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From Newton to Pierre-Simon Laplace have realized that the speed of gravity on objects is very huge. But this view is not consistent with GR. I don't know if GR is the only correct solution to gravity. If not, then the gravity model under the influence of gravitational waves provides a new way for humans to study the universe. Thanks. Tony tony1807559167@gmail.com
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Gravitational fields and gravitational waves
Tony Yuan
a)
Department of Electronic Engineering, Beihang University, No. 37 Xueyuan Road, Haidian District, Beijing
100191, China
(Received 22 December 2021; accepted 10 May 2022; published online 26 May 2022)
Abstract: The relative velocity between objects with finite velocity affects the reaction between
them. This effect is known as general Doppler effect. The Laser Interferometer Gravitational-Wave
Observatory (LIGO) discovered gravitational waves and found their speed to be equal to the speed
of light c. Gravitational waves are generated following a disturbance in the gravitational field; they
affect the gravitational force on an object. Just as light waves are subject to the Doppler effect, so
are gravitational waves. This article explores the following research questions concerning gravita-
tional waves: Is there a linear relationship between gravity and velocity? Can the speed of a gravi-
tational wave represent the speed of the gravitational field (the speed of the action of the
gravitational field upon the object)? What is the speed of the gravitational field? What is the spatial
distribution of gravitational waves? Do gravitational waves caused by the revolution of the Sun
affect planetary precession? Can we modify Newton’s gravitational equation through the influence
of gravitational waves? V
C2022 Physics Essays Publication.
[http://dx.doi.org/10.4006/0836-1398-35.2.208]
R
esum
e: La vitesse relative entre les objets
a vitesse finie affecte la r
eaction entre eux. Cet effet
est connu sous le nom d’effet Doppler g
en
eral. Le Laser Interferometer Gravitational-Wave Obser-
vatory (LIGO) a d
ecouvert des ondes gravitationnelles et a trouv
e que leur vitesse
etait
egale
ala
vitesse de la lumie`re c. Les ondes gravitationnelles sont g
en
er
ees suite
a une perturbation du champ
gravitationnel; ils affectent la force gravitationnelle sur un objet. Tout comme les ondes lumineuses
sont soumises
a l’effet Doppler, les ondes gravitationnelles le sont aussi. Cet article explore les
questions de recherche suivantes concernant les ondes gravitationnelles: Existe-t-il une relation
lin
eaire entre la gravit
e et la vitesse? La vitesse d’une onde gravitationnelle peut-elle repr
esenter la
vitesse du champ gravitationnel (la vitesse de l’action du champ gravitationnel sur l’objet)? Quelle
est la vitesse du champ gravitationnel? Quelle est la distribution spatiale des ondes gravitation-
nelles? Les ondes gravitationnelles provoqu
ees par la r
evolution du Soleil affectent-elles la
pr
ecession plan
etaire? Peut-on modifier l’
equation gravitationnelle de Newton par l’influence des
ondes gravitationnelles?
Key words: Newtonian Gravity; Doppler Effect; Gravitational Wave; Gravitational Field; LIGO; Gravitational Constant;
Precession of the Planets.
I. INTRODUCTION
Newtonian gravity
1,2
is a force that acts at a distance. No
matter how fast an object travels, gravity acts upon the object
instantaneously. Gravity is only related to the mass and dis-
tance of the object, equal to G0Mm=r2, of which the univer-
sal gravitational constant
3
G0¼6:67259 1011Nm2=kg2.
G
0
is measured when two objects are relatively stationary.
This can be regarded as a static gravitational constant. New-
tonian gravity states that the speed of the gravitational field
on an object is infinite, therefore, whether two objects are
relatively stationary or moving, both can be considered
unchanged, so there is no general Doppler effect.
4
The Laser
Interferometer Gravitational-Wave Observatory (LIGO)
5
first discovered gravitational waves
6,7
and measured their
speed. This discovery, thus, leads us to consider whether the
speed of gravitational field
8
is the same as that of the gravita-
tional wave. We know that when we put a stone into the
water, in addition to causing slow water waves, it will also
cause sound waves in the water, and the speed will be much
greater than that of water waves. So the speed of the water
waves we observe cannot represent that of sound waves in
the water. Will gravitational waves be like this?
General relativity (GR), it’s view on the speed of gravity
is different from that of Newton and Laplace. GR also
believes that the speed of gravity is equal to the speed of
light, but until now, scientists have been unable to prove this
view. But we know if the gravitational field has a finite
speed, there will be a general Doppler effect between the
gravitational field and the object. To determine the speed of
the gravitational field, we assume the speed of the gravita-
tional field is equal to the speed of light. For the convenience
of analysis, we use Xto represent the speed of the gravita-
tional field. If the planetary motions of the solar system cal-
culated under this hypothesis are consistent with
astronomical observations, the correctness of this hypothesis
can be proved, otherwise it is proved that the speed of the
gravitational field is not equal to that of light.
a)
tony1807559167@gmail.com
ISSN 0836-1398 (Print); 2371-2236 (Online)/2022/35(2)/208/12/$25.00 V
C2022 Physics Essays Publication208
PHYSICS ESSAYS 35, 2 (2022)
Since we need to analyze the speed of gravity, so we
must first figure out what is the relationship between gravity
and velocity?
II. DERIVATION OF THE RELATIONSHIP BETWEEN
GRAVITY AND VELOCITY BASED ON NEWTON’S
GRAVITY EQUATION
In a very short time slice dt, we can assume that mis sta-
tionary and the gravity received is constant. We can then
accumulate the impulse generated by the gravity on each
time slice and find the average relating to the entire time
period to obtain effective constant gravitation and determine
the relationship between the equivalent gravitation and
velocity.
Consider the influence of velocity on gravity when the
moving velocity of object mrelative to Mis not 0.
As shown in Fig. 1, there are two objects with masses M
and m, the distance between them is r,mhas a moving veloc-
ity relative to M, the speed is v, and the direction of the
velocity is depicted by the straight line connecting them.
FðtÞ¼G0Mm=ðrþvtÞ2represents the gravity on mat time
t. The Newtonian equation of gravity is used here. In any
small time dt,mcan be regarded as stationary. An accumula-
tion of the impulse dp is obtained by multiplying the gravity
and time in these small time slices. Then, the sum of the
gravitational impulse received by mwithin a certain period
can be obtained. Supposing that the gravitational impulse
obtained by mis pafter time Thas passed, the gravity is inte-
grated into the time domain
p¼ðT
0
FðtÞdt ¼ðT
0
G0Mm
ðrþvtÞ2dt ¼G0Mm
r2
ðT
0
1
ð1þvt=rÞ2dt;
p¼G0Mm
r2r=v
1þvt=r
T
0
;
p¼G0Mm
r2T
1þvT=r:(1)
For an object mwith a speed of v, the accumulated
impulse pduring time Tcan be expressed by an equivalent
constant force multiplied by time T. For the convenience of
description, we use F(v) to express this equivalent force
FðvÞ¼p=T¼G0Mm
r21þvT
r

:
There is an inverse proportional relationship between the
equivalent gravitational force and the speed v. The larger the
v, the smaller the F(v); the smaller the v, the larger the F(v).
When v¼0, it is Newtonian gravity. When vtends to infin-
ity, F(v)¼0. The Newtonian gravitational equation is based
on the premise that the gravitational field speed is infinite.
Now, we may assume that the gravitational field has a finite
speed X, therefore, the Newtonian gravitational equation is
no longer applicable and need to be modified.
Let us continue to think about the difference in the aver-
age gravitational force received by two objects at different
speeds during time T? Assuming that two objects have dif-
ferent velocities, v
1
¼v
0
dv,v
2
¼v
0
þdv, their average
gravity
Fv
1
ðÞ
¼Fv
0dv
ðÞ
¼G0Mm
r21þv0dv
ðÞ
T
r

;
Fv
2
ðÞ
¼Fv
0þdv
ðÞ
¼G0Mm
r21þv0þdv
ðÞ
T
r

;
Fv
1
ðÞ
Fv
2
ðÞ
¼G0Mm
r2
r
rþv0dv
ðÞ
T
r
rþv0þdv
ðÞ
T;
Fv
1
ðÞ
Fv
2
ðÞ
¼G0Mm
r2
2rdvT
rþv0T
ðÞ
2ðdvTÞ2
!
:
When Tis infinitesimal close to 0, Fðv1ÞFðv2Þ
¼G0Mm
r2
2dvT
r

.
Assume K¼T=r;dv¼v1v2, so the following for-
mula is obtained: Fðv1ÞFðv2Þ¼G0Mm=r2ðKdvÞ, we can
see that there is a linear relationship between average gravity
and velocity, when Tis infinitesimal close to 0, the average
gravity is the instantaneous gravity.
However, we also know that if there is relative velocity
between any two objects, there will be a general Doppler
effect between them. According to this general Doppler
effect between the object and the gravitational field, two
Doppler effect boundary conditions are introduced:
1. When an object’s velocity relative to the source of grav-
ity is 0, it is Newtonian gravity.
2. When an object’s velocity relative to the gravitational
field is 0, the gravitational force no longer acts on the
object.
As shown in Fig. 2, according to the general Doppler
effect (chase effect), using boundary conditions Fð0Þ¼
G0Mm=r2and FðXÞ¼0, it can be easily calculated
FðvÞ¼Fð0ÞþvFðXÞFð0Þ
X¼Fð0ÞXv
X
¼G0Mm
r2Xv
X:
From the above analysis, the formula of universal gravi-
tation with parameter vis as follows:
FIG. 1. Gravity model.
Physics Essays 35, 2 (2022) 209
FðvÞ¼G0Mm
r2fðvÞ;fðvÞ¼Xv
X:(2)
If it is necessary to preserve the form of Newton’s grav-
ity equation, we may write it as follows:
FðvÞ¼GðvÞMm
r2;GðvÞ¼G0Xv
X:(3)
That is, the gravitational constant becomes a function of
v,G(v). Thus, we may understand that when the gravitational
field has a different speed relative to m, the gravitational con-
stant is also different. Next, we apply the new gravitational
equation to the planetary orbit calculation to determine
whether it is consistent with actual observations.
III. CALCULATION OF THE INFLUENCE OF THE NEW
GRAVITATIONAL EQUATION ON EARTH’S ORBIT
From the above derivation, we get the gravity formula
with vas a parameter
FðvÞ¼G0Mm
r2Xv
X:
Considering that the velocity direction of the object m
may have an angle with the gravitational field, we define v
r
as the component of the speed in the direction of the gravita-
tional field and then obtain a general formula
Fv
r
ðÞ
¼G0Mm
r2Xvr
X:
The equation shows that when an object has a velocity
component in the direction of the gravitational field, that is,
there is a movement effect in the same direction between the
gravitational field and the object, the gravitational force
received decreases. When the object has a velocity compo-
nent that is opposite to the direction of the gravitational field,
that is, the two have the effect of moving toward each other,
the gravitational force received increases. This leads us to,
thus, consider what impact, under this general Doppler
effect, it may have on the planet’s orbit. Can planets main-
tain the conservation of mechanical energy in their orbits?
As shown in Fig. 3, under the new gravitational equa-
tion, as the planetary velocity has the same direction compo-
nent v
r
in the direction of the gravitational field in orbits A
and B, the gravity decreases. Therefore, the planet gains
extra force in the direction of the gravitational field. This
force travels in the same direction as v
r
. According to the
power calculation formula P¼Fv
r
>0, the planetary
mechanical energy increases.
Regarding regions C and D, as the planetary velocity has
a reverse component v
r
in the direction of the gravitational
field, the gravitational force increases. Therefore, the extra
force gained by the planet moves in the opposite direction of
the gravitational field. This force is in the same direction as
v
r
. According to the power calculation formula P¼Fv
r
>0,
the planetary mechanical energy increases.
Therefore, under the new gravitational equation, the
mechanical energy of the planet in the entire orbit continues
to increase and the mechanical energy becomes larger and
larger. This would cause the planet to gradually move away
from the Sun and eventually the solar system. Taking Earth
as an example, using the new gravitational equation, after
how many revolution cycles would Earth begin to move
away from the solar system? Below we include our theoreti-
cal analysis and calculations.
A. Introduction of polar coordinates
Let the Sun, mass M, lie at the origin. Consider a planet,
mass m, in orbit around the Sun. Let the planetary orbit lie in
the x-yplane. Let r(t) be the planet’s position vector with
respect to the Sun. The planet’s equation of motion is
m
r¼G0Mm
r2Xv
Xer;(4)
where e
r
¼r/rand v
r
¼e
r
._
r. Let r¼|r| and h¼tan
1
(y/x)be
plane polar coordinates. The radial and tangential compo-
nents of Eq. (4) are
rr_
h2¼G0M
r21_
r
X

;(5)
r
hþ2_
r_
h¼0:(6)
Equation (6) can be integrated to give
r2_
h¼h;(7)
FIG. 2. Linear relationship between gravity and velocity.
FIG. 3. (Color online) The velocity component of the planet’s gravitational
field direction.
210 Physics Essays 35, 2 (2022)
where his the conserved angular momentum per unit mass.
Equations (5) and (7) can be combined to give
rh2
r3¼G0M
r21_
r
X

:(8)
B. Energy conservation
Multiply Eq. (8) by _
r. We obtain
d
dt
_
r2
2þh2
2r2G0M
r

¼G0M_
r2
r2X(9)
or
d
dt ¼G0M_
r2
r2X0;(10)
where
¼1
2_
r2þr2_
h2

G0M
r(11)
is the energy per unit mass. Equation (10) demonstrates that
the Doppler shift correction to the law of force causes the
system to cease conserving energy. The orbital energy grows
without limit. This means that the planet will eventually
escape the Sun’s gravitational pull (when its orbital energy
becomes positive).
C. Solution of equations of motion
Let 1=r¼u½hðtÞ. It follows that
_
r¼hdu
dh;(12)
r¼u2h2d2u
dh2;(13)
thus, Eq. (8) becomes
d2u
dh2cdu
dhþu¼G0M
h2;(14)
where
c¼G0M
hX (15)
is a small dimensionless constant. To first order in c,an
appropriate solution of Eq. (14) is
uG0M
h2ð1þeexp ðchÞcos hÞ;(16)
where eis the initial eccentricity of the orbit. Thus,
rðhÞ¼ rc
1þeexp ðchÞcos h;(17)
where
rc¼h2
G0M:(18)
It can be observed that the orbital eccentricity grows
without limit as the planet orbits the Sun. Eventually, when
the eccentricity becomes unity, the planet will escape the
Sun.
D. Estimation of escape time
The planet escapes when its orbital eccentricity becomes
unity. The number of orbital revolutions, n, required for this
to happen is
eexp ðcn2pÞ¼1;(19)
where eis the initial eccentricity. Thus, n¼1=2pc ln ð1=eÞ,
c¼2pa
TX 1e2
ðÞ
1
2
;(20)
where ais the initial orbital major radius and Tis the initial
period. Hence,
n¼
TX 1e2
ðÞ
1
2ln 1
e

4p2a:(21)
For Earth, T¼3.156 10
7
s, X¼c¼2.998 10
8
m/s,
a¼1.496 10
11
m, and e¼0.0167. Hence, Earth would
escape from the Sun’s gravitational influence after
n¼
3:156107
ðÞ
2:998108
ðÞ
10:01672
ðÞ
1
2ln 1
0:0167

4p21:4961011
ðÞ
6:6103
(22)
revolutions. If each revolution takes approximately 1 year,
then the escape time is a few thousand years. However, the
age of the solar system is 4.6 10
9
years. The escape time is
smaller than this by a factor of approximately one million.
Therefore, the speed of gravitational waves cannot represent
the speed of the gravitational field. From Eq. (22), the speed
of a gravitational field Xmust be much greater than the speed
of light
9
c; this is more in line with Newton’s argument that
the force of gravity acts at a distance.
We may consider the following analogy: We use a rope
to pull a kite. When we shake it hard, the rope will fluctuate
and pass to the kite at a certain wave speed, however, when
we loosen the rope, the kite instantly loses control. It is inap-
propriate to use the wave speed of the rope to represent the
speed of the force of the rope on the kite. With this consid-
ered, how do gravitational waves affect gravity? Since the
revolution speed of the Sun will cause gravitational waves,
how are gravitational waves distributed around the Sun?
IV. THE INFLUENCE OF GRAVITATIONAL WAVES
PRODUCED BY THE SUN ON THE SURROUNDING
GRAVITY
Gravitational waves caused by the movement of the Sun
are akin to water waves caused by ships. For the convenience
Physics Essays 35, 2 (2022) 211
of explanation, we have turned the three-dimensional space
problem into a two-dimensional problem. The gravitational
influence caused by gravitational waves is different in the
direction of the Sun’s velocity and the vertical direction, as
shown in Fig. 4.
Assuming that, without considering the general Doppler
effect, the ratio of the gravitational increase caused by gravi-
tational waves to Newtonian gravitation is r
w
,we introduce a
gravitational wave influence factor of f
w
. Figure 4shows
that, due to the general Doppler effect of gravitational waves,
the energy of gravitational waves is largest in the direction
of the Sun’s velocity and the impact on gravity is the great-
est. The planet’s orbital surface is perpendicular to the direc-
tion of the Sun’s velocity, and the gravitational wave is
relatively small.
A. Calculation of the influence factor of gravitational
waves in the direction of the Sun’s velocity
We know that the revolution speed of the Sun is v
s
.
Assuming that the Sun moves from position Oto position O
after time T, the gravitational waves generated in the direc-
tion of the Sun’s velocity during this period are all located
between O’B. According to the general Doppler effect of
gravitational waves, the influence factor of gravitational
waves in this direction is as below
fw¼cþvs
c>1:0:(23)
B. Calculation of the influence factor of gravitational
waves in the vertical direction of the Sun’s velocity
The gravitational waves in the direction perpendicular to
the Sun’s velocity are located between O’A; it is only neces-
sary to calculate the ratio between O’B and O’A to determine
the gravitational wave density relationship in the two
directions
O0B¼cT vsT;(24)
O0A¼ðcTÞ2vsT
ðÞ
2
hi
1
2;(25)
thus,
fwcþvs
ccvs
cþvs

1
2¼c2v2
s
c2

1
2
:(26)
Substituting the solar revolution speed v
s
¼240 103m/
s and the gravitational wave speed c ¼2.998 108m/s, we
get O0B=O0A¼ðcvs=cþvsÞ1
20:9992. Figure 4shows
that the density of gravitational waves in the vertical direc-
tion is smaller than that in the direction of the Sun’s velocity.
The density of gravitational waves gradually decreases from
the direction of the Sun’s velocity to the vertical direction.
If the gravitational wave density is equivalent to the level of
the depression in the plane, then this gravitational wave den-
sity model is somewhat similar to the space-time depression
model described by general relativity (GR). As shown in
Fig. 5, the gravitational wave density presents a nonuniform
distribution; gravitational waves have the highest density in
the direction of the sun’s velocity (bottom of Fig. 5), and
gradually decrease upwards.
C. Calculation of the influence factor of gravitational
waves on the planetary orbital surface
We know that the planet’s orbital plane is approximately
perpendicular to the direction of the Sun’s motion; thus, the
red line in Fig. 6represents the ideal orbital plane of the
planet (completely perpendicular to the direction of the sun’s
velocity). According to formula (26), we can calculate the
influence factor of gravitational waves on the orbital surface
and thus determine that this value will be less than 1.0.
FIG. 4. (Color online) The gravitational wave model generated by the
Sun’s movement.
FIG. 5. Gravitational wave density model.
FIG. 6. (Color online) The solar gravitational wave calculation model.
212 Physics Essays 35, 2 (2022)
D. Calculation of the influence factor of gravitational waves on the reverse of the Sun’s velocity
Behind the vertical plane (to the left of the red line in Fig. 6) shows that the density of the gravitational waves will con-
tinue to decrease and reach a minimum in the opposite direction of the Sun’s velocity. At this time O0B=O0C¼cvs=cþvs,
the gravitational wave influence factor is as below
fwcþvs
ccvs
cþvs
¼cvs
c:(27)
Substituting v
s
¼cinto Eqs. (26) and (27), it can be determined that when the speed of the Sun reaches c, the orbital sur-
face of the planet perpendicular to the Sun’s velocity (the position of the red line) and the position behind it (the left side of
the red line) is no longer affected by gravitational waves.
E. Calculation of the influence factor of gravitational waves at any position
As shown in Fig. 6, assuming that the angle between O’D and the red line is h(with D at any position), then
OD2¼O0D2þOO022O0DOO0cos p
2h

;(28)
we get
O0D¼
2OO0cos p
2h

þ4OO0cos p
2h

24OO02OD2
ðÞ
hi
1
2
2
then,
O0B
O0D¼2O0B
2OO0cos p
2h

þ4OO0cos p
2h

2
4OO02OD2
ðÞ
"#
1
2
;(29)
thus
fwcþvs
c

cvs
vscos p
2h

þvscos p
2h

2
v2
sc2

"#
1
2
:(30)
F. The influence of gravitational waves on gravity
Assuming that the gravitational force of an object under the influence of gravitational waves is F
x
,F
x
can be regarded as
two parts:
Part 1: Newtonian gravity F¼G0Mm=r2.
Part 2: The gravity contributed by the gravitational wave r
x
f
x
F, where r
x
is the ratio of the gravitational increase caused
by gravitational waves to Newtonian gravitation.
Thus, we get
Fw¼FþrwfwF:(31)
Let us take the orbital position as an example to illustrate the calculation of gravity under the influence of gravitational
waves
Fw¼FþrwfwF¼F1þrwc2v2
s
c2

1
2
!
:(32)
As there is also a general Doppler effect between planets and gravitational waves, it is also necessary to consider the influ-
ence of this factor. Assuming that the speed of the planet is v
p
and the speed of the planet in the direction of the gravitational
Physics Essays 35, 2 (2022) 213
wave is v
px
, then the chase factor cvpw=cbetween the planet and the gravitational wave can be obtained and this factor is
put into Eq. (32) to get
Fw¼F1þrwc2v2
s
c2

1
2
cvpw
c
!
;(33)
substituting F, we get
Fw¼G0Mm
r21þrwc2v2
s
c2

1
2
cvpw
c
!
:(34)
Here, r
x
0.00058; this value was derived from a program simulation.
In the same way, the gravity of other positions can be calculated. We write the gravity equation of any position
Fw¼G0Mm
r21þrwcþvs
ccvs
vscos p
2h

þvscos p
2h

2
v2
sc2

"#
1
2
cvpw
c
0
B
B
@1
C
C
A
0
B
B
@1
C
C
A
:(35)
G. Gravitational waves caused by the rotation of the
Sun
The Sun’s rotation can also cause gravitational waves;
however, the Sun’s revolution speed of 240 km/s is much
greater than its rotation speed of 2 km/s. As such, this physi-
cal model does not consider the influence of gravitational
waves caused by rotation. To obtain more precise calcula-
tions, we must consider this factor.
V. ANALYSIS OF THE INFLUENCE OF GRAVITATIONAL
WAVES ON PLANETARY ORBITS
If the planet’s orbital surface is not completely perpen-
dicular to the velocity of the Sun and the orbit is split over
both sides of the red line, then the impact of gravitational
waves on planets is also irregular, which affects the orbit and
contributes part of the force to planetary precession.
10
The
closer the planet’s orbit is to the Sun, the greater the gravita-
tional wave density gradient and the more obvious the effect
of precession; the farther the distance, the less obvious. Simi-
larly, the larger the angle between the real planetary orbit sur-
face and the red line in Fig. 6, the more obvious the
precession.
In 1915, Albert Einstein published in Ref. 10, [1915, p.
839] a formula for the relativistic perihelion shift, for one
period, of
e¼24p3a2
T2c21e2
ðÞ
;(36)
where according to contemporary data Tis the orbital period
of planet, eis the eccentricity of its elliptical orbit, ais the
length of its corresponding semimajor axis, and cis the speed
of light in vacuum
d_
u¼es
T
180
p360000:(37)
Here, s¼3155814954 s is the number of seconds in one cen-
tury. We can also use a simplified calculation formula of GR
d_
u0:0383
RT :(38)
From the formulas (37) and (38), GR does not consider
the angle between the real planet’s orbital plane and the
Sun’s vertical plane (the red line in Fig. 6), and the eccentric-
ity of the orbit is not the main factor either, when calculating
the planetary precession. However, we must consider them as
the main factors in the data calculated by formula (35). These
may be the biggest differences between the two. Below, we
substitute the Rand Tvalues of each planet (see Table I) for
GR calculation.
The calculated precession data of each planet per century
are as follows:
Mercury 41.0600; Venus 8.600 ; Earth 3.8300 ; Mars 1.3400;
Jupiter 0.06200; Saturn 0.013600 ; Uranus 0.0023800 .
TABLE I. Data for the major planets in the solar system, giving the plane-
tary mass relative to that of the Sun, the orbital period in years, and the
mean orbital radius relative to that of Earth.
Planet M/M
0
T(yr) R(au)
Mercury 1.66 10
–7
0.241 0.387
Venus 2.45 10
–6
0.615 0.723
Earth 3.04 10
–6
1.000 1.00
Mars 3.23 10
–7
1.881 1.52
Jupiter 9.55 10
–4
11.86 5.20
Saturn 2.86 10
–4
29.46 9.54
Uranus 4.36 10
–5
84.01 19.19
Neptune 5.18 10
–5
164.8 30.07
214 Physics Essays 35, 2 (2022)
But we must note that when GR calculates the planet
precession deviation, it ignores the rotation of the Sun
around the center of mass of the solar system and the influ-
ence of planets on the Sun’s gravity. GR constructs an ideal-
ized 1-body model. 1-body means there is only one planet in
the solar system.
In order to maintain consistency with GR, we also made
the same omission, constructed the same 1-body ideal model,
and calculated the precession of each planet. If we want the
calculated results to be closer to the real 1-body system, we
cannot ignore the influence of the planets on the Sun, nor the
rotation of the Sun around the center of mass of the 1-body
system. We have made a clear comparison of all calculated
data in Table II . We can see that the gravitational model
constructed according to formula (35), without considering
the influence of gravitational waves (that is, classical Newto-
nian mechanics), the planet precession is zero. And consider-
ing the influence of gravitational waves, the planet
precession in the 1-body system is relatively close to the
results calculated by GR. We did not find the data of GR in
the real 1-body system, but according to the analysis of GR,
the changes in the data are very small. The data we calcu-
lated using the gravitational wave theory also reflected this.
(The precession data in the paper are all calculated after the
perihelion is projected onto the x-y plane.)
Except for Venus’s precession data of 16900 vs 8.600, the
data of other planets are relatively close to GR.
Let us examine the characteristics of Venus: Venus’s
eccentricity is abnormally low (e¼0.0068), which makes its
perihelion extremely sensitive to small disturbances. How-
ever, the angle between its orbit and the vertical plane of the
Sun is very large (3.39); thus, we have reason to believe
that gravitational waves will have a significant influence on
the orbital precession of Venus.
Why is the data of Venus (16900 vs 8.600) so different?
From formulas (37) and (38), it can be determined that GR
does not take eccentricity as the main factor and does not
consider the angle between the orbital surface and the verti-
cal surface of the Sun. Under different eccentricities and
angles, the precession data calculated by GR remain the
same. This may be the reason for the large difference
between the two.
We know that the famous Mercury Precession 4300
comes from the comparison between the calculated data of
the planetary orbit of the solar system by Newton’s classical
mechanics and the astronomical observation data. This
requires the calculation of all the planets in the solar system,
the gravitational force between the planets and the Sun, the
gravitational force between the planets, and the rotation of
the Sun around the center of mass of the solar system to con-
struct a real N-body system. Then it is necessary to calculate
the planet precession data under and without the influence of
gravitational waves. Since GR does not provide planetary
precession data under the N-body system, it cannot be com-
pared with GR. We can see that the data under the action of
gravitational waves are different (Table III). For Mercury,
the difference between the two is close to the data under the
1-body model 4300. [The initial coordinates (x, y, z) and initial
velocity (vx,vy,vz) data of the planets and the Sun used in
this paper are all from NASA’s Horizons System: https://ssd.
jpl. nasa.gov/horizons/.]
In addition, we must emphasize that the common period
of the orbits of the eight planets in the solar system is very
huge, so it is difficult for us to obtain the orbital precession
TABLE II. 1-body planetary orbit precession per century.
Real 1-body model Ideal 1-body model
Condition Planet Gravity wave ON Gravity wave OFF Gravity wave ON Gravity wave OFF GR
1.1-Body.
2.Time accuracy is 0.2 s.
3.The unit is arc seconds.
Mercury 43 0 43 0 41.06
Venus 169 0 169 0 8.6
Earth 0.35 0 0.35 0 3.83
Mars 5.6 0 5.6 0 1.34
Jupiter 1.35 0 1.35 0 0.062
Saturn 0.33 0 0.33 0 0.0136
Uranus 0.1 0 0.1 0 0.00238
Neptune 0.64 0 0.64 0
TABLE III. N-Body planetary orbit precession per century.
Condition Planet Gravity wave OFF Gravity wave ON NASA GR
1.N-Body.
2.Time accuracy is 0.2 s.
3.The unit is arc seconds.
Mercury 531 572 575 ?
Venus -200-40 400550 204 ?
Earth 10801180 10801180 1145 ?
Mars 1590 1590 1628 ?
Jupiter 6001000 6001000 655 ?
Saturn 16002200 16002200 1950 ?
Uranus 140600 140600 334 ?
Neptune 36 ?
Physics Essays 35, 2 (2022) 215
laws of planets with very small eccentricities through short-
term calculations. Through 200 years of astronomical obser-
vations, we also cannot get the periodic precession laws of
all planets, and it takes longer to observe. But for Mercury
and Mars, their eccentricity is relatively large, and we can
easily get their approximate general laws through calcula-
tions or astronomical observations.
Since the orbital data are obtained through integration in
the time domain, the averaged precession data obtained in
each orbital period has a certain range of variation. The data
in the following table are a piece of data randomly selected
after 4000 Mercury cycles. We can see that the precession
data are changing. As time increases, this change will be fur-
ther statistically averaged and gradually reduced. We can see
that the influence of gravitational waves on Mercury’s pre-
cession also changes around 3900 (Table IV).
In addition to causing planetary precession, gravitational
waves also cause planets to move away from the Sun. We
know there is also a general Doppler effect between the plan-
et’s revolution velocity and the gravitational waves caused
by the Sun. The previous 3.2 “Energy Conservation” has
analyzed the influence of the general Doppler effect on
orbital energy. Gravitational waves also cause the planetary
orbital mechanical energy to continue to increase; this causes
planets to gradually move away from the Sun.
We applied this gravitational theory to calculate the
detailed planetary orbit data (x, y, z), and used 3D technology
to draw these data, as shown in Figs. 7and 8, we can clearly
observe the orbits of the Sun and planets around the center
of mass of the solar system. As shown in Fig. 9, when we
magnify the z-axis data by ten times, we can clearly see the
angle between the planetary orbital surfaces.
GR believes that the speed of gravity is equal to the
speed of light. If this is true, then the orbits of the planets
and the Sun in the solar system should at least remain stable.
We have added the gravitational speed parameter to the pro-
gram, which can be set arbitrary, such as equal to 0.1c, equal
to c, equal to 100c, and so on. Especially in a binary star sys-
tem (1-body), when the gravitational speed parameter is set
to c, we can clearly observe that the orbits of the planets and
the Sun will no longer be stable. As shown in Fig. 10(a), the
solar orbit will spiral in a certain direction. By simulating
different gravitational speeds, we have come to the conclu-
sion that the lower the gravitational speed, the more unstable
the solar system. The theoretical basis of GR is that the speed
of gravity is equal to the speed of light, but the accuracy of
TABLE IV. Mercury precession data per century.
Condition Mercury cycles Gravity wave OFF Gravity wave ON Deviation
1.N-Body.
2.Time accuracy is 0.2 s.
3.The unit is arc seconds.
4850 532.87 572.28 39.41
4851 533.21 572.64 39.43
4852 532.62 572.14 39.52
4853 532.35 571.96 39.61
4854 532.05 571.55 39.5
4855 532.92 572.05 39.13
4856 532.78 571.88 39.1
4857 532.23 571.50 39.27
4858 531.42 570.69 39.27
4859 531.34 570.60 39.26
4860 532.39 571.36 38.97
4861 532.45 571.63 39.18
4862 531.83 570.95 39.12
4863 531.32 570.55 39.23
4864 531.51 570.62 39.11
4865 532.79 572.13 39.34
FIG. 7. (Color online) Sun rotation orbit.
FIG. 8. Planetary orbit.
FIG. 9. The angle between the planetary orbital surfaces.
216 Physics Essays 35, 2 (2022)
the orbit simulation is not reflected. Even if the space-time is
curved, the orbit will not remain stable.
As shown in Fig. 11, we have built a very simple Sun-
Earth model. If the gravitational speed is equal to the speed
of light, the gravitational force of the Sun will be delayed for
8 min, then the gravitational force of the Sun on the Earth
will come from S1 instead of the true position S0, so there
will be a component F1 that is opposite to the orbital velocity
of the Sun, under the action of F1, The Earth will move to
the left, as shown in Fig. 11(a). Finally, the Earth will follow
S1, as shown in Fig. 11(b), which is inconsistent with the
real solar system. Tom van Flandern’s paper
11
also elabo-
rated that the gravitational speed is much greater than the
speed of light.
VI. EINSTEIN–INFELD–HOFFMANN EQUATIONS
The Einstein–Infeld–Hoffmann equations
12
of motion,
jointly derived by Albert Einstein, Leopold Infeld, and
Banesh Hoffmann, are the differential equations of motion
describing the approximate dynamics of a system of point-
like masses due to their mutual gravitational interactions,
including general relativistic effects. It uses a first-order
post-Newtonian expansion and thus is valid in the limit
where the velocities of the bodies are small compared with
the speed of light and where the gravitational fields affecting
them are correspondingly weak. Given a system of N bodies,
labelled by indices A ¼1, …, N, the barycentric acceleration
vector of body A is given by
a
!A¼X
BA
GmBn
!BA
r2
AB
þ1
c2X
BA
GmBn
!BA
r2
AB v2
Aþ2v2
B4v
!Av
!B
ðÞ
3
2n
!AB v
!B
ðÞ
24X
CA
GmC
rAC
X
CB
GmC
rBC
þ1
2x
!Bx
!A
ðÞ
a
!B
ðÞ
þ1
c2X
BA
GmB
r2
AB
n
!AB 4v
!A3v
!B
ðÞ½
v
!Av
!B
ðÞ
þ7
2c2X
BA
GmBa
!B
rAB
þOc
4
ðÞ
;
where x
!Ais the barycentric position vector of body A,
v
!A¼dx
!A=dt is the barycentric velocity vector of body A,
a
!A¼d2x
!A=dt2, is the barycentric acceleration vector of
body A, rAB ¼jx
!Ax
!Bjis the coordinate distance
between bodies A and B, n
!AB ¼ðx
!Ax
!BÞ=rAB is the unit
vector pointing from body B to body A, mAis the mass of
body A, cis the speed of light, Gis the gravitational con-
stant, and the big O notation is used to indicate that terms of
order c
4
or beyond have been omitted.
The coordinates used here are harmonic. The first term
on the right hand side is the Newtonian gravitational acceler-
ation at A; in the limit as c!1, one recovers Newton’s law
of motion. It seems that this equation is perfect.
The Einstein–Infeld–Hoffmann equations are a weak
field linearized version of GR that are appropriate when the
curvature of spacetime is not too severe. This is certainly the
case in the Solar System. The full nonlinear version of GR is
only needed in situations in which the curvature of spacetime
is severe, such as in the immediate vicinity of a black hole.
The speed of gravitational effects in the Einstein–Infeld–
Hoffmann equations is the speed of light, which is why c
appears in these equations. However, the corrections to plan-
etary orbits, relative to Newtonian dynamics, predicted by
the Einstein–Infeld–Hoffmann equations are very minor.
We can simplify this equation. We only consider the
binary star system. There are only two objects A and B.
When the mass of B is much greater than the mass of A, vB
is approximately equal to 0, so the equation is simplified to
an equation with only A.
When the speed of the gravitational field is much greater
than c, then it is Newtonian gravitation. c!1, one recovers
Newton’s law of motion, there seems to be no problem.
When the speed of the gravitational field is much smaller
than c, then the result of this equation has almost nothing to
do with Newtonian gravity. c!0, force on A and B will
both be very huge, which is very illogical!
The equations in question are the first few terms on an
expansion, so obviously the equations do not give a sensible
answer when cgoes to zero because the expansion breaks
down. Therefore, I believe that the equation is a mathemati-
cal model established under the assumption that the gravita-
tional speed is equal to cto describe the planetary orbit
under N-BODY. But it does not really describe the physical
properties of gravity.
FIG. 10. (Color online) The orbit of the Sun at different gravitational speeds.
Physics Essays 35, 2 (2022) 217
Einstein thought Gravitation is the bending of space-
time, everything moving with constant speed either it be
light or anything else always follows geodesic. Then I want
to ask a question: If the light is aimed at the center of the
planet, what is the geodesic at this time? Will the light be
bent? If our Earth also bends the surrounding space-time,
then looking at the 2D space-time bending model, as shown
in Fig. 12(a), the light will never have a chance to reach the
earth, and the light will only travel along the bent space-
time, from the edge of the earth, go through. Obviously this
2D space-time bending model is incorrect, so will the 3D
space-time bending model be correct? As shown in
Fig. 12(b), it can be seen that the surrounding space is
recessed toward the center. The more severe the bending, the
greater the impact of mass on space-time. It seems that the
3D model is closer to the truth. But if we express the magni-
tude of Newtonian gravitation at different positions in space
as the distance from the source of gravity, then the final 3D
model of Newtonian gravitation is almost the same as the 3D
model of GR’s space-time bending. It seems that Newton
can also claim that universal gravitation bends space-time,
but the fact is that it is only a mathematical model that uses
distance to express the magnitude of gravitation. In this
mathematical model, there are only coordinates of the direc-
tion and the magnitude of gravity, and no space coordinates
(x, y, z), space-time has never been bent. Therefore, if a
beam of light is aimed at the center of the sun, it will con-
tinue to travel in a straight line, but will be blue shifted; if
the direction of the light and the direction of the gravitational
field are not in the same straight line, then the light will be
refracted in the gravitational field, the light will bend toward
the sun. But Einstein obviously confuses these concepts. Not
only does he use the degree of bending to express the magni-
tude of gravity, but also continues to retain the spatial coor-
dinates (x, y, z). GR is a wrong theory of gravity, which
doomed the Einstein–Infield–Hoffman equation to be wrong.
VII. CONCLUSION
The discovery of gravitational waves provides a new
way for us to understand the universe; however, the speed of
gravitational waves does not represent the speed of gravita-
tional fields. The speed of action of gravitational fields is
much greater than the speed of gravitational waves. As stated
by Newton: Gravity is an action-at-a-distance force. Gravita-
tional waves caused by the revolution of the Sun affect the
orbits of planets and provide some planetary precession data.
The general Doppler effect of gravitational waves also
causes the planetary orbital mechanical energy to continue to
increase slowly until the planet escapes from the solar sys-
tem. Gravitational waves exist; the gravitational model under
the influence of gravitational waves that we constructed was
a physical model. Through the calculation of planetary
orbital precession, the correctness of the gravity equation
under the action of gravitational waves is verified, indicating
that the gravitational physical model has research value.
FIG. 11. (Color online) Sun-Earth model at the gravitational speed equal to the speed of light.
FIG. 12. (Color online) Bending of space-time.
218 Physics Essays 35, 2 (2022)
From Newton to Pierre-Simon Laplace have realized that the
speed of gravity on objects is very huge. But this view is
inconsistent with GR. From our analysis, GR is a wrong the-
ory of gravity. The Newtonian model of universal gravitation
under the action of gravitational waves is the correct way for
humans to study the universe.
Finally, we also ask the following questions:
Is the acceleration of planetary orbits caused by the grav-
itational wave general Doppler effect related to the acceler-
ated expansion of the universe?
Is there an association between the action-at-a-distance
of the gravitational field and that in quantum mechanics?
1
F. Rohrlich, From Paradox to Reality: Our Basic Concepts of the Physical
World (Cambridge University Press, Cambridge, 1989), p. 28.
2
R. Fitzpatrick, “Newtonian gravity,” retrieved from http://farside.ph.utexas.
edu/teaching/336k/Newtonhtml/node35.html
3
J. D. Anderson, G. Schubert, V. Trimble, and M. R. Feldman, Europhys.
Lett. 110, 10002 (2015).
4
T. Henderson, “The Doppler effect—Lesson 3, waves,” Physics Tutorial,
retrieved from physicsclassroom.com (2017).
5
M. Maggiore, Gravitational Waves, Theory and Experiments Vol. 1
(Oxford University Press, Oxford, 2007).
6
LIGO Team, Astrophys. J. Lett. 915, L5 (2021).
7
B. C. Barish, “The detection of gravitational waves,” video from CERN
Academic Training Lectures, The detection of gravitational waves -
CERN Document Server, Caltech (1996).
8
J. A. Wheeler, C. Misner, and K. S. Thorne, Gravitation (W. H. Freeman
& Co., New York, 1973), p. 404.
9
M. Krizek, Math. Compu. Simul. 50, 237 (1999).
10
A. Einstein, Erkl¨Arung Der Perihel Bewegung Des Merkur Aus Der All-
gemeinen Relativi¨atstheorie (K
oniglich-Preufiische Akad. Wiss., Berlin,
1915), pp. 831–839. English translation “Explanation of the perihelion
motion of mercury from general relativity theory,” by R. A. Rydin with
comments by A. A. Vankov, pp. 1–34.
11
T. van Flandern, Phys. Lett. A 250, 1 (1998).
12
A. Einstein, L. Infeld, and B. Hoffmann, Ann. Math. Second Ser. 39,65
(1938).
Physics Essays 35, 2 (2022) 219
ResearchGate has not been able to resolve any citations for this publication.
Article
Preface Part I. At the Roof of the Endeavor: 1. Human limitations 2. Theory and the role of mathematics 3. Scientific objectivity 4. The aim of scientific theory Part II. The World of Relativity: 5. Space and time: from absolute to relative 6. Imposed consistency: special relativity 7. Gravitation as geometry: general relativity 8. Revolutions without revolutions Part III. The Quantum World: 9. The limits of the classical world 10. Concepts of the quantum world 11. From apparent paradox to a new reality 12. The present state of the art Epilogue Notes Glossary of technical terms Name index Subject index.