The Origin of Gravity in the new Vacuum Paradigm
Transcript of The Origin of Gravity in the new Vacuum Paradigm
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The Origin of Gravity in the new Vacuum Paradigm
Min Tae Kim
Institute for Theory of Everything
Daejeon 34050, Republic of Korea
E-mail: [email protected], Orcid: 0000-0002-9791-9595
Abstract: In modern physics, gravity is still an unknown physical phenomenon. We sought
to find its origin from a new concept of vacuum: The vacuum is composed of a very dense
solid medium with a regular matrix, where matter is vibration energy added to the vacuum
matrix. This vibration radiates “matter wave” to distort the vacuum matrix locally as
much as the energy. Three-dimensional symmetry of distortion develops around the
stationary matter. Any inertial movement is interpreted as the movement of the vibration
site, through which the symmetry of distortion is broken. This process yields a gradient
of the distortion in the direction of movement, which in turn sustains the inertial
movement so that the kinetic energy is preserved. When the movement is rotational, a
gradient of the distortion is also generated across the rotation orbit, attracting nearby
particles on to the plane of rotation. If the movement is spherically symmetric, like the
random vibration of a lattice point, this attraction is also spherically symmetric. This is
the origin of gravity, and mass is a collective energy of the vibrating sites gathered due to
gravity in the vacuum matrix.
Keywords: Gravity; Vacuum; General Relativity; Light deflection; Precession
1. Introduction
Gravity is one of the essential parts of Newtonian dynamics and also of the general
theory of relativity (GR theory). Gravity is known as one kind of force of nature. But its
origin is still mysterious for which only a few hypothetical theories exist.1,2 One of these
theories is called entropic gravity in that gravity is regarded as a phenomenon caused by
quantum entanglement of small bits of spacetime information.1 Entropic gravity is
sustained, as entropy is increasing with time according to the 2nd law of thermodynamics.
This theory is not yet proven and still in a lot of controversies.3 Gravity is also approached
combined with quantum mechanics. In this quantum theory of gravity, gravitons are
introduced to mediate the gravitational field in the quantum field.2 However, gravity is
nonrenormalizable, so that the theory is limited for any meaningful predictions.4
Up to now, the GR theory appears to be the one that best describes gravity. This theory
has been tested and proven in many ways: it has predicted the precession of the perihelion
of Mercury, light deflection by the Sun, gravitational redshift, etc.5 The theory also
predicts gravitational waves that carry away energy as gravitational radiation.6,7 Recently
the first observation was made as the gravitational waves radiated from a pair of merging
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black holes.8,9
Here we are thinking about how gravitational waves can move through empty spaces
and how can light waves travel in the vacuum where nothing exists. This paper considers
the possible existence of a vacuum medium which carries the gravitational and
electromagnetic waves. Thereafter, the inertial movement of a particle is characterized
under the premise that the vacuum is composed of a very dense and cold medium with
very little or no energy available and a very regular matrix. Gravity is inferred from the
characteristics of the inertial movement to give the origin of gravity in the solid vacuum
matrix. Matter waves from the vibration of the vacuum matrix are also related to the
inertial and gravitational movement, for which mass is shown to be another expression
for the vibrational energy stored in the solid vacuum matrix.
2. A new paradigm for the vacuum
Light propagation in the solid vacuum medium
There were severe debates about the nature of light in the Newtonian era. Newton was
an advocate of the particle nature of light,10 and light had been considered to be a
collection of particles until the wave characteristics of light were experimentally proven.11
If light is a wave, a medium to carry the wave was needed. So an imaginary fluid called
“ether” was hypothesized. But the existence of this hypothetical “fluid” was disproved by
Michelson and Molly,12 and light has been accepted to propagate in the vacuum without
any medium. However, if we think that a shear wave can only propagate in solid media,
and light is a shear wave, it can be imagined that light propagates through the solid
medium as a wave. In fact, there are theories that the density of vacuum is enormously
high,13,14 around 1096 kg/m3, the Planck density, incomparable to the total observable mass
of the universe of 1053 kg.15 A similar concept of vacuum was proposed by Paul Dirac in
1930, in that the vacuum consists of the infinite sea (Dirac sea) of particles with negative
energy.16 This model predicted the presence of positrons and was discovered by Anderson
in 1932.17
If light is a kind of wave and propagates in the solid vacuum medium, we may find its
similarity with the propagation of sound waves (elastic waves) in solid. Sound waves are
classified into pressure waves and shear waves. Generally, pressure waves are faster than
shear waves.18 As light is a shear wave, we consider only shear sound waves, the speed
of which is given by 19
𝑣𝑠 = √𝐺𝑠
𝜌 (1),
Here Gs is the shear modulus of the solid. We note the similarity of E = mc2, Einstein's
energy-mass equivalence, to the speed of the shear wave in solid in Eq. (1). Dividing both
sides of the equivalence equation by volume, the speed of light c is written as
𝑐 = √𝐺𝑉
𝜌𝑉 (2).
Here, GV = E/V (the energy density) and m/V = ρV (the mass density of the vacuum). Eq.
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(2) has the same form as Eq. (1). GV in Eq. (2) may be regarded as the shear modulus of
the solid vacuum. Using the Plank density 5.161094 g/cm3 for ρV, we have GV to be ca.
4.6410100 GPa, which we may call the Plank shear modulus. As the speed of sound
waves in a solid is lower when the thermodynamic energy of the solid is higher, namely
at higher temperatures and pressures,20-22 the speed of light is lower in ordinary matters
than in the solid vacuum. The refractive index is thus always more than 1 for all the
materials.23 Ordinary matters have more energy than the solid vacuum. So they are hotter
and thus less stiff for the propagation of light waves.
Figure 1. A swan moves on the water to make asymmetric waves. This picture was
adapted from an image of wikipedia at https://en.wikipedia.org/wiki/Swan.
Conservation of energy in the inertial frame (in the solid vacuum)
Here we will newly interpret the inertial movement of a particle in the solid vacuum
matrix to associate it with gravity. When a swan moves at a constant velocity over the
surface of the water, its movement will form a concentric wave that continuously spreads
in all directions, as shown in Fig. 1. The wavelength becomes shorter in the forward
direction and longer in the backward direction due to the forward movement of the swan.
Similarly, we consider that a particle moves through the solid vacuum at a constant
velocity. This particle will move without changing the velocity and direction unless
external forces are applied. This is the law of inertia, Newton's first law of motion. Unlike
the movement of a swan on the surface of the water, energy is not consumed for the
movement in the inertial frame. A moving particle has two masses, the rest mass and the
mass in the moving defined in terms of Einstein's special relativity, the relativistic mass.
The total energy of a moving particle expressed by the mass-energy equivalence is higher
than that of a stationary one. In our new vacuum paradigm, the energy of a moving particle
is the energy of distortion of the solid vacuum induced by the movement of the particle.
When a particle travels at a constant velocity, the solid vacuum is compressed in the
forward direction and relaxed in the opposite direction. Besides, the solid vacuum will
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also be distorted normal to the direction of travel. This is the Doppler effect, as shown in
Fig. 2. The kinetic energy due to the movement is then the energy of distortion of the solid
vacuum. Inversely we can say that a particle moves when there is a difference in the
distortion of the solid vacuum. It moves from the less distorted side toward the more
distorted side. When the distortion in the forward direction increases, the particle will
accelerate in that direction.
Figure 2. An image showing the Doppler’s effect. This is adapted from wikipedia at
https://ko.wikipedia.org/wiki/%EB%8F%84%ED%94%8C%EB%9F%AC_%E
D%9A%A8%EA%B3%BC (도플러 효과)
What should be the energy of distortion Ed stored in the solid vacuum due to an inertial
movement? It is deduced from the difference in the energy of the relativistic mass m and
the rest mass m0, as
𝐸𝑑 = (𝑚 − 𝑚0)𝑐2 = (𝛾 − 1)𝑚0𝑐2 (3)
Here, γ is called the Lorentz factor, a measure of the increased mass due to the movement
at the velocity v:
γ =1
√1+𝑣
𝑐√1−
𝑣
𝑐
=1
𝑧 (4)
In the regime of v << c, the distortion energy is approximated to Ed = ½mv2, being the
kinetic energy of the classical mechanics. γ can be regarded as a parameter designating
the degree of distortion of the solid vacuum associated with the movement. In Eq. (4) z,
the reciprocal of γ, is the geometric mean of 1+v/c and 1v/c, as conceptually shown the
using two symmetric right triangles in Fig. 3 in which x = 1v/c and y = 1+v/c,
respectively. x + y = 2 and z2 = xy. Regarding +v/c as the compressed distortion in the
forward direction or v/c the stretched distortion in the backward direction, the kinetic
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energy is asymmetric distortion energy stored in the solid vacuum.
Figure 3. Geometric presentation for the meaning of z, the reciprocal of γ. x and y
represent 1v/c and 1+v/c, respectively. x + y = 2 is constant and z2 = xy.
3. Origin of gravity in the solid vacuum
Interpretation of gravity in the new vacuum paradigm
As mass is another expression of energy through mass-energy equivalence, this energy
should be stored in the solid vacuum, as with the kinetic energy. As mentioned in the
previous section, the kinetic energy is the energy of asymmetric distortion of the solid
vacuum due to the movement of a particle. When a spherical particle is stationary in the
solid vacuum, the distortion will be point symmetric for which the center of symmetry is
at the center of mass of the particle. If the distortion of the solid vacuum is only originated
from the movement, then the stationary mass-energy of a particle means back-and-forth
movements (oscillation or vibration) in a random mode around a point. So the distortion
is balanced in all directions. The distortion of the solid vacuum shall be intense near the
particle and become weaker on going away from the center of mass, as the gravitational
field does.
We now consider two spherical particles, one with mass m1 and one with m2 separated
by a distance r in space, as shown in Fig. 4. There will be a difference in the intensity of
distortion of the solid vacuum in the regions between the particles and the outside of them.
The distortion at the point P will be more intense than at Q because P is nearer to m1 than
Q when the distance from P to the center of m2 is the same as that from Q to the center.
The distortion is asymmetric regarding the particle m2. The particle m2 will approach m1,
as with the movement in the inertial frame. As m2 moves toward m1, the intensity of
distortion between the two particles increases. The approach velocity further increases.
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The particle feels an acceleration force. This is the origin of gravity in the solid vacuum.
The distortion of the solid vacuum is the source of mass-energy and is also the source of
gravity. So, the gravitational mass and inertial mass are inevitably the same.
Figure 4. Two spherical particles m1 and m2 in the solid vacuum separated by a distance
r. The distance of P and Q from the center of m2 is the same. The distortion at P is
more intense than at Q due to the difference in the distance from the center of m1.
A particle moving at a constant velocity v stores energy in the solid vacuum as in Eq.
(4). Let γ1 in Eq. (3) to be the increased distortion Δ of the solid vacuum due to the
movement in the inertial frame, namely γ1 = Δ. v may be written as a function of Δ
from the relation γ1 = Δ as:
𝑣 = 𝑐√2∆+∆2
∆+1 (5)
Normally v << c and thus Δ ~ 0, so that Eq. (5) is approximated to
𝑣 = 𝑐√2𝛥 (6)
As shown in Fig. 2, which shows the Doppler effect, the solid vacuum is also distorted in
the normal direction to the direction of movement. However, since the distortion in the
normal direction is symmetric about the line of traveling of the particle, it does not affect
its movement. If the hypotenuse of the right triangle (AB) in Fig. 3 is the direction of
travel (to the left), the distortions in the upper and lower directions normal to this side are
each γ = z1 = Δ+1. These cancel each other to 0 to yield the maximum velocity in the
direction of travel. If we consider the distortion γ = Δ+1 of the upper direction only, we
have the (imaginary) normal velocity vn from Eq. (4) as
𝑣𝑛 = c√∆
∆+1 (7)
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For v << c, or Δ ~ 0, Eq. (7) is approximated to
𝑣𝑛 = 𝑐√𝛥 (8)
Comparing Eq. (6) and (8), we see that the ratio v/vn is 2 for any directions on the plane
normal to the movement. We may say that the balanced distortion of Δ on this plane has
led to the maximum gradient distortion of 2Δ to the direction of the movement. However,
if a particle makes a rotational movement instead of a linear one, the distortion of the
solid vacuum in the normal direction (radial direction of the rotation) will not cancel each
other. The balance will then be destroyed. Particles on the plane of the rotational orbit
will feel an increasing gradient in the distortion toward the center of rotation, and the
particles outside of the orbit will be attracted to the rotating particle. It means an attractive
force can be generated by the rotational movement of a massive body in our new vacuum
paradigm.
Movement of a particle in the distortional field of the solid vacuum
Now suppose a planet that revolves around the Sun at the orbital radius of rO. The mass
of the Sun is M and of the planet is mp. When this planet revolves steadily, the energy due
to the distortion of the solid vacuum surrounding the Sun (potential energy) and that due
to the imaginary outward radial velocity vR should be the same. That is
(𝛾 − 1)𝑚𝑝𝑐2 =𝐺𝑀𝑚𝑝
𝑟 (9).
The radial velocity vR is then from the definition of 𝛾 in Eq. (4)
𝑣𝑅 =𝑐√𝑟𝑆(4𝑟+𝑟𝑆)
2𝑟+𝑟𝑆 (10).
Here rS = 2GM/c2, called the Schwarzschild radius. rS of the Sun is around 2.95 km, too
much smaller than the radius of the Sun, 696,392 km,24 so that vR is approximated to
𝑣𝑅 = 𝑐√𝑟𝑆
𝑟= √
2𝐺𝑀
𝑟 (11).
Here, G is the gravitational constant. Comparing Eq. (6) and Eq. (11), 2Δ generated from
the movement in the radial direction should balance the distortion due to the presence of
mass M, and it is rS/rO at the orbital radius of rO. How can we obtain such a distortion as
if there were a movement in the radial direction? It is actually originated from the orbital
movement of the planet. As mentioned in the previous section, a rotational movement of
a particle induces a gradient in the distortion of the solid vacuum normal to the tangent
of the rotational movement. The distortion generated by the orbital velocity vO for a given
orbital radius of rO should then compensate for the distortion of the solid vacuum rS/rO in
the radial direction due to the presence of mass M. We see from Eq. (6) and (8) that the
distortion in the circumferential direction will be rS/2rO when the radial velocity is given
as Eq. (11). The circumferential velocity (orbital velocity, vO) will be then 1/√2 times the
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radial velocity, that is
𝑣𝑂 = 𝑐√𝑟𝑆
2𝑟𝑂 (12).
If the eccentricity is not large, the orbit is close to a circle. The average velocity vO of the
orbit can be approximated by the orbital period T and the semi-major axis ra as follows:
𝑣𝑂 =2𝜋𝑟𝑎
𝑇= √
𝐺𝑀
𝑟𝑎 (13).
The term on the right-hand side is obtained from Kepler's third law. Eq. (12) and (13) are
fully consistent if the orbit radius is the semi-major axis. The observed orbital velocity
and calculated values of the planets of the solar system from Eq. (12) are compared in
Fig. 5. We see that the orbital velocity estimated from our new vacuum paradigm well
coincides with the observed and that Kepler's third law can be understood in our new
vacuum paradigm.
Figure 5. Comparison of the observed orbital velocity and calculated values of the solar
system from Eq. (12).
The acceleration aR in the radial direction is obtained by differentiating the equation
(11) as follows:
𝑎𝑅 =𝑑𝑣𝑅
𝑑𝑡=
𝑑𝑟
𝑑𝑡
𝑑𝑣𝑅
𝑑𝑟= −
𝑐2
2
𝑟𝑆
𝑟2 = −𝐺𝑀
𝑟2 (14)
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Acceleration is opposite to the direction of outward radial movement. This is the law of
gravity. According to our new paradigm of vacuum, G can be regarded as an index of the
mechanical properties of the solid vacuum. Our new solid vacuum model and the
inference regarding the origin of gravity are further verified through the calculation of the
deflection of light by the Sun and the precession of Mercury. Details are shown in the
Appendices, [A1] for the light deflection, and [A2] for the precession.
4. Matter wave in the solid vacuum
In this study, we showed that the asymmetric distortion of the solid vacuum is the origin
of movements in the inertial frame. The asymmetry is sustained as long as a particle
moves. This is the law of inertia in the Newtonian dynamics. The kinetic energy of a
moving particle is stored as an energy of this asymmetric distortion in the solid vacuum.
The relativistic mass is then the rest mass plus the energy of the asymmetric distortion. If
a particle moves randomly in a spherically symmetric way, namely, it vibrates around a
point in a three-dimensional random mode, the distortion of the solid vacuum will be
developed in a spherically symmetric way around the particle. The rest mass of a particle
is then understood to be the energy of distortion of the solid vacuum stored due to its
three-dimensional random vibration. The inertial mass and the gravitational mass are
inevitably the same in the new vacuum paradigm. Thus we do not have to insist on the
equivalence principle on which the GR theory relies.
If mass is the vibrational energy of the solid vacuum (lattice point), matter will be the
source of wave generated from the vibration and propagate through the solid vacuum. We
may call it matter wave. In modern physics, there is the concept of matter wave proposed
by de Broglie in response to the duality of light (wave + particle).25 The de Broglie
wavelength λ has the following relationship with the momentum p of a moving particle.26
=ℎ
𝑝=
ℎ
𝑚𝑣 (15)
Here, h is the Planck constant. This hypothesis was validated in polyatomic molecules
such as fullerene (C60) 27 as well as electrons 28,29 and atoms,30 and thus matter is accepted
to have the duality of particle and wave. Quantum mechanics interprets this wave as a
function of probability finding a particle for a given time and space.31 How can we
interpret this matter wave in terms of our new vacuum paradigm? According to Eq. (15),
the inverse of the wavelength, the wavenumber, is proportional to the mass and velocity.
We understand the distortion of the solid vacuum is more intense near a more massive
particle moving at a higher velocity than a less massive one moving at a lower velocity.
The wavenumber can be regarded as an index for the intensity of the distortion of the
solid vacuum around a moving particle. A moving particle with the velocity v is then
steadily generating a matter wave with the wavelength given in Eq. (15) and the frequency
fm from its vibration given as 32
𝑓𝑚 =𝑚c2
ℎ=
𝛾𝑚0c2
ℎ= 𝛾𝑓𝑚0
(16).
Eq. (16) is the Plank-Einstein relation for matter wave. We apparently see from Eq. (16)
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that the vibration of the solid vacuum matrix is the origin of mass (m) and energy (mc2).
In terms of the Doppler effect,33 the frequency of a matter wave in the forward direction
𝑓𝑚+ and the backward direction 𝑓𝑚−
will be respectively
𝑓𝑚+=
1
1−𝑣
𝑐
𝑓𝑚0 ; 𝑓𝑚−
=1
1+𝑣
𝑐
𝑓𝑚0 (17).
Hence, the frequency of matter wave of a moving particle is the geometric mean of the
forward and backward frequency of matter wave:
𝑓𝑚 = √𝑓𝑚+𝑓𝑚−
= γ𝑓𝑚0 (18).
As 𝑓𝑚 − 𝑓𝑚0= (𝛾 − 1)𝑓𝑚0
from Eq. (16), we see that the asymmetric distortion of the
solid vacuum has been increased due to an increase in the frequency of matter wave of a
moving particle. The increased distortion of the solid vacuum of a moving particle ∆𝛿
corresponds to
∆𝛿 = 𝛾 − 1 = 𝑓𝑚−𝑓𝑚0
𝑓𝑚0
(19),
which is the rate of change in the frequency of matter wave due to the movement.
5. Summary - The solid vacuum as the platform for mass and wave
We have defined the vacuum as a solid medium without energy or with very low energy.
The propagation of light could then be easily understood in the new vacuum paradigm,
as similarly as that of sound waves in solids. The energy-mass equivalence is nothing but
an equation describing the speed of the propagation of light waves in the solid vacuum.
Matter is also a kind of wave, and mass is another expression of the frequency of matter
wave as apparently shown in Eq. (16). In this context, we may say that the solid vacuum
provides a platform for mass and wave. This platform has a three-dimensional matrix,
like a solid crystal with lattice. The symmetric vibration of a lattice point in the solid
vacuum yields a particle with mass corresponding to the frequency of the vibration in Eq.
(16). This vibration propagates radially outward into the whole solid vacuum matrix to
build up a spherically symmetric distortion in the solid vacuum around the particle. When
this symmetric distortion is broken, the vibration site moves to the neighboring site in the
direction of increasing distortion. This process is accompanied by the change in the
frequency of vibration in Eq. (18), as well as the change in the mass in Eq. (16)
equivalently. The inertial movement of the vibration site preserves the distortion
difference in the solid vacuum, and thus the kinetic energy is preserved. When a particle
is near to a massive body, the particle feels asymmetric distortion along the direction of
the center of the massive body due to the symmetric distortion of the solid vacuum
generated by the vibration of the body. The particle will then move toward the body,
which leads again to the increase in the distortion in the direction of the body. This is the
very origin of gravity we know.
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Appendix [A1]
Deflection of light in the distorted solid vacuum
Einstein predicted using the GR theory that light deflects while passing by the Sun, as
exaggeratedly illustrated in Figure 1. The light deflection by the Sun was demonstrated
in the observation of Edington's total solar eclipse on May 29, 1919.1 Here we calculate
the deflection angle based on our new vacuum paradigm. When light from a distance
travels toward the Sun, the movement in an infinitesimal time interval of dt can be divided
into the movement dr toward the solar center and that dq in the circumferential direction
of the Sun while advancing by cdt = dp, as shown in Figure 2. Due to the presence of the
Sun, the solid vacuum is distorted. The length in the radial direction to the solar center is
shortened by rS to give the distortion rS/r at r according to Eq. (11) of the main text. A
light wave moves in the radial direction faster than in the circumferential direction. If
there is no distortion of the solid vacuum, then dq should be equal to dq + ds, namely ds
= 0, where ds is the retarded path (compared to the path in the radial direction) of light
due to the distortion during dt. ds is given as
𝑑𝑠2 = 𝑑𝑞2 × (𝑟𝑆
𝑟)
2
= (𝑑𝑝2 − 𝑑𝑟2) (𝑟𝑆
𝑟)
2
(1).
The distance r away from the solar center is related to the distance p away from the
tangential Solar Surface as
𝑟2 = 𝑝2 + 𝑅2 → 𝑝𝑑𝑝 = 𝑟𝑑𝑟 (2).
Here R is the radius of the Sun. From Equation (2), Equation (1) is rearranged to
𝑑𝑠2 = 𝑑𝑝2 (𝑟2−𝑝2
𝑟2 ) (𝑟𝑆
𝑟)
2
= 𝑑𝑝2 𝑅2
𝑟2 (𝑟𝑆
𝑟)
2
,
so that
𝑑s =𝑟𝑆𝑅
𝑟2𝑑𝑝 =
𝑟𝑆𝑅
𝑝2+𝑅2𝑑𝑝 (3).
Figure 1. Deflection of light by the Sun. The deflection angle is highly exaggerated for
the presentation.
14
Integrating Equation (3) from∞ to 0 and from the equivalence,
∫1
𝑥2+𝑎2 =1
𝑎tan−1 𝑥
𝑎, (4),
we have the light deflection in length at the tangential Solar Surface as
s = ∫𝑟𝑆𝑅
𝑝2+𝑅2 𝑑𝑝0
−∞= 𝑟𝑆𝑅
1
𝑅(tan−1 0
𝑅− tan−1 −∞
𝑅) =
π
2𝑟𝑆 (5)
The deflection angle = ds/dp in radian at the tangential Solar Surface is given for p = 0
or r = R in Equation (3), as
=𝑑𝑠
𝑑𝑝=
𝑟𝑆
𝑅 (6).
This deflection angle becomes twice for the Earth observer since the light has the same
effect as it reaches the Earth observer. The total deflection will then be
= 2 =4𝐺𝑀
𝑐2𝑅 (7)
This equation is the same as that derived from Einstein's GR theory.2
Figure 2. Division of the light path toward the Sun: r, radial direction; q, circumferential
direction.
15
We consider, however, that the path of deflected light is asymmetric in terms of the
elapsed length of light. The maximum deflection of light from the light source at the Solar
Surface is given as Equation (5) by integrating Equation (4) from – ∞ to 0. Meanwhile,
the deflection from the Solar Surface to the Earth’s observer is calculated by integrating
Equation (4) from 0 to 215R (R is the solar radius, 695,700 km.3 The distance from Earth
to the Sun is 149,597,870 km.4). The deflection is then
s = ∫𝑟𝑆𝑅
𝑝2+𝑅2 𝑑𝑝215𝑅
0= 𝑟𝑆𝑅
1
𝑅(tan−1 215𝑅
𝑅− tan−1 0
𝑅) (8)
The light deflection from the Solar surface to Earth is around 0.4% less than that of
Equation (5). So the total deflection angle in radian should be modified to
=2𝐺𝑀
𝑐2𝑅+
1.992𝐺𝑀
𝑐2𝑅=
3.992𝐺𝑀
𝑐2𝑅 (9)
The first term in the middle of Equation (9) is the light deflection from the light source to
the Solar Surface and the second term is the light deflection from the Solar Surface to
Earth.
References of [A1]
1. Dyson, F.W., Eddington, A.S., Davidson, C.R. A Determination of the Deflection of
Light by the Sun's Gravitational Field, from Observations Made at the Solar eclipse of
May 29, 1919. Phil. Trans. Roy. Soc. A. 220, 291-333 (1920).
2. Treschman, K.J. Recent astronomical tests of general relativity. Int. J. Phys. Sci. 10,
90-105 (2015).
3. Mamajek, E.E., Prsa, A., Torres, G., et, al. (2015), "IAU 2015 Resolution B3 on
Recommended Nominal Conversion Constants for Selected Solar and Planetary
Properties", arXiv:1510.07674 [astro-ph.SR].
4. Pitjeva, E. V., Standish, E. M. (2009). "Proposals for the masses of the three largest
asteroids, the Moon–Earth mass ratio and the Astronomical Unit". Celestial Mechanics
and Dynamical Astronomy. 103 (4): 365–372. doi:10.1007/s10569-009-9203-8.
16
Appendix [A2]
Precession of the perihelion of Mercury
Figure 1. Schematic and exaggerated precession of the perihelion of Mercury.
In the solar system, the perihelion of the planet to the Sun is not fixed but rotates for
various reasons.1 This rotation is called the perihelion precession, and the main cause is
the gravitational influence of the other planets. The orbit of Mercury deviates from the
center of the Sun to be an elliptical one, and the perihelion of Mercury rotates
574.1"(arcseconds) per 100 years, as shown in Figure 1. The estimate of the precession
based on Newtonian dynamics is 532.3", about 42" different from the observed value.2 In
the GR theory, the rotation of the elliptical orbital axis of Mercury is explained in terms
of the change in the spacetime curvature due to gravity. Einstein used the GR theory to
calculate the precession ε in radian for the elliptic orbit of the planet, as 3
𝜺 = 𝟐𝟒𝝅𝟑 𝒓𝒂𝟐
𝑻𝟐𝒄𝟐(𝟏−𝒆𝟐) (1).
Here ra is the semi-major axis, e the eccentricity and T the orbital period, respectively.
For Mercury, ra is 5.79 × 1010 m, e = 0.206 and T = 87.97 days. The precession for one
revolution is then calculated to be 5.028×107 radians, corresponding to about 43.2" per
100 years, since Mercury is revolving about 415 times for 100 years. Other planets are
less dominant than Mercury regarding the precession, though they do such a perihelion
precession: 3.84" and 8.62" per 100 years for Earth and Venus, respectively.4 A recently-
discovered double pulsar system, PSR 1913+16, has a value of 4.2° per year, which is
also well explained by the GR theory.5 On the other hand, the Relativistic Newtonian
Dynamics (RND) model, a simple modification of the Newtonian dynamics, is also well
explaining the precession of the perihelion of Mercury 6,7 and that of the binary system
PSR J0737-3039A/B 8 without relying on curving spacetime in the GR theory. The RND
17
model deals with the precession only by the relativistic velocity in the inertial frame of
reference, so that it is similar to our approach to gravity from the inertial motion of a
particle in terms of the distortion of the solid vacuum. However, like the GR relativity,
this RND model did not pay any attention to the origin of gravity.
According to our new vacuum paradigm, kinetic and gravitational potential energy are
substantially the same distortion energy stored in the solid vacuum. In this context, it is
simple and clear to understand the perihelion movement of the binary system like the Sun
and Mercury in terms of the distortion of the solid vacuum. When Mercury makes one
revolution around the Sun, the associated distortion is the radial one of rS and the
circumferential one of ½rS, as evident from Eq. (11) and (12) of the main text. The total
distortion involved is 1½rS, which is 3rS in terms of the orbital length. The vacuum is
additionally distorted by 3rS from the track length 2rm, where rm is the mean radius of
the orbit of Mercury around the Sun. Hence, the precession of Mercury expressed in
Radian should be
𝜺 =𝟑𝝅𝒓𝑺
𝒓𝒎 (2).
However, as there are several mean values for the orbital movement of Mercury: three
means based on the aphelion and perihelion, and three means based on the elliptical
movement of Mercury, as listed in Table 1 and 2, it is hard to determine the right radius
for rm. Equation (2) is actually the same as Equation (1) if 𝒓𝒎 =𝟐𝒓𝑨𝒓𝑷
𝒓𝑨+𝒓𝑷, which is the
harmonic means rh of rA and rP, the aphelion and perihelion of Mercury, respectively.
Inserting the relation of Kepler’s third law, 𝑻𝟐
𝒓𝒂𝟑 =
𝟒𝝅𝟐
𝑮(𝑴+𝒎) to Equation (1) with the
definition of eccentricity, 𝒆 =𝒓𝑨−𝒓𝑷
𝒓𝑨+𝒓𝑷, we have
𝜺 =𝟔𝝅𝑮(𝑴+𝒎)
𝒓𝒂𝒄𝟐(𝟏−𝒆𝟐)=
𝟑𝝅𝒓𝑺
𝒓𝒂(𝟏−𝒆𝟐)=
𝟑𝝅𝒓𝑺
𝒓𝒉 (3).
When we apply the harmonic mean of the aphelion and perihelion, which is the
calculation by GR, we have still a difference of 1.21" in the precession between the sum
of the prediction and the observed, as shown in Table 3.
As Mercury is in the distortional field of the Sun, its rotation around the Sun will add
additional distortion in this field, acting as a dragging force, which in turn causes the
rotation of the axis connecting the center of Mercury and the Sun. The movement of
Mercury can be divided into a pure rotational one and a radial one with the Sun as the
reference point. For this reason, we may think the radius for the calculation of the
precession should be different for each type of movement. Thus the precession expressed
as Equation (2) may be modified to
휀 =2𝜋𝑟𝑆
𝑟1+
𝜋𝑟𝑆
𝑟2 (4).
The first term on the right-hand side of Equation (4) is due to the radial movement and
the second term is due to the rotational movement of Mercury, respectively. What shall
be the appropriate radius for r1 and r2 in Equation (4)? We have the least difference
18
between the estimated and observed if we apply the harmonic mean of the elliptical orbit
of Mercury to r1 and the geometrical mean of the aphelion and perihelion to r2. The
difference is given in Table 3 as 0.0067". The reason for this coincidence is not clear for
now and needs to be studied.
The accumulated distortion in the solid vacuum by one rotation will be freed through
the precession of the perihelion. One rotation plus the perihelion advance in the distorted
solid vacuum means one rotation in the solid vacuum free of additional distortion. This is
one of our interpretations of the movement of Mercury's perihelion in the regime of the
new paradigm of vacuum.
Table 1. Radii based on the aphelion and perihelion of Mercury. Radius km formula
Aphelion 69,816,900 rA
Perihelion 46,001,200 rP
Arithmetic mean 57,909,050 𝑟𝐴 + 𝑟𝑃
2
Geometric mean 56,671,520 √𝑟𝐴𝑟𝑃
Harmonic mean 55,460,436 2𝑟𝐴𝑟𝑃
𝑟𝐴+𝑟𝑃
Table 1. Radii based on the elliptical orbit of Mercury. Radius km formula
Semi-major axis 57,909,050 ra
Semi-minor axis 56,671,523 rb
Arithmetic mean 57,290,287 𝑟𝑎 + 𝑟𝑏
2
Geometric mean 57,286,945 √𝑟𝑎𝑟𝑏
Harmonic mean 57,283,604 2𝑟𝑎𝑟𝑏
𝑟𝑎+𝑟𝑏
Table 1. Radii based on the elliptical orbit of Mercury.
Cause of the precession Estimated precession
(arc seconds/100yr)
Comments
Gravitational effect of
other planets 532.3035
Distortion of space-time
due to GR theory 42.9799
The harmonic mean of the perihelion and
aphelion
Distortion of the solid
vacuum 41.7632
The geometric mean of the perihelion and
aphelion and the harmonic mean of the
elliptical orbit
Other minor effects 0.0266
Sum of the prediction 575.31 574.0933
Observed value 574.10±0.65
Difference in the
precession 1.21 0.0067
19
References of [A2]
1. Park, R.S., Folkner, W.M., Konopliv, A.S., Williams, J.G., Smith, D.E., Zuber, M.T.
Precession of Mercury’s Perihelion from Ranging to the MESSENGER Spacecraft.
Astronomical J. 153, 121-127 (2017).
2. Le Verrier, U. Lettre de M. Le Verrier á M. Faye sur la théorie de Mercure et sur le
mouvement du périhélie de cette planète. Comptes rendus hebdomadaires des séances
de l'Acadèmie des sciences (Paris) 49, 379-383 (1859).
3. Hawking, S. On the Shoulders of Giants. The Great Works of Physics and Astronomy.
Philadelphia, Pennsylvania, USA: Running Press. p. 1243, Foundation of the General
Relativity. ISBN 0-7624-1348-4.
4. Biswas, A., Mani, K.R.S. Relativistic perihelion precession of orbits of Venus and the
Earth. Cen. Euro. J. Phys. 6, 754-758 (2008).
5. Weisberg, J.M., Taylor, J.H. The Relativistic Binary Pulsar B1913+16: Thirty Years of
Observations and Analysis. Binary Radio Pulsars, Ed. F.A. Rasio and I.H. Stairs. ASP
Conference Series. 328. Aspen, Colorado, USA: Astronomical Society of the Pacific
(2005).
6. Friedman, Y. Relativistic Newtonian Dynamics under a central force. Eur. Phys. Lett.
116, 19001 (2016).
7. Friedman, Y., Steiner, J.M. Predicting Mercury’s Precession using Simple Relativistic
Newtonian Dynamics, Eur. Phys. Lett. 113, 39001 (2016).
8. Friedman, Y., Livshitz, S., Steiner, J.M. Predicting the relativistic periastron advance
of a binary without curving spacetime. V2. Eur. Phys. Lett. 116, 59001 (2016).