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The fundamental Conflict in the “Theory of Geneal Relativity” related to the observations of Gravitational Redshift

Wim Vegt

, Department of Physics, Eindhoven University of Technology, The Netherlands

Abstract
The foundation for the Theory of General Relativity is the Riemannian Mathematics for curved 4-dimensional Manifolds. Based on this mathematics Einstein predicted correctly the deflection of light by the gravitational field of the sun, observed in the Eddington Experiment in 1919 during a total solar eclipse.
In theoretical physics, the problem of time is a conceptual conflict between general relativity and quantum mechanics. Quantum mechanics regards the flow of time as universal and absolute, whereas general relativity regards the flow of time as malleable and relative.
The theory of General Relativity correctly does not predict an increase or decrease of the speed of light within a gravitational field. Because within the (gravitational) curvature of space and time, the intervals for the space coordinates and time coordinate do not change. Only the direction of the beam of light has been changed, following the (by a gravitational field caused) curved space-time path. For this reason the Theory of General Relativity predicts a constant value for the frequency of the light following a curved space-time path [37], [41]. Because the time intervals and the space interval do not change within the curvature of space and time [49]. A straight line being transformed into a circle does not get shorter or longer. Only the Lorentz timedilation and the Lorentz space contraction have to be taken into account. The Lorentz contraction and time dilation can be neglected in the experiments for Gravitational Redshift.
All publications related to the well observed effects of “Gravitational RedShift/ Blueshift” within a gravitational field explain this effect by the frequency shift due to the shift in potential energy within a gravitational field. The proportional relationship between (potential) energy and frequency is Planck’s law and has nothing to do with the theory of general relativity. The observed effects of Gravitational Redshift is in contradiction with the Theory of General Relativity. Because a curved space-time geometry does not change the time intervals and does not change the space intervals within a gravitational field. Only the direction of the beam of light will change and follow the curved 4-dimensional space-time geometry. A change of the magnitude of the speed of light has never been observed within gravitational fields.
1

The only theory which predicts Gravitational Redshift correctly, with margins smaller than 15 digits beyond the decimal point, is the new theory presented in this article.

The “Gravitational RedShift” [2, 5] between an observatory on Earth (Radius = 6 106 [m]) and a Satellite in a Galileo Orbit (Radius = 23222 103 [m]) according “Planck’s
law” (incorrectly mentioned as “General Relativity” in the publications):

 ωGR =

U = c2

0.00000000004360137706159641

The “Gravitational RedShift” [2, 5] between an observatory on Earth (Radius = 6 106 [m]) and a Satellite in a Galileo Orbit (Radius = 23222 103 [m]) according the “New
Theory”:

 ωQLT =

 URel c2

=

0.00000000004360134475689392

Both values are always within the measured Gravitational RedShift in all observations being published since the first observation of the gravitational redshift in the spectral lines from the White Dwarf which was the measurement of the shift of the star Sirius B, the white dwarf companion to the star Sirius, by W.S. Adams in 1925 at Mt. Wilson Observatory.

Keywords

Gravitational Redshift; Classical Field Theory, Gravitation and Electromagnetism, General Relativity, GEONs

Introduction

This article describes a New Electromagnetic Field Theory. The foundation of the new theory is the Universal 4-dimensional Equilibrium in the Minkowski space, expressed by the “Non-Divergence” or “Zero-Divergence” of the Stress-Energy Tensor: b Tab = f a = 0 in which f a expresses the density of the Electromagnetic Force 4-vector in the Minkowski Space (equation 45) and “a” varies from 1 until 4. [25],[26] and [27]. Equation (24) describes the Universal Equilibrium for Electromagnetic Waves propagating within time-dependent gravitational fields.

Based on the 4-dimensional equilibrium in the Minkowski space (expressed by the
Stress-Energy Tensor = 04 ), Newton’s Classical Dynamics will be unified with
Maxwell’s Electrodynamics, the 4-Dimensional Relativistic Dirac Equation (which
2

is the foundation of Relativistic Quantum Physics) represents the classical relativistic Dirac Equation (54.1) and equals the 4th component of the density of the Force 4vector in the Minkowski Space and Einstein’s General Relativity.
Extending Newton’s 3-dimensional Equilibrium in Classical Mechanics into a model of confined electromagnetic energy within a 4-dimensional Equilibrium will unify Classical Mechanics with Relativistic Quantum Physics (4-dimensional Dirac Equation) and represents the mathematical solutions (equation 31) for the in 1955 by J. A. Wheeler already announced concept of Gravitational-Electromagnetic Confined GEONs. [1]
The research has been focussed to unify the four fundamental theories in physics. Isaac Newton’s Classical Mechanics, James Clerk Maxwell’s Electrodynamics, Quantum Physics (Niels Bohr, Paul Dirac, Werner Heisenberg) and Albert Einstein’s Theory of General Relativity. These four fundamental theories also mark the different periods when they have been created and a fundamentally different way of thinking at those times. To unify these four principles of Physics, it is necessary to link electromagnetism and gravitation. Here it is new that the 2-dimensional confinement of electromagnetic radiation (a beam of light) creates its own gravitational field, and that field will interact with other gravitational fields. To illustrate the utility of this theory, an example has been discussed with regard to the measured gravitational redshift associated with Earth’s gravity. This Theory has been called “Quantum Light Theory”.
The foundation of this theory is the non-zero mass property of photons [10]. In classical physics the mass (inertia) of matter is omni-directional. The weight of an apple on a scale does not change when the apple has been rotated. The mass (inertia) of photons is bi-directional. In the direction of propagation the mass (inertia) of the photon equals zero. In the plane perpendicular to the direction of propagation, the mass (inertia) does not equal zero and has been determined by Einstein’s equation for the relationship between mass and energy (W = m c2 ). For this reason photons are not being accelerated or decelerated when they move towards to- or away from a gravitational field. The speed of light remains constant. However when the gravitational field directs in the plane perpendicular to the direction of propagation of photons, photons do have mass and interact like other particles with the gravitational field according classical mechanics.
In the first paragraph the concept of Inertia has been introduced into the Electromagnetic Field Equations. Resulting in a unification of Classical Mechanics and Electrodynamics.
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The Second paragraph introduces the concept of “4-dimensional Equilibrium” resulting in the Unification of Electrodynamics with Relativistic Quantum Physics (Relativistic Dirac Equation). The paragraph describes the confinement of Electromagnetic Radiation due to its own Gravitational-Electromagnetic field. Wheeler described these confinements in 1955 as GEONs. [1] The mathematical solutions for GEONs are also solutions of the quantum mechanical relativistic Dirac equation (Equations 25, 26, 33, 34 and 50).
The Third paragraph introduces the concept of “Gravity” into the Electromagnetic Field Equations. Resulting in a unification of Electrodynamics with General Relativity.
In this paper, a new approach to describe any arbitrary electromagnetic field, has been developed following the concept of Universal Equilibrium (Stress-Energy Tensor (44) , as proposed by Newton for classical mechanics. The outcome of which is a single equation based upon Stress-Energy Tensor (44) in 3-dimensional space, between four fundamental force densities: inertia, electric force densities, magnetic force densities and the generation of gravity. The first and last force densities are represented by additional terms in Maxwell’s equations, for which the resulting fundamental equation unifies Classical Mechanics with Electrodynamics. These two additional terms account for the electromagnetic-field’s inertia and confinement forces within Gravitational Electromagnetic Entities (GEONs) [1]. The approach of James Clerk Maxwell to describe Classical Electrodynamics is missing two fundamental terms [7].
Extending the 3-dimensional equilibrium between the 4 fundamental forces into a 4dimensional equilibrium (Energy, Time domain) results in the relativistic quantum mechanical Dirac equation, which is the fundamental foundation of Relativistic Quantum Physics.
This article starts with the introduction of the concept of “Inertia of Electromagnetic Radiation”, using Newton’s second law of motion, expressed in the inertia force density [N/ m3 ]. This concept has been presented in equation (9) and term B-1 in equation (19).
Then two classical electromagnetic equations (Coulomb and Lorentz) have been used. The same equations which James Clerk Maxwell used to build his theory of Electrodynamics,
The Coulomb force density [N/ m3] results in Equation 12 and has been expressed as term B-2 in Equation 19.
The Lorentz force density [N/ m3] results in Equation 17 and has been expressed as term B-5 in Equation 19.
4

The magnetic field terms and the electric field terms are considered to be completely symmetrically. Which results in two extra symmetrical terms. A Coulomb force density [N/ m3] expressed as a magnetic force density (term B-4) in equation 19 and a Lorentz force density [N/ m3] expressed as an electric force density (term B-3) in equation 19.
In the second paragraph the sixth term (B-6 in equation 23) has been introduced, presenting the concept “Gravity” in light (EMR). The consequence is that light (EMR) creates its own gravitational field, already predicted by John Archibald Wheeler in 1955 in his model of GEONs (Gravitational Electromagnetic Entities). [1] Wheeler could not find stable solutions for GEONs in General Relativity. Equation (23) results in stable solutions for GEONs (Equation 25). It follows from (25) that only harmonic functions are possible solutions for GEONs which results in the discrete values (quantum numbers) for the harmonic frequencies for the GEONs depending on the geometry of the Gravitational Electromagnetic Confinement and the corresponding wavelength..
Newton’s concept for a 3-dimensional equilibrium has been expanded into a universal 4-dimensional equilibrium in the 4-dimensional Minkowski space. This has been done by applying the 4-dimensional divergence on the 4-dimensional Energy-Momentum Tensor in the 4-dimensional complex Minkowski space (equation (41). This results in the complex 4-dimensioal force-density vector in the Minkowski space [N/ m3.] The requirement for the 4-force-density vector [N/ m3 ] to be zero results in equation (43) as a set of 4 equations. The 3 spatial equations are equal to equation (19) and represent the electromagnetic field equations. The equation in the 4th dimension (Space-Time domain) in the Minkowski space (equation 43) represents the conservation of electromagnetic energy.
Well-known in wave guides, there exists for harmonic functions a phase shift of 90 degrees between the electric field and the magnetic field for confined (standing) electromagnetic waves. The phase shift of 90 degrees between the electric field and the magnetic field has been expressed in equation (33) and (34) by the complex (imaginary) number “i” . Substituting the complex notation for a confined (standing) electromagnetic wave in the law for conservation of energy (32) result in the relativistic Diac Equation. [3]
The third paragraph describes the concept “Gravitational Redshift”, the impact of a gravitational field on the intensity and the frequency of light (EMR). Because the light from star constellations can be considered to be plane waves, this has been done by substituting (65) and (66) (the electromagnetic presentation for a plane EMR wave) in equation (56). To calculate the results for the experiments to measure the Gravitational
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Redshift due to the gravitational field of the earth, the proportionality between energy and frequency (Planck) has been demonstrated in equation (63). Substituting the gravitational acceleration of the earth for “g” in equation (56) results in values for the Gravitational Redshift for the experiments in [2] which are equal to 15 digits beyond the decimal point compared to the calculations in General Relativity, Equations (83) and (84). Validation of these calculations requires a sensitive and accurate observatory like the JWST or the SKA with observation accuracies better than10-16.
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1. The linear Divergence-Free “Stress-Energy Tensor” in Electrodynamics
Albert Einstein described with his “Theory of General Relativity” the interaction between Gravity and Light (EMR). His starting point was the Stress-Energy Tenor. [18]. This New Theory starts in a comparable way also with the “Stress-Energy Tensor” (44). The Stress-Energy Tensor (44) describes a Universal 4-Dimensional Equilibrium within the Electromagnetic Field [19]. The Divergence of the “StressEnergy Tensor” results in the Electromagnetic Force Vector within the Electromagnetic Field (45). A “Universal 4-Dimensional Equilibrium” within the Electromagnetic Field requires the force (densities) to be equal zero in the 4dimensional space-time coordinates (Minkowski Space). This results in equation (46). Equilibrium within the 3 spatial coordinates results in Equation (22), which describes the Electromagnetic field in an more general concept than the 4 Maxwell Equations. This more General Equation describing the Electromagnetic field can also be derived from Newton’s third law, second law (inertia) and two classical Electromagnetic Equations. (Coulomb and Lorentz).
Newton introduced the concept of “Stress-Energy Tensor (44)” in physics, when he formulated in his famous third equation Action = - Reaction. (Newton’s third law). In nowadays math the concept of “Stress-Energy Tensor (44)” has been formulated as:
𝒊=𝒏
∑ 𝑭̅𝒊 = 𝟎
𝒊=𝟎
(1)
Because the Inertia Force is a Reaction Force, the Inertia Force appears in the equation with a minus sign (Newton’s second law).
𝑖=𝑛
∑ 𝐹̅𝑖 − 𝒎𝒂̅ = 0
𝑖=0
(2)
Equation (2) is a general presentation of Newton’s famous second law of motion. In a fundamental way, Newton’s second law of motion describes the required electromagnetic equation for the Gravitational-Electromagnetic Interaction in general terms, including Maxwell’s theory of Electrodynamics published in 1865 [8] and Einstein’s theory of General Relativity published in 1911 [9]. To realize this new “Gravitational-Electromagnetic Equation”, the fundamental principle of equilibrium has been divided into 5 separate terms, as described by equation (3). Each one describes a part of the electromagnetic and inertia interaction force densities.
𝒊=𝟓
∑ 𝑩𝒊 = 𝟎
𝒊=𝟏
(3)
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The first term 𝐵1 represents the inertia of the energy (mass-) density, the terms 𝐵2 and 𝐵3 represent the electric force densities, and the terms 𝐵4 and 𝐵5 represent the magnetic force densities, of electromagnetic radiation. Fundamentally, the outcome of (3) recalls the fundamental principle of “Stress-Energy Tensor (44)”.
To apply the concept of universal equilibrium within an electromagnetic field, the electric forces 𝐹𝑖𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐, the magnetic forces 𝐹𝑗𝑀𝑎𝑔𝑛𝑒𝑡𝑖𝑐and the inertia forces will be presented separately in equation (4):
𝑖=𝑛,𝑗=𝑚
∑ 𝐹̅̅𝑖𝐸̅̅𝑙̅𝑒̅𝑐̅𝑡̅𝑟̅𝑖̅𝑐 + 𝐹̅𝑗̅𝑀̅̅𝑎̅̅𝑔̅𝑛̅̅𝑒̅𝑡̅𝑖̅𝑐 − 𝒎𝒂̅ = 0
𝑖=1,𝑗=1
(4)
1.1. The First Term (The Inertia of Light)
Albert Einstein’s “General Relativity” describes the interaction between “Gravity and Light”. It is impossible to build a framework for the interaction between Gravity and Light without describing the Inertia (mass) of Light. Because there can only be interaction between Gravity and Inertia (Mass). Without Inertia (Mass) there will be no interaction with gravity. The inertia (mass) of light is very different than the mass of objects we usually describe. The inertia (mass) of objects is always omnidirectional. It does not matter in what position we put a mass on a scale, the weight (interaction) between gravity and the object will always be the same. Reducing Equation (2) to one single Force F, equation (2) will be written in the well-known presentation:

F = ma

(5)

The right and the left term of Newton’s law of motion in equation (5) has to be divided by the Volume “V” to find an equation for the force density f related to the mass density “ρ”.

F = ma

F  V 

=

m  V 

a

(6)

f = a

8

The Inertia Force FInertia for Electromagnetic Radiation will be derived from Newton’s

second law of motion, using the relationship between the momentum vector p for

radiation expressed by the Poynting vector S :

( ) FINERTIA

=-ma =-m v=t

 mv t

=

-

 

p t

=

- 

V c2

 

S t

(7)

Dividing the right and the left term in equation (7) by the volume V results in the

inertia force density f Inertia :

( ) v  mv
FINERTIA = -m a= - m t = -  t

= -

p t

=

-

 

V c2

 S  t

FINERTIA V

=- ma =-m  v

V

V t

= -

1 V

p t

=−

 

1  c2 

S t

(8)

fINERTIA = -  a

=

-

 

1 c 2

 

S t

N/ m3 

The Poynting vector S represents the total energy transport of the electromagnetic radiation per unit surface per unit time [J / m2 s]. Which can be written as the cross
product of the Electric Field intensity E and the magnetic Field intensity H .

( ) fINERTIA

= - a = -

1  c2

  

S t

= -

1  c2

 

EH t

N/ m3 

( ) fINERTIA = -

1  c2 



EH t

N/ m3 

(9)

1.2 Second and Fourth Term in “New Theory” (Term B-2 and B-4)
An example of the Coulomb Force is the Electric Force FCoulomb acting on an electric charge Q placed in an electric field E. The equation for the Coulomb Force equals:

  FCoulomb = E Q N

(10)

Dividing the right and the left term in equation (10) by the volume V results in the
Electric force density fCoulomb :
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  FCOULOMB = E Q N

FCOULOMB V

=

E

Q V

N/ m3 

(11)

f COULOMB = E ρE N/ m3 

Substituting Gauss’s law in differential form in (11) results in Coulombs Law for
Electromagnetic Radiation for the Electric force density fCoulomb :

f COULOMB = E ρE

( ) f COULOMB = E ρE = E  . D

(12)

( ) f COULOMB = E ( . D) =  E  . E N/ m3 

In Electromagnetic Field Configurations, there is in general no preference for the electric force densities or the magnetic force densities. In general the equations for the electric field densities are universally exchangeable with the magnetic field densities.

For the magnetic field densities, equation (12) can be written as:

( ) ( ) f = E Coulomb - Electric . D =  E . E N/ m3  (Term B-2)

(13)

( ) ( ) f = H Coulomb - Magnetic . B =  H . H N/ m3  (Term B-4)

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1.3 Third and Fifth Term in “New Theory” (Term B-3 and B-5)
An example of the Lorentz Force is the Magnetic Force FLorentz acting on an electric charge Q moving with a velocity v within a magnetic field with magnetic field intensity B (magnetic induction). The equation for the Lorentz Force equals:

  FLORENTZ = Q v × B N

(14)

Dividing the right and the left term in equation (14) by the volume V results in the
Lorentz force density fLorentz

  FLORENTZ = Q v × B N

FLORENTZ V

= -

B× Qv V

N/ m3 

f LORENTZ

= - B ×

Q v V

=

- B ×

j

= - μ H ×j

N/ m3 

(15)

In which q is the electric charge, v the velocity of the electric charge, B the magnetic induction and j the electric current density. Substituting Ampère’s law in differential form in (15) results in Lorentz’s Law for Electromagnetic Radiation for the Electric
force density fLorentz :

( ) f LORENTZ = - μ H × j

( ) ( ) f LORENTZ = - μ H × j = -  H    H N/ m3 

(16)

In Electromagnetic Field Configurations, there is in general no preference for the electric force densities or the magnetic force densities. In general the equations for the electric field densities are universally exchangeable with the magnetic field densities. For the electric field densities, equation (16) can be written as:

( ) f = -  Coulomb - Electric E    E N/ m3  (Term B-3)

(17)

( ) f = -  Coulomb - Magnetic H    H N/ m3  (Term B-5)

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1.4 Fundamental Equation for Electromagnetic Interaction in “New Theory” (Term B-1 + Term B-2 + Term B-3 + Term B-4 + Term B-5)
Newton’s second law of motion applied within any arbitrary electromagnetic field configuration results in the fundamental equation (23) for any arbitrary electromagnetic field configuration (a beam of light):

NEWTON: FTOTAAL = m a represents: fTOTAAL =  a

- a +

fTOTAAL

= 0

- a +

f ELEKTRISCH

+

fMAGNETISCH

= 0

- a

+ F + F + F + F = 0 COULOMB

LORENTZ

COULOMB

LORENTZ

-

1
c2



(E   t

 )

+  0

E (  E ) −  E  (  E ) + 0

  (   ) −    (   )

0

0

=

0

B-1

B-2

B-3

B-4

B-5

(18)

Term B-4 is the magnetic equivalent of the (electric) Coulomb’s law B-2 and Term B-3 is the electric equivalent of the (magnetic) Lorentz’s law B-5. The Universal Equation for the Electromagnetic Interaction for Propagating Electromagnetic Waves (Laser Beam) and Confined Electromagnetic Waves (GEONs) has been presented in (19) and expresses the perfect equilibrium between the inertia forces (B-1), the electric forces (B-2 and B-3) and the magnetic forces (B4 and B-5) in any arbitrary electromagnetic field configuration.

-

1
c2



(E   t

) + 0

E ( 

E)

−

 E 0

( 

E)+

 0

(   )

−

  0

( 

 )

= 0

(19)

B-1

B-2

B-3

B-4

B-5

1.5 The Integration of Maxwell’s Theory of Electrodynamics in “New Theory”:

The “Fundamental Equation for Electromagnetic Interaction” (19) for any arbitrary electromagnetic field configuration can be written in the form:

-

1
c2



(E  t

) + 0

E (  E )

−  E  (  E ) + 0

  (   ) −    (   )

0

0

=

0

( ) -00







(H t

)

+

H





(E  t

)

+  E ( E ) −  E  (  E ) +   (  ) −    (   ) = 0

0

0

0

0

( ) -

0







( B t

)

+

 0

H





(D  t

)

+ E ( D ) −  E  (  E ) +  (  ) −    (   ) = 0

0

0

M-3

M-4

M-1

M-3

M-2

M-4

(20)

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The Maxwell Equations are presented in (21):

  D =  (M-1)

 B = 0

(M-2)

  E = −   (M-3) t

 

=

 D (M-4)

 t

(21)

In vacuum in the absence of any electric or magnetic charge density, it follows from (21) that all the solutions for the Maxwell’s Equations (21) are also solutions for the “Fundamental Equation for Electromagnetic Interaction” (20) for the Electromagnetic Field.

Fundamental Equation for Electromagnetic Interaction

( ) -



0





 (B) t

+

 0

H





(D  t

)

+ E ( D ) −  E  (  E) +  (  ) −    (   ) = 0

0

0

M-3

M-4

M-1

M-3

M-2

M-4

4 Maxwell's Equations   D =  (M-1)
  B = 0 (M-2)

  E = −   (M-3)  t

 

=

 D (M-4)

t

(22)

Comparing the 4 Maxwell Equations (21) with the “Fundamental Equation for Electromagnetic Interaction” (20) results in the conclusion that the 4 Maxwell equations show only the 4 parts of the “Fundamental Equation for Electromagnetic Interaction” in 4 separate terms and the 4 Maxwell equations are missing the fundamental term for inertia. For that reason it is impossible to calculate the interaction between light and gravity with the 4 Maxwell equations. To find the interaction terms between light and gravity the inertia term B-1 in (19) is necessary.

13

2. The linear Divergence-Free “Stress-Energy Tensor” in “Relativistic Quantum Physics”.
Albert Einstein described with his “Theory of General Relativity” the interaction between Gravity and Light (EMR). His starting point was the Stress-Energy Tenor. [18]. This New Theory starts in a comparable way also with the “Stress-Energy Tensor” (44). The Stress-Energy Tensor (44) describes a Universal 4-Dimensional Equilibrium within the Electromagnetic Field. [19] The Divergence of the “StressEnergy Tensor” results in the Electromagnetic Force Vector within the Electromagnetic Field (45). A “Universal 4-Dimensional Equilibrium” within the Electromagnetic Field requires the force (densities) to be equal zero in the 4dimensional space-time coordinates (Minkowski Space). This results in equation (46). Equilibrium within the 3 spatial coordinates results in Equation (22), which describes the Electromagnetic field in an more general concept than the 4 Maxwell Equations. Stress-Energy Tensor (44) in the 4th dimension (Energy-Time Domain) results in the Quantum Mechanical Relativistic Dirac Equation (47 and 49).
In physics the quantization of light has always been related to the quantization of Einstein’s concept of photons. In “The New Theory” the quantization of light has been related to a much broader concept also including Wheeler’s concept of GEONs (Ref. [1]).
In general when waves have been confined, like a sound wave on a guitar string, the phenomenon of “Standing Waves” (Stationary Waves) appears with the well known aspects of Nodes and Antinodes in all possible Modes.
The Equation in The New Theory for Confined Electromagnetic Waves is an equation for “Standing Waves” (Stationary Waves) which appear with the well known aspects of Nodes and Antinodes in all possible Modes, describing the Quantization of Electromagnetic Radiation in GEONs.
2.1 The “Quantization of Light” (Electromagnetic Radiation)
In general quantization has been determined by the match of a wavelength on a 3dimensional geometry. Because only an integer number (Natural Number) of wavelengths matches the 3-dimensional geometry. The quantization of Electromagnetic Radiation is only possible when the electromagnetic wave has been confined due to its own gravitational field and the wavelength has to match a 3dimensional geometry.
14

This is possible when light “confines” itself in a 3-dimensional stable geometry due to its own gravitational field. According John Archibald Wheeler’s “GEON’s [Ref. 1] (Gravitational Electromagnetic Entities) and Einstein’s relationship (W = m c2 ) every “confined” energy results in inertia. Inertia results in mechanical mass and mechanical mass results in a gravitational field.
To realize “quantization of light”, stable configurations have to be found of confined electromagnetic radiation in which the wavelength of the confined electromagnetic energy matches in Natural Numbers the 3-dimensional geometry of the confinement.
2.2 Realization of a stable Electromagnetic Confinement in a 3-Dimensional Configuration matching the wavelength of the confinement in Quantum Numbers.
Newton’s third law has been described for the Electromagnetic Field in 3 spatial dimensions, resulting in the “Fundamental Equation for Electromagnetic Gravitational Interaction” in The New Theory:
3-Dimensional Space Domain

B-1

B-2

B - 3

 

x3

 



-

1 c2



(E   t

 )

+

0 E

(  E ) − 0 E  (  E )

+

(23)

 

x2

 



B-4

B-5

B-6

( ( ) ( ))  

 

x1

 

+ 0

 (   ) − 0

  (   )

1 +
2

ε2

E

E + μ2 H

H

g= 



The boundary conditions for the physical experiment determines the boundary conditions for the theory to be tested. When we put a weight on a scale (thought experiment figure 11), the scale will be out of balance. The same effect can be reached by putting an electric charge under one part of the scale or a magnet. The outcome of the experiment gives no information about the origin of the disturbing force (gravitational, electric or magnetic). In equation (24) the mutual electric-, magneticor gravitational interaction force densities are exchangeable. The GravitationalElectromagnetic Confinement for the elementary structure beyond the confinement of light (electromagnetic radiation) has been presented in equation (24).

15

4-Dimensional Space Domain

 f4 

f

4

=

 

f3

 

=

 

f2

 

 f1 

T = b Tab = f a = 0

3-Dimensional Space Domain

3

b

T ab
1

= -

1 c2



(E   t

 )

+

0 E

(  E )

− 0 E  (  E ) +

(24)

+ 0  (   ) − 0   (   ) + 0 g (  g ) − 0 g  (  g ) = 

in which:

0 (  E ) = E Electric Charge Density [C/ m3  0 (   ) = M Magnetic Flux Density [Vs/ m3] or [Wb/ m3 ] 0 (  g ) = M Mass Density (Electromagnetic) [kg/ m3]

In which g represents the gravitational acceleration (as a function of space and time)
expressed in [m s-2] acting on the electromagnetic mass density of the confined electromagnetic radiation and γ0 = Gravitational Permeability of Vacuum expressed in [kg s2 m-3]. [28]

A solution for equation (24), describing a Gravitational Electromagnetic Confinement
(GEON) within a radial gravitational field with acceleration g (in radial direction),
has been represented in (25).

 G1 

 


Er E

  =

 


f

(r )

0
Sin (ω

t )

 


 E  -f (r) Cos(ω t)

 


Hr H

  =

 H 

0

ε μ

 

f

(

r )

 f (r)

Cos

(

ω

t

)

 

Sin (ω t) 

 

4



r2

 

g= 0 

 

0

 

(25)



( ) ( ) w

em

=

 

μ0 2

m . m

+ ε0 2

e

. e

 

=

ε0 f (r)2

In which the radial function f(r) equals:

16

− G1 0 0 + 8  logr

f

r

=

K

−
e

r 8 

(26)

The solution has been calculated according Newton’s Shell Theorem.

The fundamental boundary condition for the confinement of Electromagnetic radiation (GEONs) is that the energy flow (Poynting vector) S = E  H equals zero at the surcface of the confinement. This is possible at a 90 degrees Phase Shift between the Electric Field and the Magnetic Field.
Introducing the Quantum Vector Function  ,

=

 2

 H 

+

i

E 

c

 

(27)

Substituting (27) in (25) results in the quantum presentation  for the GEON:

(r,,) =

f(r)

 


r 

 


=

  

0

 2

 

1

 

f(r)

ei



t

 i 

(28)

(r, , )

=

K

e−

G1 0 0 8 π r

0

 

1

 

ei



t

 i 

2.3 GEONs with Discrete Spherical Energy Levels
A solution for equation (24) describing GEONs dependent of time and radius, presenting discrete spherical energy levels, within a radial gravitational field with
acceleration g (in radial direction), has been represented in (25).

17

 G1 

 


Er E

  = 

f

(r )

0
Sin (k r)

Sin (ω

t )

 


 E  -f (r) Cos(k r) Cos(ω t)

 


Hr H

  =

 H 



0



ε μ

 

-

f

(r )

Sin

(k

r )

Cos (ω

t

)

 

- f (r) Cos(k r) Sin (ω t)

 

4



r2

 

g= 0 

 

0

 

(29)



( ) ( ) w

em

=

 

μ0 2

m . m

+ ε0 2

e

. e

 

=

f (r)2

 

(Sin

(

k

r

)

Sin (ω t))2

+

(Cos(k r)

Cos(ω t))2

+

ε (Sin (k r)
μ

Cos (ω t))2

+

(Cos(k r)

Sin

(

ω

t

)

)2

 



In which the radial function f(r) equals:

− G1 0 0 + 8  logr

(30)

f

r

=

K

−
e

r 8 

Equation (29) presents a Standing (Confined) Electromagnetic Fields Configuration with a phase shift of 90 degrees between the electric field and the magnetic field with the corresponding Nodes and AntiNodes. [13]

Fig. 1 Nodal and Antinodal planes for Standing (Confined) Electromagnetic waves with a 90 degrees phase shift between the Electric field and the Magnetic field.
18

An essential requirement for the confinement of Electromagetic Energy is that the Poymtingvector equals zero at the surface of the confinement. For the confinement within a sphere, a standing electromagnetic wave pattern has been required which exists of concentric spheres, at every sphere an antinodal plane for E (or B) with a radius ditance between each sphere of half the wavelength of the confinement (k = n π λ and “n” a natural number and λ the wavelength).
19

2.2.1 Time and Radius dependent GEONs with discrete Energy Levels. Confinements of Electromagnetic Radiation within spherical Regions. Every concentric sphere represents an anti-nodal surface for the Electric Field or the Magnetic Field. The Poynting Vector at this spherical surface equals zero at any time and at any location at this sphere. The Electromagnetic Energy remains always within this sphere and the next concentric sphere, where all the concentric spheres have a difference in radius of one half wavelength of the electromagnetic radition within the confinement and a different discrete energy level. Every concentric sphere represents an anti-nodal suface of the electric field or the magnetic field.
Fig. 2 Nodal and Antinodal Spheres for Standing (Confined) Spherical Electromagnetic waves with a 90 degrees phase shift between the Electric field and the Magnetic field. Equation (29)
20

Fig. 3 Nodal- and Antinodal Spheres (k = 3) for Standing (Confined) Spherical Electromagnetic waves with a 90 degrees phase shift between the Electric field and the Magnetic field. Equation (29)
21

Equation (31) describes a Time and Radius dependent GEON.

E =

− G1 0 0
K e 8 r



0



 

Sin[k r] Sin[ω t]

 

 - Cos[k r] Cos[ω t]

(31)

−G1 0 0
H = K e 8 r



0



0 0

 

-SCino[sk[kr]r]CSoisn[[ωωt]t]

Equation (31) represents by the function Sin[k r] (k = 1,2,3,4….) the confinement of electromagnetic radiation between two concentric spheres. [14]

22

2.2.2 Time and Polar Angle dependent GEONs
Fig. 4 Nodal- and Antinodal Polar Angle Regions (m = 3) for Standing (Confined) Spherical Electromagnetic waves with a 90 degrees phase shift between the Electric field and the Magnetic field. Equation (29)
23

Equation (32) describes a Time and “Polar Angle” dependent GEON

E =

− G1 0 0
K e 8 r



0



 

Sin[m



]

Sin[ω

t]

 

Sin[m  ] Cos[ω t]

(32)

−G1 0 0
H = K e 8 r



0



0 0

 

Sin[m



]

Cos[ω

t]

 

 - Sin[m  ] Sin[ω t]

Equation (32) represents by the function Sin[m θ] (m = 1,2,3,4….. ) the confinement of electromagnetic radiation between two Polar Angular Regions [15].

Fig. 5 Nodal- and Antinodal Polar Angle Regions (m = 3) for Standing (Confined) Electromagnetic waves with a 90 degrees phase shift between the Electric field and the Magnetic field. Equation (29)
24

2.2.3 Time and Azimuthal Angular dependent GEONs
Fig. 6 Nodal- and Antinodal Azimuthal Angular Regions (n = 3) for Standing (Confined) Electromagnetic waves with a 90 degrees phase shift between the Electric field and the Magnetic field. Equation (29)
25

Equation (3) describes a Time and “Polar Angle” dependent GEON

E =

− G1 0 0
K e 8 r



0



 

Cos[n

]

Sin[ω

t]

 

 Cos[n ] Cos[ω t]

(33)

−G1 0 0
H = K e 8 r



0



0 0

 

Cos[n

]

Cos[ω

t]

 

 - Cos[n ] Sin[ω t]

Equation (33) represents by the function Sin[n ] (n = 1,2,3,4….. ) the confinement of electromagnetic radiation between two Azimuthal Angular Regions [16].

26

2.2.3 Time, Polar Angular and Azimuthal Angular dependent GEONs
Fig. 7 Nodal- and Antinodal Polar Angular and Azimuthal Angular Regions (n = 4 and m = 4) for Standing (Confined) Electromagnetic waves with a 90 degrees phase shift between the Electric field and the Magnetic field. Equation (29)
27

Equation (34) describes a Time “Azimuthal Angle” and “Polar Angle” dependent GEON

E =

− G1 0 0
K e 8 r



0



 

Cos[n



]

Sin[m



]

Sin[ω

t]

 

 Cos[n ] Sin[m  ] Cos[ω t]

(34)

−G1 0 0
H = K e 8 r



0



0 0

 

-

Cos[n Cos[n

] ]

Sin[m Sin[m

] ]

SCions[[ωωtt]]

Equation (34) represents by the function Cos[n ] (n = 1,2,3,4….. ) and Sin[m  ] (m = 1,2,3,4….. )the confinement of electromagnetic radiation between two Azimuthal Angular Regions [17].

28

2.2.4 Spherical Confinement of Light beween two Concentric Spheres within GEONs
Fig. 8 Nodal- and Antinodal Regions for Standing (Confined) Electromagnetic waves with a 90 degrees phase shift between the Electric field and the Magnetic field. Equation (35) Equation (35) represents the reflection of the Confined Electromagnetic Energy within the GEON between two concentric spheres while the speed of light, depending on the variable “r”, changes in magnitude and direction.
29

 −G1 0 0 E = K e 8r f t -


0

0

Cos[2

k

r]  

2 k





0



 

Sin[k

r]

Sin[ω

t]

 

 − Cos[k r] Cos[ω t]

(35)

 −G1 0 0 H = K e 8r f t -


0

0

Cos[2 k r] 

2 k





0



0 0

 

−

Sin[k

r]

Cos[ω

t]

 

 − Cos[k r] Sin[ω t]

Fig. 9 Nodal- and Antinodal Regions for Standing (Confined) Electromagnetic within two concentric spheres. Equation (35)

30

2.3 Universal Equilibrium (Stress-Energy Tensor) in the “Concept of Quantum Mechanical Probability” in “The New Theory”.
In the 4-dimensional notation of (44), the fourth term of the “Energy-Stress Tensor” [19] expresses the law of Conservation of Energy”. For an Electromagnetic Field the law for conservation of Energy has been expressed as:

.S

+

w  t

= 0

(36 )

From the equation for the “Conservation of Electromagnetic Energy” (23) the “Fundamental Equation for Confined Electromagnetic Interaction” in “The New Theory” will be derived, which equals the Relativistic Quantum Mechanical “Dirac” equation.
The “Fundamental Equation for Confined Electromagnetic Interaction” in “The New Theory” can be considered to be the relativistic version of the Quantum Mechanical Schrödinger wave equation, which equals the Quantum Mechanical Dirac Equation.

31

2.4 Confined Electromagnetic Energy within a 4-dimensional Equilibrium

The physical concept of quantum mechanical probability waves has been created during the famous 1927 5th Solvay Conference. During that period there were several circumstances which came just together and made it possible to create a unique idea of “Material Waves” (Solutions of Schödinger’s wave equation) being complex (partly real and partly imaginary) and describing the probability of the appearance of a physical object (elementary particle) generally indicated as “Quantum Mechanical Probability Waves”.
The idea of complex (probability) waves is directly related to the concept of confined (standing) waves. Characteristic for any standing acoustical wave is the fact that the Velocity and the Pressure (Electric Field and Magnetic Field in QLT) are always shifted over 90 degrees. The same principle does exist for the standing (confined) electromagnetic waves,
For that reason every confined (standing) Electromagnetic wave can be described by
a complex sum vector  of the Electric Field Vector E and the Magnetic Field Vector
B (E has 90 degrees phase shift compared to B ).

The vector functions  and the complex conjugated vector function  * will be written

as:

=

1 2 

 B + i 

E 

c

 

(37)

B equals the magnetic induction, E the electric field intensity (E has + 90 degrees phase shift compared to B ) and c the speed of light.
The complex conjugated vector function * equals:

* =

1 2 

 B 

- i

E

c

 

(38)

The dot product equals the electromagnetic energy density w:

32

.

* =

1 2 

 B 

+

i

E c

  

.

  

B

-

i

E c

  

=

1  H2 + 2

1 2

 E2 =

w

(39)

Using Einstein’s equation W = m c2 , the dot product equals the electromagnetic mass density w:



. * 1 = c2

 2

 B + i 

E c

  

 . 

B

-

i

E c

  

=

1 2



 2H2 + 1  2E2 = 2

[kg/m3 ]

(40)

The cross product is proportional to the Poynting vector (Ref. 3, page 202, equation 15).

  * =

1 2 

 B + i 

E

c

 



 B - i 

E c

  

=

i



E  H= i

S

(41)

This article presents a new “Gravitational-Electromagnetic Equation” describing Electromagnetic Field Configurations which are simultaneously the Mathematical Solutions for the Quantum Mechanical “Schrodinger Wave Equation” and more exactly the Mathematical Solutions for the “Relativistic Quantum Mechanical Dirac Equation”. The Gravitational-Electromagnetic Confinements, which are Mathematical Solutions for the “Gravitational-Electromagnetic Equation” (47) carry Mass, Electric Charge and Magnetic Spin at discrete values.

The formal mathematical way to describe the force density results from the 4dimensional divergence of the 4-dimensional energy momentum tensor, resulting in a 4-dimensional Force vector. Dividing the 4-dimensional Force vector by the Volume results in the 4-dimensional force density vector.
The 4-dimensional Electromagnetic Vector Potential has been defined by:

φ 4

=

φ4 

    

φ3 φ2 φ1

    

⎯C⎯oorC⎯dianratet⎯esiSany⎯stem⎯→

 φt 

    

φz φy φx

    

⎯⎯⎯→ Potential 4-Vector

 


i

V c

 




A z



=

A4

A   y



A x



(42)

33

In which the term φa represents the 4-dimensional electromagnetic vector potential in
the “a” direction while the indice “a” varies from 1 to 4. In a cartesian coordinate system the indices are chosen varying from the x,y,z (Magnetic Potential A) and t direction (Electric Potential V). In which the indice “t” represents the time direction which has been considered to be the 4th dimension. The 4-dimensional Electromagnetic “Maxwell Tensor” has been defined by:

Fab = b a -  a b

(43)

Where the indices “a” and “b” vary from 1 to 4.
The 4-dimensional Electromagnetic “Energy Momentum Tensor” has been defined by:

T

ab =

1  0

 

F ac

F cb

+

1 4



ab

F cd

F cd

 

(44)

The 4-dimensional divergence of the 4-dimensional Energy Momentum Tensor equals the 4-dimensional Force Density 4-vector f a :

f a = b Tab

(45)

Substituting the electromagnetic values for the electric field intensity “E” and the magnetic field intensity “H” in (71) results in the 4-dimensional representation of Newton’s third law (1):

34

Energy-Time Domain

B-7

(f4)

( ) ( ) ( )  . ( E   ) + 1  0 2

E . E + 0  t

H. H

= 0

3-Dimensional Space Domain

B-1

B-2

B-3

 

f3

 

-

1 c2



(E   t

 )

+ 0 E B-4

(  E ) − 0 E  (  E ) B-5

+

 

f 2

 

 

f1

 

+ 0  (   ) − 0   (   ) = 

(46)

In which f1 , f2 , f 3, represent the force densities in the 3 spatial dimensions and f4 represent the force density (energy flow) in the time dimension (4th dimension).
Equation (42) can be written as:

Energy-Time Domain

Conservation of Energy

B-7

(f4)

.S +

w  t

= 0

(47.1)

3-Dimensional Space Domain

(47)

B-1

B-2

B-3

 

f3

 

-

1 c2



(E   t

 )

+

0

E

(  E ) − 0 E  (  E )

+

B-4

B-5

 

f 2

 

 

f1

 

+ 0  (   ) − 0   (   ) = 

(47.2)

The 4th term in equation (45.1) can be written in the terms of the Poynting vector “S” and the energy density “w” representing the electromagnetic law for the conservation of energy (Newton’s second law of motion).

35

2.5 The 4-dimensional “Equation for Gravitational-Electromagnetic Interaction” equals the 4-Dimensional Relativistic Quantum Mechanical Dirac Equation

Substituting (40) and (41) in Equation (47.1) results in The 4-Dimensional Equilibrium Equation (48):

(x4 )

- i  . (   *) 0 0

=

-

 . *
 t

(48.1) (48)

 x3 

-

1
c2



(E   t

 )

+ E 0

(  E ) −  E  (  E ) 0

+

 

x

2

 

 x1 

+   (   ) −    (   ) = 

0

0

(48.2)

The fourth term (x4) equals the relativistic Dirac equation (49.2) which equals equation (102) page 213 in Ref.3.
Equation (45.2) represents the relativistic quantum mechanical Dirac Equation where
ψ represents the quantum mechanical probability wave function. The mathematical
evidence for the equivalent for (45.1) has been published in 1995 in the article: “A Continuous Model of Matter based on AEONs”. Equation (1) page 201 to Equation (102) page 213. (Ref. [3]).
The Electromagnetic Law for the conservation of Energy (47.1 ) and the Relativistic Dirac Equation (47.2) are identical but written in a different form.

36

The law of conservation of Electromagnetic Energy can be written in an electromagnetic form (49.1) or in an identical way in a quantum mechanical form (49.2):

Energy-Time Domain

Conservation of Energy

B-7

(f4)

 . (   *)

=

-

i c

 . *
t

(49.1)

(49)

(x4)

i m c  h



+

.



 



= -1 c

 
 t

(49.2)

The disadvantage in the Quantum Mechanical Relativistic Dirac Equation (49.2) is that the Dirac Equation is a 1-dimensional equation which is not able to describe the 4-dimensional real physical world. While Equation (46) represents a 4-dimensional Electromagnetic Vector Equation.
With the equations (40) and (41) follows from (48) the 4-Dimensional Vector-Dirac equation (50). This equation is a 4-dimensional vector equation and is coherent with the 4-dimensional physical reality.

(x4 )

 . (   *)

+

i c

 . *
t

= 0

(50)

( ) ( )  x3 

 

x

2

 

 x1 

i  (  *) c t −

  (   *) +  *  (  )

+

 (   * ) +  * (   )

= 

In which the Quantum Mechanical Complex Probability Vector Function  and the complex conjugated vector function  * equals:

 = B+ i E =H + i E

(51)

c

c

* = B - i E =  H - i E

c

c

37

The 4-Dimensional Dirac Equation (46) and (48) represents the “4-dimensional Equilibrium” in the 4-Dimensional Minkowski space en has been represented by 4 separate equations. The first one represents the well-known relativistic quantum mechanical Dirac Equation in the Time-Energy domain x4 . This represents the Energy Equilibrium in the time-domain (Conservation of Energy). The 3 quantum mechanical equations in the space-momentum domain represents the “Electromagnetic Equilibrium” for the force densities in the Space Domains (x1 ,x2 ,x3)

(x4 )

 . (   *)

+

i   . *
c t

=

0

 x3 

 

x

2

 

 x1 

( ) ( ) i  (  *)
c t

−

  (   *) +  *  (  )

+

 (   * ) +  * (   ) = 

Newton's

second law

(Inertia of Light)

1 c 2

 . * =  [kg/m3]

(Specific mass of Light)

Lorentz

Coulomb

(52)

4-dimensional Equilibrium in Space and Time

Equation (50) represents the Newtonian Equilibrium in Equation (1) in a 4dimensional Space-Time Continuum.
To transform the electromagnetic vector wave function  into a scalar (spinor or one-
dimensional matrix representation), the Pauli spin matrices  and the following
matrices (Ref. 3 page 213, equation 99) are introduced:

=

0 



0

 

and

=

ab

 

0

0

− ab

 

(53)

The Equations (52) and (53) can be written as the 4-Dimensional Relativistic Quantum Mechanical Dirac Equation:

38

(x4)

i m c  h



+



.

 



= -1 c

  t

(54.1)

x3 

-

1
c2



(E   t

 )

+

E 0

(  E ) −  E  (  E ) 0

+

(54)

 

x 2

 

(54.2)

 x1 

+   (   ) −   (   ) = 

0

0

39

3 The linear Divergence-Free “Stress-Energy Tensor in “General Relativity”.
To define the “Fundamental Equation for Electromagnetic Gravitational Interaction”, an extra term (B-6) has been introduced in equation (58). The term B-6 represents the force density of the gravitational field acting on the electromagnetic mass density.

FGRAVITY = m g [N]

Dividing both terms by the Volume V:

FGRAVITY = m g [N/ m3 ]

(55)

V

V

Results in the force density:

fGRAVITY =  g [N/ m3 ]

The specific mass “ρ” of a beam of light follows from Einstein’s equation:

W = m c2

Divinding both terms by the Volume V results in:

W = m c2

(56)

VV

which represents the energy density "w" and the specific

mass " " of the electromagnetic radiation:

w = ρ c2

which results for an expression of the specific mass :

 = 1 w=  w c2

The energy density “w” follows from the electric and the magnetic field intensities:

w = 1 ε E2 + 1 μ H2

2

2

( ) ( ) ( ) ( ) w = 1 ε E2 + μ H2 = 1 ε E . E + μ H . H

(57)

2

2

40

Substituting equation (57) in equation (56) results in the gravitational force density fGRAVITY acting on an arbitrary electromagnetic field configuration (a beam of light) with mass density ρ.

fGRAVITY =  g

( ( ) ( )) fGRAVITY

= g=  wg

=

1 2

ε2

E . E

+ μ2

H . H

g

(58)

Substituting equation (56) in equation (19) results in the fundamental equation describing the Electromagnetic-Gravitational interaction for any arbitrary electromagnetic field configuration (a beam of light):

NEWTON: FTOTAAL = m a [N]

NEWTON: Expressed in force densities: f TOTAAL =  a [N/ m3 ]

- a +

f TOTAAL

= 0

- a +

f ELEKTRISCH

+

f MAGNETISCH

+

f GRAVITY

= 0

f INERTIA

+ f + f + f + f + COULOMB

LORENTZ

COULOMB

LORENTZ

f GRAVITY

= 0

( ( ) ( )) -

1
c2

 (E   t

 )

+  0

E ( E

)− 0

E  ( 

E

) +  0



(  ) −  0

  (   ) +

1 2

ε2

E . E

+ μ2

H . H

g =0

B-1

B-2

B-3

B-4

B-5

B-6

(59)

Term B-1 represents the inertia term of the electromagnetic radiation. Term B-4 is the magnetic representation of the (electric) Coulomb’s Force B-2 and Term B-3 is the electric representation of the (magnetic) Lorentz Force B-5. Term B-6 represents the Electromagnetic-Gravitational interaction of a gravitational field with field
acceleration g acting on an arbitrary electromagnetic field configuration ( a beam of
light) with specific mass ρ.

The universal equation for the electromagnetic field (free electromagnetic waves and confined electromagnetic fields) within a gravitational field with gravity field intensity
g has been presented in (60) and expresses the perfect equilibrium between the
inertia forces (B-1), the electric forces (B-2 and B-3), the magnetic forces (B-4 and B5) and the gravitational force (B-6) in any arbitrary electromagnetic field configuration.

41

-

1
c2

 (E   t

) +  E ( E ) −  E  ( 

0

0

E)+  0

(  )

−

B-1

B-2

B-3

B-4

(60)

( ( ) ( )) −   ( ) + 1 ε2 E . E + μ2 H . H g = 0 0 2

B-5

B-6

42

3.3 Planck’s Constant in “Confined Electromagnetic Radiation”
The fundamental relationship in the interaction between gravity and light is the proportionality between energy and frequency. This effect becomes measurable in “Gravitational Red/Blue-Shift.
To describe the “proportionality between Energy and Frequency (Planck’s Constant) for confined Electromagnetic Waves, an example has been chosen of an electromagnetic wave (light) confined between two 100 % reflecting mirrors.

Figure 10 Confined Electromagnetic Radiation between two 100 % reflecting mirrors
Figure 8 represents confined electromagnetic radiation with amplitude E0 and
frequency ω0 . A general property of confined electromagnetic waves is the
proportionality between energy and frequency. As an example the confinement of light between two perfect reflecting mirrors with surface A has been chosen. When
both mirrors are being moved towards each other over a distance  x , the work  W
has been delivered:

 W = p (A  x)

(61)

In which “A” equals the surface of the mirrors and “p” equals the radiation pressure:

p = w = 1  E2 + 1  H2 [N / m2 ]

(62)

2

2

43

The Electromagnetic “Work  W ” which has been delivered equals:

W=wA x

(63)

When over a time “  t ”one mirror move towards the other with a velocity “v”, then the distance “  x ” passed over a time interval  t equals:

 x = v  t [m]

(64)

Substituting (62) in (61) results in:

 W=wA x

(65)

W

 W=wAv t=

Avt

VVolume

In which “W” equals the total energy involved in the transformation of Mechanical Energy into the Electromagnetic Energy.

When an observer moves slowly (not at a relativistic speed) towards a beam of light, a Doppler frequency shift appears:

f =





f 0

 



1+ v c
1- v c

 
- 1 =  

f 0

 

-

1

+

1 +

v c

+

1 2

 

v c

2 

+

1 2

 

v c

3 

+

3 8

 

v c

4 

+

 



v c

f0

(66)

The same happens for the moving mirror. A shifted frequency will be reflected. The
total effect of the displacement of the distance  x will be a frequency shift in the
confined electromagnetic radiation which equals:

v Δ f = c f0

v= Δf c f 0

W= W Avt= W A Δf ct

VVolume

VVolume

f 0

 

Δ W Δ f

 

=

  

W f0

  

(c  t) A
= VVolume

W nλA

 

f0

 

VVolume

= EM

(67)

44

In which λ represents the wavelength of the confined radiation and “n” the number
of wavelengths of compressions. The term EM represents an Electromagnetic equivalent for Planck’s constant.
45

3.2 “Gravitational RedShift/ BlueShift in “Light (EMR)” due to “Electromagnetic Gravitational Interaction”

The “Stress-Energy Tensor (44) Equation” for Electromagnetic Waves interacting
with a Gravitational Field with acceleration g has been presented in Equation (64)

-

1 c2



(E   t

 )

+

0 E

(  E ) − 0 E  (  E )

+

0  (   )

(68)

( ( ) ( )) − 0   (   )

1 +
2

ε2

E

E + μ2 H

H

g= 

A fundamental Solution for Equation (64) within a gravitational constant gravitational
field g in the z-direction, equals for the electric field: [6]

( E =

ex 

 

e

y

 

=

 e

-

1 2

g

z

μ 0

ε 0



h

 

ω e - g z μ0

ε0



0

t -

 ez 

 

0



)   z

 

 










(69)

And for the Magnetic Field:



Hx  

H

=

 

H

y

 

=

 

 Hz  



0

( 1
ε0 μ0

e-

1 2

g

z μ0

ε 0

h

 

ω e - g z μ0 ε0

t -

0





)   z

 



(70)

The term EM (Electromagnetic equivalent for Planck’s constant) defines a
proportionality between the energy density and the frequency ω . The solutions (65)
and (66) represent a “Constant Speed of Light” “c”. The “Gravitational Red-Shift” equals:

ω = ω e Gravitational Redshift

- g z μ0 ε0 0

(71)

In which ω0 equals the original frequency of the beam of light propagating in the direction of the constant gravitational field in the z-direction and the exponential term demonstrates the Gravitational Redshift when the beam of light moves away from the Gravity Source (Solar System or Black Hole). Including the “Doppler RedShift the total frequency shift equals:
46

ω = ω e Gravitational Redshift

- g z μ0 ε0 0

v 1 +
c
v 1 -
c

(72)


Gravitational Redshift


Doppler Redshift

Equation (72) represents the “Gravitational Redshift” for a Constant Gravitational Field (with a constant gravitational acceleration “g”). The “Electromagnetic Equilibrium Equation” (60) in “The New Theory” represents possible solutions for all “Time- and Space Depending Gravitational Fields” and Equation (60) is the fundamental equation to be “tested” relative to Einstein’s theory of General Relativity. Measurements of “Gravitational Red-Shift/ Blue-Shift from World Wide Observatories could falsify or confirm “The New Theory”.

For simplicity 3 distinguishments have been made for the different kinds of gravitational fields. The first one is a function with a linear increment to the source of the gravitational field (The center of the of the Milky Way Galaxy to the Edge of the Milky Way Galaxy) in which the gravitational acceleration “g”. and the corresponding solution for equation (60) have been presents in equation (73.1) . The second gravitational field is proportional to 1 / r2 (The gravitational field around the edge of the Milky Way Galaxy) in which “r” equals the distance (Equation 73.2). The third type of Gravitational Field is a constant Gravitational Field over Long Distances far away from sources of gravity like Black Holes (73.3).

3.3 Mathematical Calculations for Gravitational RedShift/ BlueShift.
When a ball is thrown upwards from Earth, its kinetic energy is converted into a potential energy, the higher it goes the kinetic energy reduces according to the ball’s inertia and the planet’s gravitation. Accounting for EMR inertia if a beam of light (e.g., a laser beam) is directed radially away from the Earth’s surface, something comparable happens. The EM Field Intensity becomes exponentially lower (the higher the beam shines upwards) but the speed of light does not change. The speed of light remains constant. For a Laser Beam shining downwards on earth, the opposite happens. The intensity becomes (exponentially) higher and the observed frequency higher (Blue Shift).
Equation (71) represents 3 solutions for 3 different kinds of Gravitational Fields. The Gravitational RedShift (GRS) term in QLT depends on the type of Gravitational Field:

47

 (ωRS = ωGRS Doppler )

Gravitational Field Type 1

g = G1 z

ω RS

=

ω e−

1 2

G 1

z 2

0

0

0

1+ v c

1 -

v c

Gravitational Field Type 2

g =

G1 z2

ω NLGR

= ω e−

G 1

0 z

0

0

1+ v c
1- v c

Gravitational Field Type 3

g = G1

e ω = ω NLGR

0

- g z μ0 ε0

1+ v c
1- v c

(73.1) (73)
(73.2)
(73.3)

The Mathematical Solutions for the Gravitational Redshift term in The New Theory
(GRQLT) differ from the Gravitational Redshift in General Relativity (GRGR) with a
factor  . Within a constant gravitational field over a distance of 36000 [km] (average
Satellite height) and an estimated gravitational acceleration of g = 9.8 [m s-2]. [2]

( )  =

GRQLT

- (GRGR )

=

-

1 2

g 



- 2 10−9

(74)

The Mathematical Solutions (73) for equation (60) have been calculated in [5]. The mathematical solutions in Equation (60) for a gravitational field proportional to zn ,
equals:

48

Gravitational Field Type zn g = G1 zn

ω NLGR

- G1 z1 + n ε0 μ0
= ω0 e 2 (1 + n)

v 1 +
c 1- v
c

(75)

In general the mathematical solutions for Equation (60) can be represented as:

Gravitational Field as a function "h(z)" along the z-axis between "object" and observer

g = G1 g[z]

ω NLGR

e  = ω0

-

1 2

ε 0

μ 0

G1

z g(h)
0

d h

v 1 +
c
v 1 -

(76)

c

frequency original

Gravitational Doppler

being

frequency Redshift

Redshift

observed

49

3.4 Deviations in Gravitational Redshift calculated in General Relativity and The New Theory

The Mathematical Solutions for the Gravitational Redshift term in The New Theory
(GRQLT) differ from the Gravitational Redshift in General Relativity (GRGR) with a
factor  .

In General Relativity the frequency change in Gravitational Redshift is proportional to the change of the Electromagnetic (Potential Energy) U of the radiation

 f

U c2

(77)

An accurate way to test “General Relativity” and “The New Theory” theory is the “Gravitational RedShift” within a well-known gravitational field like the gravitational field of the Earth. Both theories give different outcomes in the values for the “Gravitational RedShift” within the gravitational field of the Earth between an observatory on Earth (Radius = 6 106 [m]) and a Satellite in a Galileo Orbit (Radius = 23 222 106 [m]). The Galileo Space Segment comprises a constellation of a total of 30 Medium Earth Orbit (MEO) satellites, of which 3 are spares, in a so-called Walker 27/3/1 constellation.

“Gravitational RedShift” [2, 5] between an observatory on Earth (Radius = 6 106 [m]) and a Satellite in a Galileo Orbit (Radius = 23222 103 [m]) according Planck’s law
(incorrectly mentioned as General Relativity in the publications):

U  ωGR = c2 = 0.00000000004360137706159641

(78)

“Gravitational RedShift” [2, 5] between an observatory on Earth (Radius = 6 106 [m]) and a Satellite in a Galileo Orbit (Radius = 23222 103 [m]) according “The New
Theory”:

 ωQLT =

 URel c2

=

0.00000000004360134475689392

(79)

In “The New Theory” the result of the calculations has been based on a “Perfect Equilibrium in Space and Time, taking into account the effect of the change in Electromagnetic Mass due to the potential energy level .

50

In [2] a factor  has been defined which presents the measured deviation  between
the predicted Gravitational RedShift by General Relativity and the Measured Gravitational RedShift.

( )  =  ωMEASURED -  ωGR = 2.2  1.6  10−5

(80)

A comparable factor  can be used to determine which theory (GR or QLT) has the
closest approach to the experimental data. Highly accurate measuring experiments are required with an accuracy higher than 16 digits beyond the decimal point (78 and 79).

4.1 Experiments to test “The linear Divergence-Free “Stress-Energy Tensor
Experiments are required to validate “The New Theory” in relation to the fundamental Theories: Electrodynamics, Quantum Physics and General Relativity.
4.1 Experiments to test the mew theory:
A fundamental experiment to test any arbitrary theory related to Electrodynamics is the projection of a slide on a screen. The projection of a slide (or a film) in the zdirection, represents an arbitrary intensity division f(x,y) in the (x,y) plane perpendicular to the z-axis (axis of propagation):

 f[x,y] g[z - c t]

E

=

 

0

 



0



(81)

H =



0



μ ε

 

f[x,y]

g[z 0

-

c

t]

 

The force density in the 3 spatial directions for this Electromagnetic Field Configuration equals:

f

fx 

=

 

f y

 

=

-

1
c2

 (E 
 t

) +  0

E (  E ) −  E  (  E ) + 0

  (   ) −    (   )

0

0

 fz 

(82)

Substituting (79) in (80) and when the propagation speed c equals:

51

c= 1

(83)



The resulting force density equation (82) equals:

f

fx   = fy 

 0

=

-

1
c2

 (E 
 t

) + 0

E (  E ) −  E  (  E ) + 0

 0



( 

)

−

 0



( 

)

=

 

 0

 fz 

 0

(84)

Exactly at the speed of light a perfect equilibrium exists in the x-, y, and z-direction.

This represents the required evidence that any arbitrary slide can be projected on a

screen because the electromagnetic field configuration is always a solution of the

Electromagnetic Field Equation in “The New Theory” for any arbitrary function f(x,y).

The only requirement in (82) is that the light propagates exactly with the speed

of light c =

1 

.

The projection of a slide (or a film) in the z-direction on a screen, represents an arbitrary intensity division f(x,y) in the (x,y) plane perpendicular to the z-axis. An infinite amount of functions f(x,y) is not divergence free. And for that reason not a solution of the Maxwell equations (83-a) and (83-b).

  D =  (a)

 B = 0

(b)

  E = −   (c) t

  =

 D (d)

 t

(85)

There is no slide (intensity division proportional to the arbitrary function f(x,y)2) which cannot be projected. It is a fundamental requirement for any electromagnetic field equation that the arbitrary function f(x,y) is part of the solution set for this Electromagnetic Field Equation.

The Electromagnetic Field Equation (82) fulfils this essential requirement. The Maxwell equations (83-a) and (83-b) do not fulfil that essential requirement and for that reason cannot be used to describe an electromagnetic field configuration. QED

52

4.2. Experiments to test the new theory

The 4-dimensional Equilibrium Equation represents the relativistic quantum mechanical Dirac equation in the 4th dimension (Energy time domain):

(x4)

i m c  h



+

 .



 



= -1 c

  t

(86.1)

x3 

-

1
c2



(E   t

 )

+

E 0

(  E ) −  E  (  E ) 0

+

(86)

 

x 2

 

(86.2)

 x1 

+   (   ) −    (   ) = 

0

0

Every electromagnetic confinement has to be part of the solution set of equation (86). GEONs (Electromagnetic Confinements by Gravitational Electromagnetic Interaction) are part of the solution set and are for that reason part of the solution set for the Dirac Equation. Like elementary particles, GEONs carry mass, charge and spin in discrete values (the wavelength of the confinement has to match the geometry of the confinement in discrete values) and have common properties with elementary particles. Further research is required to investigate the relationship between elementary particles and GEONs.

53

4.3 Experiments to test the new theory with General Relativity/ (Planck’s law)

An accurate way to test “General Relativity” (Planck’s law) with the new theory is the “Gravitational RedShift” within a well-known gravitational field like the gravitational field of the Earth. Both theories give different outcomes in the values for the “Gravitational RedShift” within the gravitational field of the Earth between an observatory on Earth (Radius = 6 106 [m]) and a Satellite in a Galileo Orbit (Radius = 23 222 106 [m]). The Galileo Space Segment comprises a constellation of a total of 30 Medium Earth Orbit (MEO) satellites, of which 3 are spares, in a so-called Walker 27/3/1 constellation.

The “Gravitational RedShift” [2, 5] between an observatory on Earth (Radius = 6 106 [m]) and a Satellite in a Galileo Orbit (Radius = 23222 103 [m]) according “Planck’s
law” (incorrectly mentioned as General Relativity in the publications):

U  ωGR = c2 = 0.00000000004360137706159641

(87)

The “Gravitational RedShift” [2, 5] between an observatory on Earth (Radius = 6 106 [m]) and a Satellite in a Galileo Orbit (Radius = 23222 103 [m]) according “The New
Theory”:

 ωQLT =

 URel c2

=

0.00000000004360134475689392

(88)

In “The New Theory” the result of the calculations has been based on a “Perfect Equilibrium in Space and Time, taking into account the effect of the change in Electromagnetic Mass due to the potential energy level  . URel

In Ref. 2 (Gravitational Redshift Test Using Eccentric Galileo Satellites, Sven
Herrmann, Felix Finke, Martin Lülf, Olga (et. Al.), 20218) a factor  has been defined which presents the measured deviation  between the predicted Gravitational
RedShift by General Relativity and the Measured Gravitational RedShift.

( )  =  ωMEASURED -  ωGR = 2.2  1.6  10−5
(89)
A comparable factor  can be used to determine which theory (GR or QLT) has the
closest approach to the experimental data. Highly accurate measuring experiments are required with an accuracy higher than 17 digits beyond the decimal point (80 and 81).

54

4.4 The Weight of Light (Photons) calculated with Gravitational RedShift
The boundary conditions for the physical experiment determines the boundary conditions for the theory to be tested. In this thought experiment two identical boxes, with at the inside 100 % reflecting mirrors, have been placed on a balance scale. One box has been filled with light (Electromagnetic Radiation) with energy density “w” and the other box is empty and has not been filled with light. With Gravitational Redshift the weight of the Light will be calculated

Figure 11 In a “Thought Expiriment”: Measuring the weight of Confined Light (Electromagnetic Radiation) in one of both boxes with 100 % reflecting mirrors at the inside
The Radiation Pressure of the confined Light (Electromagnetic Radiation) equals:

p = w [N/ m2 ]

[90]

Inside the box with 100 % reflecting mirrors (with dimensions: Width x Length x Height) the upwards oriented Force FU equals:

FU (z + H) = p(z + H)  A = w(z + H)  W  L [N]

[91]

The downwards oriented Force FD equals:

FD (z) = p(z)  A = w(z)  W  L [N] 55

[92]

The energy density of the confined electromagnetic radiation is proportional to the frequency (Equation 67)

e ω GR

= ω0

- g L μ0 ε0

[93]

e w GR

= w0

- g L μ0 ε0

According equation 71.3 the Weight (Gravitational Force: FGRAV) of the Confined Light (Electromagnetic Radiation) with energy density w[z] in the box within a constant gravitational field equals:

FGRAV = FD - FU = ( w(z) - w(z + H)) W L

(94)

( ) FGRAV = w[z] - w[z] e - g H μ0 ε0 W L

( ) FGRAV = w[z] 1 - e - g H μ0 ε0 W L

Applying series development on the exponentional function results in:

( ) FGRAV = w[z] 1 - e - g H μ0 ε0 W L

FGRAV

= w[z]1


-

1 - g H μ0

ε 0

+

1 2

(g

H

μ 0

)ε0 2

-

1 6

(

g

H

μ

0

)ε0 3

+ ..........

 

  

W L

(95)

Which results in a first order approach into Einstein’s well-known equation W = m c2 : [21], [22], [46]

FGRAV

= w[z]

g H μ0

ε 0

-

1 2

(

g

H

μ 0

)ε0 2

+

1 6

(

g

H

μ 0

)ε0 3

+ ..........

W L

(96)

FGRAV



w[z] g μ0 ε0 H W L =

1 c2

Wg=mg

56

4.5 Quantum Transitions in GEONs

GEONs can drop back in a lower energy state under the emission of light of a specific frequency or jump to a higher energy level under the absorption of light of a specific frequency.

According equation (67) the Energy Level of GEONs has been related to the fundamental frequency of the GEON:

Δ f =

v c

f0

Δ f v= c
f 0

W

W Δf

W=

Av t=

A ct

VVolume

VVolume

f 0

 

Δ W Δ f

 

=

  

W f0

  

(c  t) A
= VVolume

W nλA

 

f0

 

VVolume

= EM

(97)

According equation (35) for GEONs:

E =

−G1 0 0
K e 8 r

 


0



Sin[k

r]

Sin[ω

t]

 

 − Cos[k r] Cos[ω t]

(98)

−G1 0 0
H = K e 8r



0



0 0

 

−

Sin[k

r]

Cos[ω

t]

 

 − Cos[k r] Sin[ω t]

When an GEON drops back to a lower energy level, the radius of the sphere will become smaller with a lower frequency.

57

−G1 00
Figure 12 Plot of the function: H = e 8r Sin[k r] at t = π / 2 ω

Figure 13 Plot of the function: t= 3π/2ω

−G1 00
E = e 8r Sin[k r] at t = π / 2 ω and at

58

Figure 14 Plot of the concentric spherical Areas of Anti-Node of the Gravitational-Electromagnetic Confinements (Standing Spherical Waves) of the Electromagnetic Energy.
59

When a GEON drops back to a lower energy level, the discrete amount of energy, confined between 2 concentric spheres (orange area of confinement between two concentric spheres) will be emitted as a single electromagnetic wave package (photon).
Figure 15 2-dimensional Plot of the concentric spherical Areas of Anti-Node of the Gravitational-Electromagnetic Confinements (Standing Spherical Waves) of the Electromagnetic Energy during an Energy Transition of a GEON .
60

4.6 Gravitational Waves
Waves transport energy through space. Water waves transport potential energy over the water surface. Electromagnetic waves transport “Electromagnetic Energy” through space and Gravitational Waves transport “Gravitational Energy” through space.

4-Dimensional Space Domain

 f4 

f

4

=

 

f3

 

=

  

f2 f1

  

T = b Tab = f a = 0

3-Dimensional Space Domain

b

3
T ab
1

= -

1 c2



(E   t

 )

+ 0 E (  E )

− 0 E  (  E ) +

+ 0  (   ) − 0   (   ) +  0 g (  g ) −  0 g  (  g ) = 

in which:


0

( 



E ) = E

Electric Charge Density

[C/ m3 

0 (   ) = M Magnetic Flux Density [Vs/ m3] or [Wb/ m3 ]

 0 (  g ) = M Mass Density (Electromagnetic) [kg/ m3]

Electric Energy Density:

w E

= 1  E2
20

Magnetic Energy Density:

w M

=

1 2

0

H 2

Gravitational Energy Density:

w G

=

1  g2
20

(99)

61

4.7 The Gravitational and Electromagnetic Wave Equations
The transport of energy through space an time is always a transport in a multidimensional space-time continuum. An electric field cannot be transported in space like a magnetic field cannot be transported in space without its relativistic counterpart. Electric fields and Magnetic fields are each other’s counterpart. For this reason a transport of electric- or magnetic energy will always be observed as an electromagnetic wave. In a comparable way a gravitational field cannot be transported in space without its relativistic counterpart. The relativistic counterpart of gravitational fields has not been discovered yet but according this theory this we be matter of time.
4
Electromagnetic Fields are described by the Electromagnetic Potential 4-vector A related to the Liénard–Wiechert potential. Expressed in Cartesian Coordinates:

4
A =

 

i



V

c

 

 Ax 

 

Ay A z

 

(100)

E = -V - B t

B= A
The New Theory (Quantum Light Theory) has been based on a 4-dimensional universal Equilibrium in Fields.
1) Electric Fields form a “4-dimensional universal Equilibrium” with “Lorentz Transformed Magnetic Fields [ 0   (   ) ]” which are Electric Fields
2) Magnetic Fields form a “4-dimensional universal Equilibrium” with “Lorentz Transformed Electric Fields [ 0 E  (  E ) ]” which are Magnetic Fields
3) Gravitational Fields form a “4-dimensional universal Equilibrium” with “Lorentz Transformed Gravitational-L Fields [0 gL  (  gL ) =  ]” which are Gravitational Fields
62

4) Gravitational-L Fields form a “4-dimensional universal Equilibrium” with “Lorentz Transformed Gravitational Fields [0 g  (  g ) =  ]” which are Gravitational-L Fields

4.7.1 Electromagnetic Wave Equation Electromagnetic Waves have been controlled by the Electromagnetic Wave Equation:

Inertia of Electromagnetic Energy

Electric-Electric Interaction

ELTMI

-

1 c2



(E   t

 )

+

0 E (  E ) − 0 E  (  E ) +

Magnetic-Magnetic Interaction

MLTEI

+

0  (   ) − 0   (   ) = 


0

(



E ) = E

Electric Charge Density

[C/ m3 

in which: 0 (   ) = M Magnetic Flux Density [Vs/ m3] or [Wb/ m3]

Electric Energy Density:

w E

= 1  E2
20

Magnetic Energy Density:

w M

=

1  H2
20

(101)

63

4.7.2 Calculations Electromagnetic Waves

A beam of light with an arbitrary intensity division g[x,y] in the x-y plane and propagating with the speed of light “c” in the z-direction equals:

  E = gx,y g t - z 0 0  , 0, 0

  H =

0 0

0, gx,y g t - z

0 0  , 0

(102)

Substituting (102) in equation (103) results in the total sum of the 5 force densities and equals zero in every direction of the chosen coordinates (x,y,z).

  E = gx,y g t - z 0 0  , 0, 0

  H =

0 0

0, gx,y g t - z

0 0  , 0

(103)

f (x,y,z)

=

-

1 c2



(E  t

 )

+ 0

E

(  E ) − 0 E  (  E )

+

0

 (   )

− 0

  (   ) = 0

64

4.7.3 Gravitational Fields in the 4-dimensional Minkowski Space
4
Gravitational Fields are described by the Gravitational Potential 4-vector G related to the Liénard–Wiechert potential. Expressed in Cartesian Coordinates:

4
G=

 

i



VG c

 


 Gx 

 

Gy G z

 

(104)

gL = -  VG

-

g t

g = × G

The New Theory (Quantum Light Theory) has been based on a 4-dimensional universal Equilibrium in Fields.
1) Electric Fields form a “4-dimensional universal Equilibrium” with “Lorentz Transformed Magnetic Fields [ 0   (   ) ]” which are Electric Fields
2) Magnetic Fields form a “4-dimensional universal Equilibrium” with “Lorentz Transformed Electric Fields [ 0 E  (  E ) ]” which are Magnetic Fields
3) Gravitational Fields form a “4-dimensional universal Equilibrium” with “Lorentz Transformed Gravitational-L Fields [0 gL  (  gL ) =  ]” which are Gravitational Fields
4) Gravitational-L Fields form a “4-dimensional universal Equilibrium” with “Lorentz Transformed Gravitational Fields [0 g  (  g ) =  ]” which are Gravitational-L Fields

65

4.7.4 Four Independent Wave Equations

In classical Physics interactions are being described by the interactions between

particles and fields. In this new physics interactions are only being described by the

interactions between fields. Electric Fields only interact with Electric Fields.

Magnetic Fields ( Lorentz Transformed Electric Fields ) only interact with Magnetic

Fields ( Lorentz Transformed Electric Fields ). Gravitational Fields only interact

with Gravitational Fields. Lorentz Transformed Gravitational Fields g only

L
interact with Lorentz Transformed Gravitational Fields g . This effect results in 4

independent Wave Equations.

L

Equation (101) expresses the Perfect Equilibrium between Gravitational-ElectroMagnetic Energy resulting in the propagation of Gravitational-Electro-Magnetic waves.

It is possible to derive from equation (101) four independent wave equations. One wave equation for the separate electric field, one wave equation for the separate magnetic field (Lorentz Transformed Electric Field), one wave equation for the separate gravitational field and one wave equation for the separate Lorentz Transformed gravitational field. The four separate fields are coupled and propagate with the speed of light but remain in a certain way independent.

66

4.7.5 Electric Wave Equation Electric Waves have been controlled by the Electric Wave Equation:

Inertia of GravitationalElectromagnetic Energy

-

1 2 c2



(E   t

 )

+ 0 E (  E )

−

0   (   ) = 

(105)

In which:


0

(



E ) = E

Electric Charge Density

[C/ m3 

− 0   (   ) Electric Field (Lorentz Transformed Magnetic Field)

Electric Energy Density:

wE

=

1 2

 0

E2

In which “S” [ W/ m2] represents the “Electric Energy Flux” per second through a surface of 1 [m2 ] and “E” the Electric Field Intensity.

67

4.7.6 Calculations Electric Waves

A beam of light with an arbitrary intensity division g[x,y] in the x-y plane and propagating with the speed of light “c” in the z-direction equals:

  E = gx,y g t - z 0 0  , 0, 0

  H =

0 0

0, gx,y g t - z

0 0  , 0

(106)

Substituting (108) in equation (109) results in the total sum of the 3 Electric force densities and equals zero in every direction of the chosen coordinates (x,y,z).

  E = gx,y g t - z 0 0  , 0, 0

  H =

0 0

0, gx,y g t - z

0 0  , 0

( ) fELECTRIC x,y,z

=

-

1 2 c2



(E  t

 )

+ 0

E

(  E ) − 0   (   ) = 0

Calculations Mathematica (term1 + term2 + term5)

(107)

68

4.7.7 Magnetic Wave (Lorentz Transformed Electric Wave) Equation

Magnetic Waves (Lorentz Transformed Electric Waves) have been controlled by the Magnetic Wave Equation:

Inertia of Gravitational-

Electromagnetic Energy

-

1 2 c2



(E   t

 )

+ 0  (   ) −

0 E  (  E ) = 

(108)

in which:


0

( 



 ) = M

Magnetic Flux Density [Vs/ m3] or [Wb/ m3

]

− 0 E  (  E )

Magnetic Field (Lorentz Transformed Electric Field)

Magnetic Energy Density: wM

=

1  H2
20

In which “S” [ W/ m2] represents the “Magnetic Energy Flux” per second through a surface of 1 [m2 ] and “H” the Magnetic Field Intensity.

69

4.7.8 Calculations Magnetic Waves

A beam of light with an arbitrary intensity division g[x,y] in the x-y plane and propagating with the speed of light “c” in the z-direction equals:

  E = gx,y g t - z 0 0  , 0, 0

  H =

0 0

0, gx,y g t - z

0 0  , 0

(109)

Substituting (111) in equation (112) results in the total sum of the 3 Magnetic force densities and equals zero in every direction of the chosen coordinates (x,y,z).

  E = gx,y g t - z 0 0  , 0, 0

  H =

0 0

0, gx,y g t - z

0 0  , 0

( ) fMAGNETIC x,y,z =

-

1 2 c2



(E   t

 )

+

0



(   )

−

0 E  (  E ) = 0

(110)

Calculations Mathematica (term1 + term3 + term4)

70

4.7.9 Gravitational Wave Equation

Gravitational Waves have been controlled by the Gravitational Wave Equation:

Inertia of Gravitational-

Energy

- 1  (g  gL ) 2 c2  t

+

 0 g (  g ) −

 0 gL  (  gL ) = 

(111)

in which:

 0 (  g ) = M Mass Density (Gravitational) [kg/ m3]

Gravitational Energy Density:

w G

=

1  g2
20

In which “S” [ W/ m2] represents the “Gravitational Energy Flux” per second through a surface of 1 [m2 ] and “g” the Gravitational Acceleration [m/ s2].

4.7.10 Lorentz Transformed Gravitational Wave Equation

Lorentz Transformed Gravitational Waves have been controlled by the Lorentz Transformed Gravitational Wave Equation:

Inertia of Lorentz Transformed

Gravitational - Energy

- 1  (g  gL ) c2  t

+

L gL

(  gL ) −

 L g  (  g ) = 

(112)

in which:

 L (  gL ) = LM Mass Density (Lorentz Transformed Gravitational) [kg/ m3]

Lorentz Transformed Gravitational Energy Density:

w LG

=

1 2

 L

gL2

In which “S” [ W/ m2] represents the “Lorentz Transformed Gravitational Energy Flux” per second through a surface of 1 [m2 ] and “g” the Gravitational Acceleration [m/ s2].

71

5. Conclusions
Based on the assumption of the zero rest mass of photons, General Relativity describes the interaction between Gravity and Light within a 4-dimensional curvature in Space and Time due to a gravitational field. Light follows a path defined by this curved 4-dimensional Space and Time geometry.
The “New Theory” has been based on an unchanged “Space-Time” geometry and describes a perfect 4-dimensional equilibrium in the “Electric, Magnetic and Gravitational” domain. The New Theory results in values for the observed “Gravitational RedShift” within the accuracy range of all the observations for “Gravitational RedShift” since 1925. This is in contradiction with the “Theory of General Relativity” which predicts the absence of “Gravitational Redshift/ Blueshift” (Gravitational redshift/ blueshift = 0). Only Planck’s law describing the proportional relationship between the “(Potential Gravitational) Energy” and the frequency of light also predicts the “Gravitational Redshift” within the accuracy range of the observations. [2,5]. The experimentally proven physical effect of Gravitational Redshift/ Blueshift is in contradiction with the predictions in General Relativity. The only theory which predicts this physical effect has been presented in this article.
The new theory presented here, describes a bi-directional separation between mass and inertia for light (photons). Inertia only in the direction of propagation of the beam of light (photons) which determines the speed of light. Mass of the beam of light (photons) only in the plane perpendicular to the direction of propagation, which determines the deflection of a beam of light (photons) by a gravitational field in the plane perpendicular to the direction of propagation.
GEONs (Gravitational-Electromagnetic Confinements) are fundamental solutions of the relativistic quantum mechanical Dirac equation. The inertia of light has been represented by term (1) and the non-zero rest-mass has been presented by term B-6 in equation (23).
Within a 4-Dimensional Equilibrium and taking into account the inertia- and the gravitational- force densities within the electromagnetic field configurations, Gravitational Electromagnetic Confinements (GEONs) are a physical reality and are solutions of the Relativistic Quantum Mechanical Dirac Equation and present spherical confinements with discrete separate energy levels. .
72

5.1 Data Availability
All the Data and all the Calculations to provide evidence to “The New Theory” have been published at:
https://quantumlight.science/
References
[1] Wheeler; John Archibald; GEONs, Physical Review Journals Archive, 97, 511, Issue 2, pages 511-526, Published 15 January 1955, Publisher: American Physical Scociety, DOI: 10.1103/PhysRev.97.511: [2] Sven Herrmann, Felix Finke, Martin Lülf, Olga (et. Al.) l; Test of the Gravitational Redshift with Galileo Satellites in an Eccentric Orbit ;Phys. Rev. Lett. 121, 231102 – Published 4 December 2018; Gravitational Redshift Test Using Eccentric Galileo Satellites, DOI: 10.1103/PhysRevLett.121.231102 [3] Vegt, J. W. A Continuous Model of Matter based on AEONs, Physics Essays, Volume 8, Number 2, 1995, DOI: 10.31219/osf.io/ra7ng [4] Mathematical Solutions for the Propagation of Light in Quantum Light Theory, Calculations in Mathematica 13.1: https://community.wolfram.com/groups//m/t/2576692?p_p_auth=mTldHX3v [5] Gravitational RedShift between two Atomic Clocks, Calculations in Mathematica 13.1: https://community.wolfram.com/groups/-/m/t/2622560?p_p_auth=EC8QO0Xz [6] Propagation of Light within a Gravitational Field in Quantum Light Theory, Calculation in Mathematica 13.1: https://community.wolfram.com/groups//m/t/2576537?p_p_auth=iljE3giH [7] Raymond J. Beach; A classical Field Theory of Gravity and Electromagnetism; Journal of Modern Physics; 2014, 5, 928-939 [8] Maxwell; James Clerk; A dynamical theory of the electromagnetic field; 01 January 1865; https://royalsocietypublishing.org/doi/10.1098/rstl.1865.0008 [9] A. Einstein; On the Influence of Gravitation on the Propagation of Light; Annalen der Physik (ser. 4), 35, 898–908, http://myweb.rz.uniaugsburg.de/~eckern/adp/history/einstein-papers/1911_35_898-908.pdf [10] Mahendra Goray, Ramesh Naidu Annavarapu, Rest mass of photon on the surface of matter, Results in Physics 16 (202) 102866, January 2020, DOI: 10.1016/j.rinp.2019.102866 [11] Genova, A., Mazarico, E., Goossens, S. et al. ; Solar system expansion and strong equivalence principle as seen by the NASA MESSENGER mission; Nat Commun 9; 289 (2018). DOI: 10.1038/s41467-017-02558-1
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