Download Periodic Universal Gravitation Resulting in the Phenomenon of Dark

Periodic Universal Gravitation Resulting in the Phenomenon of Dark Matter and Dark Energy
Periodic Universal Gravitation Resulting in the Phenomenon
of Dark Matter and Dark Energy
Xiang Li1
Chengdu Miracle Consulting and Management Co. Ltd., Chengdu 610041, PR China
[Abstract]
The research on the periodic universal gravitation aims to explain the phenomenon of dark
matter and dark energy through periodic universal gravitation. Periodic gravitation is different from
Newton’s universal gravitation, since the coefficient of periodic gravitation is the difference of a
constant and a cosine function of cosmic age, while the coefficient of Newton’s universal gravitation
is a constant. Periodic gravitation and Newton’s universal gravitation are not contradictory, and the
latter is approximately correct for a period of time which is decided by acceptance of errors. In the
first half of period of cosmic life, space expands from the big bang to the maximum. In the later half
of cosmic life, cosmic contraction occurs. Energy provided by periodic gravitation is the energy
behind cosmic expansion or contraction. Such energy stops being an unknown dark energy once it is
identified. The space expansion acceleration between the star and the galaxy center is always
negative, and then at any time, between the star and the galaxy center exists a mutually-approaching
contraction force, which is equivalent to dark matter in the center of galaxy. Because expanding
acceleration is negative, the expanding speed of distance between two stars is always decreasing, so
the star under observation is always pulling away from the view direction. Periodic universal
gravitation results in periodic cosmic evolution, therefore the end of a period of universe is
beginning of the new one.
【Keywords】
Universal gravitation, dark matter, dark energy
1
Email address: [email protected]
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Periodic Universal Gravitation Resulting in the Phenomenon of Dark Matter and Dark Energy
1.
Introduction
The rule of Newton’s universal gravitation F=G m1m2/ r2 is known to all [1]. But problems arise in
the application of Newton’s universal gravitation rule. Such troubling problems include: The speed of
stars moving around the galactic center is faster than the speed calculated by Newton’s universal
gravitation during the observation of distant galaxies; the speed of the outer stars moving around the
galactic center is faster than that of the inner stars, which is inconsistent with the calculation result of
Newton’s universal gravitation. Therefrom comes the theory of dark matters existing in the center of
galaxy [2-4]. Spectral acceleration redshift is observed among the distant stars, indicating the stars are
speeding away from the earth, which cannot be explained through Newton’s universal gravitation [5].
Therefrom comes the theory of certain dark energy resulting in the mutual repelling of matters. Both dark
matter and dark energy occur far away from the earth. Thus we can only imagine the form, quantity of
them and opinions vary a lot among researchers.
The author believes that the universal gravitation demonstrates period change. And the period
universal gravitation will be able to explain the phenomenon of dark matter and dark energy. Periodic
universal gravitation causes the periodic cosmic evolution; therefore the end of a period of universe is
beginning of the new one.
The paper mainly discusses the physical principles and no observatory data is provided due to
occupation of the author. Those who have relevant data and are interested in periodic universal gravitation
are welcomed to verify the phenomenon and equations proposed here. Before that, the author hasn’t
found any report on periodic universal gravitation.
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Equation of Periodic Universal Gravitation
The equation of periodic universal gravitation is:
2
F  G0(1 cos t) m1m
,
2
r
  2 / T0 ,
(1)
where F is periodic universal gravitation, with the same direction and unit as Newton’s universal
gravitation, T0 is time difference from the beginning to the end of the universe, i.e. universal life, G 0
(=4.915×10-11Nm2/kg2) is part of the coefficient of periodic universal gravitation, t is cosmic life, i.e.
time from the beginning of periodic universe to a certain time point (t n stands for current universal life),
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Periodic Universal Gravitation Resulting in the Phenomenon of Dark Matter and Dark Energy
m1 and m2 are the mass of subject 1 and 2, respectively, and r is the distance between the centers of
subject 1 and 2 (rn stands for the current distance).
We further assume
G(t)  G0(1  cos t) ,
(2)
where G(t) is the coefficient of periodic universal gravitation. The curve of periodic gravitation G(t)
changing along cosmic age t is shown in Figure 1.
Fig. 1 Relationship between periodic universal gravitation coefficient and cosmic age
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Periodic Universal Gravitation Leading to Cosmic Expansion or Contraction
3.1 Cosmic Expansion Equation
Based on equation (1), assuming the increase of periodic universal gravitation F within the time
period dt is dF. Under the effect of periodic universal gravitation, the increase on the direction r within the
period of dt is dr, the work increase of universal gravitation is:
 F  dr  r  dF  r  [
G0 m1m2 sin t
F
F
dr 
dt ]  2 F  dr 
dt
r
t
r
Substituting equation (1), the calculation result is:
dr 
sin t
 
r
dt
3 (1  cos t )
d 2r
dt
2
(3)

1
 ( ) 2  (1 
)r
3
1  cos t
(4)
Since d2r/dt2 < 0, dr/dt is monotonously decreasing. The relationship between
equation (3) is as illustrated in Figure 2.
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sin t
and t in
(1  cos t )
Periodic Universal Gravitation Resulting in the Phenomenon of Dark Matter and Dark Energy
Fig.2
The diagram of cosmic expansion speed
From equation (3) and (4), we can know that:
1) dr/dt conducted from the periodic universal gravitation on two objects m1 and m2 is independent
of m1 and m2, indicating the dr/dt is the speed of cosmic expansion. So equation (3) is referred to as the
cosmic expansion equation.
2) The cosmic expansion acceleration d2r/dt2 is negative. A vacuum force on the same direction as
the periodic universal gravitation exists between any two objects and is in positive correlation with the
distance between the two objects. The vacuum force pointing from m2 to m1 is Fv2 =m2.d2r/dr2; the
vacuum force pointing from m1 to m2 is Fv1 =m1.d2r/dr2.
3) In the section (0, T 0/2), dr/dt>0 indicates cosmic expansion. When t approaches 0 from right, r≠0,
and dr/dt approaches infinity. At this singularity, all the known physical laws fail [6]. When t=T0/2, dr/dt
=0, and r is at maximal value, and cosmos expands to the maximum. And the effect of the shock wave
from the big bang terminates here, leaving us a cosmic microwave background radiation mark with the
temperature of 3K.
4) In section (T0/2, T0), dr/dt <0 indicates cosmic contraction. When t approaches T0, r≠0, and the
contraction speed approaches infinity, and the space at the end of the periodic universe contracts to a tiny
quantum space [7]. When observing two galaxies or two stars in one galaxy in the contracting section,
their distance is decreasing. According to the research result of NASA, our galaxy will collide into
Andromeda Galaxy after 4 billion years, possibly integrating into one galaxy. t = T 0 is the time of
termination of this periodic universe, as well as the beginning of the next, for which, the big bang at the
beginning will be as described in 3).
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Periodic Universal Gravitation Resulting in the Phenomenon of Dark Matter and Dark Energy
3.2 Star’s Radial Recession, Cosmic Expansion and Dark Energy
For the same star, assuming the distance between the star and the earth in the past time t is r and the
distance at current time tn is rn. Integrate the equation (3) on both sides and we will get:
rn dr tn 2
sin t
 
dt

r r
t 3T0 (1  cos t )
r  ( 1  cos t )1 / 3  rn
1  cos tn
(5)
Assuming V as the radial velocity of the distant star, we will get:
V  dr t  dr tn  2 [ sint  ( 1  cos t )1 / 3  sin tn ]  rn
3T0 1  cos t 1  cos t n
1  cos tn
dt
dt
(6)
Take H (t )  2 [ sint  ( 1  cos t )1 / 3  sintn ] ,
3T0 1  cos t 1  cos tn
1  cos tn
(7)
where H(t) is Hubble coefficient, which is not a constant.
From equation (4) we know dr/dt always decreases as t increases, therefore when t < t m, V = dr/dt|t dr/dt|tn > 0. So when observing from the earth, the star is always pulling away from the view, and the
pulling speed is directly proportional to the distance r n. Taking derivative of tn from V, we get:
dV  ( ) 2  (1 
1
)r
3
1  cos tn n
dtn
(8)
From equation (8) we know dV/dtn >0, so along the increase of cosmic age tn, the star is pulling away
from the earth at accelerating speed. The longer the distance, the faster the acceleration is.
The energy from the periodic universal gravitation leads to the cosmic expansion and contraction,
causing the accelerating radial recession of stars under observation. By identifying the energy as the
energy from periodic universal gravitation, it is no longer an unknown dark energy. Since the distance of
any two objects on earth is too small, the dark energy phenomenon is too feeble to be observed.
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Hubble Law and Cosmic Life
4.1 Cosmic Life
The temperature observed in the cosmic microwave background radiation is -270℃ [7]. The cosmic
microwave background radiation is the mark of cosmic expansion to the maximum level. In equation (3)
take dr/dt = 0, we get when t = T 0/2, r has a maximal value. That’s when the cosmic expands to the
maximum.
Bring dr/dt|T0/2=0 into equation (6), the cosmic microwave background radiation is equivalent to the
recessing speed of the earth V, which is:
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Periodic Universal Gravitation Resulting in the Phenomenon of Dark Matter and Dark Energy
V  dr
dt
T0/ 2
 dr
  dr tn   2 sintn  rn
dt tn
dt
3T0 (1  cos tn)
Take H (T0 / 2)  
(9)
2 sintn   2.094 sintn , where H(T /2) is Hubble constant, which
0
3T0 (1  cos tn)
T0 (1  cos tn)
changes along with the current cosmic age tn.
The latest Hubble constant measured during observation of cosmic microwave background radiation
is 71km/Mpc, therefore the cosmic age is 13.7 billion years. Insert this number into H(T 0/2), we will get
the cosmic life T0. Solve the equation:

2.0944 sin(2  tn / T0 )
 Mpc  71 ,
T0[1  cos(2  tn / T0 )]
where Mpc = 3.26×106 light year. The result is the cosmic life is about 19.8 billion years.
4.2 Application of the Hubble Law
When 0 < t < T0/2
2 [ sint  ( 1  cos t )1 / 3  sintn ]  H (T / 2)
0
3T0 1  cos t 1  cos tn
1  cos tn
According to equation (6), the distance of stars calculated with Hubble constant is usually larger than the
actual distance.
When T0/2 < t <tn
2 [ sint  ( 1  cos t )1 / 3  sintn ]  H (T / 2)
0
3T0 1  cos t 1  cos tn
1  cos tn
According to equation (6), the distance of stars calculated with Hubble constant is usually smaller than
the actual distance.
We can solve the distance of star rn using the following equation set.
（10）
rn = C(tn-t)
V  2 [ sint  ( 1  cos t )1 / 3  sintn ]  rn
3T0 1  cos t 1  cos tn
1  cos tn
（6）
The conditions are tn=13.7 billion years, T0=19.8 billion years, the observed value V and light speed C.
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Periodic Universal Gravitation Leads to Dark Matter
Assuming the mass of the center of a distant galaxy is M 1, and the mass of the star moving around it
is M2, M2 generates light at the time that can be observed, at which time the periodic universal gravitation
coefficient is G(t). The distance between M1 and M2 is r. Then the periodic universal gravitation of star M2
from the center of galaxy M1 is:
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Periodic Universal Gravitation Resulting in the Phenomenon of Dark Matter and Dark Energy
F  G(t ) 
M 1M 2
r2
The periodic universal gravitation leads to the cosmic expansion or contraction. Based on equation (4),
the space acceleration between M1 and M2 is:
d 2r
dt
2

1
 ( ) 2  (1 
)r
3
1  cos t
Since the acceleration is negative, it is equivalent of exerting a vacuum force of F v(t,r) on M2 of the same
direction towards M1 as the universal gravitation.

1
FV (r , t )  ( ) 2  (1 
)r  M2
3
1  cos t
（11）
The star M2 is under the combined force from the center of galaxy which is the sum of the universal
gravitation and the vacuum force, i.e.
F  Fv (r , t )  G(t ) 
M 1M 2
r
2

1
 ( ) 2  (1 
)r  M2
3
1  cos t
（12）
From equation (12), we can deduce the mass of the dark matter existing at the center of the galaxy
equivalent to M1 is:
 2 r
1
M 1 ( dark )  ( ) 
 (1 
)
3
G (t )
1  cos t
3
（13）
The mass of the dark matter is irrelevant to the mass M 1 of visible matter at the center of galaxy and the
mass M2 of the star moving around it.
Assuming a star moving around the center of a certain galaxy is found at 200 million light years
from the earth, the distance between the star and the galaxy center is R, and the dark matter with the mass
equivalent to N earths exists at the galaxy center due to the vacuum force. The calculation result is shown
in Table 1.
Table 1
Distance between the star to
Number of earth with the equivalent mass to
the galaxy center R
the dark matter at the galaxy center N
100 million light years
3.9×1022
100,000 light years
3.9×1013
100 light years
3.9×104
Dependence of mass of dark matter on the distance between the star to the galaxy center
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Periodic Universal Gravitation Resulting in the Phenomenon of Dark Matter and Dark Energy
Observing from the earth, the mass of the dark matter at the center of the sun is 9.8×108kg.
From equation (11), the expression of the vacuum force, we can see that:
1) The vacuum force Fv(t,r) is of the same direction as the universal gravitation, and increases along
the distance between M1 and M2, which is equivalent to adding additional gravitational mass at the galaxy
center. Such additional gravitational mass is the mass of currently unknown dark matter. The outer stars in
a galaxy are relatively farther from the galaxy center and the vacuum force pointing towards the galaxy
center is larger than that of the inner stars, and the moving speed is faster than that of inner stars.
2) In section (0, T0/2), G(t) decreases along with t, while 1/(1-cosω t) increases as t decreases. The
older the stars in a galaxy under observation are, the greater the vacuum force from the galaxy center is.
And the dark matter at the galaxy center is more obvious.
3) The vacuum force is at minimum when t = T 0/2. Expressed as Fvmin, the dark matter at the galaxy
center is still observable.
4) In section (T0/2, tn), G(t) decreases as t increases, while 1/(1-cosω t) increases as t increases. The
dark matter caused by vacuum force greater than F vmin can be observed.
5) The mass of dark matter is irrelevant to the mass of the visible matter at the center of galaxy.
Therefore it is normal that the mass of dark matter at the galaxy center is several times, or even dozens or
hundreds of times of the mass of visible matter, depending on the mass of the visible matter at the galaxy
center.
6) The dark matter is only visible when observing the relative movement which is easily to assume
that the dark matter is hidden in distant galaxies. But in fact, the dark matter is distributed all over the
universe.
7) Since the distance of any two objects on earth is too small, the dark energy phenomenon is too
feeble to be observed.
Periodic universal gravitation causes cosmic expansion acceleration, which causes vacuum force,
which leads to the dark matter phenomenon. Only the phenomenon of dark matter but no dark matter
exists at the center of distant galaxies.
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Correctness of Newton’s Universal Gravitation
When measured at different times, the universal gravitation between two objects m 1 and m2 with the
same distance r in between varies (assuming the distance is near and no vacuum force is considered).
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Periodic Universal Gravitation Resulting in the Phenomenon of Dark Matter and Dark Energy
Taking the instances as follows:
The ratio between the universal gravitation at the year 2013 and the year 13 AD (2000 years ago) is:
[1  cos(
2 137 108 )] /[1  cos( 2  (137 108  2000))]  0.999999563 .
198 108
198 108
And the ratio between the universal gravitation at the year 2013 and the year 4013 (2000 years later) is
1.000000437.
Therefore, within a certain period (depending on the acceptable error), Newton’s universal
gravitation law is almost correct. The tiny change in universal gravitation has almost no influence
whatsoever to people’s life and production. But it plays a decisive role in our study of the cosmic
evolution, dark matter and dark energy.
The relations between Newton’s universal gravitation constant G and the coefficient of periodic
universal gravitation G0 is G  G0 [1  cos( 2  t n / T0 )] , the calculation result is G0 = 0.737G = 4.915×
10-11Nm2/kg2.
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Cause of Periodic Universal Gravitation
The coefficient of periodic universal gravitation G(t) is an objective existence just like any other
natural laws that are already recognized. It determines the timetable of cosmic evolution, marking the
beginning and end of periodic universe and is not subject to any elements, natural or artificial.
G(t) causes the periodic universal gravitation and determines the magnitude of it. And the energy
from periodic universal gravitation causes cosmic expansion or contraction. The periodic universal
gravitation, cosmic expansion or contraction are all responses to G(t) and couldn’t change G(t) in return.
Therefore the coefficient of periodic universal gravitation G(t) is the cause while periodic universal
gravitation, cosmic expansion or contraction is effect.
G(t) is an intermediate cause, for which the cause is yet unclear.
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Conclusion
The coefficient of periodic universal gravitation leads to the periodic cosmic evolution: the end of a
period of universe is beginning of the new one. The phenomenon of dark matter is caused by negative
cosmic expansion acceleration. Dark energy is the energy from periodic universal gravitation. Dark
matter doesn’t exist. Dark energy is a recognizable energy. The phenomenon of dark matter and dark
energy is proof of periodic universal gravitation. In a life cycle of a periodic universe, cosmic expansion
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Periodic Universal Gravitation Resulting in the Phenomenon of Dark Matter and Dark Energy
occurs in the first half while cosmic contraction occurs in the second half. The universe we are in now is
contracting, therefore when observed form the earth, distant stars are always speeding away. When
observed from the earth, the distance between two galaxies with the distance no longer than 3.8 billion
light years from the earth (t > T 0/2), or the distance between any two stars in one galaxy is slowly
decreasing. Newton’s universal gravitation is almost correct in a certain time period. Periodic universal
gravitation is supplementation and development of Newton’s universal gravitation.
Reference
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[2] V. Trimble, Ann. Rev. Astron. Astrophys 25: 425-472, 1987
[3] D. Raine, T. Thomas, An Introduction to the Science of Cosmology, IOP Publishing, 2001
[4] G. Bertone, D. Merritt, Mod Phys. Lett. A, 20 (14): 1021-1036, 2005
[5] R. V. Pound, G. A. Rebka, Jr. Phys. Rev. Lett. 3: 439-441, 1959
[6] Wang Yuguo, Tang Shanlin, Zhao Baoqun, University Physics Vol. II, Science Press, 2013
[7] Steven Hawking, Leonard Mlodinow, The Grand Design, Bantam Books, 2010
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