Quantum Gravitational Shielding

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https://hal.archives-ouvertes.fr/hal-01074607v3

Preprint submitted on 28 Apr 2015

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Quantum Gravitational Shielding

Fran de Aquino

To cite this version:

Fran de Aquino. Quantum Gravitational Shielding. 2014. �hal-01074607v3�

Professor Emeritus of Physics, Maranhao State University, UEMA.

Titular Researcher (R) of National Institute for Space Research, INPE Copyright © 2014 by Fran De Aquino. All Rights Reserved.

We propose here a new type of Gravitational Shielding. This is a quantum device because results from the behaviour of the matter and energy on the subatomic length scale. From the technical point of view this Gravitational Shielding can be produced in laminas with positive electric charge, subjected to a magnetic field sufficiently intense. It is easy to build, and can be used to develop several devices for gravity control.

Key words: Gravitation, Gravitational Mass, Inertial Mass, Gravitational Shielding, Quantum Device.

1. Introduction

Some years ago [1] I wrote a paper where a correlation between gravitational mass and inertial mass was obtained. In the paper I pointed out that the relationship between gravitational mass, , and rest inertial mass, , is given by

mg 0

mi

( )

1 1

1 2 1

1 1

2 1

1 1

2 1

2 2

2 0 2

2 0 0

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⎟ −

⎟

⎠

⎞

⎜⎜

⎝ +⎛

−

=

⎪=

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⎟ −

⎟

⎠

⎞

⎜⎜

⎝ +⎛

−

=

⎪=

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⎟⎟ −

⎠

⎜⎜ ⎞

⎝ +⎛ Δ

−

=

=

c Wn

c m

Un

c m

p m

m

r i

r i i

g

ρ χ

where

is the variation in the particle’s kinetic momentum; is the electromagnetic energy absorbed or emitted by the particle; is the index of refraction of the particle;

is the density of energy on the particle ;

Δp U

n

r

( ^{J /} W ^kg )

^ρ

^is

the matter density ( ^{kg m}

³

) ^{and is the}

speed of light.

c

Also it was shown that, if the weight of a particle in a side of a lamina is P

r

m

_g

g

r

=

( perpendicular to the lamina) then the weight of the same particle, in the other side

of the lamina is , where

gr

g m P

r′=

χ

_gr

0 i

g m

=m

χ

(

m_g

and m

_i0

are respectively,

the gravitational mass and the inertial mass of the lamina). Only when

χ =1

, the weight is equal in both sides of the lamina. The lamina works as a Gravitational Shielding.

This is the Gravitational Shielding effect.

Since

P′=χP=

( )

χm_g g=m_g

(

χg

) , we can consider that

m′_g =χm_g

or that

g′=χg

. In the last years I have proposed several types of Gravitational Shieldings.

Here, I describe the Quantum Gravitational Shielding. This quantum device is easy to build and can be used in order to test the correlation between gravitational mass and inertial mass previously obtained.

2. Theory

Consider a conducting spherical shell with outer radius

r

. From the subatomic viewpoint the region with thickness of

φe

(diameter of an electron) in the border of the spherical shell (See Fig.1 (a)) contains an amount, , of electrons. Since the number of atoms per , , in the spherical shell is given by

Ne

m3 n_a

( )

2 0

s a s

A

n N ρ

=

where , is

the Avogadro’s number;

kmole atoms

N₀ =6.02214129×10²⁶ /

ρs

is the matter density of the spherical shell (in kg/m

³

) and

As

is the molar mass ( ) . Then, at a volume

−1 kmole kg.

φS

of the spherical shell, there are atoms per , where

Na m³

2

r

(a)

E = q / 4πε0 r² = V / r Nh = q / e

V

(b)

Fig.1 – Subatomic view of the border of the conducting spherical shell.

+ φe

ρs

As

+

φe

x =φa /2 + +

+

+

( )

4 S

n N_e = _eφ

By dividing both sides of Eq. (3) by , given by Eq. (4), we get

Ne

( )

5

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

a a e

e N

n N n

Then, the amount of electrons, in the border of the spherical shell, at the region with thickness of φe is

( )

⁰ S

( )

6

N N A S N n

N _e

a e s

e s e e

e ρ φ

φ

φ _⎟⎟

⎠

⎜⎜ ⎞

⎝

= ⎛

=

Assuming that in the border of the spherical shell, at the region with thickness of

a 2

x≅φ (See Fig.1 (b)), each atom contributes with approximately Z 2electrons (Zis the atomic number). Thus, the total number of electrons, in this region, is^N_e( )^{x =}

( )

^{Z 2N}_e^e

( )

^φ_e ^.

Thus, we can write that

( ) ( ) ( )

7

2 2

0 S

N N A Z N Z N

x

N _e

a x e s e s

e

e ρ φ

φ _⎟⎟

⎠

⎜⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

=⎛

⎟⎠

⎜ ⎞

⎝

=⎛

where

(

^N_e ^N_a

)

_x ^{≅Z 2.}

Now, if a potential V is applied on the spherical shell an amount of electrons, , is removed from the mentioned region. Since

Nh

e q

N_h = and E=q 4πε_rε₀r², then we obtain

( )

⁴ 2 0 0

_{( )}

₈

e E S

e E

N_h πr ε^rε ε^rε

=

=

Thus, we can express the matter density,ρ, in the border of the spherical shell, at the region with thickness of x≅φ_a 2, by means of the following equation

( ) ( )

0 0 2 0

0 0

2 2

2

e a r a e s

s

a e h e e h e

e m E A

N Z

S m N x N Sx

m N x N

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ⎟⎟⎠−

⎜⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

= ⎛

⎟ =

⎠⎞

⎜⎝

⎛ −

⎟ =

⎠⎞

⎜⎝

⎛ −

=

φ ε ε φ φ ρ ρ φ

or

( )

9

2 2 ⁰

0 2 0

e a r a e s

s m

re V A

Z N

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ⎟⎟−

⎠

⎜⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

= ⎛

φ ε ε φ φ ρ ρ

since

E=V r

.

a

and covered with a thin layer (20μm) of Barium titanate^* (BaTiO3), whose relative permittivity at 20^°C is ε_r =1250, then Eq. (9) gives

(

^343106851038 ^227250331021

)

² 0

^{( )}

¹⁰ e

e− × V m

×

= . φ .

ρ

Assuming that the electron is a sphere with radius and surface charge , and that at an atomic orbit its total energy is equal to the potential electrostatic energy of the surface charge,

re −e

2 0c m E≅ _e r

e

E_pot= ² 8πε₀ [2], then these conditions determine the radius r≡r_e:

m c

m e

r_e = ² 2.4πε_{0 e0} ² ≅ 1.4×10^{−15 †}, which is equal to the radii of the protons and neutrons. Thus, we can conclude that in the atom, electrons, protons and neutrons have the same

radius.

Thus, substitution of

m r_e

e=2 =2.8×10⁻15

φ

into Eq. (10) gives

(

^960699181023 ^{22725033 10}21

)

² 0

^{( )}

¹¹ me

× V

−

×

= . .

ρ

For

V =422.7493 volts

, Eq. (11) gives

(

⁶⁸×¹⁰14

)

² 0=¹²×¹⁰−¹⁵ −³

^{( )}

¹²

= . m_e . kg.m

ρ

Note that the voltage

V=422.7493 volts

is only a theoretical value resulting from inaccurate values of the constants present in the Eq. (11), and that leads to the critical value

6.8×10¹⁴

shown in Eq. (12), which is fundamental to obtain a low density,

ρ

.

However, if for example, ,

then the critical value increases to (more than 100,000 times the initial value) and,

therefore the system shown in

volts V =422.7 1020

1 1. ×

* Dielectric Strength: 6kV mm, density: 6,020kg/m³.

† The radius of the electron depends on the circumstances (energy, interaction, etc) in which it is measured. This is because its structure is easily deformable. For example, the radius of a free electron is of the order of 10⁻¹³m [3], when accelerated to 1GeV total energy it has a radius of 0.9×10⁻¹⁶m[4].

4

60 cm

30 cm 30 cm

Pair of Helmholtz Coils (Bmax = 20mT)

B

Vmax = 425 volts Vv

V i

g

Fig.2 – Quantum Gravitational Shielding produced in the border of a Lithium Spherical Shell with positive electric charge, subjected to a magnetic field B.

+ + +

+ + χ²g

+ + +

χg χg χ²g

-P

P

-2P -3P

0 Lithium Spherical Shell (with outer diameter 20 cm

and inner diameter 19 cm)

Quantum Gravitational Shielding

Mechanical dynamometer

Barium titanate (20μm thickness)

value

6.8×10¹⁴

or a close value, must be found by using a very accurate voltage source in order to apply accurate voltages around the value

V=422.7493 volts

at ambient temperature of 20°C.

Substitution of the value of

ρ

(density in the border of the Lithium Spherical Shell,

at the region with thickness of x≅φ_a 2

), given by Eq. (12), into Eq. (1) yields

(

^{93 10}

)

¹

^{( )}

¹³

1 2

1 ^{3 2}

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ ⎥

⎦

⎢ ⎤

⎣

⎡ + × −

−

= . ⁻ W

χ

Substitution of

( )

2 0 2 0

2 1 2 0 2 2 1 0 2 2 1 0 2 2

1ε E μ H ε c E B μ B μ

W= + = + =

into Eq. (13) gives

( )

14 1

10 4 5 1 2

1 ^{7 4}

⎭⎬

⎫

⎩⎨

⎧ ⎥⎦⎤

⎢⎣⎡ + × −

−

= . B

χ

Therefore, if a magnetic field

B=0.020T

passes through the spherical shell (See Fig.

(2)) it produces a Gravitational Shielding (in the border of the Lithium Spherical Shell,

at the region with thickness of x≅φ_a 2)

with a value of

χ

, given by

( )15

−3 χ≅

Also, it is possible to build a Flat Gravitational Shielding, as shown in Fig. 3.

Consider a cylindrical or hexagonal container, and a parallel plate capacitor, as shown in Fig.

3(a). When the capacitor is inserted into the container the positive charges of the plate of the capacitor are transferred to the external surface of the container (Gauss law), as shown in Fig. 3(b).

Thus, in the border of the container, at the region with thickness of x≅φ_a 2 the density,ρ, will be given by Eq. (9), i.e.,

0 0 2 0

2 _a 2 ^e

r a e s

s m

e E A

N Z

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ⎟⎟−

⎠

⎜⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

= ⎛

φ ε ε φ φ ρ ρ

where

( )

( )

16

0 0

0

Sd V A

S CV

S q

E

r c

r

r r

r

ε ε

ε ε ε

ε ε

ε σ

=

=

=

=

=

Thus, we obtain

Therefore, if the container is made of Lithium (Z=3,ρ_s =534kg.m⁻³,A_s =6.941kg/kmole,

a 10m

10 04

3 × ⁻

= .

φ ) and, if the dielectric of the

capacitor is Barium titanate (BaTiO3), whose relative permittivity at 20^°C is ε_r =1250, and the area of the capacitor is A=S, and d=1mm, then Eq. (17) gives

(

^960699181023 ^227250331023

)

² 0

^{( )}

¹⁸ me

× V

−

×

= . .

ρ

For V =4.227493 volts, Eq. (18) gives

(

⁶⁸×¹⁰14

)

² 0=¹²×¹⁰−¹⁵ −³

^{( )}

¹⁹

= . m_e . kg.m

ρ

Substitution of this value into Eq. (1) gives

( )

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ ⎥

⎦

⎢ ⎤

⎣

⎡ + × −

−

= 1 2 1 9.3 10⁻³W ² 1 χ

This is exactly the Eq. (13), which leads to

⎭⎬

⎫

⎩⎨

⎧ ⎥⎦⎤

⎢⎣⎡ + × −

−

= 1 2 1 5.4 10⁷B⁴ 1 χ

Therefore, if a magnetic field B=0.020T passes through the Lithium container, it produces a Quantum Gravitational Shielding (in the border of the container, at the region with thickness of

a 2

x≅φ ) with a value of χ, given by

−3 χ≅

(a)

(b)

Fig. 3 – Flat Gravitational Shielding or Flat Gravity Control Cell (GCC).

S E q

r r

ε

0

ε ε

0

ε

σ

=

=

Lithium Container

V d

+q

−q

V εr(c)

A

S

( )

( )

A d V CV

q= =ε_{r c} ε₀

−q +q

6

Fig. 4 – Flat Gravity Control Cell - Experimental Set-up. (BR Patent Number: PI0805046-5, July 31, 2008).

V

Digital force gauge with external sensor

(

^{±20N; 0.01N}

)

http://www.andilog.com/graphical-force-gauge-centor-easy-with-external-sensor.html?category_id=13&tab=spec

Lithium: 1mm thick Sample Testing

200 g

Barium titanate (BaTiO3); 1mm thick

DC 1-10ref: fixed voltage reference, providing an 1.000000V and an 10.000000 V output.

http://www.stahl-electronics.com/voltage-supplies.html External load cell

888888

Coil: 18mm x 18mm; 10 x 10 = 100 turns, #14 AWG Max. 5cm

Min. 70cm

18 mm Air

Dielectric base

References

[1] De Aquino, F. (2010) Mathematical Foundations of the Relativistic Theory of Quantum Gravity, Pacific Journal of Science and Technology, 11 (1), pp. 173-232.

[2]Alonso, M., and Finn, E., (1967) Foundations of University Physics. Portuguese version (1977), Vol. II, Ed. Blucher, SP, p.149.

[3]Mac Gregor, M. H., (1992) The Enigmatic Electron, Boston, Klurer Academic; Bergman, D. L., (2004) Foundations of Science, (7) 12.

[4]Caesar, C., (2009) Model for Understanding the Substructure of the Electron, Nature Physics 13 (7).