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Journal of High Energy Physics, Gravitation and Cosmology, 2020, 6, 357-367

https://www.scirp.org/journal/jhepgc

ISSN Online: 2380-4335

ISSN Print: 2380-4327

DOI:

10.4236/jhepgc.2020.63029 Jul. 8, 2020 357 Journal of High Energy Physics, G

ravitation and Cosmology

New Mechanism and Analytical Formula for

Understanding the Gravity Constant G

Nader Butto

Rabin Medical Centre, Petah-Tikva, Israel

Abstract

The nature of gravitation and

G

is

not well understood. A new gravitation

mechanism is proposed that explains the origin and essence of the gravita-

tional constant,

G

. Based on general relativity, the vacuum is considered to be

a superfluid with measurable density. Rotating bodies drag vacuum and cre-

ate a vortex with gradient pressure. The drag force of vacuum fluid flow in

the arm of the vortex is calculated relative to the static vacuum and a value

that is numerically equal to that of

G

is obtained. Using Archimedes’ princi-

ple, it is determined that

G

is the volume of vacuum displaced by a force

equivalent to its weight which is equal to the drag force of the vacuum. It is

concluded that the gravitational constant

G

expresses the force needed to dis-

place a cubic metre of vacuum that weighs

one kg in one second. Therefore,

G

is not a fundamental physical constant but rather is an expression of the re-

sistance encountered by the gravitational force in the vacuum.

Keywords

Gravitational Constant, Vacuum Density, Drag Force, Vortex Formation,

Specific Volume Flow, Archimedes’ Principle

1. Introduction

The gravitational constant denoted by the letter

G

, is an empirical physical con-

stant pivotal in the calculation of gravitational effects in Newton’s law of univer-

sal gravitation and in Albert Einstein’s general theory of relativity.

Gravity is most accurately described by the general theory of relativity (pro-

posed by Albert Einstein in 1915), which describes gravity not as a force but as a

consequence of the curvature of space-time caused by the uneven distribution of

mass. The most extreme example of this curvature of space-time is a black hole,

from which nothing can escape once past its event horizon, not even light [1].

How to cite this paper:

Butto, N. (2020

)

New Mechanism and Analytical Formula

for Understanding the Gravity Constant

G

.

Journal of High Energy Physics

,

Gravit

a-

tion and Cosmology

,

6

, 357-367.

https://doi.org/10.4236/jhepgc.2020.63029

Received:

April 10, 2020

Accepted:

July 5, 2020

Published:

July 8, 2020

Copyright © 20

20 by author(s) and

Scientific

Research Publishing Inc.

This work is licensed under the Creative

Commons Attribution International

License (CC BY

4.0).

http://creativecommons.org/licenses/by/4.0/

Open Access

N. Butto

DOI:

10.4236/jhepgc.2020.63029 358 Journal of High Energy Physics, G

ravitation and Cosmology

In 1687, Newton published Principia, which hypothesises the inverse-square

law of universal gravitation [2].

Newton’s law states that every object in the universe attracts every other ob-

ject with a force which, for any two bodies, is proportional to their mass and

varies inversely as the square of the distance between them. This statement is

expressed mathematically by the following well-known equation:

( )

2

12g

F Gm m r= ⋅

, (1)

where

m

1 and

m

2 are the interacting masses and

r

is their relative distance vec-

tor. The Newtonian constant of gravitation

G

, is typically assumed to be a uni-

versal constant whose measured value is

( )

11 3 1 2

6.67408 31 10 m kg s

− −−

× ⋅⋅

[3].

The numerical value of

G

was initially determined by English physicist Cavendish

in 1798 through the measurement of the attractive force between two spheres

with the aid of a torsion balance.

More than three hundred and fifty years after the discovery of gravity by New-

ton, there is still no theoretical explanation for the mechanism of gravity. As a re-

sult, the true nature of gravity and the essence of

G

are not understood. It is un-

known whether the origin of

G

can be described using an analytical formula [4].

The current spread of values is approaching 0.05%, which is more than 10 times

the uncertainties on each measurement, and it therefore appears that we know

G

only to three significant figures! This is very poor compared with other physical

constants, many of which have uncertainties of the order of parts in 108 [5].

Determining the ultimate physical origin of gravity and its associated constant

G

could provide important insights into a fundamental understanding of the

universe.

In this article, the origin and essence of

G

is described and a mathematical

formula is derived to calculate its value.

The starting point is the superfluid vacuum that has calculable density based

on universe expanding measurements and Hubble constant determination. Then

gravity is described as the result of the dragging force of the rotating planet that

generates a vortex that curves space and time and generates pressure gradient

and vacuum flow to the centre of the vortex. Calculation of the dragging force of

the flow with the vacuum gives us the same value of constant

G

.

The superfluid nature of the vacuum

Models that describe the theory of gravity based on the idea that the physical

space could be “filled” with a constitutive continuum medium characterised by

specific properties have been previously proposed. The vacuum could therefore

be comprised of a fundamental substrate (on the quantum scale) such as an elas-

tic solid-state medium, a fluid, or a Higgs condensate [6] [7] [8] [9].

According to the superfluid theory of vacuum, the physical vacuum is de-

scribed as a quantum superfluid, which behaves like a fluid with minimal viscos-

ity and with extremely high thermal conductivity. It is a perfect fluid in the sense

that it is non-particulate and has no structural memory. If perturbed, it has no

tendency to revert to its former physical state.

N. Butto

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10.4236/jhepgc.2020.63029 359 Journal of High Energy Physics, G

ravitation and Cosmology

The superfluid vacuum theory proposes a mass generation mechanism that

may replace or supplement the electroweak Higgs mechanism. It has been shown

that the masses of elementary particles could arise because of their interaction

with a superfluid vacuum. This phenomenon is similar to the gap generation

mechanism in superconductors [10].

Furthermore, the first postulate of general relativity states that the source of a

gravitational field is the stress-energy tensor of a perfect fluid [11]. This

‘stress-energy tensor’ contains four non-zero components,

i.e.

, the density of the

perfect fluid and the pressure of the perfect fluid in each of the three physical

axes. According to general relativity, a perfect fluid is defined as a fluid with no

viscosity or heat conduction.

2. Density of the Vacuum

Although there is no consensus regarding vacuum density, its value primarily

depends on general relativity. It is therefore possible to measure the energy den-

sity of the vacuum through astronomical observations that determine the curva-

ture of space-time and the expansion of the universe.

The measurement of universal expansion based on the relation between galaxy

velocity (

v

) and its distance (

d

) [12]

v Ho d

= ×

. (2)

This relation is the well-known Hubble Law. It indicates a constant expansion

of the cosmos, where galaxies recede from each other at a constant speed per

unit distance; thus, more distant objects move faster than nearby ones.

The expansion of the universe has been studied by several different methods.

The Wilkinson Microwave Anisotropy Probe (WMAP) mission completed in

2003, represents a major step toward precision in determining the expansion of

the universe and calculating vacuum density [13].

Another method is using the Baryon Oscillation Spectroscopic Survey (BOSS)

[14] by studying more than 140,000 extremely bright galaxies known as quasars,

which serve as a “standard ruler”, scientists map density variations in the uni-

verse. By nearly tripling the number of quasars previously studied, as well as im-

plementing a new technique, the scientists were able to calculate the expansion

rate to 42 miles (68 kilometres) per second per 1 million light-years with greater

precision, while looking farther back in time.

It is important to note that the study of the expansion rate of the universe has

shown that the universe is close to critical density. Critical density is the value at

which the Universe is balanced and expansion is halted.

The density is typically expressed as a fraction of the density required for the

critical condition to be fulfilled through the use of a parameter known as omega

(Ω) where

critical

ρρ

Ω=

. The limiting critical density is described by

1Ω=

.

For a value of omega less than 1 (known as an “open universe”), the final fate of

the universe is a “cold death”. In this case, the universe expands forever, albeit at

an ever-decreasing rate. For omega greater than 1, the universe is “closed” and

N. Butto

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10.4236/jhepgc.2020.63029 360 Journal of High Energy Physics, G

ravitation and Cosmology

will at some point collapse in on itself and end in a “big crunch”. For omega

equal to 1, the universe is called “flat”; this universe has a critical density and

expansion is halted only after an infinite time. Currently, the estimated sum of

the contributions to the total density parameter, Ω0, is

0

1.02 0.02

Ω= ±

, which

indicates that the universe is close to critical density.

Hence, the critical density that defines the boundary condition between the

open and closed solutions of the standard cosmological model is [15]

( )

2 2 29 3

0

3 8 1.88 10 g cm

cr

HG h

ρ

−

= π = ×

, (3)

where

ρcr

is the critical density,

H

is the current value of the Hubble constant and

11

0

71 km sec MpchH

−−

≡ ⋅⋅

.

11

0

71 km s MpcH

−−

= ⋅⋅

(WMAP value for the

Hubble parameter [16]).

The infrared camera was installed on the Hubble Telescope in 2009, and the

astronomical measurements used to calculate the Hubble constant obtained a

slightly higher value with narrow error bars.

The Hubble Space Telescope can determine the distances to Milky Way Ce-

pheids (a type of variable star) through accurately separating their spectra from

the bluer stars that tend to surround cepheids. In a recent publication, Riess [17]

reported that

11

073.24 km s Mpc

H−−

= ⋅⋅

,

where Mpc is equal to 3.09 × 1019 km.

The most recent result published in this year (2017) [18], and the cosmologi-

cal density

ρc

(with small uncertainty) is therefore calculated to be

( )

2 27 3

,0 0

3 8 11.11 1.05 10 kg m

c

HG

ρ

−

= = ±×π

. (4)

3. Mechanism of Gravitation

According to general relativity, the gravitational attraction observed between

masses results from the warping of space and time by these masses. The gravita-

tional potential generated by a mass, which depends on the radial distance from

the mass, affects the running rate of clocks, the measurement of distances and

the velocity of light. This fact is theoretically explained within general relativity

and supported by strong experimental evidence. Nevertheless, there is no de-

scription of causation as a curvature of space.

In this work, a new model that explains the gravitational force and the curvature

of space-time is proposed. The interaction of rotating masses within the vacuum

leads to a drag effect, vortex formation, space-time curvature and resultant gravi-

tational force. All particles and celestial bodies are immersed in this fluid vacuum.

All particles of matter spin and exert a drag effect on the surrounding vacuum.

This creates a small vortex around the particle even if it is located within a station-

ary body. The sum of the effects due to all the constituent particles creates a gravi-

tational force arising from the mass. In a rotating celestial body, the rotating parti-

cles have translational speed that drags the surrounding vacuum. It follows from

N. Butto

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ravitation and Cosmology

the case of the “normal” drag force for an object moving in a linear fashion, such

that rotational motion creates a vortex around the celestial body.

The vortex resulting from the rotating mass creates a dynamic pressure that is

lowest in the core region closest to the vortex core. This pressure increases with

distance from the centre of the vortex. This behaviour is in accordance with

Bernoulli’s principle, which states that for an inviscid flow, an increase in the

speed of the fluid occurs simultaneously with a decrease in pressure or a de-

crease in the fluid’s potential energy. The gradient of this pressure forces the

fluid to curve around the central axis of the vortex. This dynamic pressure

causes the gravitational force and is proportional to the square of the distance

“r,” from the axis according to the inverse-square law. This law states that a

specified physical quantity or intensity is inversely proportional to the square of

the distance from the source of that physical quantity.

The flow in the arms of the vortex (created by the gradient pressure) travels at

the speed of light relative to the static surrounding vacuum. This creates a drag

between static vacuum and vortex arm flow.

Drag is a common term which refers to any force that opposes motion. When

a fluid moves in a fluid such as a vortex, it experiences two forms of drag force.

Forces normal to the motion are referred to as pressure drag and shear forces

due to flow along surfaces “parallel” to the motion (edges) are referred to as vis-

cous drag.

Pressure drag is caused by molecules hitting a surface and returning. This

causes a change in linear momentum and results in normal force. Viscous drag

results from the attraction between molecules due to the relative velocity be-

tween flux and static fluid.

The drag force on any object is proportional to the density of the fluid and the

square of the relative flow speed between the moving object and static fluid, ac-

cording to the following formula:

2

1

2

dD

F v AC

ρ

=

, (5)

where

Fd

is the drag force (which is defined as the force component in the direc-

tion of flow velocity [19]),

ρ

is the mass density of the fluid,

v

is the flow velocity

relative to the object,

A

is the interaction area and

CD

is the drag coefficient.

In fluid dynamics, the drag coefficient is a dimensionless quantity that is used

to quantify the viscous drag or shear forces on an object in a fluid environment.

The drag coefficient is the ratio of drag force to the product of area and the force

generated by dynamic pressure.

2

1

2

Dd

C F vA

ρ

=

. (6)

However, if there are no data on the drag coefficient and drag force, the above

mentioned formula cannot be used to determine the drag coefficient.

Another method of calculating the drag coefficient is by using the Reynolds

number. The drag coefficient of an object is regarded as a function of the Rey-

N. Butto

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ravitation and Cosmology

nolds number, based on the relative velocity between the rotating object and

surrounding fluid. Reynolds number is the ratio of inertial (resistance to change

or motion) forces to viscous forces [20].

( )

( )

2

Re u u L

ρµ

=

,

(7)

Re uL

ρµ

=

,

(8)

Re uL

ν

=

,

(9)

where

Re =

Reynolds Number (non-dimensional),

ρ

=

density (kg/m3),

u

=

is the

mean velocity of flow relative to vacuum (speed of light; SI units: m/s),

μ

=

dy-

namic viscosity (Ns/m2),

L

=

characteristic length (m)

and

ν

=

μ

/

ρ

=

kinematic

viscosity (m2/s).

In general, the drag coefficient is not an absolute constant for a given body

shape. Larger velocities, larger objects and lower viscosities (such as that in this

case) contribute to larger Reynolds numbers [21].

While there is no theorem that relates the Reynolds number to turbulence,

flows at Reynolds numbers larger than 5000 are typically turbulent, while those

at low Reynolds numbers are laminar.

As the Reynolds number increases, inertial forces become stronger than vis-

cous forces, and a laminar boundary layer is generated. Therefore, the drag coef-

ficient decreases as the Reynolds number increases.

The graph of

CD

vs.

Re

is shown in Figure 1.

According to Figure 1, the value of the drag coefficient can be estimated.

Planets and large celestial bodies have large Reynolds numbers. It can therefore

be deduced that these objects have drag coefficients that lie between 0.1 and 0.2.

4. Drag Force of the Vacuum

If the vacuum is considered as a liquid that travels at the speed of light and given

its density, the drag force can be calculated by applying the drag force formula.

The drag force equation is transformed to the pressure equation by dividing

both sides by area to obtain:

2

1

2

dD

F A P vC

ρ

= =

, (10)

Figure 1.

CD

vs.

Re

for a sphere. The dashed curve represents theoretical results for small

values of

Re

[22].

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ravitation and Cosmology

where

P

is the pressure gradient generated by drag. The drag pressure of the

vacuum is constant because it is derived through conservation of momentum

using density and velocity.

If the vacuum density is 11.11 × 10−27 kg/m3,

8

3 10 m svc= = ×

and the drag

coefficient is between 0.1 and 0.2.

Substituting for

ρ, v

2

and

CD

(0.13349) in Equation (9) gives:

11 2 2

6.67383255 10 kg m s or N mP−

= ×⋅

The resulting answer has the same numerical value as

G

,

i.e.

, (6.67384 ±

0.00080) × 10−11 m3/kg∙s or N⋅m2/kg2.

This drag pressure is the pressure required to move the equivalent specific

volume per second (specific volume rate). In fluid mechanics, a fluid is displaced

when an object is immersed in it

i.e.

, the object displaces the fluid and occupies

its space. In this case, drag force generated by gradient pressure is what displaces

the vacuum fluid. The volume of vacuum that is displaced is equivalent to the

volume of fluid flow generated by gradient pressure.

In thermodynamics, specific volume is defined as the number of cubic metres

occupied by one kilogram of a particular substance or the ratio of a substance’s

volume to its mass. It is the reciprocal of density and it is an intrinsic property of

matter. The standard unit of specific volume is the cubic metre per kilogram

(m3∙kg−1). Specific volume rate is defined as the number of cubic metres occu-

pied by one kilogram of a particular substance that flows, per unit time. This is

equivalent to the flow rate per unit weight because m3/s represents flow rate.

According to Archimedes’ principle, the weight of a displaced fluid is directly

proportional to its volume. The magnitude of force required to counteract flow

is equal to the weight of the displaced fluid. Therefore, the rate of displacement

of specific volume per kg per second (

G

constant) would be proportional to the

pressure needed to move the weight of that volume, which is equivalent to the

drag force.

Therefore,

G

is the resistance force that gravity must overcome in order to

move the weight of one cubic metre of vacuum.

5. Discussion

In this work, a new mechanism of gravity related to the rotation of a planet in a

superfluid vacuum has been proposed. Rotating planets drag vacuum energy

around their particles and curve space-time by creating vortex flow. Although

this idea is not new because general relativity predicts that the rotation of planets

drags superfluid vacuum and ‘warps’ space-time, there is no explanation for the

curvature of space-time. In the proposed model, curved space-time is the effect

and not the cause of gravity.

According to the proposed model, a rotating planet drags vacuum from all

directions toward its centre and creates a vortex with a pressure gradient that at-

tracts the vacuum fluid from the periphery to the centre of the vortex. Therefore,

the vacuum superfluid flow from the periphery to the centre of the vortex

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ravitation and Cosmology

“curves” space-time and generates the pressure that pushes the objects forward

to the centre of the vortex. Objects obstructing the flow will be pushed to the

centre of the vortex. This is the origin of the gravitational force.

The vortex model is not new. A large number of philosophers used the idea of

cosmic vortices in their explanation of creation. In the ancient world, these phi-

losophers included Empedocles, Leucippus, Democritus and Aristotle. During

the Renaissance, this idea was developed by R. Descartes, J. MacCullagh, J. J.

Thomson and W. Thomson (Lord Kelvin). However, these scientists could not

formulate their ideas in a rigorous and mathematical form and instead their

findings were formulated as philosophical speculations. In recent years, different

theories proposed the existence of ether vortex mechanisms as the cause of grav-

ity, such as the vortex gravitation model [23]. However, there is no explanation

of the mechanism of vortex formation. Furthermore, the theory does not predict,

calculate, or describe the essence of

G

.

In this model, the drag force created by the interaction between a mass and

the surrounding vacuum is considered to be the origin of vortex formation.

The drag theory of gravity was originally proposed by Nicolas Fatio de Duillier in

1690 and later by Georges-Louis Le Sage in 1748 [24]. They proposed a me-

chanical explanation for Newton’s gravitational force in terms of streams of

minute unobservable particles impacting all material objects from all directions.

According to this model, any two material bodies partially shield each other

from impinging corpuscles. This results in a net imbalance in the pressure ex-

erted by the impact of the corpuscles on the bodies, which tends to drive the

bodies together. This mechanical explanation for gravity was not widely ac-

cepted, although it continued to be studied occasionally by physicists until the

beginning of the 20th century. However, by this time it was considered to be de-

finitively discredited [25].

According to general relativity, planet rotation is dragged by an unknown

force. Such a “drag” implies that there is friction in the motion of space-time

with respect to a mass where “inertial dragging” occurs. General relativistic for-

mulations show the requirement of tangential motion when the continuum is

assumed to be a superfluid. The explanation of inertial dragging does not pro-

vide an identifying cause based on a fundamental theory [26]. The inertial drag-

ging is explained by the theory of vortex space rotation. This states that the

gravitational force is independent of the masses and densities of bodies. The

masses of the planets have therefore been determined based on the law of angu-

lar momentum conservation as they were created at the centres of space torsions

through matter accumulation. The same force that created celestial bodies con-

tinues to exist and it exerts inertial dragging that maintains their rotation.

This theory does not exclude a drag force as the causes the rotation of celestial

bodies. However, vortex formation caused by a rotating mass is proposed. In

both scenarios, there is vortex formation and associated pressure gradient that

attracts vacuum fluid to the centre of the planet as demonstrated by experiments

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ravitation and Cosmology

of rotating spheres in liquid.

In order to obtain

G

, the drag force of the flowing vacuum fluid is calculated

relative to the static surrounding vacuum that depends primarily on the density

of the vacuum and the square of velocity of the flow according to dragging force

formula.

6. Conclusions

The results presented here strongly suggest that

G

is

related to the structure and

properties of the physical vacuum, where the vacuum is considered as a medium

characterised by specific properties such as density, viscosity and speed. The den-

sity of vacuum is calculated based on the current value of the Hubble constant.

A new mechanism of gravity is proposed according to which gravitational

force is the result of the gradient pressure of the vacuum that is generated by the

drag force of a rotating planet. The drag force of a vortex created by a rotating

body is calculated, and this is found to have the same numerical value as

G

.

It is therefore concluded that the gravitational force is a “push force” that de-

livers a part of its momentum to a mass upon colliding with it and pushes it

forward toward the vortex centre. However, the gravitational force is diminished

by the

G

value that represents the resistance of the vacuum to the gradient flow of

the vortex. Therefore,

G

is not a fundamental physical constant instead it is an ex-

pression of the resistance encountered by the gravitational force in the vacuum.

This paper proposes a new approach to understand gravity. Hence, it will sig-

nificantly contribute to understand the mechanism of gravitation.

Limitation

The density value used to calculate the drag force is based on recent astronomi-

cal measurements. This value, which has been determined as 11.11 × 10−27 kg/m3

still has an associated uncertainty of (±1.05), which can result in a significant

change in the value of G. Furthermore, although the drag coefficient value used

to calculate the drag force was within the range values, it is an estimated value.

Further research is therefore needed to confirm this theory.

Acknowledgements

The author would like to thank Enago (https://www.enago.com/) for the English

language review.

This research did not receive any specific grant from funding agencies in the

public, commercial, or not-for-profit sectors.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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