222(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
2/3
+4π ×
6.674×10
−11
π ×68.32265222(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×m
1
×m
2
2016(m
1
+m
2
)
2
+
59×(m
1
×m
2
)
2
(m
1
+m
2
)
4
6.674×10
−11
π ×68.32265222(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
4/3
,
π × 3755.061951 ==
96
(
6.674×10
−11
)
5/3
(
m
1
×1.989×10
30
)(
m
2
×1.989×10
30
)
(π× 92.72359945)
11/3
5×(2.998×10
8
)
5
(m
1
×1.989×10
30
+m
2
×1.989×10
30
)
1/3
×
1 −
743
336
+
11×m
1
×m
2
4(m
1
+m
2
)
2
6.674×10
−11
π ×92.72359945(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
2/3
+4π ×
6.674×10
−11
π ×92.72359945(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×m
1
×m
2
2016(m
1
+m
2
)
2
+
59×(m
1
×m
2
)
2
(m
1
+m
2
)
4
6.674×10
−11
π ×92.72359945(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
4/3
,
π × 11831.23042 ==
96
(
6.674×10
−11
)
5/3
(
m
1
×1.989×10
30
)(
m
2
×1.989×10
30
)
(π× 136.3582345)
11/3
5×(2.998×10
8
)
5
(m
1
×1.989×10
30
+m
2
×1.989×10
30
)
1/3
×
1 −
743
336
+
11×m
1
×m
2
4(m
1
+m
2
)
2
6.674×10
−11
π ×136.3582345(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
2/3
+4π ×
6.674×10
−11
π ×136.3582345(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
+
34103
18144
+
13661×m
1
×m
2
2016(m
1
+m
2
)
2
+
59×(m
1
×m
2
)
2
(m
1
+m
2
)
4
6.674×10
−11
π×136 .3582345(m
1
+m
2
)×1.989×10
30
(2.998×10
8
)
3
4/3
,
{m
1
, 6, 130} , {m
2
, 6, 130} , GridLinesStyle → Directive[Dashed], Axes → False,
GridLinesStyle → Directive[Dashed], PlotLegends → Placed[{“Peak(45.376Hz,559.200Hz/s)”,
“Peak(58.445Hz,1142.134Hz/s)”, “Peak(79.318Hz,2747.764Hz/s)”, “Peak(116.644Hz,8657.494Hz/s)”,
“Trough(42.731Hz,418.092Hz/s)”, “Trough(53.046Hz,764.196Hz/s)”, "Trough(68.323Hz,1560.827Hz
2
)",
"Trough(92.724Hz,755.062Hz
2
)", "Trough(136.358Hz,11831.2Hz
2
)"
, {0.76, 0.60}
,
Ticks → {Range[−10, 10, 2]}, ContourStyle → {{Hue[0.1], AbsoluteThickness[3]},
{Hue[0.2], AbsoluteThickness[3]}, {Hue[0.3], AbsoluteThickness[3]},
{Hue[0.4], AbsoluteThickness[3]}, {Hue[0.5], AbsoluteThickness[3]},
{Hue[0.60], AbsoluteThickness[3]}, {Hue[0.7], AbsoluteThickness[3]},
{Hue[0.8], AbsoluteThickness[3]}, {Hue[0.9], AbsoluteThickness[3]}},
FrameLabel → {"m
1
(m
⊙
)", "m
2
(m
⊙
)"} , AspectRatio → 0.86]
Latest findings
Dongfang Modified Equations of Molecular Dynamics
The unitary principle test of the pressure equation of ideal gas gives a negative conclusion, so I systematically revise the basic equation of molecular dynamics. The specific logic and conclusions are as following. The classical molecular dynamics theory established the wrong physical model of uniform motion of molecules under the action of an equivalent constant force, and the classical equations of ideal gas pressure and temperature derived from this model that violates the principle of mechanics is all incorrect. A variety of physical models of molecular interaction in accordance with the mechanical principle is established, and the correct equation of ideal gas pressure is derived consistently. It is proved that the pressure of an ideal gas is equal to the molecular energy per unit volume, and the thermodynamic temperature of an ideal gas is equal to the quotient of molecular average kinetic energy and Boltzmann constant. Various inferences of different models are consistent, so they comply with the unitary principle. Finally, I introduce the problem of the definite solution of the gas molecular rate distribution function that meets the limit condition of the speed of light, and put forward experimental suggestions to verify the theoretical gas temperature correction equation.