1

The Origin of Gravity in the new Vacuum Paradigm

Min Tae Kim

Institute for Theory of Everything

Daejeon 34050, Republic of Korea

E-mail: kimnogravity@gmail.com, 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.

3

(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

5

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.

6

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)

7

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

8

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)

9

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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11

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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).