generally covariant unified field theory - Atomic Precision

GENERALLY COVARIANT
UNIFIED FIELD THEORY
THE GEOMETRIZATION OF PHYSICS III
Myron W. Evans
March 11, 2006

i
I dedicate this book to
”Objective Natural Philosophy”

Preface
In the third volume of this comprehensive monograph, generally covariant unified
field theory is developed for some key areas and phenomena which do not
have a satisfactory explanation in the standard model. This continues the systematic
coverage of all key areas of physics begun in preceding volumes. Over
the past two years Einstein Cartan Evans (ECE) field theory has been attracting
unprecedented attention from essentially all institutions worldwide which
have an interest in physics, chemistry, mathematics or engineering. This can
be seen objectively using the feedback software for our two sites, www.aias.us
and www.atomicprecision.com. In addition there is major interest from leading
corporations, government departments, military facilities, health organizations
and individuals from approximately ninety countries worldwide. The theory has
therefore been accepted as mainstream physics because it has been evaluated
and studied exhaustively by leading theoretical physicists and mathematicians
of relevance to the subject matter. ECE theory provides for the first time a
generally covariant theory of all physics: of the four fundamental fields thought
to exist in nature and of matter fields. It is the unified field theory sought for
by Einstein from the early twenties of the twentieth century, and is based on
standard Cartan geometry, taught in all good universities. ECE theory also
produces for the first time a generally covariant wave mechanics (see chapter
ten) and so unifies generally relativity and quantum mechanics in an objective
manner. Unification of the four radiated fields and of matter fields is achieved
using standard Cartan geometry, the fundamental field is the tetrad (vierbein).
Unification of quantum mechanics and general relativity is achieved using the
basic mathematical fact that a vector field is independent of the way it is written.
This fact is known in standard Cartan geometry as the tetrad postulate.
The ECE Lemma and wave equation follow directly by covariant differentiation
of the tetrad postulate. It is now known experimentally in several independent
ways that the Heisenberg uncertainty principle has been refuted. The causal
objective physics indicated by Einsteinian relativity has on the other hand been
proven experimentally to the greatest achievable accuracy in many ways for
more than eighty years. ECE theory is the final form of Einsteinian relativity
theory. It claims to be no more and no less than that.
In chapter one ECE theory is applied to the Sagnac effect, which is explained
straightforwardly by rotating tetrad fields in general relativity. In the standard
model the Sagnac effect has found no satisfactory explanation since it was discovered
in 1913. In chapter two the influence of gravitation on the Sagnac effect
is considered. ECE allows the influence of gravitation on any effect in optics
and electrodynamics to be evaluated systematically. This is not possible in the
standard model. In chapter three, ECE theory is put into dielectric formaliii

PREFACE
ism, which is simpler to understand and which is written in vector notation
for engineers. Some effects of gravitation on spectra and electrodynamics are
considered in chapter four. In the following chapter five contemporary cosmology
is systematically re-evaluated with the new ECE theory and several basic
assumptions of the standard model in cosmology found to be unsatisfactory.
In chapter six, comprehensive mathematical details of ECE theory are summarized
prior to coding for numerical solution. The ECE field and wave equations
are given in form, tensor, vector and dielectric notation. In chapter seven the
d’Alembert, Proca and superconductivity wave equations are put in generally
covariant form in order to evaluate the effects of gravitation. This again is not
possible in the standard model. Chapters eight and nine give the theoretical
formulae needed to develop two major new industries: the production of electric
power from ECE space-time and the production of counter gravitational devices
for the aerospace industry. The key realization in chapters eight and nine is the
role of resonance. Resonance solutions to the ECE field equations are used to
suggest how these industries may develop in future and some simple analytical
solutions are given prior to the expected development of more complete numerical
solutions. Finally in chapter ten, several aspects of generally covariant wave
mechanics are summarized. This is a new subject area in physics, one that does
not exist in the standard model.
In March 2005 the author was appointed to the Civil List by Parliament and
Queen Elizabeth II in recognition of outstanding contributions to British science.
In the area of chemical physics the predecessors on the Civil List were Michael
Faraday and John Dalton. The author wishes to thank the Prime Minister, Mr
Tony Blair, for this high honor and appointment, and also the Royal Society and
Royal Society of Chemistry and external international referees for advising the
Prime Minister. Franklin Amador is thanked for meticulous and highly accurate
typesetting of all three volumes to date, and Linda Caravelli for producing
excellent and useful indices to the highest professional standards. Last but not
least the voluntary and part time staff of the Alpha Foundations’s Institute for
Advanced Study (A.I.A.S.) and other scientists and colleagues are thanked for
many interesting discussions and for continuously refereeing this work.
Craigcefnparc, Wales
February 2006
Myron W. Evans
The British and Commonwealth Civil List scientist
iv

Contents
1 Evans Field Theory Of The Sagnac Effect 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The Rotating Tetrad Fields . . . . . . . . . . . . . . . . . . . . . 2
1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Einstein Cartan Evans (ECE) Unified Field Theory: The Influence
Of Gravitation On The Sagnac Effect 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Rigorous ECE Theory . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Inferences From Dielectric Theory Of ECE Spacetime . . . . . . 16
3 Dielectric Theory Of ECE Spacetime 23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Homogeneous ECE Field Equation in Dielectric Theory . . . . . 24
3.3 Homogeneous Field Equation In Terms Of Magnetization And
Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Spectral Effects Of Gravitation 33
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Extension Of The Faraday Law Of Induction . . . . . . . . . . . 36
4.3 The Effect Of Gravitation On The Wave Properties Of A Light
Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Cosmological Anomalies: EH Versus ECE Field Theory 47
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Red Shift By The Relative Permeability of ECE Spacetime . . . 50
5.3 Absorption And The Beer Lambert Law In ECE Cosmology . . . 57
5.4 Relation Between Power Absorption and Mass . . . . . . . . . . 63
5.5 Gravitational Anomalies Within The Solar System . . . . . . . . 66
6 Solutions Of The ECE Field Equations 73
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2 Details In The Derivation Of The ECE Field Equations, Form,
Tensor, Vector And Dielectric Notation . . . . . . . . . . . . . . 74
6.3 Dielectric ECE Theory, Analytical And Numerical Solutions . . . 88
v

CONTENTS
7 ECE Generalization Of The d’Alembert, Proca And Superconductivity
Wave Equations: Electric Power From ECE Space-
Time 97
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.2 Derivation Of The Wave Equation . . . . . . . . . . . . . . . . . 99
7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8 Resonance Solutions Of The ECE Field Equations 109
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2 The Resonance Equations . . . . . . . . . . . . . . . . . . . . . . 112
8.3 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9 Resonant Counter Gravitation 129
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.2 The Resonance Equations Of ECE Field Theory . . . . . . . . . 131
9.3 Basic Definitions And Conventions For Numerical Solutions . . . 138
10 Wave Mechanics And ECE Theory 153
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
10.2 Lagrangian Formulation Of Generally Covariant Wave Mechanics 156
10.3 Planck Constant, Planck-Einstein And de Broglie Equations, And
The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . 160
10.4 The Aspect Experiment And Quantum Entanglement . . . . . . 164
10.5 Phase Velocity Of ECE Waves . . . . . . . . . . . . . . . . . . . 169
vi

Chapter 1
Evans Field Theory Of The
Sagnac Effect
by
F. Amador, P. Carpenter, A. Collins, G. J. Evans, M. W. Evans,
L. Felker, J. Guala-Valverde, D. Hamilton, J. B. Hart, J. Heidenreich,
A. Hill, G. P. Owen and J. Shelburne.
Alpha Foundation’s Institute for Advance Study (A.I.A.S.).
(emyrone@aol.com, www.aias.us, www.atomicprecision.com)
and
Energy Secretariat of the Argentine Government,
Fundacion Julio Palacios,
Alderete 285 (8300) Neuquen,
Argentina.
Abstract
The Sagnac effect is described straightforwardly in Evans unified field theory
as an example of rotational relativity. The circulating light in a Sagnac interferometer
at rest is a rotation of spacetime described by a tetrad field. This
is multiplied by the scalar valued vector potential magnitude A (0) to produce
vector potentials rotating at an angular frequency
ω 1 = c r
where r is the radius of the circular platform. The additional mechanical spinning
of the platform results in a time delay which is the Sagnac effect. The
time delay is that between the light circulating with the spinning platform and
the light circulating against the spinning platform. This is observed as a frame
independent phase shift. Thus the Sagnac effect is an example of general or
rotational relativity in optics and electrodynamics.
Keywords: Evans field theory; general and rotational relativity in optics and
electrodynamics; the Sagnac effect.
1

1.1. INTRODUCTION
1.1 Introduction
There have been many attempts to explain the well known Sagnac effect using
special relativity and gauge theory [1]– [3]. The effect is observed as a phase
shift in a mechanically rotated Sagnac interferometer and has been developed
into a high accuracy ring laser gyro. The rotational motion implies the use of
general relativity to explain the effect theoretically. Thus the many attempts
over more than ninety years based on special relativity are not valid because
the latter theory does not deal with the accelerations automatically introduced
by rotation. Barrett [4] has offered an explanation based on gauge theory and
non-simply connected topology. This is in the spirit of general relativity, but
is transitional towards the fully developed Evans field theory [5]– [30], which
is a straightforward extension of the Einstein field theory of gravitation to the
unified field.
In section 1.2 the effect is understood straightforwardly as one of general or
rotational relativity [31]– [38] in electrodynamics and optics. The rotating light
beam of the static Sagnac interferometer sets up a rotating tetrad field and a
rotating potential. The angular frequency of rotation (radians per second) is:
ω 1 = c r
(1.1)
where c is the vacuum speed of light and r the radius of the circular platform, of
area πr 2 . The vacuum speed of light is a universal constant of general relativity
[39]. The radius r can be thought of as a Thomson or photon radius. Its inverse
is a wavenumber:
κ = 1 r . (1.2)
The mechanical rotation of the platform at an angular frequency ω 1 produces
phase shifts in the circulating tetrad fields of the Evans field theory, and from
these shifts a time delay can be calculated and compared with the experimental
result. The time delay is:
∆t = 2π
( )
1
ω 1 − Ω − 1
ω 1 + Ω
= 4πΩ
ω1 2 , (1.3)
− Ω2
and is the delay between a beam rotating with the spinning platform and a beam
rotating against the spinning platform. This is the Sagnac effect and is a clear
experimental proof to very high precision of the fact that the electromagnetic
field in general relativity is spinning spacetime [5]– [30]. These concepts do not
exist in the standard model, which is based on special relativity, notably the
Lorentz covariant Maxwell Heaviside equations. These are T invariant, where
T is the motion reversal operator, and so cannot describe the Sagnac effect, or
any type of rotational relativity such as the Faraday disc effect [31]– [38].
1.2 The Rotating Tetrad Fields
Consider the rotation of a beam of light of any polarization around a circle of
area πr 2 in the XY plane at an angular frequency ω 1 to be determined. The
rotation is a rotation of spacetime described by the rotating tetrad field [5]– [30]:
q (1) = 1 √
2
(i − ij) e iω1t (1.4)
2

CHAPTER 1.
EVANS FIELD THEORY OF THE SAGNAC EFFECT
i.e. rotation around the rim of the circular platform of the static Sagnac interferometer
with the beam of light. The Evans Ansatz [5]– [30] converts the
geometry into physics as follows:
A (1) = A (0) q (1) . (1.5)
The geometry is Cartan geometry, or Riemann geometry with torsion. Thus
Eq.1.5 describes a vector potential field rotating around the rim of the circular
Sagnac platform at rest. Rotation to the left is described by:
and rotation to the right by:
A (1)
L
A (1)
R
= A(0)
√
2
(i − ij) e iω1t (1.6)
=
A(0)
√
2
(i + ij) e iω1t . (1.7)
When the platform is at rest a beam going around left-wise takes the same time
to reach its starting point on the circle as a beam going around right-wise. The
time delay between the two beams is:
( 1
∆t = 2π − 1 )
= 0. (1.8)
ω 1 ω 1
Note carefully that Eqs.1.6 and 1.7 do not exist in special relativity because
electromagnetism is thought of as an entity superimposed on a passive or static
frame which never rotates.
Now consider the beam 1.6 rotating left-wise and spin the platform left-wise
at an angular frequency ω. The result is an increase in the angular frequency
of the rotating tetrad, (because the spacetime is spinning more quickly):
ω 1 → ω 1 + Ω. (1.9)
Similarly consider the beam 1.6 rotating left-wise and spin the platform rightwise
at the same angular frequency Ω. The result is a decrease in the angular
frequency of the rotating tetrad, (because the spacetime is spinning more
slowly):
ω 1 → ω 1 − Ω. (1.10)
The time delay between a beam circling left-wise with the platform and a beam
circling left wise against the platform is:
( )
1
∆t = 2π
ω 1 − Ω − 1
(1.11)
ω 1 + Ω
and this is the Sagnac effect.
In order to calculate the angular frequency ω 1 we use the well known experimental
result [1]– [4], so:
where:
∆t = 4ΩAr
c 2 = 4πΩ
ω 2 1 − Ω2 (1.12)
Ar = πr 2 (1.13)
3

1.3. DISCUSSION
for a circular platform. If:
it is found that
Ω ≪ ω 1 (1.14)
ω 1 = c = cκ (1.15)
r
Q.E.D. This is the angular frequency of the rotating tetrad, or rotating spacetime.
1.3 Discussion
The well known features of the Sagnac effect are all described by this analysis
of general relativity. The effect is observed as a phase shift which is frame
independent, so is the same to an observer on or off the platform. The time
delay itself is frame dependent but the phase is frame independent, being a
scalar. The time delay is not observed directly. The effect is independent of the
optical properties of the fiber that carries the beam of light around the circle,
and is the same if the beam of light is guided by mirrors instead of a fiber.
The reason is that the effect is due to mechanical rotation of spacetime itself,
i.e of the frame of reference itself. Analogously gravitation is the bending of
spacetime itself. The Sagnac effect is therefore similar [3] to the well known
Tomita Chiao effect - a phase shift observed in a light beam traversing a helical
optical fiber. The Sagnac effect can be thought of as a Tomita Chiao effect using
a circle rather than a helix. This is usually referred to as a topological phase
shift similar to the class of Berry phases. These have been shown to originate
in the Evans phase of the unified field theory [5]– [30], which also gives the
result 1.12 [3]. These phase shifts all originate therefore in general or rotational
relativity and not in special relativity.
The Sagnac effect can be influenced [1]– [4] by gravitational or Coriolis type
forces or centripetal type forces in dynamics. In the Evans field theory this
type of influence is due to the fact that gravitation affects electromagnetism
through the homogeneous current governing the homogeneous field equation
[5]– [30]. In the absence of gravitation the current is zero, in the presence of
gravitation it may be non-zero for the general spin connection. Thus solutions to
the homogeneous field equation are changed by gravitation, and in consequence
solutions to the rotating potential fields are changed, giving a shift in the Sagnac
effect due to the influence of central gravitational forces or non-central Coriolis
and centripetal forces on electromagnetism. A calculation of these effects in
general must be numerical.
Closely related to the Sagnac effect is the class of geometrical phases such
as that first observed by Tomita and Chiao [40]. The root cause of all geometrical
phases is parallel transport [41], a basic property of general relativity.
In the Evans field theory geometrical phases are due to tetrad fields. In the
Tomita Chiao effect the geometrical phase manifests itself by the passage of
light through a fiber wound into a helix. Nothing else is required to produce the
phase, which is evidently therefore a property of spacetime itself - a property of
general relativity. In special relativity (Maxwell Heaviside field theory) no such
effect is predicted, contrary to the experimental data. The reason is that in
special relativity the electromagnetic field is an entity which is superimposed on
a passive frame of reference in flat or Minkowski spacetime. Therefore in special
4

CHAPTER 1.
EVANS FIELD THEORY OF THE SAGNAC EFFECT
relativity there is no tetrad field because the tetrad is by definition the matrix
which links two frames of reference [39]. In gravitational theory the upper index
of the tetrad (the a index) denotes the Minkowski tangent spacetime at a point
P - tangent to the base manifold indexed µ. In the Evans unified field theory
the concept is extended to optics and electromagnetism. The upper index is
that of the complex circular basis:
and the lower index is that of the cartesian basis:
a = (1), (2), (3) (1.16)
µ = X, Y, Z. (1.17)
The existence of spin in electromagnetic theory is therefore defined by ((1), (2), (3))
superimposed on (X, Y, Z). More generally this is the way that spin is treated
[5]– [30] within the Evans field theory for any radiated or matter field. This concept
can be applied to give a straightforward explanation of the Tomita Chiao
effect (and all geometrical phases [1]– [4]) as follows.
The geometrical phase of Tomita and Chiao is usually expressed [40] as:
φ = e (2πi(1− p s )) (1.18)
where p is the pitch of the helix and s is its length. A helix turns 2π radians in
a pitch p, so p is the wavelength λ. Align s in Z. Therefore:
using
φ = e (2πi(1− λ Z )) = e (2πi) e (−2πi λ Z ) (1.19)
= e (−2πi λ Z ) (1.20)
e 2πi = cos 2π + i sin 2π = 1. (1.21)
In general relativity (Evans field theory) the vector potential of the light
traversing the helical path is a rotating and translating tetrad field q (1) multiplied
by A (0) :
A (1) = A (0) q (1) (1.22)
where:
In general:
q (1) = 1 √
2
(i − ij) e (i(ωt−κZ)) . (1.23)
ω = v r , κ = 1 , ω = κv. (1.24)
r
Here ω is an angular frequency in radians per second, r is a distance in meters,
κ is a wave-number in inverse meters, and v is a velocity in meters per second.
Thus Eq.1.23 denotes in general a propagating and circularly polarized wave of
space-time, and can be expressed as:
q (1) = √ 1
“
(i − ij) e 2
−i b Z
r
”
(1.25)
where:
Ẑ = Z − vt. (1.26)
5

1.3. DISCUSSION
This is a wave of spacetime which propagates in the same way in the absence
of matter. Therefore by general relativity theory it propagates at the speed of
light c. Thus:
v = c. (1.27)
This inference is supported experimentally by the fact that the geometrical
phase is independent of the type of glass or dielectric making up the fiber.
If the optical fiber were dispensed with completely and the beam of light were
guided in a helical path by a system of mirrors in a high vacuum, the geometrical
phase would be the same, Eq.1.19. Therefore the geometrical phase must be
a property of spacetime itself, i.e. must be the tetrad field [24] of the Evans
field theory. As in the Sagnac effect, this is clear experimental proof of the fact
that optics and electrodynamics are governed by general relativity and not by
special relativity.
Comparing Eqs1.19 and 1.25:
In the special case:
r = ZẐ
2πλ . (1.28)
Z = Ẑ = λ (1.29)
then:
r = λ
2π = 1 κ . (1.30)
In special relativity (Maxwell Heaviside field theory) the electromagnetic field
is a nineteenth century entity separate from the frame of reference. Therefore
there is no propagating wave of space-time since space-time in special relativity
is the flat Minkowski space-time [39]. In the Maxwell Heaviside field theory
a light beam traveling in a helix is predicted to have the same (dynamical)
phase as a light beam traveling in a straight line, contrary to the experimental
observation of the geometric phase. The latter is due to parallel transport of
space-time and there is no parallel transport of space-time in special relativity.
The parallel transport methods of gauge theory [39] use an abstract gauge space
superimposed on Minkowski space-time. The abstract gauge space is purely
mathematical in nature and so is extraneous to general relativity. In Evans field
theory this procedure, introduced by Yang and Mills, is replaced entirely by
Cartan geometry, which is general relativity itself. Cartan geometry is rigorously
equivalent to the most general type of Riemann geometry. Thus, by Okham’s
Razor, Evans field theory is preferred to Yang Mills field theory. The Evans field
theory gives all the results [5]– [30] of the Yang-Mills field theory of the weak
and strong forces, but using geometry alone, as demanded by general relativity.
The conventional Yang Mills field theory is a theory of special relativity.
Acknowledgment The British Government is thanked for a Civil List pension
to MWE. Franklin Amador is thanked for meticulous typesetting, Ted
Annis, Tony Craddock and John B. Hart for funding, and the staff of AIAS and
others for many interesting discussions.
6

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Electromagnetic Theory (World Scientific, Singapore, 1993).
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7

BIBLIOGRAPHY
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P. Owen and J. Shelburne, Experiments to test the Evans Unified Field
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and the B(3) Field (World Scientific, Singapore, 2001).
8

BIBLIOGRAPHY
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[32] J. Guala-Valverde and P. Mazzoni, Am. J. Phys., 64, 147 (1996).
[33] J. Guala-Valverde and P. Mazzoni Apeiron, 8, 41 (2001).
[34] J. Guala-Valverde, Phys. Scripta, 66, 252 (2002).
[35] J. Guala-Valverde, P. Mazzoni and R. Achilles, Am. J. Phys., 70 1052
(2001).
[36] J. Guala-Valverde, Apeiron, 11, 327 (2004).
[37] J. Guala-Valverde, communications to A.I. A.S. (2005, posted on
www.aias.us and www.atomicprecision.com).
[38] A. Shadowitz, Special Relativity, (Dover, 1968, pp.125 ff.).
[39] S. P. Carroll, Lecture Notes in General Relativity (a graduate course at
Harvard, UC Santa Barbara and Univ Chicago, public domain arXiv: gr -
gc 973019 v1 1997).
[40] A. Tomita and R. Y. Chiao, Phys. Rev. Lett., 57, 937 (1986).
[41] www.mi.infrn.it/manini/berryphase.html
9

BIBLIOGRAPHY
10

Chapter 2
Einstein Cartan Evans
(ECE) Unified Field
Theory: The Influence Of
Gravitation On The Sagnac
Effect
by
F. Amador, P. Carpenter, G. J. Evans, M. W. Evans,
L. Felker, J. Guala-Valverde, D. Hamilton, J. B. Hart, J. Heidenreich,
A. Hill, G. P. Owen and J. Shelburne.
Alpha Foundation’s Institute for Advance Study (A.I.A.S.).
(emyrone@aol.com, www.aias.us, www.atomicprecision.com)
and
Energy Secretariat of the Argentine Government,
Fundacion Julio Palacios,
Alderete 285 (8300) Neuquen,
Argentina.
Abstract
Einstein Cartan Evans (ECE) unified field theory is an extension of the original
Einstein Hilbert gravitational field theory of 1915 to all radiated and matter
fields. This allows the systematic investigation of the mutual interaction of
gravitation and electromagnetism (EMG coupling) and allows the development
of a generally covariant quantum field theory in which the wave-function is the
tetrad. The latter is the fundamental field of the Palatini variation of general
relativity. The ECE field theory is used to investigate the effect of gravitation
on the Sagnac effect through a non-zero current of the homogeneous field equation.
A frequency shift of the Sagnac effect is predicted due to gravitation, in
accord with experimental observation.
11

2.1. INTRODUCTION
Keywords : Einstein Cartan Evans (ECE) unified field theory; effect of gravitation
on the Sagnac effect.
2.1 Introduction
The original and well known theory of general relativity was developed independently
by Einstein and Hilbert in 1915 and was applied to the gravitational field.
It is now known [1] to be precise to about one part in one hundred thousand
for the solar system using contemporary Eddington experiments. The Einstein
Cartan Evans (ECE) unified field theory was initiated in 2003 and has been
tested with a variety of experimental data [2]– [30]. The ECE field theory is
based on the most general type of Riemann geometry, developed by Cartan into
differential geometry. ECE field theory is a theory of general relativity and
thus of rigorously objective physics applied to all radiated and matter fields,
it is not confined to the gravitational field. The fundamental wave equation
of ECE theory is based on the Evans Lemma [2]– [30], a subsidiary proposition
to the Evans wave equation, and an identity of Cartan geometry based on
the well known tetrad postulate [31]. Within a C negative potential magnitude
A (0) , the electromagnetic field has been identified as the torsion form of
Cartan, and the electromagnetic potential as the tetrad form of Cartan. The
field and potential are inter-related through the first structure equation of Cartan,
an inter-relation which involves the spin connection. The latter is missing
from special relativity and the Maxwell Heaviside field theory of the standard
model. The field equations of electromagnetism have been identified as the first
Bianchi identity of Cartan geometry, and the way electromagnetism interacts
with gravitation (EMG coupling) has been recognised as being governed entirely
by Cartan geometry. The fundamental equations governing this interaction are
the two Cartan structure equations and the two Bianchi identities [2]– [31]. The
first Cartan structure equation defines the torsion form as the covariant exterior
derivative of the tetrad, and the second Cartan structure equation defines the
Riemann form as the covariant exterior derivative of the spin connection. The
first Bianchi identity relates the covariant exterior derivative of the torsion form
to the Riemann form, and the second Bianchi identity asserts that the covariant
exterior derivative of the Riemann form vanishes identically.
The Einstein Hilbert (EH) field theory of gravitation has been identified as
a limit of the ECE field theory, the limit where the torsion form vanishes. In
this case the covariant exterior derivative of the tetrad vanishes and there is no
torsion form present in the geometry.
This limit defines the approximation to Riemann geometry used by Einstein
and Hilbert to derive their well known field equation of 1915. In consequence
of this approximation the EH field theory cannot describe EMG coupling because
the electromagnetic field is undefined without the torsion form. In ECE
field theory the electromagnetic field is self-consistently defined within general
relativity as the spinning of spacetime, the gravitational field as the curving of
spacetime. The tetrad of ECE field theory is the fundamental field. The weak
and strong fields are tetrads in the appropriate representation space [2]– [30].
The ECE field theory is therefore a straightforward unification scheme based
directly on well known and standard Cartan geometry. The latter is rigorously
12

CHAPTER 2. EINSTEIN CARTAN EVANS (ECE) UNIFIED FIELD . . .
equivalent [2]– [31] to the most general type of Riemann geometry, Riemann
geometry in which the torsion tensor and the Riemann tensor are both nonzero,
and in which the connection is in general asymmetric in its lower two
indices. For rotational motion [2]– [31] the connection of Riemann geometry is
anti-symmetric in its lower two indices, the spin connection of Cartan geometry
is dual to the tetrad, and the Riemann form is dual to the torsion form.
In consequence the homogeneous current of the ECE theory vanishes for pure
rotational motion [2]– [30]. For pure translational motion governed by centrally
directed forces the EH theory applies, the connection is symmetric in its lower
two indices, the torsion form vanishes, and the homogeneous current again vanishes.
In this paper the effect of gravitation on the Sagnac effect in ECE theory
is evaluated with a non-zero homogeneous current. In this case the connection
in Riemann geometry is asymmetric in its lower two indices and rotational and
translational motion are in consequence of this mutually influential. The latter
influence is shown to produce a frequency shift of the Sagnac effect, a shift
that can be looked for with a high precision ring laser gyro. In Section 2.2 the
rigorous theory is given, and in Section 2.3 the rigorous theory is approximated
in terms of the vector notation used in electrical engineering, classical electrodynamics
and dielectric theory. It is shown that the main prediction of the
paper, a frequency shift of the ring laser gyro due to gravitation, is supported
by known experimental evidence [32]. Contemporary ring laser gyro technology
is much more precise. This type of frequency shift is due to a non-zero homogeneous
current of ECE field theory. This is expected to be very tiny and is a
novel classical effect different from the well known bending of light by gravity
in an Eddington experiment. The latter is a quantum effect which relies on
the mass of the photon interacting through pure gravitation (EH theory) with
another mass such as that of the sun. The Eddington effect and contemporary
developments thereof [1] are described in the torsion free limit when ECE field
theory reduces to EH field theory. In order to describe the effect of gravitation
on an optical effect such as the Sagnac effect, the rigorous ECE unified field
theory is needed. The available data [32] strongly suggest that gravitation does
indeed cause a broadening (unresolved frequency shift) in the Sagnac effect, an
important verification of ECE field theory if the data [32] are found to be reproducible
and repeatable. If after further precise experimental testing it is found
that there is no shift of the Sagnac effect in a gravitational field it would mean
simply that the mutual influence is too small to be measured by the contemporary
ring laser gyro. Many other types of experimental verification of ECE
theory are already available [2]– [30].
2.2 Rigorous ECE Theory
The effect of gravitation on the Sagnac effect is given in general by the homogeneous
field equation [2]– [30] of the ECE theory:
d ∧ F a = µ 0 j a (2.1)
where
F a = d ∧ A a + ω a b ∧ A b , (2.2)
13

2.2. RIGOROUS ECE THEORY
A a = A (0) q a . (2.3)
Here F a is the vector valued electromagnetic field two-form, d∧ denotes the
exterior derivative where ∧ is the wedge product, µ 0 is the vacuum permeability,
j a is the homogeneous current three-form, A a is the electromagnetic potential
one-form, ω a b is the spin connection, and qa is the tetrad one-form. The Sagnac
effect is due [30] to a rotating tetrad, so gravitation affects q a and therefore the
Sagnac effect when:
j a ≠ 0. (2.4)
Some early experimental support for such an effect is given in ref. (32). Conversely
the observation of the influence of gravitation on the ring laser gyro
would be a confirmation of the existence of the homogeneous current . The
latter is defined by [2]– [30]:
j a = A(0)
µ 0
(
R
a
b ∧ q b − ω a b ∧ T b) (2.5)
where R a b
is the tensor valued Riemann or curvature two-form. For pure electromagnetism
or for pure gravitation [2]– [30]:
j a = 0. (2.6)
In order for j a to be non-zero a spin connection ω a b
is needed which is neither
symmetric nor anti-symmetric in its a and b indices. This is the geometrical
condition needed for gravitation to influence electromagnetism.
In tensor notation Eq.2.1 is [2]– [30]:
∂ µ F a νρ + ∂ ν F a ρµ + ∂ ρ F a µν = A (0) (R a µνρ + R a νρµ + R a ρµν
−ω a µbT b νρ − ω a νbT b ρµ − ω a ρbT b µν )
(2.7)
the Hodge dual of which is:
∂ µ ˜F aµν = µ 0˜j aν = A (0) ( ˜Ra
µν
µ − ω a µb ˜T bµν) . (2.8)
Note that the Hodge dual current is a vector valued one-form. In EH theory:
˜R a µν
µ = ˜T bµν = 0 (2.9)
because the first Bianchi identity (or cyclic identity) in EH theory vanishes:
R a b ∧ q b = 0 (2.10)
and in EH theory there is no torsion. For pure electromagnetism uninfluenced
by gravitation:
R a b = ɛ a bcT c , ω a b = ɛ a bcq c , (2.11)
in consequence of which [2]– [30]:
˜R a µν
µ = ω a µb ˜T bµν (2.12)
So in the absence of EMG coupling the homogeneous current vanishes and:
∂ µ ˜F aµν = 0. (2.13)
14

CHAPTER 2. EINSTEIN CARTAN EVANS (ECE) UNIFIED FIELD . . .
In vector notation Eq.2.13 becomes two simultaneous equations:
∇ · B a = 0, (2.14)
∇ × E a + ∂Ba
∂t
= 0, (2.15)
where B a is the magnetic flux density (in tesla or weber per meter squared) and
E a is the electric field strength (in volts per meter). Eq.2.14 for each polarization
index a is the Gauss law applied to magnetism, and Eq.2.15 is the Faraday law
of induction. Therefore Eqs.2.14 and 2.15 are true when the ring laser gyro is
not influenced by gravitation.
In the presence of EMG coupling Eqs.2.14 and 2.15 become:
∇ · B a = µ 0˜j a , (2.16)
where:
∇ × E a + ∂Ba
∂t
= µ 0˜ja , (2.17)
˜j aν =
(˜j a ,˜j a) . (2.18)
Thus EMG coupling produces a shift in the frequencies of B a and E a through
the scalar current ˜j a and vector current ˜j a . This means that the frequency of
A a and q a are also shifted, and there is a shift (Section 2.3) in the Sagnac effect
or ring laser gyro. This is the qualitative explanation of such a shift in ECE
theory.
A quantitative or numerical calculation of such a shift requires the definitions
of the scalar and vector currents appearing in Eqs.2.16 and 2.17. The scalar
current is defined by the indices:
and so:
ν = 0, µ = 1, 2, 3 (2.19)
˜j a = A(0)
µ 0
(
˜Ra
i0
i − ω a ib ˜T bi0) , i = 1, 2, 3. (2.20)
In Eq.2.20 summation is implied over repeated indices i as usual. The scalar and
vector currents are non-zero if and only if Eq.2.11 is not true, i.e. if and only if
the spin connection is not dual to the tetrad and if and only if the curvature form
is not dual to the spin connection. This means that EMG coupling originates
in an influence of spacetime curving on spacetime spinning and vice-versa. If
the ring laser gyro can detect this influence, as suggested by ref. [32], then
novel technologies could emerge. The most important of these would be the
acquisition of electric power from ECE spacetime. The vector current is defined
by [2]– [30]:
˜ja = ˜j a X i + ˜j a Y j + ˜j a Z k, (2.21)
where the three components are given by:
˜j a X = A(0) a 10 a 12 a 13
( ˜R 0 + ˜R 2 + ˜R 3
µ 0
−ω a 0b ˜T b10 − ω a 2b ˜T b12 − ω a 3b ˜T b13 )
(2.22)
15

2.3. INFERENCES FROM DIELECTRIC THEORY OF ECE . . .
˜j a Y
= A(0) a 20 a 21 a 23
( ˜R 0 + ˜R 1 + ˜R 3
µ 0
−ω a 0b ˜T b20 − ω a 1b ˜T b21 − ω a 3b ˜T b23 )
(2.23)
˜j a Z = A(0) a 30 a 31 a 32
( ˜R 0 + ˜R 1 + ˜R 2
µ 0
−ω a 0b ˜T b30 − ω a 1b ˜T b31 − ω a 2b ˜T b32 ).
(2.24)
Therefore a quantitative calculation would require knowledge of the scalar elements
in Eq.2.21. In general it can be seen that the vector current depends
on the spin connection, curvature and torsion. These are not mathematically
independent quantities because they are related by the fundamentals of Cartan
geometry, the two Cartan structure equations:
and the two Bianchi identities:
T a = D ∧ q a = d ∧ q a + ω a b ∧ q b , (2.25)
R a b = D ∧ ω a b = d ∧ ω a b + ω a c ∧ ω c b, (2.26)
D ∧ T a = R a b ∧ q b , (2.27)
D ∧ R a b = 0. (2.28)
The tetrad is always defined by the Evans Lemma [2]– [30]:
where
□q a µ = Rq a µ, (2.29)
R = −kT, (2.30)
is the scalar curvature, where T is the index contracted energy - momentum
tensor and k is Einstein’s constant [2]– [30]. In ECE theory Eq.2.30 applies to
all radiated and matter fields as intended by Einstein [33]. Thus for a given
T the eigenvalues of the tetrad can be calculated. The spin connection obeys
the second Bianchi identity 2.28 and is related to the gamma connection by the
tetrad postulate [2]– [30]:
D µ q a ν = 0. (2.31)
From a knowledge of the spin connection and tetrad, the torsion and Riemann
forms can be found, and finally the scalar and vector currents.
2.3 Inferences From Dielectric Theory Of ECE
Spacetime
It is necessary to prove using dielectric theory that Eq.2.17 results in a frequency
shift of the Sagnac effect. It is first shown that Eq.2.17 can always be rewritten
as:
∇ × D a + 1 ∂H a
c 2 = 0, (2.32)
∂t
where
D a = ɛ 0 E a + P a , (2.33)
16

CHAPTER 2. EINSTEIN CARTAN EVANS (ECE) UNIFIED FIELD . . .
H a = 1 µ 0
B a − M a . (2.34)
Here D a is the displacement, H a the magnetic field strength, P a the polarization,
M a the magnetization, and ɛ 0 the vacuum permittivity [34] for each
polarization index a. using Eqs.2.33 and 2.34 in Eq.2.32:
∇ × (ɛ 0 E a + P a ) + µ 0 ɛ 0
∂
∂t
( 1
µ 0
B a − M a )
, (2.35)
where:
ɛ 0 µ 0 = 1 c 2 . (2.36)
It follows that:
where:
∇ × E a + ∂Ba
∂t
˜ja = ∂Ma
∂t
= µ 0˜ja
(2.37)
− 1
ɛ 0 µ 0
∇ × P a (2.38)
Q.E.D. Thus Eq.2.38 is an expression for the vector current of ECE spacetime
in terms of dielectric polarization and magnetization. Thus EMG coupling can
be understood in terms of a polarization and magnetization of ECE spacetime.
If the EMG coupling is weak, as expected experimentally, then:
Thus:
˜ja −→ 0. (2.39)
M a
∂t − 1
ɛ 0 µ 0
∇ × P a ∼ 0. (2.40)
Now introduce the permeability µ and permittivity ɛ of the ECE spacetime. In
standard dielectric theory the polarization is related to the electric field strength
as follows:
P a = (ɛ r − 1) ɛ 0 E a (2.41)
and the magnetization is related to the magnetic flux density by
M a = 1 ( ) κ
B a (2.42)
µ 0 1 + κ
Here
is the relative permittivity or dielectric constant and
ɛ r = ɛ
ɛ 0
(2.43)
µ r = µ µ 0
= 1 + κ (2.44)
is the relative permeability, κ being the volume magnetic susceptibility (not
to be confused with the wavenumber κ). It follows that the polarization and
magnetization both vanish when:
ɛ = ɛ 0 , (2.45)
µ = µ 0 , (2.46)
17

2.3. INFERENCES FROM DIELECTRIC THEORY OF ECE . . .
i.e. vanish in the vacuum, for which ˜j a also vanishes. In a dielectric (the ECE
spacetime):
˜ja = κ ∂Ba − (ɛ r − 1) (1 + κ) ∇ × E a (2.47)
∂t
so ECE spacetime is a dielectric with relative permittivity ɛ r and relative permeability
µ r . Thus EMG coupling changes ɛ 0 to ɛ and changes µ 0 to µ. This
means that EMG coupling changes the refractive index defined [35] by:
n 2 = ɛµ
ɛ 0 µ 0
= ɛ r µ r . (2.48)
In the presence of absorption the refractive index and relative permittivity and
permeability become complex valued with frequency dependent real and imaginary
parts [36]. In general therefore novel optical and spectroscopic effects are
expected from EMG coupling. The phase velocity of the electromagnetic wave
is changed by EMG coupling from its vacuum value c to:
v = c n . (2.49)
The phase of the Sagnac effect is shifted in consequence to
φ = 4ΩAr
λv
(2.50)
from
φ = 4ΩAr
λc . (2.51)
The time delay [30] of the Sagnac effect is shifted by EMG coupling to:
∆t = 4ΩAr
v 2 (2.52)
from
∆t = 4ΩAr
c 2 (2.53)
in the absence of EMG coupling. Here Ar is the area of the Sagnac platform, λ is
the wavelength of the light, and Ω the angular rotation frequency of mechanical
rotation of the Sagnac platform.
Acknowledgments The British Government is thanked for a Civil List pension
(2005) to MWE. Franklin Amador is thanked for meticulous typesetting,
Ted Annis, Tony Craddock and John B. Hart for funding, and the staff of AIAS
and others for many interesting discussions.
18

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[11] M. W. Evans, Explanation of the Eddington Experiment in the
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19

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to the Generally Covariant Dirac Equation, (2005, preprint on
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and Quantum Electrodynamics, (2005, preprint on www.aias.us and
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S tensor, (2005, preprint on www.aias.us and www.atomicprecision.com).
[22] M. W. Evans, Explanation of the Faraday Disc Generator in the
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J. Guala-Valverde, D. Hamilton, J. B. Hart, J. Heidenreich, A. Hill,
G. P. Owen and J. Shelburne, Experiments to Test the Evans Unified
Field Theory and General Relativity in Classical Electrodynamics, Found.
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www.atomicprecision.com).
[24] F. Amador, P. Carpenter, A.Collins, G. J. Evans, M. W. Evans, L.
Felker, J. Guala-Valverde, D. Hamilton, J. B. Hart, J. Heidenreich, A.
Hill, G. P. Owen and J. Shelburne, Evans Field Theory of the Sagnac Effect,
Found. Phys. Lett., submitted (2005, preprint on www.aias.us and
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[25] M. W. Evans, (ed.), Modern Non-Linear Optics, a special topical issue of
I. Priogine ad S. A. Rice (eds.), Advances in Chemical Physics (Wiley-
Interscience, New York 2001, 2nd ed.), vols. 119(1) to 119(3).
[26] M. W. Evans and L. B. Crowell, Classical and Quantum Electrodynamics
and the B (3) Field (World Scientific, Singapore, 2001).
[27] M. W. Evans, J.-P. Vigier et alii, The Enigmatic Photon (Kluwer, Dordrecht,
1994 - 2002, hardback and softback) vols. One to Five.
[28] M. W. Evans and A. A. Hasanein, The Photomagneton in Quantum Field
Theory (World Scientific, Singapore, 1994).
20

BIBLIOGRAPHY
[29] M. W. Evans and S. Kielich (eds.), first edition of ref. (25) (Wiley-
Interscience, New York, 1992, reprinted 1993, softback 1997), vols. 85(1)
to 85(3).
[30] M. W. Evans, Physica B, 182, 227, 237 (1992), the original papers on the
B (3) spin field, and papers in Found. Phys. Lett. and Found. Phys., 1994
to present.
[31] S. P. Carroll, Lecture Notes in General Relativity (a graduate course given
at Harvard, UC Santa Barbara and Univ Chicago, public domain), arXiv :
gr - gc 973019 v1 1997).
[32] W. Tan, X. Song and Z Gu, Chinese Journal of Lasers, 10, 327 (1983).
[33] A. Einstein, The Meaning of Relativity (Princeton Univ. Press, 1921 - 1954
eds.).
[34] P. W. Atkins, Molecular Quantum Mechanics (Oxford Univ Press, 1983,
2nd ed.).
[35] J. D. Jackson, Classical Electrodynamics (Wiley, New York, 3rd ed., 1998).
[36] M. W. Evans, G. J. Evans, W. T. Coffey and P. Grigolini, Molecular Dynamics
and the Theory of Broad Band Spectroscopy (Wiley - Interscience,
New York, 1982).
21

BIBLIOGRAPHY
22

Chapter 3
Dielectric Theory Of ECE
Spacetime
by
F. Amador, P. Carpenter, G. J. Evans, M. W. Evans,
L. Felker, J. Guala-Valverde, D. Hamilton, J. B. Hart, J. Heidenreich,
A. Hill, G. P. Owen and J. Shelburne.
Alpha Foundation’s Institute for Advance Study (A.I.A.S.).
(emyrone@aol.com, www.aias.us, www.atomicprecision.com)
and
Energy Secretariat of the Argentine Government,
Fundación Julio Palacios,
Alderete 285 (8300) Neuquen,
Argentina.
Abstract
The Einstein Cartan Evans (ECE) unified field theory is developed using the
methods of dielectric spectroscopy to show that the effect of gravitation on
electromagnetism is in general to create a spectrum. Therefore a light beam
reaching a telescope after having passed through regions in which such interaction
takes place will in general have spectral properties such as dispersion,
refraction and absorption. An objective unified field theory is needed to realize
this, and so these effects are not present in the standard model despite being
routinely observable as anomalous shifts and quantized shifts.
Keywords: ECE unified field theory; interaction of gravitation and electromagnetism;
spectral effects of ECE theory; dielectric spectroscopy and ECE field
theory.
3.1 Introduction
The Einstein Cartan Evans (ECE) unified field theory is based directly on Riemann
geometry as developed by Cartan and completes the well known work
of Einstein and Cartan. In consequence it becomes possible for the first time
23

3.2. HOMOGENEOUS ECE FIELD EQUATION IN DIELECTRIC . . .
to study the detailed spectral effects of gravitation on electromagnetism using
both classical and quantum theory. A spectrum is created by ECE spacetime,
which is a four-dimensional spacetime with curvature and torsion [1]- [30]. The
spectrum is generated by the homogeneous and inhomogeneous currents of ECE
field theory. In this paper the homogeneous field equation of ECE field theory
is developed in section 3.2 as an equation in the electric displacement D a and
magnetic field strength H a of ECE spacetime, and in Section 3.3 as an equation
in magnetization M a and polarization P a of ECE spacetime. The latter is
therefore a dielectric or ponderable medium analogous in nature to a gas, liquid,
solid, liquid crystal and so on. A dielectric is characterized by a spectrum [31]
so ECE spacetime is also characterized by a spectrum, one which originates
in the effect of gravitation on electromagnetism such as a light beam from a
cosmological object reaching a telescope. In general this spectrum consists of a
frequency dependent power absorption coefficient [31] which is directly related
to the dielectric loss, and a frequency dependent refractive index which is directly
related to the dielectric dispersion. The spectra of ECE spacetime may
be as varied as those of any dielectric, consisting in general of many peaks, or
anomalous shifts or quantized shifts [32]- [33] such as those observed experimentally
in cosmology. These are a mystery to the standard model because
the latter is not a truly objective unified field theory of general relativity, i.e.
cannot describe the effect of gravitation on electromagnetism.
3.2 Homogeneous ECE Field Equation in Dielectric
Theory
The homogeneous field equation of ECE field theory [1]- [30] derives directly
from the first Bianchi identity of Cartan geometry using the Evans Ansatz:
A a = A (0) q a (3.1)
where A (0) is a scalar valued, C negative, potential magnitude, where q a is the
tetrad form and A a the potential form. The tetrad [1]- [30] is the fundamental
field of the Palatini variation of general relativity. In ECE field theory it is the
fundamental field for matter and all kinds of radiation. It is seen from Eq.3.1
that Cartan geometry becomes unified field theory in physics directly through
the Evans Ansatz. Since Cartan geometry is the most general type of Riemann
geometry the ECE field theory is directly grounded in well known and accepted
and rigorously self-consistent mathematics [1]- [30]. The well known Einstein
Hilbert theory (EH theory) of 1915 used the same Riemann geometry but assumed
that the torsion tensor is absent. This is adequate for some situations in
gravitational theory but not for unified field theory. Cartan elegantly developed
Riemann geometry into his well known theory of differential forms and anticipated
in the nineteen twenties that the electromagnetic field may be his torsion
form within a C negative factor. Despite thirty years of effort neither Einstein
nor Cartan successfully pursued this to its logical conclusion. The main reason
for this is that they did not have available at that time the inverse Faraday
effect [34] which is the magnetization of matter by electromagnetic radiation.
This effect was not inferred until the mid fifties and was not confirmed experimentally
until the mid sixties. It was not until 1992 [35] that the effect was
24

CHAPTER 3.
DIELECTRIC THEORY OF ECE SPACETIME
first explained using what is now known to be the Evans spin field, which is the
archetypical signature of general relativity in electromagnetism [1]- [30]. The
Evans spin field cannot be described in the standard model, which incorrectly
uses special relativity for the electromagnetic sector. The ECE field theory [1]-
[30] has confirmed this prediction of Cartan’s using the Evans Ansatz (1), which
implies:
F a = A (0) T a (3.2)
where F a is the electromagnetic field form and T a the torsion form of Cartan.
Using Eq.3.2 the first Bianchi identity of Cartan geometry:
d ∧ T a = R a b ∧ q b − ω a b ∧ T b (3.3)
translates directly into the homogeneous field equation of ECE theory:
d ∧ F a = µ 0 j a = A (0) ( R a b ∧ q b − ω a b ∧ T b) . (3.4)
The homogeneous current is also defined directly by the Cartan geometry:
j a = A(0)
µ 0
(
R
a
b ∧ q b − ω a b ∧ T b) . (3.5)
Here R a b is the Riemann or curvature form, ωa b
is the spin connection, d∧
denotes the Cartan exterior derivative and µ 0 is the vacuum permeability in
S.I. units. Eq.3.4 splits into two equations in vector notation:
∇ · B a = µ 0˜j a (3.6)
and
∇ × E a + ∂Ba = µ 0˜ja
∂t
where the homogeneous current is the four vector:
(3.7)
˜j aν =
(˜j a ,˜j a) (3.8)
and where a is the polarization index [1]- [30].
The equivalents of Eqs.3.6 and 3.7 in the standard model are:
∇ · B = 0 (3.9)
and
∇ × E + ∂B
∂t = 0 (3.10)
in which the homogeneous current is incorrectly missing and where the polarization
index is implied . The basic failure of the standard model is that it is not
rigorously objective, being a theory of special relativity where general relativity
is needed. Eq.3.9 is known as the Gauss law applied to magnetism and Eq.3.10
is known as the Faraday law of induction. These laws work only in particular
experimental situations. They are not able to account for the effect of gravitation
on electromagnetism, not even in a qualitative way. The correctly objective
laws of general relativity [1]- [30] are given by Eqs.3.6 and 3.7, which are derived
from exactly the same principles and same type of Riemann geometry as used
25

3.2. HOMOGENEOUS ECE FIELD EQUATION IN DIELECTRIC . . .
in 1915 for gravitation by Einstein, and independently, Hilbert. As mentioned
however, the Einstein Hilbert (EH) theory neglects the torsion form.
Therefore the logical choice is to accept ECE theory or reject general relativity.
It has been shown [24] that Eq.3.7 may be written as:
∇ × D a + µ 0 ɛ 0
∂H a
if the homogeneous current is defined as:
∂t
= 0 (3.11)
˜ja = ∂Ma
∂t
− c 2 ∇ × P a . (3.12)
Here ɛ 0 is the vacuum permittivity, defined in S.I. units by the speed of light c:
ɛ 0 µ 0 = 1 c 2 . (3.13)
Eq.3.11 translates the homogeneous current into the permittivity ɛ and permeability
µ of ECE spacetime using the standard relations [36, 37] between the
displacement and polarization and the magnetic field strength and magnetization:
D a = ɛ 0 E a + P a = ɛE a (3.14)
B a = µ 0 (H a + M a ) = µH a (3.15)
Here
ɛ r = ɛ/ɛ 0 (3.16)
µ r = µ/µ 0 (3.17)
are the relative permittivity and permeability respectively.
If it is assumed for the sake of simplicity and illustration that:
then:
∂B a
∂t
ɛ ∼ constant, (3.18)
µ ∼ constant, (3.19)
+ µɛ
µ 0 ɛ 0
∇ × E a = 0. (3.20)
The refractive index is defined in standard dielectric theory [36, 37] as:
n 2 = µɛ
µ 0 ɛ 0
(3.21)
so in this particular approximation it is seen clearly that the homogeneous
current is equivalent to a dielectric with refractive index 3.21. In consequence
ECE spacetime is in general a dielectric and not a vacuum with permeability
µ 0 and ɛ 0 permittivity.
One solution of Eq.3.20 consists of a plane wave with phase velocity v, and
phase:
φ = ωt − n 2 κZ (3.22)
26

CHAPTER 3.
DIELECTRIC THEORY OF ECE SPACETIME
where ω is the angular frequency and where κ is the wavenumber. If this plane
wave were traversing a vacuum its refractive index would be:
n 2 = 1 (3.23)
and so the ECE spacetime SHIFTS the wavenumber κ to n a κ. Recall that
this is a special case defined by Eq.3.18 and 3.19, but this case is enough to
show that ECE spacetime produces cosmological shifts due to the interaction of
gravitation with electromagnetism. From Eq.3.22 the phase velocity is:
v = 1 c. (3.24)
n2 The electric susceptibility κ E of ECE spacetime and its volume magnetic susceptibility
κ m are defined by the standard [36, 37] equations:
ɛ r = 1 + κ E (3.25)
µ r = 1 + κ m . (3.26)
ECE spacetimes for which κ m < 0 are diamagnetic and ECE spacetimes for
which κ m > 0 are paramagnetic. In the presence of absorption both ɛ and µ are
complex:
ɛ = ɛ ′ + iɛ ′′ (3.27)
µ = µ ′ + iµ ′′ (3.28)
so the wavenumber shift and the phase velocity become spectral quantities dependent
on the nature of ECE spacetime, i.e. dependent on Cartan geometry.
In consequence many types of shifts are expected, as observed in anomalous
shifts and quantized shifts [32,33]. As can be seen from Eq.3.5, the ECE spacetime
is in general a complicated function of differential forms and of the spin
connection and there is reason to expect a rich spectrum of shifts, again as
observed experimentally [32, 33] in contemporary cosmology.
In the next section it is shown that Eq.3.11 is a limiting case of a more
general equation, and so Section 3.3 makes the dielectric theory a little more
exact, but this section illustrates the major conclusion - that ECE field theory
explains cosmological shifts as being due to the spectral effects of gravitation
on electromagnetism. This concept is missing entirely from the standard model
and contemporary cosmology, yet is a logical outcome of the work of Einstein
and Cartan.
3.3 Homogeneous Field Equation In Terms Of
Magnetization And Polarization
The polarization and magnetization of ECE spacetime can be expressed in terms
of the electric field strength E a and the magnetic flux density B a as:
P a = (ɛ − ɛ 0 ) E a (3.29)
( 1
M a = − 1 B
µ 0 µ)
. (3.30)
27

3.3. HOMOGENEOUS FIELD EQUATION IN TERMS OF . . .
So the homogeneous field equation becomes:
∂M a ( 1
+ − 1 ) ( ) 1
∇ × P a = µ
∂t µ 0 µ ɛ − ɛ 0˜ja . (3.31)
0
In the weak interaction limit defined by:
and
Eq.3.31 reduces to:
∂M a
∂t
ɛ −→ ɛ 0 , (3.32)
µ −→ µ 0 , (3.33)
˜ja −→ 0 (3.34)
( 1
+ − 1 ) ( ) 1
∇ × P a ∼ 0 (3.35)
µ 0 µ ɛ − ɛ 0
which is a soluble wave equation. When interaction between gravitation and
electromagnetism (EMG coupling) is entirely absent:
ɛ = ɛ 0 , (3.36)
µ = µ 0 , (3.37)
P a = 0, (3.38)
M a = 0, (3.39)
and we recover:
∇ × E a + ∂Ba = 0. (3.40)
∂t
This is the Faraday law of induction for each polarization index a. Eq.3.31
makes no assumptions about the current so this development is more general
than that in Section 3.2. In the presence of absorption the refractive index is
complex:
n (ω) = n ′ (ω) + in ′′ (ω) (3.41)
and the power absorption coefficient [31] is:
α (ω) = ωɛ′′ (ω)
n ′ (ω)c . (3.42)
A spectrum is defined as a graph of α against wavenumber. This graph can be
built up from geometry of ECE spacetime and in general is what is observed in a
telescope as cosmological shifts. In the standard model these shifts are explained
simplistically either with special relativity (Doppler red shift due to assumed
universal expansion - Big Bang) or by the gravitational shifts of EH theory using
a hybrid mix of photon mass and classical theory. Thus ECE theory is more
self consistent and far richer in predictive power than the standard model. This
is what is expected from a truly objective unified field theory in classical and
quantum mechanics.
Acknowledgments The British Government is thanked for a Civil List pension
(2005) to MWE. Franklin Amador is thanked for meticulous typesetting,
Ted Annis, Tony Craddock and John B. Hart for funding, and the staff of AIAS
and others for many interesting discussions.
28

Bibliography
[1] M. W. Evans, Found. Phys. Lett., 16, 367, 507 (2003).
[2] M. W. Evans, Found. Phys. Lett., 17, 25, 149, 267, 301, 393, 433, 535, 663
(2004).
[3] M. W. Evans, Found. Phys. Lett., 18, 139, 259, 519 (2005).
[4] M. W. Evans, Generally Covariant Unified Field Theory (in press, preprint
on www.aias.us and www.atomicprecision.com).
[5] L. Felker, The Evans Equations of Unified Field Theory (in press, preprint
on www.aias.us and www.atomicprecision.com).
[6] M. W. Evans, The Objective Laws of Classical Electrodynamics, the Effect
of Gravitation on Electromagnetism, preprint on www.aias.us and
www.atomicprecision.com.
[7] M. W. Evans, First and Second Order Aharonov Bohm Effects
in the Evans Unified Field Theory, preprint on www.aias.us and
www.atomicprecision.com
[8] M. W. Evans, The Spinning of Spacetime as Seen in the Inverse Faraday
Effect, preprint on www.aias.us and www.atomicprecision.com.
[9] M. W. Evans, On the Origin of Polarization and Magnetization, preprint
on www.aias.us and www.atomicprecision.com.
[10] M. W. Evans, Explanation of the Eddington Experiment in the Evans Unified
Field Theory, preprint on www.aias.us and www.atomicprecision.com
.
[11] M. W. Evans, The Coulomb and Ampère Maxwell Laws in the
Schwarzschild Metric: A Classical Calculation of the Eddington Effect
from the Evans Unified Field Theory, preprint on www.aias.us and
www.atomicprecision.com.
[12] M. W. Evans, Generally Covariant Heisenberg Equation from
the Evans Unified Field Theory, preprint on www.aias.us and
www.atomicprecision.com.
[13] M. W. Evans, Metric Compatibility and the Tetrad Postulate, preprint on
www.aias.us and www.atomicprecision.com.
29

BIBLIOGRAPHY
[14] M. W. Evans, Derivation of the Evans Lemma and Wave Equation
from the First Cartan Structure Equation, preprint on www.aias.us and
www.atomicprecision.com.
[15] M. W. Evans, Proof of the Evans Lemma from the Tetrad Postulate,
preprint on www.aias.us and www.atomicprecision.com.
[16] M. W. Evans, Self-Consistent Derivation of the Evans Lemma and Application
to the Generally Covariant Dirac Equation, preprint on www.aias.us
and www.atomicprecision.com.
[17] M. W. Evans, Quark-Gluon Model in the Evans Unified Field Theory,
preprint on www.aias.us and www.atomicprecision.com.
[18] M. W. Evans, The Origin of Intrinsic Spin and the Pauli Exclusion Principle
in the Evans Unified Field Theory, preprint on www.aias.us and
www.atomicprecision.com.
[19] M. W. Evans, General Covariance and Coordinate Transformation in
Classical and Quantum Electrodynamics, preprint on www.aias.us and
www.atomicprecision.com .
[20] M. W. Evans, The Role of Gravitational Torsion in General Relativity: the
S Tensor, preprint on www.aias.us and www.atomicprecision.com.
[21] M. W. Evans, Explanation of the Faraday Disc Generator in the Evans Unified
Field Theory, preprint on www.aias.us and www.atomicprecision.com.
[22] F. Amador, P. Carpenter, G. J. Evans, M. W. Evans, L. Felker, J. Guala-
Valverde, D. Hamilton, J. B. Hart, J. Heidenreich, A.Hill, G. P. Owen
and J. Shelburne, Experiments to Test the Evans Unified Field Theory
and General Relativity in Classical Electrodynamics, Found. Phys. Lett.,
submitted (preprint on www.aias.us and www.atomicprecision.com.
[23] F. Amador, P. Carpenter, G. J. Evans, M. W. Evans, L. Felker, J. Guala-
Valverde, D. Hamilton, J. B. Hart, J. Heidenreich, A. Hill, G. P. Owen and
J. Shelburne, ECE Field Theory of the Sagnac Effect, Found. Phys. Lett.,
submitted (2005, preprint on www.ais.us and www.atomicprecision.com.
[24] F. Amador, P. Carpenter, G. J. Evans, M. W. Evans, L. Felker, J. Guala-
Valverde, D. Hamilton, J. B. Hart, J. Heidenreich, A. Hill, G. P. Owen and
J. Shelburne, Einstein Cartan Evans (ECE) Field Theory: The Influence of
Gravitation on the Sagnac Effect, Found. Phys. Lett., submitted (preprints
on www.aias.us and www.atomicprecision.com.
[25] M. W. Evans (ed.), Modern Non-Linear Optics, a special topical issue of
I. Prigogine and S. A. Rice (eds.), Advances in Chemical Physics (Wiley
Interscience, New York, 2001, 2nd ed.), vols. 119(1), 119(2) and 119(3).
[26] M. W. Evans and L. B. Crowell, Classical and Quantum Electrodynamics
and the B (3) Field (World Scientific, Singapore, 2001).
[27] M. W. Evans, J.-P. Vigier et alii, The Enigmatic Photon (Kluwer, Dordrecht,
1994 - 2002, hardback and softback) vols. 1 - 5.
30

BIBLIOGRAPHY
[28] M. W. Evans and A. A. Hasanein, The Photomagneton in Quantum Field
Theory (World Scientific, Singapore, 1994).
[29] M. W. Evans and S. Kielich (eds.), first edition of ref. (25) (Wiley-
Interscience, New York 1992, reprinted 1993, softback 1997), vols. 85(1),
85(2) and 85(3).
[30] M. W. Evans, papers in Found. Phys. Lett. and Found. Phys., 1994 to 2005.
[31] M. W. Evans, G. J. Evans, W. T. Coffey and P. Grigolini, Molecular Dynamics
and the Theory of Broad Band Spectroscopy (Wiley-Interscience,
New York, 1982).
[32] R. Furth, Phys. Lett., 13, 221 (1964).
[33] D. F. Crawford, Nature, 277, 633 (1979), papers in Apeiron, books by
Apeiron Press.
[34] reviewed by R. Zawodny in ref. (25), vol. 85(1). See also ref. (8).
[35] M. W. Evans, Physica B, 182, 227, 237 (1992).
[36] P. W. Atkins, Molecular Quantum Mechanics (Oxford Univ. Press, 1983,
2nd ed.).
[37] J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998, 3rd ed.).
31

BIBLIOGRAPHY
32

Chapter 4
Spectral Effects Of
Gravitation
by
F. Amador, P. Carpenter, G. J. Evans, M. W. Evans,
L. Felker, J. Guala-Valverde, D. Hamilton, J. B. Hart, J. Heidenreich,
A. Hill, G. P. Owen and J. Shelburne.
Alpha Foundation’s Institute for Advance Study (A.I.A.S.).
(emyrone@aol.com, www.aias.us, www.atomicprecision.com)
and
Energy Secretariat of the Argentine Government,
Fundación Julio Palacios,
Alderete 285 (8300) Neuquen,
Argentina.
Abstract
It is argued that the various cosmological shifts observed routinely are due to the
spectral effects of gravitation on electromagnetism. These effects originate in
the homogeneous current of ECE field theory and the homogeneous field equation.
In this paper the latter is re-expressed as a wave equation in the refractive
index of the ECE spacetime considered as a ponderable medium. Gravitation
can change the electric and magnetic field strengths of electromagnetic radiation
traversing a region of ECE spacetime, which acts like a dielectric of complex
permittivity and permeability. In consequence a spectrum is observable as the
result of the homogeneous current. The frequency, wave-number and phase velocity
of a monochromatic wave are changed by gravitation. If a polychromatic
light beam is considered, a rich variety of effects is expected in general.
Keywords: Spectral effects of gravitation, cosmological consequences of ECE
field theory.
33

4.1. INTRODUCTION
4.1 Introduction
A straightforward field unification scheme has recently been developed [1]– [30]
based on the philosophy of general relativity, that physics is an objective subject.
This was originally proposed by Francis Bacon in the sixteenth century and developed
into the theory of relativity at the turn of the twentieth century. Many
contributed to relativity theory prior to Einstein’s well known special theory
of 1905. The latter is routinely applied to dynamics and electrodynamics but
does not apply to gravitation or rotational motion. This is self contradictory
because electrodynamics is based on rotational motion, rotating electric and
magnetic fields implying automatically rotational accelerations. Special relativity
contains no accelerations yet is routinely applied to electrodynamics in the
Maxwell Heaviside field theory. The incorporation of acceleration into dynamics
(but not electrodynamics) was finally achieved within general relativity in 1915
by Einstein, and independently by Hilbert using a different lagrangian approach
to the problem. Both obtained the same field equation:
R µν − 1 2 Rg µν = kT µν (4.1)
where R µν is the Ricci tensor, R is the scalar curvature, g µν is the symmetric
metric, k is the Einstein constant and T µν is the canonical energy-momentum
tensor. Index contraction of Eq.4.1 [31] produces a particularly clear correlation
between geometry and physics:
R = −kT. (4.2)
Eq.4.1 was derived by Einstein from the cyclic equation of Ricci:
R σµνρ + R σνρµ + R σρµν = 0 (4.3)
where R σµνρ is the Riemann tensor with lowered indices. Hilbert derived the
field equation 4.1 by constructing a lagrangian.
The Ricci cyclic equation is often known as a Bianchi identity, but it is
neither an identity nor was it first derived by Bianchi. Eq.4.3 follows from the
assumption:
Γ κ µν = Γ κ νµ (4.4)
for the Christoffel symbol [32], and if this assumption is not true Eq.4.3 is not
true and the field equation 4.1 is not true. The torsion tensor [32] of Riemann
geometry is defined as:
T κ µν = Γ κ µν − Γ κ νµ (4.5)
so the assumption 4.4 is equivalent to assuming that the torsion tensor is zero.
This assumption severely restricts the validity of the famous theory of general
relativity, even within the restricted gravitational context for which the 1915
theory was developed and first tested. A consideration of torsion is best achieved
using Cartan geometry [32]. This geometry is fully equivalent to the most
general type of Riemann geometry but is more elegant and easier to work with.
Cartan geometry is essential for the development of a unified field theory and
for the development of a truly objective physics. Cartan himself was the first
to suggest that the electromagnetic field tensor is the Cartan torsion, more
34

CHAPTER 4.
SPECTRAL EFFECTS OF GRAVITATION
accurately the torsion form. The Einstein Cartan Evans (ECE) field theory [1]–
[30] develops this suggestion by Cartan into an objective unified field theory
applicable to the whole of physics and chemistry and related subject areas of
natural science.
In order to make the language of Cartan more understandable a few preliminary
definitions are given first before summarizing each section of this paper,
whose main aim is to develop a simple and easily understandable version of ECE
theory based on dielectric theory. The dielectric version of ECE field theory is
used to show that the well known cosmological shifts [33] are due to the effect
of gravitation on light in general relativity. They cannot be Doppler shifts in
special relativity because as just argued, special relativity cannot be applied
to electromagnetism, and cannot therefore be applied to the objective unified
field theory sorely needed to explain basic cosmological facts. This argument is
simple, but unfortunately, the Maxwell Heaviside field theory is still widely used
as the basis for an expanding universe through a Doppler type explanation of
red shifts in special relativity. The basic assumption of an expanding universe
theory has been refuted experimentally in many ways. For example it has been
shown [34] that different red shifts occur for objects equidistant from the Earth,
contradicting the simple Hubble Law. However the Big Bang theory (expanding
universe) is still widely taught despite the facts.
Cartan geometry is based on differential forms [1]– [30, 32]. The torsion
form is a two-form and its tensorial representation is a rank two antisymmetric
tensor. In addition there is an a index representing the tangent spacetime at a
point P to the base manifold. Therefore the torsion form is known as a vector
valued two-form and can be thought of as representing spin in a four dimensional
spacetime. The torsion form is represented in Cartan geometry by:
T a µν = −T a νµ . (4.6)
The antisymmetric Greek indices represent the base manifold. Therefore this
is the basic representation of electromagnetism as proposed by Cartan and in
consequence a unified field theory cannot be obtained without considerations of
torsion. The ECE field theory flows from this basic assumption [1]– [30].
In Section 4.2 the first structure equation of Cartan is used to derive the
extended Faraday Law of induction in the ECE theory, in which the influence
of gravitation on electromagnetism is accounted for objectively [1]– [30]. As
shown in previous papers the extended law can be expressed as a simple wave
equation in which appears the refractive index n. In general the latter is a
complex quantity, and can be used to generate a spectrum consisting as usual
of frequency dependent power absorption coefficients and frequency dependent
refractive indices. In general therefore the effect of gravitation on a light beam
reaching the Earth from a distant cosmological source is to create spectral effects
of various kinds. These cannot be the velocity dependent red shifts of special
relativity as asserted in the standard model because the latter is not objective
physics as argued. In other words general relativity is needed for a self consistent
description both of electromagnetism and gravitation. The fundamental
field of ECE theory is an eigenfunction, the tetrad governed by the ECE wave
equation [1]– [30, 32], and in consequence the refractive index n is quantized in
general. The nature of cosmological shifts is therefore defined by the regions of
ECE spacetime through which the light beam has traveled before reaching the
35

4.2. EXTENSION OF THE FARADAY LAW OF INDUCTION
Earthbound spectrometer. These are diverse because the universe is obviously
richly structured.
Finally in Section 4.3 the effects of gravitation on the angular frequency,
wavenumber and phase velocity of the light beam are defined.
4.2 Extension Of The Faraday Law Of Induction
In the standard model the Faraday law of induction is well known to be:
∇ × E + ∂B
∂t = 0 (4.7)
where E is the electric field strength in volt/m and where B is the magnetic
flux density in tesla. In ECE theory [1]– [30] Eq.4.7 is extended to:
n∇ × E a + 1 ∂B a
= 0 (4.8)
n ∂t
where a is the polarization index and n is the refractive index of ECE spacetime
considered as a dielectric or ponderable medium. The refractive index is defined
[35, 36] as:
n 2 = µɛ
µ 0 ɛ 0
(4.9)
where µ is the permeability and where ɛ is the permittivity of ECE spacetime,
and where µ 0 and ɛ 0 are their vacuum values. In general:
n = n ′ + in ′′ , (4.10)
µ = µ ′ + iµ ′′ , (4.11)
ɛ = ɛ ′ + iɛ ′′ , (4.12)
are complex quantities [35]– [37] whose real parts characterize dispersion and
whose imaginary parts characterize absorption or dielectric loss. The power
absorption coefficient is defined [37] by:
α(ω) = ωɛ′′ (ω)
n ′ (ω)c . (4.13)
Therefore the effect of gravitation on electromagnetism is measured by these
spectral quantities, for example by a graph of α(ω).
It has been shown [1]– [30] that Eq4.8 is equivalent to:
∇ × E a + ∂Ba
∂t
= ˜j a (4.14)
where ˜j a is the homogeneous current of ECE theory, a current defined by:
j a = A(0)
µ 0
(
R
a
b ∧ q b − ω a b ∧ T b) (4.15)
in standard differential form notation [32]. In Eq4.15 R a b
is the Riemann or
curvature form, q b is the tetrad form, ω a b is the spin connection and T b is the
36

CHAPTER 4.
SPECTRAL EFFECTS OF GRAVITATION
torsion form. When there is no influence of gravitation on electromagnetism [1]–
[30]:
R a b ∧ q b = ω a b ∧ T b (4.16)
and
n 2 = 1 (4.17)
Eqs.4.16 and 4.17 define the validity of the Faraday law of induction of the
standard model. The latter is further restricted by the subjective, nineteenth
century, assumption that electromagnetism is an entity superimposed on a passive
or static frame of reference, whereas in ECE theory and general relativity
electromagnetism is described objectively by Cartan geometry. Loosely writing,
electromagnetism in an objective physics is a spinning frame, gravitation in an
objective physics is a curving frame.
Using the two Cartan structure equations [1]– [30, 32]:
and
it is seen that Eq.4.16 implies:
T a = d ∧ q a + ω a b ∧ q b (4.18)
R a b = d ∧ ω a b + ω a c ∧ ω c b (4.19)
ω a b = κɛ a bcq c , (4.20)
R a b = κɛ a bcT c , (4.21)
where κ has the units of inverse meters or wave-number. Eqs.4.20 and 4.21
show that ω a b is dual to qc and that R a b is dual to T c , and these two equations
define the validity of the Faraday law of induction of the standard model for
each polarization index [1]– [30]:
Eqs.4.20 and 4.21 are equivalent to:
a = (1), (2) and (3). (4.22)
˜ja = 0, (4.23)
n 2 = 1, (4.24)
µ = µ 0 , (4.25)
ɛ = ɛ 0 . (4.26)
In Eqs.4.20 and 4.21 the spin connection ω a b and Riemann form Ra b
are antisymmetric
in their a and b indices. The wave-number in Eqs.4.20 and 4.21 is
defined as
κ = ω (4.27)
c
where ω is the angular frequency (radians / s) and where c is the vacuum speed
of light (m / s).
The effect of gravitation is to produce:
˜ja ≠ 0, (4.28)
n 2 ≠ 1, (4.29)
37

4.2. EXTENSION OF THE FARADAY LAW OF INDUCTION
and from Eq.4.8:
µ ≠ µ 0 , (4.30)
ɛ ≠ ɛ 0 , (4.31)
∇ × (nE a ) + ∂ ∂t (Ba /n) = 0, (4.32)
i.e. to produce the shifts:
E a → nE a ,
B a → Ba
n . (4.33)
In form notation Eq.4.33 means that the field two-form F a is changed by gravitation.
The Evans Ansatz [1]– [30]:
implies that:
where:
F a = A (0) T a (4.34)
F a = d ∧ A a + ω a b ∧ A b (4.35)
A a = A (0) q a (4.36)
is the potential one-form. The latter is defined by the ECE wave equation:
□A a = RA a (4.37)
in which R is defined by Eq.4.2, the fundamental and objective correlation
between geometry and physics in relativity theory. Therefore gravitation shifts
both the potential and the field from their values in the vacuum to their values in
the presence of gravitation. The interaction of gravitation and electromagnetism
changes the symmetry of the spin connection and Riemann form from antisymmetric
to asymmetric in a and b, so the duality conditions 4.20 and 4.21 no
longer hold. The interaction affects R because the tetrad is affected, and so in
consequence T is also affected. These geometrical changes manifest themselves
in observable cosmological spectra currently referred to as shifts. These effects
can be summarized [1]– [30] as a polarization (P a ) and magnetization (M a ) of
ECE space-time through the homogeneous current:
˜ja = ∂Ma
∂t
− 1
µ 0 ɛ 0
∇ × P a . (4.38)
Therefore in general gravitation has several, hitherto unknown, effects on electromagnetism.
The problem of describing the various observable cosmological shifts is described
objectively in ECE theory using Cartan geometry and the fundamental
Ansatz 4.2 [1]– [30]. Rigorous objectivity is given by the structure of Cartan
geometry. One way of approaching the problem is to define a model for T or R
and to solve for A a from the eigenequation 4.37. Conversely the eigenfunction
A 0 may be modeled to give the set of eigenvalues R or T . Eq4.37 is an equation
of wave mechanics, i.e. of rigorously objective quantum electrodynamics influenced
by gravitation. Its solutions are eigenvalues which translate into spectral
lines or in contemporary parlance quantized cosmological shifts. The latter are
now routinely observable and cannot be explained by the red shift to distance
38

CHAPTER 4.
SPECTRAL EFFECTS OF GRAVITATION
correlation of the Hubble Law. It makes no sense to try to describe these spectra
with a complete universe somehow expanding in sudden jumps, thus vividly
exposing the fallacy in the Big Bang theory. The spectral patterns from the
ECE wave equation are governed as in the Schrödinger equation (a well defined
limit of the ECE wave equation) by the model chosen for T or the hamiltonian.
This argument is analogous to the unequally spaced lines of atomic spectra, for
example, where the Coulomb law is used in the model hamiltonian, or to the
equally spaced lines given by a harmonic oscillator or Hooke’s law model for the
hamiltonian at the root of quantum electrodynamics and photon number theory.
Recall that these models are being applied conceptually here to ECE space-time
in order to model the effect of gravitation on electromagnetism. The rigorously
complete problem is defined by Cartan geometry and probably requires numerical
methods of solution. However some simple analytical examples of solution
are possible, and have been given in this series of papers and books [1]– [30].
Without immediate recourse to supercomputers it is possible as in this paper
to fit observable cosmological shifts of any kind with a model for the refractive
index n of ECE spacetime. This makes a map of a region of spacetime in the
universe in terms of n. The spectrum defined as a graph of power absorption coefficient
against angular frequency is the Fourier transform [37] of the rotational
velocity correlation function:
c (t) = 〈 ˙µ (t) · ˙µ (0)〉 (4.39)
where µ is the dipole moment. In the far infra red for example a statistical
model such as Mori theory [37] can be used to build up the relevant correlation
functions, which can also be obtained from molecular dynamics computer
simulation [38]. Arguing by analogy, the same type of modelling can be used
to build up the refractive index n in ECE theory. Yet another approach would
be to build up the spin connection from perturbation theory and proceed from
there.
4.3 The Effect Of Gravitation On The Wave Properties
Of A Light Beam
Gravitation changes the fundamental wave equation of the light beam from:
∇ × E a + ∂Ba
∂t
= 0 (4.40)
to:
n∇ × E a + 1 ∂B a
= 0. (4.41)
n ∂t
Therefore the phase of the light wave (or radio frequency wave) is changed from:
φ = ωt − κZ (4.42)
to
φ = ω t − nκZ. (4.43)
n
The fundamental S.I. equation:
E (0) = cB (0) (4.44)
39

4.3. THE EFFECT OF GRAVITATION ON THE WAVE . . .
is changed to:
E (0) = vB (0) (4.45)
where
v = c
n 2 . (4.46)
The angular frequency of the beam is changed as follows by gravitation:
ω −→ ω n
(4.47)
and the wavenumber of the beam is changed by gravitation as follows:
κ −→ nκ. (4.48)
Finally the phase velocity of the beam is changed by gravitation as follows:
c −→ c
n 2 . (4.49)
When there is no absorption the refractive index is a real quantity [35]– [37]
and is well known to cause the phenomenon of refraction. Thus gravitation in
ECE space-time causes refraction, in other words changes the path of the light
beam as observed in the Eddington effect. Thus ECE theory gives the required
classical explanation of this well known effect. In the contemporary standard
model the explanation of the Eddington effect is still the semi-quantum model
proposed by Einstein in terms of the gravitational effect of an object on the
photon mass. ECE theory gives this explanation [1]– [30], because Einstein’s
theory of general relativity is a limit of ECE theory, but the latter also gives
the much needed classical explanation in terms of refraction.
In any red shift ECE theory shows that the frequency is lowered according to
Eq.4.47 by n > 1. Thus the no blue shift rule of cosmology is given immediately
by ECE field theory through the fact that the refractive index is almost always
greater than one in nature. Only under very special conditions can n appear
to be less than unity in some manufactured composites, and even then this is
not a fundamental property. So shifts are almost always red shifts. This well
known observation of astronomy has nothing to do with an assumed expanding
universe or Big Bang theory. The latter is a subjective construct made in the
absence of an objective unified field theory. Some parts of ECE spacetime (the
universe) may be locally expanding, but other parts may be locally contracting.
It is vanishingly unlikely that such a big and complicated place such as the
universe will all expand uniformly from an assumed single initial condition at
an assumed single time and place, or event. Some parts of the universe may
contract locally to a very dense condition, then re-expand to produce features
such as galaxies, stars and planets. In a simple red shift without absorption
the frequency appears to be lowered as in Eq.4.47. Thus a simple red shift
is explained in ECE theory as diffraction, i.e. the Eddington effect. More
generally, when there is absorption then the refractive index becomes complex
valued [37]:
n = n ′ + in ′′ (4.50)
so the effect of gravitation on a light beam in this case is as follows
ω −→ (n′ − in ′′ )
n ′2 ω (4.51)
+ n
′′2
40

CHAPTER 4.
SPECTRAL EFFECTS OF GRAVITATION
The real and physical part of ω is thus shifted as follows:
A red shift occurs because:
Re (ω) −→
n′ ω
n ′2 + n ′′2 (4.52)
n ′2 + n ′′2 > n ′ (4.53)
but this red shift depends on the real and imaginary parts of the refractive index
and is no longer a simple red shift.
The definitions of this section are for a monochromatic source - in a polychromatic
source there will be different shifts for different frequencies, thus building
up a SPECTRUM of shifts in an Earth-bound or Earth orbiting telescope.
Acknowledgments The British Government is thanked for a Civil List pension
(2005) to MWE. Franklin Amador is thanked for meticulous typesetting,
Ted Annis, Tony Craddock and John B.Hart for funding, and the staff of AIAS
and others for many interesting discussions.
41

4.3. THE EFFECT OF GRAVITATION ON THE WAVE . . .
42

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[2] M. W. Evans, Found. Phys. Lett., 17, 25, 149, 267, 301, 393, 433, 535, 663
(2004).
[3] M. W. Evans, Found. Phys. Lett., 18, 139, 259, 519 (2005), see also M. W
Evans, papers in Found. Phys. Lett. And Found. Phys., 1994 to 2002.
[4] M. W. Evans, Generally Covariant Unified Field Theory (in press 2005,
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[5] L. Felker, The Evans Equations of Unified Field Theory (in press, preprint
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[6] M. W. Evans, The Objective Laws of Classical Electrodynamics, the Effect
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[7] M. W. Evans, First and Second Order Aharonov Bohm Effects
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[8] M. W. Evans, The Spinning of Space-time as Seen in the Inverse Faraday
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on www.aias.us and www.atomicprecision.com.
[10] M. W. Evans, Explanation of the Eddington Experiment in the Evans Unified
Field Theory, preprint on www.aias.us and www.atomicprecision.com.
[11] M. W. Evans, The Coulomb and Ampère Maxwell Laws in the
Schwarzschild Metric: A Classical Explanation of the Eddington Effect
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[12] M. W. Evans, Generally Covariant Heisenberg Equation from
the Evans Unified Field Theory, preprint on www.aias.us and
www.atomicprecision.com.
[13] M. W. Evans, Metric Compatibility and the Tetrad Postulate, preprint on
www.aias.us and www.atomicprecision.com.
43

BIBLIOGRAPHY
[14] M. W. Evans, Derivation of the Evans Lemma and Wave Equation
from the First Cartan Structure Equation, preprint on www.aias.us and
www.atomicprecision.com.
[15] M. W. Evans, Proof of the Evans Lemma from the Tetrad Postulate,
preprint on www.aias.us and www.atomicprecision.com.
[16] M. W. Evans, Self-Consistent Derivation of the Evans Lemma and Application
to the Generally Covariant Dirac Equation, preprint on www.aias.us
and www.atomicprecision.com.
[17] M. W. Evans, Quark-Gluon Model in the Evans Unified Field Theory,
preprint on www.aias.us and www.atomicprecision.com.
[18] M. W. Evans, The Origin of Intrinsic Spin and the Pauli Exclusion Principle
in the Evans Unified Field Theory, preprint on www.aias.us and
www.atomicprecisoin.com).
[19] M. W Evans, General Covariance and Co-ordinate Transformation in
Classical and Quantum Electrodynamics, preprint on www.aias.us and
www.atomicprecision.com.
[20] M. W. Evans, The Role of Gravitational Torsion in General Relativity: the
S Tensor, preprint on www.aias.us and www.atomicprecision.com.
[21] M. W. Evans, Explanation of the Faraday Disc Generator in the Evans Unified
Field Theory, preprint on www.aias.us and www.atomicprecision.com.
[22] F.Amador, P. Carpenter, G. J. Evans, M. W. Evans, L. Felker, J. Guala-
Valverde, D. Hamilton, J. B. Hart, J. Heidenreich, A. Hill, G. P. Owen
and J. Shelburne, Experiments to Test the Evans Unified Field Theory
and General Relativity in Classical Electrodynamics, Found. Phys. Lett.,
submitted (preprint on www.aias.us and www.atomicprecision.com.
[23] F. Amador, P. Carpenter, G. J. Evans, M. W. Evans, L. Felker, J. Guala-
Valverde, D. Hamilton, J. B. Hart, J. Heidenreich, A. Hill, G. P. Owen and
J. Shelburne, ECE Field Theory of the Sagnac Effect, Found. Phys. Lett.,
submitted (2005, preprint on www.aias.us and www.atomicprecision.com.
[24] F. Amador, P. Carpenter, G. J. Evans, M. W. Evans, L. Felker, J. Guala-
Valverde, D. Hamilton, J. B. Hart, J. Heidenreich, A. Hill, G. P. Owen and
J. Shelburne, Einstein Cartan Evans (ECE) field Theory : The Influence of
Gravitation on the Sagnac Effect, Found. Phys. Lett., submitted (preprints
on www.aias.us and www.atomicprecision.com.
[25] F. Amador, P. Carpenter, G. J. Evans, M. W. Evans, L. Felker, J. Guala-
Valverde, D. Hamilton, J. B. Hart, J. Heidenreich, A. Hill, G. P. Owen and
J. Shelburne, Dielectric Theory of ECE Spacetime, Found. Phys. Lett.,
submitted (preprints on www.aias.us and www.atomicprecision.com.
[26] M. W. Evans (ed.), Modern Non-Linear Optics, a special topical issue in
three parts of I. Prigogine and S. A. Rice (eds.), Advances in Chemical
Physics (Wiley Interscience, New York, 2001, 2nd ed..) vols. 119(1) - 119(3).
44

BIBLIOGRAPHY
[27] M. W. Evans and L. B. Crowell, Classical and Quantum Electrodynamics
and the B (3) Field, (World Scientific, Singapore, 2001).
[28] M. W. Evans and J.-P. Vigier, The Enigmatic Photon (Kluwer, Dordrecht,
1994 - 2002, hardback and softback), vols. 1 - 5.
[29] M. W. Evans and A A. Hasanein, The Photomagneton in Quantum Field
Theory (World Scientific, Singapore, 1994).
[30] M. W. Evans and S. Kielich (eds.), first edition of ref. (26) (Wiley Interscience,
New York, 1992, reprinted 1993, softback 1997), vols. 85(1) -
85(3).
[31] A. Einstein, The Meaning of Relativity (Princeton Univ. Press, 1921 -
1954).
[32] S. P. Carroll, Lecture Notes in General Relativity ( graduate course at
Harvard, UCSB and Univ Chicago, public domain) arXiv : gr - gc 973019
v1 1997).
[33] R. Keys (ed.), www.redshift.vif.com
[34] R. Keys (ed.), papers and books of the Apeiron publishing house (Montreal).
[35] P. W. Atkins, Molecular Quantum Mechanics (Oxford Univ. Press, 1982,
2nd ed.).
[36] J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998, 3rd ed.).
[37] M. W. Evans, G. J. Evans, W. T. Coffey and P. Grigolini, Molecular Dynamics
and the Theory of Broad Band Spectroscopy (Wiley Interscience,
New York, 1982).
[38] M. W. Evans in I. Prigogine and S. A Rice (series eds.), Advances in Chemical
Physics (Wiley Interscience, New York, 1991), vol.
45

BIBLIOGRAPHY
46

Chapter 5
Cosmological Anomalies:
EH Versus ECE Field
Theory
by
M. W. Evans
Alpha Foundation’s Institute for Advance Study (A.I.A.S.).
(emyrone@aol.com, www.aias.us, www.atomicprecision.com)
Abstract
Some anomalies of contemporary cosmology are discussed by considering the advantages
of Einstein Cartan Evans (ECE) unified field theory over the original
Einstein Hilbert (EH) un-unified gravitational field theory of 1915. A classical
explanation of diffraction in the Eddington effect is given, including ring
diffraction. A new explanation is given of the anomalies that apparently lead
to missing mass using the Beer Lambert law to account for observable intergalactic
absorption and to redefine the relation between intrinsic luminosity,
distance and mass of a cosmological object. ECE theory allows a self-consistent
explanation of differing red shifts for equidistant objects, something which is
not allowed for by the Hubble law. Red shifts in ECE field theory are due to
the relative permeability of ECE spacetime, so that rotational as well as translational
motion is taken into account in ECE theory. Only translational motion
is considered in the EH theory underpinning the concept of Big Bang. It is
shown that Big Bang is riddled with anomalies which can be addressed by the
required ECE unified field theory.
Keywords: Cosmological anomalies, Einstein Hilbert (EH) field theory, Einstein
Cartan Evans (ECE) field theory, Beer-Lambert law, Eddington effect, diffraction,
red-shift, missing mass anomaly, diffraction rings in cosmology, quantized
red shifts, equidistant anomalies in the Hubble law, permeability in ECE field
theory, gravitational dependence of permeability.
47

5.1. INTRODUCTION
5.1 Introduction
A theory of the unification of gravitation with electromagnetism and other radiated
and matter fields has recently been suggested [1]– [31] and has rapidly
become the main field theory in physics [32] because it is predictive, relatively
simple, and much more powerful than other attempts at field unification such
as the unpredictive and unphysical string theories of pure mathematics. The
new unified field theory has been well tested experimentally and is based on
the ideas of relativity theory and thus of rigorously objective physics. Such a
unified field theory is needed to describe cosmology - a subject based on the
spectroscopic observations of astronomy and therefore on the properties of electromagnetic
radiation. A cosmological theory should therefore be based on the
interaction of light and electromagnetic radiation with gravitation, but the prevailing
Big Bang theory is purely gravitational, being based on the Einstein
Hilbert (EH) theory of 1915 [33], a theory that does not deal self consistently
with electromagnetism. The Einstein Cartan Evans (ECE) unified field theory
[1]– [31]accounts self consistently for the effect of gravitation on light and
electromagnetic radiation.
In Section 5.2 the homogeneous field equation of the dielectric version of ECE
theory is applied to give an explanation of red shifts as spectral phenomena,
not necessarily due to an expanding universe. The origin of shifts to lower
frequency (red shifts) in cosmology is traced to the relative permeability of ECE
spacetime, a four dimensional spacetime with torsion as well as curvature [34].
In EH field theory there is no torsion, and in consequence EH theory loses a
great deal of information. For example, EH theory does not have the ability
to describe classically refraction effects in cosmology. The most well known
refraction effect is the Eddington effect [35] and this is described using a semiclassical
theory in which a photon is captured by the gravitational field of the
sun. The path of the photon is therefore an orbit. This is a kinematic and
central theory in which the photon is assumed to have a mass which is centrally
attracted to the sun’s mass. The resulting orbit in EH theory is found to produce
twice the deflection of Newtonian theory, the weak field limit of EH theory.
However, the assumed photon mass does not appear in the final expression for
the orbit of the photon and so the photon mass cannot be estimated from the
Eddington experiment. Despite the known accuracy of this central theory (one
part in one hundred thousand [35]) it is therefore only partly successful, in
particular there is no classical electrodynamics in the EH theory by definition.
ECE theory on the other hand builds in classical electrodynamics and is able
to describe the Eddington effect classically [1]– [31] in terms of refraction in an
inhomogeneous dielectric whose refractive index is a function of ECE spacetime.
This refraction is accompanied by a red shift in frequency ω/µ r where µ r is the
relative permeability of EEC spacetime. Both the refraction and the red shift
are caused in ECE theory by the interaction of gravitation with electromagnetic
waves on the classical level. The functional dependence of the refractive index
n on ct, X, Y and Z defines the observed orbit of the light. The EH theory
is the well defined zero torsion (or central) limit of ECE theory and so the
Eddington effect can also be described by the central limit of ECE theory, its
kinematic part. Therefore ECE theory gives both the classical and the well
known semi-classical theory of the Eddington effect. The orbit of the light is
therefore defined to one part in one hundred thousand by the central or semi-
48

CHAPTER 5. COSMOLOGICAL ANOMALIES: EH VERSUS ECE . . .
classical part of ECE and so the functional dependence of the refractive index
on ct, X, Y, and Z is also defined classically and accurately to one part in one
hundred thousand.
ECE theory is however capable of giving much more classical information
from its homogeneous field equation. The explanation in Section 5.2 is confined
to refraction in the absence of optical absorption. In section 5.3 optical
absorption is considered classically using the Beer Lambert law and by using a
complex-valued refractive index. The latter is a function of the power absorption
coefficient of the Beer Lambert law [36]. Using this information it is shown
in Section 3 that various type of red shifts are possible due to the absorption and
simultaneous refraction or dispersion of light from a distant cosmological object
in inter-stellar or inter-galactic spacetime. One possible cause of absorption is
the interaction of the light with the gravitation of well observed [37] inter galactic
dust particles. It is well known experimentally [38] that radio frequencies
are absorbed in inter galactic and inter stellar regions. In general the light from
equidistant sources in different parts of the sky is absorbed in different ways
on its journey to the observing telescope, giving rise to different red shifts for
equidistant objects as observed by Arp [39]. These findings invalidate the simple
Hubble law, both experimentally (Arp) and theoretically (ECE theory). Red
shifts alone cannot therefore be used to measure the distance of an object from
a telescope. The only valid method is direct parallax measurements currently
possible only for objects relatively close to the observing telescope.
The arguments in Sections 5.2 and 5.3 cast further doubt [40] on the basic
premise underpinning the Big Bang model - that inter-stellar and inter-galactic
space is a vacuum, and that the observed red shift of galaxies is due to expansion
of the metric of the universe according to the EH field equation. The ECE
field theory shows that the underlying red shift could be due to the absorption
coefficient and not the expansion of the metric. Assis [40] has argued that if the
power absorption coefficient of the Beer Lambert law is H/c then the Hubble
law follows. Here H is the Hubble constant and c is the vacuum speed of light.
There may therefore be a constant background absorption in the observable
universe which means a constant background refractive index greater than the
unity of the vacuum. Assis has also shown [40] that the Beer Lambert law leads
straightforwardly to a more accurate value of the 2.7 K background radiation
than Big Bang, and Vigier et al. [41] have shown that the tired light model
is more accurate in several ways than Big Bang. The tired light model may
also be obtained from the Beer Lambert law [40]. In Big Bang cosmology
the Beer Lambert law is not considered, despite the contrary evidence of the
Eddington effect. ECE theory shows that superimposed on the cosmological red
shift there may be a theoretically infinite number of different types of red shift
due to different types of absorption in different parts of the universe, a big and
complicated place. Examples are the data given by Arp [40], the quantized red
shifts and the diffraction rings observed in contemporary gravitational lensing.
It is shown in Section 5.4 that these arguments also cast grave doubt on the way
in which luminosity is related to mass and distance in conventional cosmology,
so the arguments for missing mass and dark matter may be false. They could
equally well be due to absorption. Also, it has been argued already that the
criteria used to measure the distance of far galaxies by red shift also depend
on the assumption that inter stellar and inter galactic spacetime is a vacuum,
whereas this assumption has already been invalidated by data [40] - for example
49

5.2. RED SHIFT BY THE RELATIVE PERMEABILITY OF ECE . . .
the existence of inter galactic and inter stellar dust [37] and the absorption of
radio waves [38] in these regions. Finally in Section 5.4 it is argued that the
apparent acceleration of far galaxies is due to a changing refractive index of EEC
spacetime in these regions. However, as soon as doubt is cast on the Hubble
law the actual distance of these objects becomes unknown. The reason is that
this distance cannot be measured by parallax and the Hubble law cannot be
used to estimate these distances from red shifts alone. So it is not even known
whether far galaxies are in fact more distant than near galaxies, they only
APPEAR to be more distant because of what has happened to the light on the
journey from source to observing telescope. The overall conclusion is that the
universe is as likely to be in a non-expanding steady state as expanding. There
may be local regions of expansion giving rise to the formation of galaxies, stars
and planets, and to the distribution of the elements as argued by Pinter [42]
in convincing detail. In Section 5.5 it is argued that recent spacecraft data
show many gravitational anomalies within the solar system. Jensen [43] has
argued that these anomalies are due to a mass dependent permeability. The
latter is incorporated into ECE theory in Section 5.5 using the fact that in
pure gravitational theory the torsion is not in general zero. Incorporation of
torsion leads to a mass dependent permeability as used by Jensen [43] to explain
numerous gravitational anomalies in EH theory within the solar system.
5.2 Red Shift By The Relative Permeability of
ECE Spacetime
It has been shown [1]– [31] that the homogeneous field equation of ECE theory
gives the Faraday law of induction in the required objective form, it must
be an equation of general relativity rather than special relativity as must all
equations of physics in the theory of relativity. The requirement for objectivity
in physics goes back to Bacon in the sixteenth century. Thus, objectivity in
physics has the major advantage of giving a classical equation for the effect of
gravitation on electromagnetism. A mechanism is therefore deduced that has
not been hitherto considered in cosmology and astronomy for explaining the
well known cosmological red shift in terms of the effect of gravitational fields on
electromagnetic radiation emanating from a far distant source. The frequency
of a feature such as the sodium D line is decreased to ω/µ r as argued in the
introduction and this effect does not necessarily imply that distant cosmological
objects such as galaxies are moving away from the observer. The objective (i.e.
generally relativistic) Faraday law of induction is [1]– [31]:
∇ × E a + ∂Ba
∂t
= µ 0˜ja . (5.1)
Here E a is the electric field strength (volt / m) and B a the magnetic flux density
(in tesla) of electromagnetic radiation such as a light beam emitted by a distant
source and observed in a telescope. In Eq.5.1 µ 0 is the vacuum permeability
and ˜j a is the homogeneous current [1]– [31] of ECE theory. It has been shown
that Eq.5.1 can be re-written as:
∇ × D a + µ 0 ɛ 0
∂H a
∂t
= 0 (5.2)
50

CHAPTER 5. COSMOLOGICAL ANOMALIES: EH VERSUS ECE . . .
where D a is the displacement, H a the magnetic field strength and ɛ 0 the vacuum
permittivity. In these equations the index a denotes the state of polarization of
the light beam. For example in the complex circular basis [1]– [31], the space-like
polarizations are denoted:
a = (1), (2), (3) (5.3)
where (1) and (2) indicate complex conjugate transverse states and (3) indicates
a longitudinal state of polarization. In general the permittivity and permeability
are functions of spacetime:
n 2 = ɛµ/ɛ 0 µ 0 (5.4)
and so the refractive index, n, is also a function of spacetime.
In Eq.5.2 the following definitions are used:
D a = ɛ 0 E a + P a = ɛE a (5.5)
H a = 1 µ 0
B a − M a = 1 µ Ba (5.6)
where P a is the polarization of ECE spacetime and M a is the magnetization
of ECE spacetime considered as a ponderable medium or dielectric with properties
different from the vacuum. Here ɛ is the absolute permittivity and µ the
absolute permeability of this dielectric. In order to obtain Eq.5.2 from Eq.5.1
the homogeneous current must be defined as:
˜ja = ∂Ma
∂t
− 1
µ 0 ɛ 0
∇ × P a (5.7)
and is the mechanism responsible for the interaction of gravitation with the
light beam as the latter travels from source to telescope, a distance Z. Over
this immense distance it is certain that the light beam encounters myriad species
of gravitational field before reaching the telescope and the observer. However
weak these fields may be in inter stellar and inter galactic ECE spacetime, the
enormous path length Z amplifies the current ˜j a to measurable levels, and appears
in the telescope as a red shift. This inference is analogous to the well
known fact that the absorption coefficient in spectroscopy depends on the path
length - the greater the path length the greater the absorption of the light beam
and the weaker the signal at the detector. Therefore what is always observed
in astronomy is the effect of gravitation on light through the current of Eq.5.7
- in general an absorption (or dielectric loss) accompanied by a dispersion (a
change in the refractive index). It is also well known in spectroscopy that the
more dilute the sample the sharper are the spectral features (the effect of collisional
broadening is decreased by dilution). Since inter stellar and inter galactic
spacetime is very tenuous (or dilute) the stars and galaxies appear sharply defined.
This does mean at all that the spacetime is empty or void as in Big Bang
theory [40]. The empty inter stellar and inter galactic spacetime of Big Bang is
defined by EH theory alone, without any classical consideration of the classical
effect of gravitation on a light beam. The red shifts are defined in Big Bang
by a particular solution to the EH field equations using a given metric. No
account is taken of the homogeneous current ˜j a and so the effect of gravitation
on light is not considered classically. These are major omissions, leading to the
apparent conclusion that the universe is expanding - simply because the metric
51

5.2. RED SHIFT BY THE RELATIVE PERMEABILITY OF ECE . . .
demands this conclusion. This is however a circular argument - the conclusion
(expanding metric deduced) is programmed in at the beginning (expanding
metric assumed).
The homogeneous current of ECE field theory [1]– [31] is defined by the
Bianchi identity of Cartan geometry and in standard differential form notation
[34] is:
j a = A(0)
µ 0
(
R
a
b ∧ q b − ω a b ∧ T b) . (5.8)
Here A (0) is the fundamental vector potential magnitude of ECE field theory,
R a b is the curvature or Riemann form, T a is the torsion form, ω a b
is the spin
connection and q a is the tetrad form, the fundamental field of ECE theory. It
is seen from Eq.5.8 that the interaction of the light beam with gravitation is
governed by geometry as required in relativity theory [34]. Without going in
to the details of the geometry and without using supercomputers we can go far
using the dielectric version of ECE field theory.
Using Eqs.5.5 and 5.6 Eq. [2] can be written as:
∇ × (ɛ r E a ) + ∂ ( ) B
a
= 0 (5.9)
∂t µ r
where the relative permittivity ɛ r , and relative permeability µ r are defined as
ɛ r = ɛ/ɛ 0 , µ r = µ/µ 0 . (5.10)
Therefore the effect of gravitation on a light beam is summarized by:
E a → ɛ r E a (5.11)
B a → 1 µ r
B a . (5.12)
The various changes to the light beam caused by gravitation are given by solutions
of Eq.5.9, either analytical or numerical. With a powerful enough computer
we can calculate these changes from the original geometry of Cartan but
proceed here without loss of insight or generality using the summary structure
of Eq.5.9. Cosmological red shifts and the Eddington type of experiments [35]
show that gravitation influences light and this influence is described classically
for the first time by Eq.5.9. Such an influence may also be detectable on the opposite
microscopic scale in the close vicinity of an electron in a circuit. Near an
electron, spacetime is curved considerably and electric and magnetic fields are
intense. These are ideal conditions for the generation of the homogeneous current
˜j a . Therefore electric power can be obtained from ECE spacetime through
the current ˜j a . If harnessed technologically this power is of clear importance.
The laboratory conditions under which the Faraday law of induction is usually
tested are intermediate between the macroscopic domain of cosmology and
the microscopic domain near one electron. This is the reason why the special
relativistic Faraday law of induction:
∇ × E + ∂B
∂t = 0 (5.13)
appears to be adequate. Under these conditions gravitation is very weak in
comparison with electromagnetism, and this limit is described by:
ɛ r → 1, µ r → 1, n → 1, (5.14)
52

CHAPTER 5. COSMOLOGICAL ANOMALIES: EH VERSUS ECE . . .
˜ja → 0. (5.15)
Therefore the homogeneous current vanishes in this limit, and the ECE spacetime
reduces to that of the vacuum. The Faraday law of induction under these
conditions is written conventionally without reference to the polarization index
a, and is a law of electromagnetism uninfluenced by gravitation. In this limit the
ECE spacetime reduces to a Minkowski spacetime and the equations of electromagnetism
are covariant under the Lorentz transformation only. More generally
they must be covariant under the general coordinate transformation [1]– [31,34]
as for any equations of general relativity. These are therefore the limits of validity
of the well known Maxwell Heaviside field equations. The Faraday law of
induction 5.13 is one of these equations. In the elegant differential form notation
of Cartan geometry [34] the Maxwell Heaviside equations are well known to be:
d ∧ F = 0 (5.16)
d ∧ ˜F = µ 0 J (5.17)
where J is the inhomogeneous current and where d∧ is the exterior derivative
of Cartan geometry. The tilde denotes the Hodge dual transform [1]– [31, 34].
In ECE field theory Eqs.5.16 and 5.17, respectively its homogeneous and inhomogeneous
field equations, evolve to:
d ∧ F a = µ 0 j a (5.18)
d ∧ ˜F a = µ 0 J a (5.19)
and Eq.5.1 is part of Eq.5.18 in vector notation rather than differential form
notation. Eqs.5.18 and 5.19 are capable of describing the effect of gravitation
on electromagnetism, whereas Eqs.5.16 and 5.17 are not.
The well known plane waves [44, 45] of light are exact transverse solutions
to Eq.5.13 for each index a. For example, for
the transverse electric and magnetic plane waves are:
a = (1) (5.20)
E (1) = E(0)
√
2
(i − ij) e (i(ωt−κZ)) , (5.21)
B (1) = B(0)
√
2
(ii + j) e (i(ωt−κZ)) . (5.22)
Here ω is the angular frequency at an instant of time t of electromagnetic radiation
propagating with a phase velocity c in the vacuum, and κ is the wave
vector magnitude of the radiation at a point Z, which is the axis of propagation
being considered. Here i and j are unit vectors in the X and Y axes. The unit
vectors of the complex circular basis are defined by [1]– [31]:
e (1) = e (2)∗ = 1 √
2
(i − ij) , (5.23)
e (3) = k, (5.24)
53

5.2. RED SHIFT BY THE RELATIVE PERMEABILITY OF ECE . . .
and the light beam described by Eqs.5.21 and 5.22 is circularly polarized. The
Evans spin field of the radiation propagates with it and is defined [1]– [31] by:
B (3) = B (3)∗
= −igA (1) × A (2)
= B (0) k
(5.25)
where
A (1) = A (2)∗ (5.26)
is the vector potential of the wave. Here:
g =
κ
A (0) (5.27)
The spin field is observed experimentally in the inverse Faraday effect [1]– [31]-
the magnetization of matter by circularly polarized electromagnetic radiation -
and the spin field is the key property of light and electromagnetic radiation that
shows that the latter is a phenomenon not of special relativity but of general
relativity [34]. In special relativity the Evans spin field is undefined but in
general relativity it is well defined because light is realized to be the spinning of
ECE spacetime. This is an important inference of ECE field theory, an inference
which overhauls the theory of light from the nineteenth to twenty first centuries.
In the Cartan representation of the nineteenth century Maxwell Heaviside field
theory the relation between the field and potential of light is [1]– [31]:
In ECE field theory of the twenty first century it is:
F = d ∧ A. (5.28)
F a = d ∧ A a + ω a b ∧ A b (5.29)
where the spin connection self consistently defines the spinning frame and the
Evans spin field [1]– [31] of general relativity. Eq.5.29 cures a problem that
plagued field theory throughout the twentieth century: gravitation was considered
to be a curving spacetime (the EH spacetime) whereas electromagnetism
was considered to be still the pre-relativistic nineteenth century theory: essentially
an abstract entity (the electromagnetic field) superimposed on a separate
frame - the passive and flat Minkowski spacetime of special relativity. This
severe self inconsistency prevented the unification of gravitation with electromagnetism
for over one hundred and fifty years and therefore blocked the development
of cosmology. In the unified ECE field theory [1]– [31] the true nature
of cosmology can be seen to be the interaction of gravitation with light reaching
telescopes in astronomy. The interaction is precisely defined by the currents
j a and J a . In this paper we restrict attention to j a and to simple analytical
solutions of Eq.5.9.
The simplest solution of all can be deduced by noting from Eq.5.22 that:
∂B (1)
∂t
= iωB (1) = −ω B(0)
√
2
(i − ij) e (i(ωt−κZ)) (5.30)
54

CHAPTER 5. COSMOLOGICAL ANOMALIES: EH VERSUS ECE . . .
and from Eq.5.21 that:
∣ ∣∣∣∣∣ i j k
∇ × E (1) = E(0)
∂ ∂ ∂
√ 2
∂X ∂Y ∂Z
e iφ −ie iφ 0 ∣
(5.31)
= −i E(0)
√ κ (ii + j) e iφ
2
where
Therefore
From Eqs.5.30 and 5.31, Eq.5.33 is true if:
φ = ωt − κZ. (5.32)
ωB (0) = κE (0) . (5.33)
E (0) = cB (0) (5.34)
i.e. if:
and if the phase velocity is:
ω
κ = E(0)
B (0) (5.35)
c = ω κ . (5.36)
In order to find the required simplest possible solution of Eq.5.9, identify the
phase as:
φ = ω µ r
t − ɛ r κZ. (5.37)
If for the sake of simplicity we assume that the relative permeability µ r is a
function of X, Y and Z but not a function of t then:
∂
∂t eiφ = iω
µ r
e iφ . (5.38)
It has also been assumed implicitly that the wave-vector component κ is κ Z ,
i.e. defined by:
κ = κ z k, (5.39)
because the wave is propagating in Z. So it follows that:
The required simplest possible solutions are then:
∂
∂Z eiφ = −iɛ r κe iφ . (5.40)
E (1) = E(0)
√
2
(i − ij) e (i( ω µr t−ɛrκZ)) (5.41)
and
provided that
B (1) = B(0)
√
2
(ii + j) e (i( ω µr t−ɛrκZ)) (5.42)
ω
µ r
B (0) = ɛ r κE (0) . (5.43)
55

5.2. RED SHIFT BY THE RELATIVE PERMEABILITY OF ECE . . .
From Eq.5.43 we identify the RED SHIFT:
ω → ω µ r
(5.44)
where µ r is in general a function of X, Y and Z. The red shift is therefore
caused by the relative permeability of ECE spacetime. The phase velocity v of
the light beam is also defined by Eq.5.43 and is less than c. The light beam is
slowed by its interaction with gravitation to:
v = c
n 2 = c . (5.45)
µ r ɛ r
In the next section this will be shown to be the root cause of the tired light
theory, a theory shown conclusively by Vigier et al. [41] to be preferred experimentally
to Big Bang. The slowing of light by a dielectric (ECE spacetime)
is analogous to the well known refraction of a light beam by a medium such
as water or glass. The refraction causes the path of the light to be changed,
and this is exactly what happens in the Eddington effect. Historically the latter
has been addressed with great accuracy by the central part of ECE theory as
described in the introduction. The central part is EH theory, and EH theory is
recovered when the torsion vanishes, leaving only the curvature. In the central
limit the first Bianchi identity of Cartan geometry [1]– [31, 34]:
reduces to the Ricci cyclic equation:
d ∧ T a + ω a b ∧ T b = R a b ∧ q b (5.46)
R a b ∧ q b = 0 (5.47)
because the torsion form T a is zero. The Ricci cyclic equation 5.47 implies that
the Christoffel connection of Riemann geometry is symmetric:
Γ κ µν = Γ κ νµ (5.48)
and this assumption pervades the whole of EH field theory. The latter can
therefore describe central effects with great accuracy, but cannot describe rotational
accelerations involving torsion. EH theory was applied historically to
the bending of light by the sun by considering the interaction of photon mass
with the sun’s mass. It happens to be that this theory is accurate to one part
in one hundred thousand for the sun, but as discussed in the introduction, this
is a semi-classical theory without any consideration of the electric and magnetic
fields of the light. The Eddington effect in ECE field theory can be calculated
to one part in one hundred thousand accuracy [35] from its central part (EH
theory) and ALSO recognized as being the red shift:
ω →
ω
µ r (X, Y, Z)
(5.49)
where µ r is defined by the trajectory of the photon around the sun (an orbit).
So the ECE theory describes the gravitational pull of the sun on the photon
defined as a particle with mass, and also defines the interaction of gravitation
with electromagnetism inherent in the process but unrecognized in EH theory
and Big Bang. One cannot have a photon without an electromagnetic wave and
56

CHAPTER 5. COSMOLOGICAL ANOMALIES: EH VERSUS ECE . . .
the orbit of the photon is described by the fact that the relative permeability
is a function of X, Y and Z. This function is the classical description of the
orbit as a REFRACTION, or gradual change in path of a light beam in an
inhomogeneous dielectric (one in which the relative permeability is a function
of X, Y and Z).
5.3 Absorption And The Beer Lambert Law In
ECE Cosmology
As argued in Section 5.2 it is important to account for absorption processes in
inter stellar and inter galactic regions because absorption means that the light
reaching a telescope is not related straightforwardly in general to the intrinsic
luminosity of a cosmological object. Luminosity in astronomy is defined in watts,
the rate at which energy of all types is radiated in all directions. This is the
area integral of the power density I in watts per meter. Luminosity in physics
is defined as the density of luminous intensity in a given direction, i.e. watts
per square meter per steradian, or power density per steradian (the measure of
solid angle). All that can actually be measured experimentally in astronomy is
the power density or intensity of light reaching a telescope from an object such
as a star or galaxy:
I =
L
4πZ 2 (5.50)
where L is the assumed intrinsic astronomical luminosity of an object and where
Z is the distance from object to telescope. It is almost always assumed that
there is no absorption along the entire length Z - despite the fact that this
may be an immense distance. Even the nearest star is millions of kilometers
away from the telescope. It is almost certain however that the light from any
cosmological object has been absorbed in many different ways by many different
absorbing mechanisms before it reaches the telescope. The absorption is defined
by the well known Beer Lambert law:
I = I 0 e −αZ (5.51)
where α is the power absorption coefficient in S. I. units of neper per meter
[46] and where I and I 0 are respectively the power density or intensity of the
radiation after and before absorption. The assumption of no absorption means
that:
I = I 0 , (5.52)
α = 0, (5.53)
and this is the assumption used in standard astronomy to measure the intrinsic
luminosity of a given object such as a star. This assumption is vanishingly
unlikely to be true.
The intensity I must be measured with a detector. The first type was a
photographic plate. The film was exposed to a light source of known intensity
and the total energy required to produce the image was calculated for a given
57

5.3. ABSORPTION AND THE BEER LAMBERT LAW IN ECE . . .
exposure time and given telescope aperture. This gave a baseline for the measurement
of I from an object such as a star. The distance Z in Eq.5.50 can only
be estimated if the intrinsic luminosity L of the object is known independently.
Vice versa L can only be calculated if Z is known independently. Only I can
be measured experimentally, and as argued this is vanishingly unlikely to be
I 0 . Therefore L is vanishingly unlikely to be the true intrinsic luminosity of an
object in astronomy. These simple facts question the experimental basis of the
whole of cosmology. Before the advent of photography the method of intensity
could not be used to estimate the object to telescope distance Z. A method
such as parallax could be used, a method which directly estimates the distance
of a star by trigonometry. Contemporary electronic methods such as the helium
cooled Rollin detector [46] can measure I accurately, but the assumption that I
is I 0 is almost always used. This false assumption makes the accuracy of measurement
of I irrelevant. Trigonometric parallax in contemporary astronomy
may be used for nearby stars and is accurate to a thousandth of an arc second.
This is therefore the only reliable measure of distance Z in astronomy. Even
then, the intrinsic luminosity of an object has to be based on the assumption
of no absorption. Therefore L is not known with any certainty even when Z
has been measured by parallax. The various assumptions of cosmology, such
as dark matter, missing mass, and universal expansion are based on extrapolations
from data of this kind. It is assumed that there is a relation between the
intrinsic luminosity L and the mass of an object. This is a theoretical model,
not a law of nature. Above all, as argued in Section 5.2, cosmology is based on
EH cosmology, which is not the required unified field theory, and therefore has
shaky foundations.
Assis [40] has also argued for the presence of absorption in inter-stellar and
inter-galactic regions. He uses the Beer Lambert law for the total electromagnetic
energy density U (joules per cubic meter), which is related to the power
density I (watts per square meter) by:
Thus
I = cU (5.54)
U = U 0 e −αZ (5.55)
Assis assumes that the beam of light is monochromatic, (made up of n photons
at the same frequency), so that the following equation is valid:
U = ω/V. (5.56)
The red shift in this monochromatic beam is therefore defined directly from the
Beer Lambert law as follows:
ω = ω 0 e −αZ . (5.57)
The observed red shift in this simple monochromatic model can therefore be
deduced to be due to absorption and not due to the expansion of the universe.
Different red shifts may occur for physically linked objects equidistant from the
observer because the light reaching the telescope has simply been differently
absorbed. Standard cosmology cannot explain this simple observation by Arp
et al. [39].
58

CHAPTER 5. COSMOLOGICAL ANOMALIES: EH VERSUS ECE . . .
The extra input to the Assis theory given by ECE theory is that light has
been absorbed by the residual gravitational field in a universe produced by n
gravitating objects. As for any absorption process [46] a spectrum (frequency
dependent power absorption coefficient) is produced at a detector (a telescope),
accompanied by frequency dependent dispersion and refraction. The mechanism
for absorption is given by Cartan geometry and the homogeneous current j a .
The light may be thought of as a source, and the gravitational field through
which the light travels may be thought of as a sample of absorbing dielectric.
The absorption of light by gravitation produces heat, and heat is governed by
the laws of thermodynamics. The absorption process is therefore accompanied
by an increase in entropy in the universe as observed [40]. The interaction of
light with the gravitational field produces heat from the homogeneous current j 0
and this is radiated as the background black body radiation of the universe at an
observed temperature of 2.7 K [40]. This is the origin of cosmic radiation, which
is well known experimentally to arrive at the earth from outside our galaxy, and
to pervade the known observable universe. The 2.7 K temperature is described
by Assis [40] as the average temperature of the whole universe, made up of stars,
galaxies, other objects and also the background radiation.
Regener [47] measured a temperature of 2.8 K and later Penzias and Wilson
later [48] confirmed Regener’s much earlier result using radio astronomy, a
different method. Regener considered cosmic rays to be black body radiation
consisting of many frequencies. The black body radiation from an ensemble of
N Planck oscillators (photons) is described [44] by:
( )
dU
dv = 8πhv3 e
−hv/kT
c 3 1 − e −hv/kT
(5.58)
where v is the frequency, h is the Planck constant, k is the Boltzmann constant,
and T the temperature. The total electromagnetic energy density (U) of a black
body such as the 2.7 K background radiation is obtained by integrating Eq.5.58
over all frequencies contained within the radiator. Thus:
U =
∫ ∞
8πhv 3
c 3 ( e
−hv/kT
1 − e −hv/kT )
dv
0
= 4 σ (5.59)
c T 4
where
σ = 2π5 k 4
15c 2 h 3 = 1 ( π 2 k 4 )
4 15c 2 3
is the Stefan-Boltzmann constant [44]. Thus:
(5.60)
and at 2.7 K [44]:
I = cU = 4σT 4 (5.61)
U = 4.02 × 10 −14 Jm −3 . (5.62)
Regener [47] and others [49] measured I and obtained T through the Stefan-
Boltzmann law 5.61. In ECE theory the further insight given is that the origin of
I (cosmic ray intensity) is the interaction of light (or electromagnetic radiation)
with the residual gravitational field of the universe. This is also the origin of all
59

5.3. ABSORPTION AND THE BEER LAMBERT LAW IN ECE . . .
observed red shifts. The numerous different types of red shift now observable [50]
are due to different mechanisms of absorption. In un-unified EH theory and
standard cosmology these mechanisms are absent because there is no current
j a . Red shifts in standard cosmology are described using an assumed expansion
of the universe and an assumed Hubble law [40] which does not work in general.
As argued by Assis the tired light model [40] can be obtained, for monochromatic
radiation only, from an assumed absorption coefficient:
α = H/c (5.63)
where H is the so called Hubble constant. In ECE theory a more rigorous and
more generally applicable derivation of the red shift [1]– [31] is given in terms
of permeability as described in Section 5.2. ECE theory applies to black body
radiation (many frequencies) as well as monochromatic radiation (one frequency
only). Neither the tired light model nor the Big Bang model is applicable in
general because the permeability of ECE space time is a complicated function
of Cartan geometry, of the curvature form, the torsion form, the spin connection,
and the tetrad. The tired light model is the limit of ECE theory where
the power absorption is a constant. More generally, as for any spectrum, the
power absorption coefficient is a richly structured function of frequency, not a
constant. Thus red shifts are given by ECE theory which are given neither by
standard cosmology nor Big Bang theory. Observed examples include anomalous
shifts near the sun [51], quantized red shifts [52], and different red shifts
for equidistant objects [39]. The fact that red shifts are observed to increase
near the limits of observation of the known universe [53] is a property of absorption,
a spectral phenomenon, and not due to the accelerations of far galaxies
as presently thought. There is no reason to think that the power absorption
coefficient generated by the interaction of light with residual gravity must be a
constant, as in Eq.5.63, and data have shown conclusively that H is not in fact
a constant.
The standard procedure in cosmology used to link the intrinsic luminosity or
emitted bolometric power (L in watts) of a cosmological object to its observed
power density at a telescope (I in watts per square meter) is:
I 0 = L 0
4πZ 2 (5.64)
where Z is the object to telescope distance. The area 4πZ 2 is the surface
area of a sphere with origin at the object and radius Z. As argued here and
independently by Assis [40] the Beer Lambert law changes Eq.5.64 to:
I = L 0
4πZ 2 e−αZ . (5.65)
Assis calculates the average total flux 〈I〉 of an Universe of infinite radius consisting
of n objects per m 3 in a sphere of radius Z (in units of watts m 2 ) as
follows:
∫ ∞
L 0
〈I〉 =
4πZ 2 · 4πZ2 ndZ = L 0n
α . (5.66)
0
We note that this procedure is an integration over a sphere in spherical polar
coordinates. The surface area of a sphere in spherical polar coordinates is given
60

CHAPTER 5. COSMOLOGICAL ANOMALIES: EH VERSUS ECE . . .
by [53]
S =
∫ 2π
0
dθ
∫ π
0
r 2 sin φdφ = 4πr 2 (5.67)
where the three cartesian distances along an axis (x, y, and z) for a radius r are
defined by:
x = r sin φ cos θ (5.68)
y = r sin φ sin θ (5.69)
z = r cos φ. (5.70)
Here φ is the angle between r and the z axis, and θ the angle between the
projection of r on the xy plane and the x axis [53]. We therefore find that:
∫ 2π ∫ π
0
0
sin φdφ = 4π (5.71)
and this is the relation implicitly used by Assis in arriving at his important
equation 5.66. The volume of a sphere is found by integrating over its surface
area:
V =
∫ r
0
4πr 2 dr = 4 3 πr3 (5.72)
and the infinitesimal angle element in spherical polar coordinates [53] is:
Therefore more generally, Assis’ Eq.5.66 is:
〈I〉 = n
∫ ∞
0
dΩ = sin φdφdθ. (5.73)
r 2 Idr
∫ 2π
0
dθ
∫ π
0
sin φdφ (5.74)
which is an integration over a sphere as argued. The equivalent integration over
a solid angle or cone depends on the limits taken for the integrals, in general:
∫ r
〈I〉 = nL 0 e −αr 1
dr ·
4π
0
∫ θ
0
dθ
∫ φ
0
sin φdφ. (5.75)
Eq.5.74 is applicable for integration over the whole universe and gives the value
of 〈I〉 measured experimentally by Regener [47] and others [49]. From this the T
= 2.7 K background is found using the Stefan-Boltzmann law. Assis’ equation
5.66 or more generally Eq.5.75, are basically important because they give the
correct background radiation temperature as confirmed (but not discovered) by
Penzias and Wilson [48]. It has been found that the inter-stellar regions of our
own galaxy are at this same temperature. The origin of this temperature in
ECE theory is black body emission from the residual gravitational field of the
universe, as argued already. Thus COSMIC ABSORPTION is the root cause of
this temperature, not an assumed expansion as in Big Bang. Light reaching a
telescope has always been absorbed in some way on its immensely long journey
from the source (for example a star) to the telescope. The power absorption
coefficient must always be calculated from a unified field theory (ECE), and not
a theory of gravitation only (EH).
The night sky is dark because the mean temperature of the whole universe
is only 2.7 K above absolute zero (colder than liquid helium). In contrast the
61

5.3. ABSORPTION AND THE BEER LAMBERT LAW IN ECE . . .
sun is very bright, even at a distance of ninety three million miles, because its
surface temperature is very high. In standard (EH) cosmology inter stellar and
inter galactic regions are assumed to be ”void”, i.e, a vacuum, and red shifts
are assumed to be due to an initial singularity. Therefore in the Big Bang light
is never absorbed on its journey from source to telescope. It is well known
experimentally [40] that this assumption is incorrect, as argued in this paper
already. The correct ECE cosmology of unified field theory produces red shifts
from Cartan geometry, which leads [1]– [31] to the homogeneous field equation
5.9. The simplest possible solution to this equation gives the red shift rule:
ω → ω µ r
. (5.76)
In general the permeability of ECE spacetime is complex valued:
µ r = µ ′ r + iµ ′′
r (5.77)
so the power absorption coefficient is defined from Eq.5.76 by [46]:
√
2ωµ
′′
r (ω)
α(ω) =
(5.78)
c(µ ′ r + (µ ′2 r + µ ′′2 r ) 1/2 ) 1/2
and is in general richly structured, i.e can generate spectral features such as the
diffraction rings observed in gravitational lensing, quantized red shifts and so
forth. Spectral shifts also depend on whether the ECE spacetime is diamagnetic,
paramagnetic , ferromagnetic or superconducting [44]– [45] because permeability
depends on these properties in the theory of magnetism. None of these concepts
exist in EH theory.
For a given volume V the flux 〈F 〉 of Eq.5.66 can be related to a mean
measured power density 〈I〉 and temperature through the Stefan-Boltzmann
law 5.61:
〈α〉 = nL 0 / 〈I〉 ,
(5.79)
n = n 0 /V.
Therefore the power absorption coefficient is defined by:
For black body radiation at 2.7 K [44]:
The volume V for a radius Z is from Eq.5.72:
〈α〉 = n 0L 0
4σV T 4 . (5.80)
〈I〉 = 1.205 × 10 −5 wattsm −2 . (5.81)
V = 4 3 πZ3 (5.82)
so the mean power absorption coefficient for absorption by the average gravitational
field of the universe is:
〈α〉 = 3n 0L 0
16σZ 3 T 4 . (5.83)
It is seen that this is inversely proportional to the cube of distance Z and
inversely proportional to the fourth power of the mean temperature of the universe,
T = 2.7 K. Here the universe has been considered to be of infinite extent
62

CHAPTER 5. COSMOLOGICAL ANOMALIES: EH VERSUS ECE . . .
and spherical. If there were no absorption there would be no mean intrinsic
luminosity 〈L 0 〉 of the universe, meaning that if there were no absorption of
light by the average gravitational field of the universe, there would be no heat
created, and there would be no black body radiation, or cosmic radiation. This
is contrary to experimental evidence, cosmic radiation is well known, but this
is also the result of standard EH cosmology in which 〈α〉 is zero. Thus ECE
cosmology is preferred experimentally to EH cosmology. ECE unified field theory
correctly predicts the 2.7 K background temperature without use of Big
Bang. The latter is replaced by a cosmology in which there may be local expansions
occurring (leading to the evolution of planets and elements) but no overall
expansion from an unphysical singularity or arbitrarily assumed initial event.
5.4 Relation Between Power Absorption and Mass
In this section the equation relating the mass of an object to its intensity is
derived, a derivation based on the fact that the intensity absorbed by an object
of mass M and radius R (surface area of 4πR 2 ) is the same as the intensity the
object absorbs from the universe around it:
The absorbed intensity in watts per square meter is:
I absorbed = I emitted . (5.84)
∫ r
〈I absorbed 〉 = nL 0 e −αr 1
dr ·
4π
0
∫ θ
0
dθ
∫ φ
0
sin φdφ (5.85)
where L 0 is the intrinsic luminosity of a surface area 4πr 2 of the whole universe,
and n is the number of objects in the equivalent volume 4 3 πr3 . Assis [40] has
considered the limit where integration in Eq.5.85 is carried out over a sphere of
infinite radius. In this limit:
r → ∞, θ → 2π, φ → π. (5.86)
This average is described by Assis [40] as the total flux received by an object
from the whole universe. The flux is a power density or intensity of black body
radiation in watts per square meter, with n defined [40] by:
n = ρ/M. (5.87)
Here ρ is the mean density of matter in the universe (10 −27 kgmm −3 ), and M is
the average mass of an object (kgm). Therefore the units of n are inverse cubic
meters.
Therefore:
〈I absorbed 〉 = ρL 0
Mα
(5.88)
and so L 0 is interpreted as the average luminosity of an astronomical object of
average mass M. Assis considers n such objects in an infinite spherical universe
which is considered to be in a steady state. Therefore an astronomical object
emits the same intensity of radiation as it absorbs (Eq.5.84) because it is in
63

5.4. RELATION BETWEEN POWER ABSORPTION AND MASS
thermodynamic equilibrium with the heat bath. If the radius of the object is R
its emitted power density or intensity is [40]:
and it is assumed that:
I emitted = L 0
4πR 2 (5.89)
〈I absorbed 〉 = I emitted . (5.90)
This assumption means that the emitted intensity is the same as the average
〈I absorbed 〉 absorbed by the object (for example a star or galaxy) from the rest
of the universe. From Eqs.5.88 and 5.89:
and using the Stefan-Boltzmann law:
α = 4πR 2 ρ/M (5.91)
I = L 0
4πR 2 = 4σT 4 . (5.92)
Here T is interpreted by Assis [40] as being the average background temperature
of the universe, and is the measured temperature of the all pervasive background
radiation. This has been measured experimentally to be 2.725 ± 0.002K.
Therefore the average power absorption coefficient of the steady state universe
is:
α = ρL 0
4MσT 4 . (5.93)
If R can be measured independently by parallax L 0 can be found from Eq.5.92.
Given the mean density ρ of the universe (about 10 −27 kgm per cubic meter)
the mass M can be expressed as being inversely proportional to α:
( ) ρL0 1
α =
4σT 4 M . (5.94)
Conventional cosmology assumes that the power absorption coefficient is the
Hubble constant within a factor c [40]:
α = H/c, (5.95)
and therefore the mean mass M of an object can be found given R, T and H.
From this model Assis [40] finds that:
L 0
M ∼ 10−5 wattskg −1 , (5.96)
L 0
R 2 ∼ 4 × 10−5 wattsm −2 , (5.97)
which is in order of magnitude agreement with experimental data for most
galaxies. For the Milky Way for example [40]:
L 0
M ∼ 2.5 × 10−5 wattskg −1 , (5.98)
L 0
R 2 ∼ 4 × 10−5 wattsm −2 . (5.99)
64

CHAPTER 5. COSMOLOGICAL ANOMALIES: EH VERSUS ECE . . .
For a perfect black body at 2.7 K the intensity radiated from the Stefan Boltzmann
law is 1.2 × 10 −5 W m −2 [45]. In a steady state model for the universe
therefore the mean intensity of radiation from the mean temperature of the universe
is 1.2 × 10 −5 W m −2 . This is only about an order of magnitude less than
the mean intensity of most galaxies so the background radiation makes up a
large part of the radiated heat of the steady state universe.
ECE theory is an objective unified field theory, unlike the purely gravitational
hot Big Bang and steady state cosmologies of EH theory, so ECE theory
is a cosmology in which there are unified fields, not isolated component fields
such as gravitation, electromagnetism, weak and strong. Therefore in ECE cosmology
electromagnetism can be changed into gravitation and vice versa. One
field is interacting with the other. The homogeneous equation governing this
interaction is Eq.5.9. The relative permittivity is defined through the refractive
index and the relative permeability by:
ɛ r = n 2 /µ r (5.100)
and is general a complex quantity. The power absorption is defined [46] conventionally
as:
α =
ωɛ′′ r
n ′ (5.101)
(ω)c
Therefore Eq.5.95 is true if
i.e.
ωɛ ′′
r
cn ′
= H c
(5.102)
ɛ ′′
r = n′ H
(5.103)
ω
which is the mean or background dielectric loss in a steady state universe. This
dielectric loss is due to the mean ρ of the universe heating up after absorbing
light or electromagnetic radiation through the homogeneous current j a . However
it is known experimentally [40] that α is not H/c in general because there
are many types of anomalous red shift now known experimentally as discussed
already in this paper. The parameter H varies in general and is not a constant
of cosmology, therefore as for any absorption coefficient α may be structured
in general, i.e. may have spectral features. In other words both α and H can
only be described as averages or backgrounds on which are superimposed many
other features. This is an experimental result. In hot Big Bang H is asserted
incorrectly to be a constant and absorption is unconsidered, inter stellar and
inter galactic absorption is ignored because it is assumed that such regions are
essentially high vacua. This assumption is known experimentally to be false [40].
Effectively, hot Big Bang claims to predict the background temperature of 2.725
± 0.002 K to within one thousandth of a degree after fifteen billion years of complicated
evolution. In reality hot Big Bang uses several parameters to fit data
and is not a predictive theory from first principles. Its initial conditions are
obviously not physics - infinite mean temperature and density within zero volume
for the universe. There are other well known problems in hot Big Bang
- flatness, isotropy, dark matter and accelerated rate of expansion, so there is
an urgent need to replace hot Big Bang with ECE theory, a correctly unified
field theory without unphysical initial conditions and groundless claims. ECE
theory correctly introduces absorption α from Cartan geometry.
65

5.5. GRAVITATIONAL ANOMALIES WITHIN THE SOLAR SYSTEM
It is therefore clear from these simple arguments alone that the mean mass
M from Eq.5.94 is meaningful only in an order of magnitude approximation. It
is unsurprising that estimates of M are not the same as those obtained from
other data (the ”missing mass” problem). This does not mean that there is
missing mass in the universe - it means simply that absorption has not been
correctly accounted for in EH field theory because it is not a unified field theory.
The conclusion of this section is that there is no evidence for dark matter in the
universe, but there is plenty of experimental evidence for absorption processes
neglected in hot Big Bang.
5.5 Gravitational Anomalies Within The Solar
System
Jensen [54] has discussed small gravitational anomalies in flight paths of vehicles
such as Odysseus, Galileo, Pioneer 6, 10 and 11, Polar lander and Climate
orbiter. Jensen assumes a mass dependent permeability and in this section it is
demonstrated that this concept is inherent in ECE theory.
We start with the Bianchi identity in the form [1]– [31]:
where the homogeneous current is defined by:
d ∧ T a = j a (5.104)
j a = R a b ∧ q b − ω a b ∧ T b . (5.105)
Eq.5.104 is the homogeneous field equation linking torsion and curvature in ECE
theory. EH theory is the limit:
The tensor form of Eq.5.104 is:
where the torsion tensor is defined by:
⎡
˜T aµν =
⎢
⎣
R a b ∧ q b = T a = 0. (5.106)
∂ µ ˜T aµν = ˜j aν (5.107)
0 −T a1 s −T a2 s −T a3 s
T a1 s 0 T a3 c T a2 c
T a2 s −T a3 c 0 T a1 c
T a3 s T a2 c −T a1 c 0
⎤
⎥
⎦ . (5.108)
The homogeneous field equation 5.107 leads to the following vector equation:
∇ × (ɛ r T a 0) + 1 (
∂ 1
T
c ∂t µ
s)
a = 0 (5.109)
r
which is the direct analogue of Eq.5.9. In this equation the relative permeability
µ r and relative permittivity ɛ r are defined as in Eq.5.9 as those of ECE spacetime.
In general the permeability is a function of X, Y and Z and therefore of
mass, as postulated by Jensen [54], i.e.:
µ r (X, Y, Z) = n2
ɛ r
. (5.110)
66

CHAPTER 5. COSMOLOGICAL ANOMALIES: EH VERSUS ECE . . .
In the Einstein Hilbert limit the relative permeability approaches unity:
µ r −→ 1,
ɛ r −→ 1,
n −→ 1.
(5.111)
As argued by Jensen [54] very tiny departures due to Eq.5.110 can account for
the flight path anomalies in the above space probes. Under the condition 5.110,
following the argument by Jensen [54], the mass appearing in the Newtonian
limit becomes the total proximal mass, and not the mass of the object in motion.
The proximal mass becomes larger as the object approaches the sun or a
planet. In consequence less potential energy is changed into kinetic energy and
the orbital velocity increases at a slightly lower rate than in the Newton equations.
Accelerating away from the sun again, the probe would require slightly
less energy to achieve a greater acceleration and to return the probe to its predicted
path. This is what is observed in Pioneer 6 as it passes near the limb of
the sun. Observed solar wind effects on Odysseus and Galileo also follow this
model. Therefore the argument by Jensen [54] is effectively is that more potential
energy than in the Newtonian limit is stored in a more massive environment
described by a particular model of the ECE permeability5.110. This means for
example that probes to Venus are proportionally slowed and probes to Mars
are proportionally accelerated. Therefore the mass of Venus is overestimated
and the mass of Mars is underestimated in orbital calculations used by NASA.
This in turn would cause negative gravity anomalies near the mountain peaks
on Venus and positive gravity anomalies near valley floors on Venus. The equivalent
anomalies on Mars would have opposite sign. These probes give gravity
maps of Mars from 300 to 800 km but the maps at 300 km are not consistent
with those at 800 km, the 300 km data showing greater anomalies. The moment
of inertia for Mars is apparently different if ranging data to the Pathfinder and
Viking probes are used rather than the inertial moment necessary to explain the
gravity anomalies. All of the Martian probes have landed at higher velocities
than predicted from Newtonian equations, and also entered at higher attitudes.
All descent trajectory models have required a thinner than expected upper atmosphere
and higher than expected surface winds. Additionally the data from
the Huygens probe can be better modeled, according to Jensen [54], with a
mass dependent permeability as defined in Eq.5.110. These include data for
rocks, craters, strata and Doppler descent data. There are no indications from
the latest data from the Huygens probe of the wind shear of -230 km per hour
necessary to model the Huygens descent from a pure Newtonian theory. Furthermore,
MRO will provide gravity maps at 150 km which would be expected
to show even greater anomalies from Eq.5.110. Finally [54] careful mapping
effects of Saturn’s moons on the NASA Cassini probe should provide further
confirmation of Eq.5.110 as developed by Jensen [54].
Acknowledgments The British Government is thanked for a Civil List pension
(2005) to MWE. The staff of AIAS are thanked for many interesting discussions.
67

5.5. GRAVITATIONAL ANOMALIES WITHIN THE SOLAR SYSTEM
68

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[42] P. A. Pinter private communications and website (2002 to present).
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[45] P. W. Atkins, Molecular Quantum Mechanics, (Oxford Univ.Press, 1983,
2nd. ed).
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Interscience, New York, 1983).
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[50] J.-P. Vigier, IEEE Trans. Plasma Sci., 18, 1 (1990).
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[52] H. C. Arp, Apeiron, 5, 7 (1989).
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Nature, 346, 807 (1990).
[54] This section is based on a communication of Sept. 2005 from J. Jensen to
C. R. Keys.
72

Chapter 6
Solutions Of The ECE Field
Equations
by
M. W. Evans
Alpha Foundation’s Institute for Advance Study (A.I.A.S.).
(emyrone@aol.com, www.aias.us, www.atomicprecision.com)
Abstract
Solutions of the ECE field equations are given in the dielectric formulation of
the theory. The effect of gravitation on electromagnetism is to change the amplitudes
of the plane wave solutions of free electromagnetism, to change the
phase velocity and to shift the frequency. In the presence of gravitation the in
the free electric field strength E a and magnetic flux density B a become plane
waves in the displacement D a and magnetic field strength H a .
Keywords: Solutions of the ECE field equations, unified field theory, interaction
of gravitation and electromagnetism
6.1 Introduction
The dielectric formulation of the Einstein Cartan Evans (ECE) unified field
theory [1]– [32] has recently been developed in order to provide a framework
for relatively straightforward numerical solutions without having to go immediately
into the full details of Cartan geometry. In this paper it is shown that in
a specific approximation, the effect of gravitation on the plane waves of the free
electromagnetic field is to produce plane waves in the displacement (D a ) and
magnetic field strength (H a ) instead of the free space electric field strength (E a )
and the magnetic flux density (B a ). In this approximation the ECE field equations
have a well defined analytical solution which can be used to test computer
code before embarking on the numerical solution of the dielectric formulation
of the ECE field equations. In Section 6.2 the stages involved in deriving the
dielectric formulation are summarized. Some important mathematical details
73

6.2. DETAILS IN THE DERIVATION OF THE ECE FIELD . . .
are given, starting from the form notation [33] of Cartan geometry on which
ECE unified field theory is directly based. These details are not easily found
elsewhere, but are important for coding purposes. In section 6.3 the dielectric
ECE field equations are solved in a well defined approximation to give an analytical
solution. In general the ECE field equations must be solved numerically,
and so enough mathematical detail is given in this paper to help to achieve this
aim.
6.2 Details In The Derivation Of The ECE Field
Equations, Form, Tensor, Vector And Dielectric
Notation
The most elegant statement of the ECE field equations in mathematics uses
the form notation of differential or Cartan geometry [1]– [33]. However in field
theory in physics the tensor notation is more often used and in engineering
the vector notation is used. In chemistry the dielectric formulation is often
used. All these descriptions are interchangeable and equivalent, so it is useful
to summarize them in this section and to give sufficient mathematical detail for
coding up the equations to give graphs and animations.
The geometrical fundamentals of ECE theory are the well known fundamentals
of standard Cartan geometry: the two structure equations, the two Bianchi
identities, and the tetrad postulate. Two ansatzen are used to transform the
geometry into an objective unified field theory in general relativity [1]– [32].
The two Cartan structure equations are sometimes known in contemporary
mathematics as the master equations of differential geometry. The first Cartan
structure equation defines the torsion form (T a ) as the covariant exterior
derivative of the tetrad (q a ), the fundamental field of ECE theory:
T a = D ∧ q a = d ∧ q a + ω a b ∧ q b . (6.1)
The second structure equation of Cartan defines the Riemann or curvature form
(R a b ) in terms of the spin connection (ωa b ):
R a b = D ∧ ω a b = d ∧ ω a b + ω a c ∧ ω c b. (6.2)
Here d∧ is the exterior derivative of Cartan. The covariant exterior derivative
[33] is the operator:
D∧ = d ∧ +ω ∧ . (6.3)
It can be seen that the fundamental variables are the tetrad, (the fundamental
field), and the spin connection, which defines the way the frame is curved and /
or spun in ECE spacetime. The Latin indices are those of the tangent spacetime
at P to the base manifold. The latter is indexed with Greek letters, and the
convention [33] of standard differential geometry has been followed. In this
convention the Greek indices are omitted because they are always the same on
each side of an equation. If however the Greek indices are temporarily restored
to equations 6.1 and 6.2 for the sake of instruction, they become the differential
form equations
T a µν = (d ∧ q a ) µν
+ ( ω a b ∧ q b) (6.4)
µν
74

CHAPTER 6.
SOLUTIONS OF THE ECE FIELD EQUATIONS
R a bµν = (d ∧ ω a b) µν
+ (ω a c ∧ ω c b) µν
(6.5)
The tetrad is a vector valued one-form, a rank two mixed index tensor, so has
only one Greek subscript, the torsion form is a vector valued two-form which is
antisymmetric in its Greek indices:
The Riemann form is a tensor valued two-form:
T a µν = −T a νµ . (6.6)
R a bµν = −R a bνµ. (6.7)
The spin connection is a tensor valued one-form, but is not a tensor because
it does not transform as a tensor [33] under coordinate transformation. This
property is analogous to that of the Christoffel connection [33], which is not a
tensor for the same reason. In order to be able to transform these form equations
to tensor equations the following fundamental definitions are needed.
The exterior derivative of the differential form A [33] is defined in general
by:
(d ∧ A) µ1···µ p+1
= (p + 1) ∂ [µ1 A µ2···µ p+1]. (6.8)
Thus the exterior derivative of a one-form is:
and the exterior derivative of a two-form is:
(d ∧ A) µ1µ 2
= (d ∧ A) µν
= 2∂ [µ A ν]
= ∂ µ A ν − ∂ ν A µ
(6.9)
(d ∧ A) µ1µ 2µ 3
= 3∂ [µ1 A µ2µ 3] = ∂ µ A νρ + ∂ ν A ρµ + ∂ ρ A µν (6.10)
The wedge product of a form A and a form B is defined in general by [33]:
(A ∧ B) µ1···µ p+q
=
(p + q)!
A µ1···µ
p!q!
p
B µp+1···µ p+q
. (6.11)
Therefore the wedge product of two one-forms is defined by:
p = 1, q = 1, µ 1 = µ, µ 2 = ν (6.12)
and is:
(A ∧ B) µν
= 2!
1!1! A [µB ν] = A µ B ν − A ν B µ . (6.13)
The wedge product of a one-form and a two-form is given by:
p = 1, q = 2, µ 1 = µ, µ 2 = ν, µ 3 = ρ (6.14)
and is:
(A ∧ B) µ1µ 2µ 3
= 3!
2!1! A [µ 1
B µ2µ 3] = 3A [µ B νρ]
= A µ B νρ + A ν B ρµ + A ρ B µν .
(6.15)
75

6.2. DETAILS IN THE DERIVATION OF THE ECE FIELD . . .
The form notation of Cartan is more elegant than the notation of tensor analysis,
but both are equivalent. The advantage of Cartan geometry is that it allows
much more insight into the fundamental structure of equations than the more
complicated tensor notation. However the latter may be more useful for coding
and the standard vector notation used in engineering is almost always derived
from the tensor notation. Form notation and tensor notation are rarely if ever
used in engineering.
The Bianchi identities of Cartan geometry [1]– [33] are the rigorous generalizations
of the Ricci cyclic equation and the Bianchi identity of the type of
Riemann geometry used in Einsteinian general relativity, i.e. used in the Einstein
Hilbert (EH) field theory of gravitation proposed in 1915 [34]. The EH
field theory is a famous landmark of twentieth century physics but is restricted
by its omission from consideration of the torsion tensor. In Cartan geometry
the first Bianchi identity is:
and the second Bianchi identity is:
D ∧ T a = d ∧ T a + ω a b ∧ T b := R a b ∧ q b (6.16)
D ∧ R a b = d ∧ R a b + ω a c ∧ R c b − R a c ∧ ω c b
:= 0.
(6.17)
In Eqs.6.16 and 6.17 we have reverted to the standard notation in which the
Greek subscripts are omitted [33]. The Bianchi identities involve the exterior
derivatives of the torsion and Riemann forms, and using Eqs.6.1 and 6.2 can be
written as differential equations in the tetrad and the spin connection:
d ∧ ( d ∧ q a + ω a b ∧ q b) + ω a b ∧ ( d ∧ q b + ω b c ∧ q c)
:= (d ∧ ω a b + ω a b ∧ ω c b) ∧ q b (6.18)
and
d ∧ (d ∧ ω a b + ω a c ∧ ω c b) + ω a c ∧ ( d ∧ ω a b + ω c d ∧ ω d )
b
− ( d ∧ ω a c + ω a d ∧ ω d c)
∧ ω
c
b := 0.
(6.19)
In references [1]– [32] the equivalents of the structure equations and Bianchi
identities have been derived in the most general type of Riemann geometry, i.e.
the form notation has been translated into tensor notation.
Translation from differential form to tensor notation requires the well known
tetrad postulate, which can be proven in several complementary and instructive
ways [1]– [33], each proof giving the same result (the tetrad postulate) and
each proof reinforcing the other complementary proofs. The most fundamental
meaning of the tetrad postulate is perhaps the fact that the same vector field
can be expressed equivalently in different coordinate systems. Here vector field
means that a vector is defined by its vector components in base coordinate
elements such as unit vectors, spinors, or Pauli matrices [33]. The vector field
expressed in cartesian or spherical polar coordinates for example is the same
vector field but expressed in different coordinates. In Cartan geometry [1]– [33]
it follows that the covariant derivative of the tetrad vanishes:
D ν q a µ = 0 (6.20)
76

CHAPTER 6.
SOLUTIONS OF THE ECE FIELD EQUATIONS
and this fundamental property is known conventionally as the tetrad postulate.
However, nothing is postulated (i.e. nothing is needed to derive Eq.6.20) other
than the fact that a vector field in different coordinates is the same vector
field. (If it were not the same vector field then the coordinate system would
not be a valid coordinate system.) In order to correctly define the covariant
derivative of the tetrad it is necessary to define the covariant derivative of a
mixed index rank two tensor, a tensor whose upper index is a and whose lower
index is µ. In order to do this the fundamental definition of covariant derivative
is needed [33]. Examples are given here for clarity of exposition. In general the
covariant derivative of a tensor of any rank is defined by [33]:
D σ T µ1µ2···µ k ν1ν 2···µ l
= ∂T µ1µ2···µ k ν1ν 2···µ l
+ Γ µ1 σλ T λµ2···µ k ν1ν 2···µ l
+ Γ µ2 σλ T µ1λ···µ k ν1ν 2···µ l
+ · · ·
− Γ λ σν 1
T µ1µ2···µ k
λν2···µ l
− Γ λ σν 2
T µ1µ2···µ k
ν1λ···µ l
− · · · .
(6.21)
For mixed Greek and Latin indices the gamma connection is replaced by the spin
connection. Therefore the covariant derivative of the tetrad is, from Eq.6.21:
D µ q a λ = ∂ µ q a λ + ω a µbq b λ − Γ ν µλq a ν . (6.22)
Similarly the covariant derivative of a contravariant four vector is:
the covariant derivative of a covariant vector is:
and the covariant derivative of a tensor is:
D µ V ν = ∂ µ V ν + Γ ν µλV λ , (6.23)
D µ V ν = ∂ µ V ν − Γ λ νµV λ , (6.24)
D µ T νρ = ∂ µ T νρ + Γ ν µλT λρ + Γ ρ µλ T νλ . (6.25)
The covariant derivative of a rank three tensor is:
D µ (D µ q a ν ) = ∂ µ (D µ q a ν ) + Γ µ µλ Dλ q a ν + ω a µbD µ q b ν − Γ λ µνD µ q a λ. (6.26)
The above general formulae allow one to rewrite the first Bianchi identity 6.16
in tensor notation as follows:
∂ µ T a νρ + ∂ ρ T a µν + ∂ ν T a ρµ + ω a µbT b νρ + ω a ρbT b µν + ω a νbT b ρµ
= R a bµνq b ρ + R a bρµq b ν + R a bνρq b µ
(6.27)
= R a µνρ + R a ρµν + R a νρµ
and this is the basic tensorial structure of the field equations of ECE theory [1]–
[32]. The geometry 6.27 is transformed into the field equation using the ansatz:
F a µν = A (0) T a µν (6.28)
where A (0) is a scalar valued potential magnitude and where:
F a µν = −F a νµ (6.29)
77

6.2. DETAILS IN THE DERIVATION OF THE ECE FIELD . . .
is the anti-symmetric field tensor of electromagnetism influenced by gravitation.
The basic field equation of ECE theory is therefore:
∂ µ F a νρ + ∂ ρ F a µν + ∂ ν F a ρµ
= A (0) ( R a µνρ + R a ρµν + R a νρµ − ω a µbT b νρ − ω a ρbT b µν − ω a νbT b ) (6.30)
ρµ
and is rewritten as follows to define the homogeneous current of ECE field
theory [1]– [32]:
∂ µ F a νρ + ∂ ρ F a µν + ∂ ν F a ρµ = µ 0
(
j
a
µνρ + j a ρµν + j a νρµ)
. (6.31)
In differential form notation Eq.6.31 is:
d ∧ F a = µ 0 j a (6.32)
and is the homogeneous field equation of ECE theory. The way in which gravitation
influences electromagnetism is defined by the current j a . In Section 6.3
we give analytical solutions to the homogeneous field equation and its Hodge
dual [1]– [32], the inhomogeneous field equation. Firstly in this section enough
mathematical detail is given to develop Eq.6.31 into two vector equations, and
to derive the Hodge dual of Eq.6.31. This detail is again needed for coding
purposes.
The general Hodge dual of a tensor is defined [33] by:
where
à µ1···µ n−p
= 1 p! ɛν1···νp µ 1···µ n−p
A ν1···ν p
(6.33)
ɛ µ1µ 2···µ n
= |g| 1/2 ɛ µ1µ 2···µ n
(6.34)
is the Levi-Civita tensor. The latter is defined as the square root of the modulus
of the determinant of the metric multiplied by the Levi-Civita symbol:
1 for even subscript permutation
ɛ µ1µ 2···µ n
=
−1 for odd subscript permutation
∣ 0 otherwise
∣ . (6.35)
Using the metric compatibility condition [33]:
it is seen that:
D µ g νρ = 0 (6.36)
D µ |g| 1/2 = ∂ µ |g| 1/2 = 0 (6.37)
because the determinant of the metric is made up of individual elements of the
metric tensor. The covariant derivative of each element vanishes by Eq.6.36, so
we obtain Eq.6.37. The premultiplier |g| 1/2 is a scalar, and in deriving Eq.6.37
we have used the definition [33]:
D ρ S = ∂ ρ S (6.38)
where S is any scalar. The Hodge dual of Eq.6.31 may now be defined using
the general formula 6.33 and used: a) to obtain the vector formulation of the
78

CHAPTER 6.
SOLUTIONS OF THE ECE FIELD EQUATIONS
homogeneous field equation and b) to obtain the inhomogeneous ECE field
equation from the homogeneous field equation.
The first step in obtaining the vector formulation is to prove that Eq.6.31
can be rewritten as:
∂ µ ˜F aµν = µ 0˜j aν (6.39)
in which:
˜F aµν = 1 2 |g|1/2 ɛ µνρσ F a ρσ (6.40)
˜j aσ = 1 6 |g|1/2 ɛ µνρσ j a µνρ (6.41)
are Hodge duals. To prove Eq.6.39 consider individual tensor elements such as
those defined by ν = 1, µ = 0, 2, 3. In this case:
∂ 0 ˜F a01 + ∂ 2 ˜F a21 + ∂ 3 ˜F
a31
= 1 2 |g|1/2 ɛ µ1ρσ ∂ µ F a ρσ
= 1 ( 2 |g|1/2 ɛ 01ρσ ∂ 0 F a ρσ + ɛ 21ρσ ∂ 2 F a ρσ + ɛ 31ρσ ∂ 3 F a )
ρσ
(6.42)
= |g| 1/2 (∂ 0 F a 23 + ∂ 2 F a 30 + ∂ 3 F a 02 )
which is a special case of the general result:
∂ µ ˜F aµν → |g| 1/2 ( ∂ µ F a νρ + ∂ ρ F a µν + ∂ ν F a )
ρµ . (6.43)
Consider Eq.6.41 for σ = 1 to obtain:
˜j a1 = 1 6 |g|1/2 (ɛ 0231 j a 023 + ɛ 0321 j a 032
+ɛ 2031 j a 203 + ɛ 3021 j a 302
+ɛ 2301 j a 230 + ɛ 3201 j a 320)
(6.44)
Similarly, the other two current terms:
= 1 3 |g|1/2 (j a 023 + j a 302 + j a 230)
˜j aσ = 1 6 |g|1/2 ɛ ρµνσ j a ρµν (6.45)
and
˜j aσ = 1 6 |g|1/2 ɛ νρµσ j a νρµ (6.46)
give Eq.6.44 two more times. So the right hand side of Eq.6.31 for ν = 1 is:
˜j a1 = |g| 1/2 (j a 023 + j a 302 + j a 230) . (6.47)
Finally we use Eq.6.37 to find that:
)
∂ µ
(|g| 1/2 F a νρ = |g| 1/2 ∂ µ F a νρ (6.48)
and so derive Eq.6.39 from Eq.6.31, Q.E.D. Note that the |g| 1/2 premultiplier
cancels out either side of Eq.6.39. The vector formulation of Eq.6.39 follows by
standard methods [1]– [33] and is:
∇ · B a = µ 0˜j a0 (6.49)
79

6.2. DETAILS IN THE DERIVATION OF THE ECE FIELD . . .
∇ × E a + ∂Ba = µ 0˜ja
(6.50)
∂t
where the four-current is defined by:
) (˜j
˜j aν a0
=
c ,˜j a . (6.51)
The currents terms in Eq.6.31 are defined by:
j a µνρ = A(0) (
R
a
µ µνρ − ω a µbT b )
νρ
0
(6.52)
and so on. Since R a µνρ and T b νρ are antisymmetric in their last two Greek
indices they have Hodge duals defined by:
˜R a µν
τ = 1 2 |g|1/2 ɛ µνρσ R a τρσ (6.53)
˜T aµν = 1 2 |g|1/2 ɛ µνρσ T a ρσ . (6.54)
The four-current of the homogeneous ECE field equation is therefore given in
terms of these Hodge duals as follows:
˜j aν = A(0)
µ 0
(
˜Ra
µν
µ − ω a µb ˜T bµν) , (6.55)
and defines the way in which gravitation affects the Gauss law applied to magnetism
and the Faraday law of induction.
The inhomogeneous field equation is derived from the homogeneous field
equation by taking the Hodge duals term by term of each two-form in the
homogeneous equation:
d ∧ F a = µ 0 j a = A (0) ( R a b ∧ q b − ω a b ∧ T b) . (6.56)
The two-form in this equation are: F a µν , R a bµν , and T b µν . Writing out each
two-from in tensor notation, the three Hodge duals are:
˜F aαβ = 1 2 |g|1/2 ɛ αβµν F a µν , (6.57)
˜R a αβ
b
= 1 2 |g|1/2 ɛ αβµν R a bµν, (6.58)
˜T bαβ = 1 2 |g|1/2 ɛ αβµν T b µν , (6.59)
and each Hodge dual is equivalent to an anti-symmetric rank two tensor. Therefore
the inhomogeneous ECE field equation [1]– [32] is:
d ∧ ˜F a = µ 0 J a = A (0) ( ˜Ra b ∧ q b − ω a b ∧ ˜T b) (6.60)
and the pre-multiplier |g| 1/2 cancels out either side of the equation. In tensor
notation, Eq.6.60 is:
∂ µ ˜F
a
νρ + ∂ ρ ˜F
a
µν + ∂ ν ˜F
a
ρµ = µ 0
(
J
a
µνρ + J a ρµν + J a νρµ
)
. (6.61)
80

CHAPTER 6.
SOLUTIONS OF THE ECE FIELD EQUATIONS
In summary, the homogeneous and inhomogeneous ECE field equations are:
d ∧ F a = µ 0 j a (6.62)
d ∧ ˜F a = µ 0 J a . (6.63)
In differential form notation, the Maxwell Heaviside (MH) field equations of the
standard model are well known to be [33, 34]:
d ∧ F = 0, (6.64)
d ∧ ˜F = µ 0 J. (6.65)
It is seen that the field form and its Hodge dual appear in the MH equations, but
the homogeneous current is missing, indicating that there is no mechanism in
MH theory for considering the effect of gravitation on electromagnetism. Also,
the inhomogeneous current J of MH theory is introduced empirically (i.e. from
experiment), and not from the first theoretical principles of Cartan geometry
and generally covariant unified field theory as required in objective physics. The
ECE field equations 6.62 and 6.63 identify the source of j a and J a in geometry.
The properties of the Hodge dual can be checked with the Schwarzschild
metric (SM) [1]– [33]. The SM is a solution of the famous EH field equation of
1915. In EH theory:
R a b ∧ q b = 0, (6.66)
In tensor notation Eq.6.66 is the Ricci cyclic equation:
and Eq.6.67 is:
T a = 0. (6.67)
R σµνρ + R σρµν + R σνρµ = 0 (6.68)
T κ µν = Γ κ µν − Γ κ νµ = 0 (6.69)
where Γ κ µν is the Christoffel connection [33]. In the SM the non-zero elements
of the Riemann tensor are:
R 0 101, R 1 212, R 1 313, R 2 323, R 0 202, R 0 303 ≠ 0, (6.70)
so Eq.6.68 is true automatically in the SM because the last three subscripts of
the Riemann tensors appearing in the Ricci cyclic equation must be all different,
i.e. occur in cyclic permutation. However no such elements are non-zero in the
SM. The relevant Hodge dual of Eq.6.66 is defined by:
i.e. by:
Therefore, upon taking Hodge duals such as:
R a b ∧ q b −→ ˜R a b ∧ q b (6.71)
˜R αβµν = 1 2 |g|1/2 ρσ
ɛµν R αβρσ . (6.72)
˜R 0123 = |g| 1/2 R 0101 , (6.73)
˜R 0231 = |g| 1/2 R 0202 , (6.74)
81

6.2. DETAILS IN THE DERIVATION OF THE ECE FIELD . . .
˜R 0312 = |g| 1/2 R 0303 , (6.75)
it is concluded that
i.e.:
˜R 0123 + ˜R 0231 + ˜R 0312 ≠ 0 (6.76)
˜R a b ∧ q b ≠ 0. (6.77)
Eq.6.77 means, importantly, that the inhomogeneous current J a can be very
large even if the homogeneous current may be vanishingly small. This result
has been illustrated here for the SM, but is true for any metric. This result of
ECE field theory explains why the homogeneous current can become zero (as
in MH theory), while the inhomogeneous current can become very large. It is a
property of geometry.
In deriving Eq.6.63 from Eq.6.62 we have used the Hodge dual of a twoform
in four dimensional space-time, a special case of the general Hodge dual
formula 6.33. In this case the result is another two-form as argued. The currents
in Eqs.6.31 and 6.61 are three-forms, whose Hodge duals in four dimensional
space-time are one-forms as we have argued. Denoting the Hodge dual of j a
by ˜j a , and the Hodge dual of J a by ˜J a , then the tensorial homogeneous and
inhomogeneous field equations of ECE theory, Eqs.6.31 and 6.61, become [1]–
[32]:
∂ µ ˜F aµν = µ 0˜j aν , (6.78)
∂ µ F aµν = µ 0 J aν . (6.79)
The homogeneous and inhomogeneous currents in tensor notation are:
˜j aν = A(0)
µ 0
(
˜Ra
µν
µ − ω a µb ˜T bµν) , (6.80)
˜J aν = A(0)
µ 0
(
R
a µν
µ − ω a µbT bµν) . (6.81)
The current four-vectors are defined in S.I. units by:
( ) 1
˜j aν =
c˜j a0 ,˜j a , (6.82)
˜J aν =
( 1
c ˜J a0 , ˜J a )
, (6.83)
The two field equations 6.78 and 6.79 in vector notation therefore become the
four objective laws of classical electrodynamics in general relativity:
∇ · B a = µ 0˜j a0 , (6.84)
∇ × E a + ∂Ba
∂t
= µ 0˜ja , (6.85)
∇ · E a = µ 0 c ˜J a0 , (6.86)
∇ × B a − 1 ∂E a
c 2 = µ 0
∂t c ˜J a . (6.87)
Unlike the MH theory, the laws 6.84 to 6.87 can describe the effect of gravitation
on electromagnetism. This was a major aim of both Einstein and Cartan.
82

CHAPTER 6.
SOLUTIONS OF THE ECE FIELD EQUATIONS
Eq.6.84 is the Gauss law applied to magnetism; Eq.6.85 is the Faraday law of
induction; Eq.6.86 is the Coulomb law; and Eq.6.87 is the Ampère-Maxwell law.
It is important to realize that the four laws are now written in the presence of a
gravitational field, whereas the familiar MH laws are for electromagnetism without
consideration of the gravitational field. This is of course the fundamental
aim of a unified field theory.
The relevant S.I. units [35] are as follows:
B a = JsC −1 m −2 = Tesla, (6.88)
E a = JC −1 m −1 = V m −1 , (6.89)
µ 0 = 4π × 10 −7 Js 2 C −2 m −1 , (6.90)
˜j a0 = Cs −1 m −2 = Am −2 , (6.91)
˜j a0 /c = Cm −3 , (6.92)
1
c˜j a0 = 1 c ˜J a0 = charge density, (6.93)
˜ja = ˜J a = current density. (6.94)
Eqs.6.84 to 6.87 use the fundamental S.I. relation [35] in free space:
E (0) = cB (0) . (6.95)
In the limit of zero gravitation the electromagnetic component of the unified
ECE field splits off, and is referred to as the free electromagnetic field. The
Cartan geometry of the free electromagnetic field is defined [1]– [32] by the fact
that in this limit the homogeneous current j a vanishes, so it follows that:
A solution of Eq.6.96 is:
R a b ∧ q b = ω a b ∧ T b . (6.96)
R a b = κɛ a bcT c (6.97)
ω a b = κɛ a bcq c , (6.98)
where the scalar κ has the units of wave-number (inverse meters). It is important
to understand that there may be a Riemann form for a spinning frame. The
Riemann form is the curvature form only for the free gravitational field. For
rotational motion (i.e. the spinning of the free electromagnetic field) Eqs.6.97
and 6.98 show that for each µ and ν, the Riemann form is the antisymmetric
tangent space-time tensor corresponding to the axial vector T c . Similarly, for
each µ, the spin connection is the antisymmetric tensor corresponding to the
axial vector q c . The Hodge dual of Eq.6.96 is:
and in consequence, for the free electromagnetic field:
˜R a b ∧ q b = ω a b ∧ ˜T b , (6.99)
˜j aν = ˜J aν = 0. (6.100)
Therefore, for the free electromagnetic field, the four laws 6.84 to 6.87 simplify
to:
∇ · B a = 0 (6.101)
83

6.2. DETAILS IN THE DERIVATION OF THE ECE FIELD . . .
∇ × E a + ∂Ba = 0
∂t
(6.102)
∇ · E a = 0, (6.103)
∇ × B a − 1 ∂E a
c 2 = 0. (6.104)
∂t
These equations have analytical solutions [1]– [32] and describe the electromagnetic
field in the hypothetical limit of vanishing mass. These are sometimes
known as source-free fields in the standard literature on MH theory, because a
source, by definition, must be a radiating electron whose mass is not zero.
The converse limit of zero electromagnetism, (the free gravitational field), is
defined [1]– [32] by zero torsion:
F a = A (0) T a = 0, (6.105)
R a b ∧ q b = 0. (6.106)
Polarization and magnetization are defined [35] in ECE theory by a straight
forward extension of their MH counterparts to include the polarization index a.
Thus:
D a = ɛE a = ɛ 0 E a + P a , (6.107)
H a = 1 µ Ba = 1 µ 0
B a − 1 µ 0
M a , (6.108)
where D a is the electric displacement (Cm −2 ) and H a the magnetic field strength
(Am −1 ). Here P a is the polarization, M a the magnetization, ɛ the dielectric
permittivity, µ the magnetic permeability, ɛ 0 the vacuum permittivity and µ 0
the vacuum permeability. The relative permittivity and permeability are therefore
[35]:
ɛ r = ɛ/ɛ 0 , (6.109)
and the refractive index is
µ r = µ/µ 0 , (6.110)
n 2 = ɛ r µ r . (6.111)
In general ɛ r and µ r are inhomogeneous functions of spacetime:
ɛ r = ɛ r (ct, X, Y, Z) (6.112)
µ r = µ r (ct, X, Y, Z) , (6.113)
and in the presence of absorption may become complex valued [35, 36]:
ɛ r = ɛ ′ r + iɛ ′′
r , (6.114)
µ r = µ ′ r + iµ ′′
r . (6.115)
The power absorption coefficient (neper m −1 ) is defined [36] by:
and by the Beer Lambert law:
α = ωɛ′′ r
n ′ c , (6.116)
I = I 0 e (−αZ) . (6.117)
84

CHAPTER 6.
SOLUTIONS OF THE ECE FIELD EQUATIONS
Here Z is the sample length, I 0 the intensity of incident and I the intensity
of absorbed radiation. The effect of classical gravitation on the classical electromagnetic
field is therefore in general to refract, reflect, diffract and absorb
electromagnetic radiation, and this is developed in Section 6.3. In other words
gravitation acts as a dielectric material which may be a reflector, an absorber,
a polarizable medium, a magnetizable medium, a conductor, a superconductor
and so forth. Finally in this Section 6.2 the wave equation of ECE field theory
is discussed and cross-checked mathematically prior to computation.
The basic wave equation of ECE field theory [1]– [32] is derived straightforwardly
form the tetrad postulate (6.20) through a lemma, or subsidiary geometric
proposition, the ECE Lemma. The fundamental structure of the latter
is:
D µ (D µ q a ν ) := 0 (6.118)
and is seen from Eq.6.20 to be an identity of Cartan geometry. From Eq.6.38
the lemma is seen to be:
∂ ν (D µ q a ν ) := 0 (6.119)
i.e.
∂ µ ( ∂ µ q a λ + ω a µbq b λ − Γ ν µλq a ν
)
:= 0. (6.120)
The dAlembertian operator is defined [33, 34] as:
so Eq.6.120 is:
□ := ∂ µ ∂ µ (6.121)
□q a λ = ∂ µ ( Γ ν µλq a ν − ω a µbq b λ)
. (6.122)
Now define the scalar curvature:
R := q λ a∂ ν ( Γ ν µλq a ν − ω a µbq b )
λ
and use the fundamental [33] inverse identity of tetrads:
to deduce the ECE Lemma [1]– [32]:
To check this derivation use Eq.6.20 in the form:
to find:
(6.123)
q a λq λ a = 1 (6.124)
□q a λ := Rq a λ. (6.125)
Γ ν µλq a ν − ω a µbq b λ = ∂ µ q a λ (6.126)
R = q λ a∂ µ ∂ µ q a λ = q λ a□q a λ (6.127)
Q.E.D.
The ECE wave equation [1]– [32] is obtained from the ECE lemma by using
the Einstein Ansatz:
R = −kT (6.128)
where k is the Einstein constant and T the index contracted energy-momentum
tensor. Einstein [37] asserted that the ansatz 6.128 must be applied to all the
radiated and matter fields of physics, not only the gravitational field. However
85

6.2. DETAILS IN THE DERIVATION OF THE ECE FIELD . . .
until the emergence of ECE theory in 2003 [1]– [32] the ansatz necessarily had
to be restricted to gravitation. In gravitational theory Eq.6.128 can be deduced
directly from the EH field equation:
using the inverse metric definition [1]– [32, 37]:
R µν − 1 2 Rg µν = kT µν . (6.129)
g µν g µν := 4. (6.130)
Multiply both sides of Eq.6.129 by g µν to obtain Eq.6.128 [37], as first shown by
Einstein. Here R µν is the symmetric Ricci tensor of EH field theory, R the scalar
curvature of EH field theory, g µν the symmetric metric of EH field theory, and
T µν the symmetric canonical energy-momentum tensor of EH field theory. In
the more general unified ECE theory the Einstein Ansatz 6.128 has been proven
in several ways. The key point is that the Einstein Ansatz in ECE field theory
applies to all radiated and matter fields, i.e., in logic, to the unified field. There
is only ONE unified field by definition, and so Einstein’s fundamental link of
physics and geometry must apply to that unified field. All other fields in nature
are components of the unified field: the gravitational, electromagnetic, weak,
strong and matter fields are variations of the tetrad field [1]– [33] in the Palatini
formulation of general relativity, and thus of causal and objective physics. This
is a major philosophical advance of ECE field theory from the standard model.
Therefore from Eqs.6.125 and 6.128 the ECE wave equation is:
(□ + kT ) q a µ = 0 (6.131)
and is the archetypical wave equation of objective physics, i.e. of general relativity
applied to the unified field.
The Dirac equation for the fermionic matter field, for example, is the linear
limit of Eq.6.131 defined by:
( me c
) 2
kT →
(6.132)
where m e is the mass of the fermion, is the reduced Planck constant and c
the vacuum speed of light, a universal constant of all relativity theory. In the
linear limit, as the name suggests, the ECE field equation linearizes, because its
eigenvalues are no longer intrinsically functions of the tetrad. More generally the
ECE wave equation is non-linear because R depends on the tetrad as in Eq.6.123.
Numerical methods are needed therefore to solve the ECE wave equation in
general. The Dirac equation is therefore:
(
□ +
( me c
) ) 2
q a µ = 0 (6.133)
where the tetrad defines the Dirac four-spinor [1]– [32]. Using the ECE Ansatz
in the form:
A a µ = A (0) q a µ (6.134)
we define the electromagnetic potential field. The governing wave equation of
the electromagnetic part of the unified field (the electromagnetic field for short)
is therefore:
(□ + kT ) A a µ = 0. (6.135)
86

CHAPTER 6.
SOLUTIONS OF THE ECE FIELD EQUATIONS
In the linear limit 6.132 we obtain the Proca equation from Eq.6.135:
kT →
( mp c
) 2
(6.136)
where m p is the mass of the photon, a boson. It is important to understand
that a different representation space [1]– [32] is used for the tetrads defining
the fermion and boson. Similarly a different representation space of the tetrad
is used for gluons and quarks [1]– [32], but the fundamental field is always the
tetrad field. Thus quantum electrodynamics in ECE theory proceeds by solving
Eqs.6.131 and 6.135 simultaneously with exchange of photons between two
electrons [1]– [32]. Similarly quantum chromodynamics proceeds by setting up
simultaneous ECE field equations with exchange of gluons between two quarks.
These procedures must be carried out numerically to avoid singularities and
renormalization. The Feynman calculus and the unobjective path integral formalism
[34]are by-passed completely by the numerical methods of ECE field
theory. Singularities do not occur in nature, and do not occur in the theory
of relativity and in objective and causal physics. In Feynman’s path integral
formalism the electron ”can do anything it likes”, ”go backwards in time”, and
so on [34]. These hypothetical trajectories are essentially summed to give what
APPEARS superficially to be an accurate result for the anomalous magnetic
moment of the electron and so forth. These ideas of quantum electrodynamics
are obviously and diametrically at odds with a causal and objective relativity
theory such as ECE field theory, wherein each event must be preceded by a
cause, as in Newtonian natural philosophy. The claimed accuracy of quantum
electrodynamics and quantum chromo-dynamics has more to do with the selective
use of several parameters than with a first principles theory of physics such
as ECE field theory or EH field theory.
The derivation of the ECE Lemma can be cross checked in at least two ways.
Apply the Leibnitz Theorem to Eq.6.118:
and to Eq.6.119:
D µ (D µ q a ν ) = (D µ D µ ) q a ν = 0 (6.137)
∂ µ (D µ q a ν ) = (∂ µ D µ ) q a ν + D µ (∂ µ q a ν ) = 0. (6.138)
Therefore Eq.6.118 is:
D µ ( ∂ µ q a λ + ω a µbq b λ − Γ ν µλq a )
ν
(D µ ∂ µ ) q a λ + ( D µ ω a µb)
q
b
λ − ( D µ Γ ν µλ)
q
a
ν = 0,
(6.139)
where we have used Eq.6.20 again. Now use the results:
D µ ∂ µ = □ + Γ µ µλ ∂λ , (6.140)
D µ = g µν D ν , (6.141)
∂ µ = g µν ∂ ν , (6.142)
to find:
D µ ∂ µ = g µν D ν g µν ∂ ν = 4D µ ∂ µ . (6.143)
87

6.3. DIELECTRIC ECE THEORY, ANALYTICAL AND . . .
From Eq.6.143 in Eq.6.139:
4 (D µ ∂ µ ) q a λ + ( D µ ω a µb)
q
b
λ − ( D µ Γ ν µλ)
q
a
ν = 0 (6.144)
Now use the Leibnitz Theorem again:
to find:
(D µ ∂ µ ) q a λ = D µ (∂ µ q a λ) + ∂ µ (D µ q a λ) (6.145)
4 ( D µ (∂ µ q a λ) + ∂ µ (D µ q a λ) + ( D µ ω a µb)
q
b
λ − ( D µ Γ ν µλ)
q
a
ν
)
= 0. (6.146)
Comparing Eqs.6.146 and 6.119:
i.e.
which is:
4D µ (∂ µ q a λ) + D µ ω a µb − ( D ν Γ ν µλ)
q
a
ν = 0 (6.147)
D ν ( ∂ µ q a λ + ω a µbq b λ − Γ ν µλq a ν
)
= 0 (6.148)
D µ (D µ q a λ) = 0 (6.149)
implying self-consistently the tetrad postulate 6.20, Q.E.D.
Secondly the ECE Lemma may be cross-checked using the general formula
6.20 for the covariant derivatve of any tensor. Regarding D µ q a ν as a rank three
mixed index tensor with two upper indices, µ and a, and one lower index, ν,
Eq.6.21 gives:
D µ (D µ q a ν ) = ∂ ν (D µ q a ν ) + Γ µ µλ Dλ q a ν
+ ω a µbD µ a b ν − Γ λ µνD µ q a λ = ∂ µ (D µ q a ν )
(6.150)
where we have used Eq.6.20 again, Q.E.D.
6.3 Dielectric ECE Theory, Analytical And Numerical
Solutions
In this Section an analytical solution is given in a well defined approximation
of the simultaneous equations 6.85 and 6.87; in general these must be solved
numerically along with the other two equations 6.84 and 6.86. First develop the
free fields E a and B a as follows using Eqs.6.107 and 6.108:
From Eqs.6.151 and 6.152 in Eq.6.85 we obtain:
1
∇ × D a ∂H a
+ µ 0
ɛ 0 ∂t
Therefore if the homogeneous current is defined as:
E a = 1 ɛ 0
(D a − P a ) (6.151)
B a = µ 0 (H a + M a ) (6.152)
= µ 0˜ja + 1 ∇ × P a ∂M a
− µ 0 . (6.153)
ɛ 0 ∂t
˜ja := ∂Ma
∂t
− c 2 ∇ × P a , (6.154)
88

CHAPTER 6.
SOLUTIONS OF THE ECE FIELD EQUATIONS
then we obtain:
∇ × (ɛ r E a ) + ∂ ( ) 1
B a = 0, (6.155)
∂t µ r
which can be reexpressed as:
∇ × D a + 1 ∂H a
c 2 = 0. (6.156)
∂t
Eqs.6.155 and 6.156 are true if and only if Eq.6.154 is true. However, the homogeneous
current can always be expressed as a combination of polarization and
magnetization as in Eq.6.154. The latter can therefore serve as a general definition
of the homogeneous current. In other words there is no loss of generality
in the derivation of Eqs.6.155 and 6.156 from Eq.6.85.
Similarly, using Eqs.6.151 and 6.152 in Eq.6.87, we obtain:
∇ × H a 1 − ∂Da 1
∂t
Therefore if we define the inhomogeneous current as:
(
)
˜J a := c ∇ × M a 1 + ∂Pa 1
∂t
we obtain the equation:
= 1 c ˜J a − ∇ × M a 1 − ∂Pa 1
∂t . (6.157)
(6.158)
∇ × H a 1 − ∂Da 1
= 0. (6.159)
∂t
This equation can be expressed in terms of the relative permittivity ɛ r1 and
permeability µ r1 as:
( ) B
a
∇ × − 1 ∂
µ r1 c 2 ∂t (ɛ r1E a ) = 0. (6.160)
Therefore the analytical and computational problem has been reduced to solving
the simultaneous equations 6.155 and 6.160. It is important to note [38] that
ɛ r1 is in general different from ɛ r , and that µ r1 is in general different from µ r .
The reason is that the current ˜J a is in general different from the current ˜j a .
Therefore the input parameters for the numerical solution of the simultaneous
equations 6.155 and 6.160 are ɛ r , ɛ r1 , µ r and µ r1 .
In the special case:
ɛ r = ɛ r1 , µ r = µ r1 (6.161)
analytical solutions can be obtained of the simultaneous equations 6.155 and
6.160, because in this special case:
giving the simultaneous equations:
D a 1 = D a , (6.162)
H a 1 = H a , (6.163)
∇ × D a + 1 ∂H a
c 2 = 0, (6.164)
∂t
∇ × H a − ∂Da
∂t
= 0. (6.165)
89

6.3. DIELECTRIC ECE THEORY, ANALYTICAL AND . . .
These can be written as:
∇ × (cD a ) + ∂ ( ) H
a
= 0, (6.166)
∂t c
∇ × ( Ha
c ) − 1 ∂
c 2 ∂t (cDa ) = 0, (6.167)
and so have the same structure as:
∇ × E a + ∂Ba
∂t
= 0, (6.168)
∇ × B a − 1 ∂E a
c 2 = 0. (6.169)
∂t
The plane wave solutions of Eqs.6.168 and 6.169 are well known. For example,
for a = (1) in the complex circular basis [1]– [32], the plane wave solutions are:
E (1) = E(0)
√
2
(i − ij) e (ωt−κZ) , (6.170)
B (1) = B(0)
√
2
(ii + j) e i(ωt−κZ) . (6.171)
It follows that Eqs.6.164 and 6.165 have solutions such as:
D (1) = D(0)
√
2
(i − ij) e (ωt−κZ) , (6.172)
H (1) = H(0)
√
2
(ii + j) e i(ωt−κZ) , (6.173)
where:
Now use:
H (0) = cD (0) . (6.174)
D (0) = ɛE (0) , (6.175)
H (0) = 1 µ B(0) , (6.176)
to find:
E (0) = vB (0) = c
n 2 B(0) (6.177)
where the refractive index is:
n 2 = ɛ r µ r (6.178)
and the phase velocity is:
v = c
n 2 . (6.179)
In the special case 6.161 there are also the simultaneous equations:
˜ja = ∂Ma − c 2 ∇ × P a , (6.180)
∂t
)
˜J a = c
(∇ × M a + ∂Pa , (6.181)
∂t
where ˜J a and ˜j a are linked by Hodge duality (Section 6.2). Therefore in the
special case 6.161 the effect of gravitation on electromagnetism can be deduced
analytically from ECE theory as follows.
90

CHAPTER 6.
SOLUTIONS OF THE ECE FIELD EQUATIONS
1. Gravitation changes the amplitudes of the plane waves:
E (0) −→ D (0) , (6.182)
B (0) −→ H (0) . (6.183)
2. Gravitation changes the phase velocity of the free space plane waves from
c to v, causing diffraction as in the Eddington effect [1]– [32].
3. Gravitation causes a red shift in angular frequency for a given κ because
the phase velocity is defined by
v = ω κ
(6.184)
and has been decreased from c to v if the refractive index is greater than
unity.
Acknowledgments The British Government is thanked for a Civil List pension
and the AIAS group and environment for many interesting discussions.
91

6.3. DIELECTRIC ECE THEORY, ANALYTICAL AND . . .
92

Bibliography
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[2] M. W. Evans, Found. Phys. Lett., 17, 25, 149, 267, 301, 393, 433, 535, 663
(2004).
[3] M. W. Evans, Found. Phys. Lett., 18, 139, 259, 519 (2005), and papers in
Found. Phys. Lett. and Found. Phys. (1994 to present).
[4] M. W. Evans, Generally Covariant Unified Field Theory (in press 2005,
preprints on www.aias.us and www.atomicprecision.com).
[5] L. Felker, The Evans Equations of Unified Field Theory (in press, preprint
on www.aias.us and www.atomicprecision.com).
[6] M. W. Evans, The Objective Laws of Classical Electrodynamics, the Effect
of Gravitation on Electromagnetism, (2005, preprint on www.aias.us and
www.atomicprecision.com).
[7] M. W. Evans, First and Second Order Aharonov Bohm Effects in
the Evans Unified Field Theory, (2005, preprint on www.aias.us and
www.atomicprecision.com).
[8] M. W. Evans, The Spinning of Spacetime as Seen in the Inverse Faraday
Effect, (2005, preprint on www.aias.us and www.atomicprecision.com).
[9] M. W. Evans, On the Origin of Polarization and Magnetization, (2005,
preprint on www.aias.us and www.atomicprecision.com).
[10] M. W. Evans, Explanation of the Eddington Experiment in the
Evans Unified Field Theory, (2005, preprint on www.aias.us and
www.atomicprecision.com).
[11] M. W. Evans, The Coulomb and Ampère Maxwell Laws in the
Schwarzschild Metric: A Classical Explanation of the Eddington Effect
from the Evans Field Theory, (2005, preprint on www.aias.us and
www.atomicprecision.com).
[12] M. W. Evans, Generally Covariant Heisenberg Equation from the
Evans Unified Field Theory, (2005, preprint on www.aias.us and
www.atomicprecision.com).
[13] M. W. Evans, Metric Compatibility and the Tetrad Postulate, (2005,
preprint on www.aias.us and www.atomicprecision.com).
93

BIBLIOGRAPHY
[14] M. W. Evans, Derivation of the Evans Lemma and Wave Equation from
the First Cartan Structure Equation, (2005, preprint on www.aias.us and
www.atomicprecision.,com).
[15] M. W. Evans, Proof of the Evans Lemma from the Tetrad Postulate, (2005,
preprint on www.aias.us and www.atomicprecision.com).
[16] M. W. Evans, Self-Consistent Derivation of the Evans Lemma and Application
to the Generally Covariant Dirac Equation, (2005, preprint on
www.aias.us and www.atomicprecision.com).
[17] M. W. Evans, Quark-Gluon Model in the Evans Unified Field Theory,
(2005, preprint on www.aias.us and www.atomicprecision.com).
[18] M. W. Evans, The Origin of Intrinsic Spin and the Pauli Exclusion Principle
in the Evans Unified Field Theory, (2005, preprint on www.aias.us and
www.atomicprecision.com).
[19] M. W. Evans, General Covariance and Co-ordinate Transformation in Classical
and Quantum Electrodynamics, (2005, preprint on www.aias.us and
www.atomiprecision.com).
[20] M. W. Evans, The Role of Gravitational Torsion : the S Tensor, (2005,
preprint on www.aias.us and www.atomicprecision.com).
[21] M. W. Evans, Explanation of the Faraday Disc Generator in the
Evans Unified Field Theory, (2005, preprint on www.aias.us and
www.atomicprecision.com).
[22] M. W. Evans et al., (AIAS Author Group), Experiments to Test the Evans
Unified Field Theory and General Relativity in Classical Electrodynamics,
(preprints on www.aias.us and www.atomicprecision.com).
[23] M. W. Evans et al., (AIAS Author Group), ECE Field Theory of the Sagnac
Effect (preprints on www.aias.us and www.atomicprecision.com).
[24] M. W. Evans et al., (AIAS Author Group), ECE Field Theory: the Influence
of Gravitation on the Sagnac Effect, (preprints on www.aias.us and
www.atomicprecision.com).
[25] M. W. Evans et al., (AIAS Author Group), Dielectric Theory of ECE
Spacetime (preprints on www.aias.us and www.atomicprecision.com).
[26] M. W. Evans et al., (AIAS Author group), Spectral Effects of Gravitation,
(preprints on www.aias.us and www.atomicprecision.com).
[27] M. W. Evans, Cosmological Anomalies: EH Versus ECE Field Theory,
(preprints on www.aias.us and www.atomicprecision.com).
[28] M. W. Evans, (ed.), Modern Non-Linear Optics, in I. Prigogine and S. A.
Rice (eds.), Advances in Chemical Physics, (Wiley-Interscience, New York,
2001, 2nd ed.), vols. 119(1)-119(3).
[29] M. W. Evans and L. B. Crowell, Classical and Quantum Electrodynamics
and the B (3) Field, (World Scientific, Singapore, 2001).
94

BIBLIOGRAPHY
[30] M. W. Evans and J.-P. Vigier, The Enigmatic Photon (Kluwer, Dordrecht,
1994 - 1002, hardback and softback).
[31] M. W. Evans and A. A. Hasanein, The Photomagneton in Quantum Field
Theory (World Scientific, Singapore, 1994).
[32] M. W. Evans and S. Kielich (eds.), first edition of ref. (28) (Wiley-
Interscience, New York, 1992, reprinted 1993, softback 1997), vols. 85(1) -
85(3).
[33] S. P. Carroll, Lecture Notes in General Relativity (graduate course in the
public domain, Harvard, UCSB and Chicago, arXiv : gr - gc 973019 v1
1997).
[34] L. H. Ryder, Quantum Field Theory (Cambridge Univ Press, 1996, 2nd
ed.).
[35] P. W. Atkins, Molecular Quantum Mechanics (Oxford Univ. Press, 2nd
ed., 1983).
[36] M. W. Evans, G. J. Evans, W. T. Coffey and P. Grigolini, Molecular Dynamics
and the Theory of Broad Band Spectroscopy (Wiley-Interscience,
New York, 1982).
[37] A. Einstein, The Meaning of Relativity (Princeton Univ. Press, 1921 -
1954).
[38] H. Eckardt, personal communication, (Oct. 2005)
95

BIBLIOGRAPHY
96

Chapter 7
ECE Generalization Of The
d’Alembert, Proca And
Superconductivity Wave
Equations: Electric Power
From ECE Space-Time
by
M. W. Evans
Alpha Foundation’s Institute for Advance Study (A.I.A.S.).
(emyrone@aol.com, www.aias.us, www.atomicprecision.com)
Abstract
The well known d’Alembert, Proca and superconductivity wave equations are
shown to be special cases of the wave equations which can be constructed from
the homogeneous and inhomogeneous field equations of Einstein Cartan Evans
(ECE) unified field theory. One of the important practical consequences is that
a material can become a superconductor by absorption of the inhomogeneous
and homogeneous currents of ECE spacetime. This means that in a well designed
circuit or material, the output voltage or power can exceed by orders of
magnitude the input power needed to run the circuit. This phenomenon has
been observed experimentally and is reproducible and repeatable. An array of
such circuits would in principle produce a new type of electric power for general
use on a sufficiently large scale to be significant.
Keywords: Einstein Cartan Evans (ECE) unified field theory; electric power
from ECE space-time; generally covariant d’Alembert, Proca and superconductivity
equations.
97

7.1. INTRODUCTION
7.1 Introduction
Rigorous objectivity in science requires general relativity to be applied to all
the main equations of physics, chemistry and the natural sciences. The equations
must all retain their form under the general coordinate transformation,
and the philosophy of relativity requires that every event have a cause, science
must be causal as well as objective. A plausible theoretical structure for
causal and objective natural science has recently been suggested using standard
Riemann geometry in its most general form [1]– [34]. This type of geometry
was developed by Cartan into well known differential geometry, a concise and
elegant formulation of the general Riemann geometry of the early nineteenth
century. In well known letters to Einstein, Cartan suggested in the early twenties
of the twentieth century that the electromagnetic field be the torsion form
of Cartan geometry. From 2003 onwards this suggestion led to ECE unified
field theory [1]– [33]. The gravitational field is represented by the Riemann or
curvature form of Cartan geometry. The torsion form represents the spinning
of four-dimensional space-time (the electromagnetic, weak and strong fields,
fermion and boson matter fields), and the curvature form the curving of fourdimensional
space-time (the gravitational field, gravitons and gravitinos). This
was a notable advance upon the Einstein Hilbert field theory of 1915, which
was the first theory to successfully apply general relativity to gravitation. In
so doing however general Riemann geometry was specialized into Riemann geometry
without torsion. Only the curvature of space-time was considered, the
spinning of space-time was neglected by use of the symmetric or Christoffel connection
[34], sometimes also known as the Riemann or Levi-Civita connection.
In general Riemann geometry the connection is asymmetric in its lower two indices.
For a given upper index the connection is therefore a matrix which is the
sum of a symmetric and anti-symmetric component. In Cartan’s well known
formulation of general Riemann geometry, the torsion and curvature forms are
in general non zero, and defined respectively by the Cartan structure equations,
known in contemporary, standard, differential geometry as the master equations.
A differential two-form translates into an anti-symmetric tensor, both the
torsion and Riemann tensors are anti-symmetric in their last two indices. The
Riemann form is a tensor valued two-form, the torsion form is a vector valued
two-form [34]. As inferred by Cartan, the latter becomes the electromagnetic
field, for example, through the Evans Ansatz [1]– [33]:
F a = A (0) T a (7.1)
where A (0) is a vector potential magnitude. The electromagnetic potential is
the fundamental field, the tetrad, or fundamental field of ECE theory:
A a = A (0) q a . (7.2)
In Cartan geometry the torsion form is the covariant exterior derivative of the
tetrad:
T a = d ∧ q a + ω a b ∧ q b (7.3)
and the curvature form is defined by the spin connection as follows:
R a b = d ∧ ω a b + ω a c ∧ ω c b. (7.4)
98

CHAPTER 7. ECE GENERALIZATION OF THE D’ALEMBERT, . . .
The torsion and curvature forms are inter-related by the first Bianchi identity:
d ∧ T a + ω a b ∧ T b := R a b ∧ q b (7.5)
and the curvature form always obeys the second Bianchi identity:
D ∧ R a b := 0. (7.6)
Using the Evans Ansatz 7.1 or 7.2 links the electromagnetic field to the electromagnetic
potential as follows:
F a = d ∧ A a + ω a b ∧ A b (7.7)
using Eq.7.3. The first Bianchi identity 7.5 produces with the Ansatz the homogeneous
field equation of ECE theory [1]– [33]:
d ∧ F a = µ 0 j a (7.8)
where the homogeneous current of ECE field theory is defined by:
j a = A(0)
µ 0
(
R
a
b ∧ q b − ω a b ∧ T b) . (7.9)
Hodge duality transformation fo Eq.7.9 [1]– [33] produces the inhomogeneous
field equation of ECE field theory:
where the inhomogeneous current is defined by:
d ∧ ˜F a = µ 0 J a (7.10)
J a = A(0)
µ 0
(
˜Ra b ∧ q b − ω a b ∧ ˜T b) . (7.11)
In Section 7.2 Eqs.7.7 and 7.10 are translated into tensor notation and the ECE
generalization obtained of the d’Alembert, Proca and superconductivity wave
equations [35, 36]. This means (Section 7.3), that superconductivity can be
understood in terms of general relativity, a conclusion which indicates the possibility
of designing a material or circuit that becomes a superconductor from
absorbing j a and J a from ECE space-time. The generally covariant London
equation and Meissner effect follow from the ECE superconductivity equation.
This conclusion of ECE field theory has been verified experimentally in reproducible
and repeatable experiments [37].
7.2 Derivation Of The Wave Equation
The d’Alembert wave equation in the standard model [35, 36] is derived using
tensor notation. This exercise is given here in preparation for deriving the
equivalent wave equation in ECE theory. The electromagnetic field tensor in
the standard model is defined by:
F µν = ∂ µ A ν − ∂ ν A µ (7.12)
99

7.2. DERIVATION OF THE WAVE EQUATION
and the inhomogeneous wave equation of the standard model is given by:
∂ µ F µν = µ 0 J ν . (7.13)
A wave equation may be constructed by substituting Eq.7.12 into Eq.7.13 to
give:
∂ µ (∂ µ A ν − ∂ ν A µ ) = µ 0 J ν (7.14)
i.e.
The standard model uses the Lorentz condition [35]:
to give the d’Alembert wave equation:
□A ν = ∂ ν (∂ µ A µ ) + µ 0 J ν . (7.15)
∂ µ A µ = 0 (7.16)
□A ν = µ 0 J ν (7.17)
whose solutions are the Liennard-Wiechert potentials. The Proca equation of
the standard model is [1]– [33]:
(
□ +
( mp c
) ) 2
A ν = 0 (7.18)
where m p is the mass of the photon, c the speed of light and the reduced
Planck constant. Eqs.7.17 and 7.18 imply that photon mass may be understood
as a charge current density [1]– [33]:
J µ = − 1 ( mp c
) 2
(7.19)
µ 0
For all practical purposes in the laboratory:
giving the free space d’Alembert equation:
m p −→ 0 (7.20)
□A µ = 0. (7.21)
In the standard model the space-time used for electrodynamics is Minkowski
space-time, or flat space-time. In consequence electrodynamics in the standard
model cannot be unified with gravitational theory [35] because the equations of
electrodynamics are not generally covariant, they are Lorentz covariant. In ECE
theory the equations of electrodynamics (Section 7.1) are generally covariant and
unified with the equations of all other fields, including gravitation. In both the
standard model and ECE field theory indices are raised and lowered with the
metric tensor [34]. Thus for example:
⎫
∂ µ = g µρ ∂ ρ , A ν = g νρ A ρ , ⎪⎬
∂ µ A ν = g µρ g ρσ ∂ ρ A σ ,
(7.22)
⎪⎭
F µν = g µρ g νσ F ρσ ,
and
}
F µν = ∂ µ A ν − ∂ ν A µ ,
F µν = ∂ µ A ν − ∂ ν A µ .
100
(7.23)

CHAPTER 7. ECE GENERALIZATION OF THE D’ALEMBERT, . . .
From the ECE Lemma [1]– [33] the correct and complete structure of the wave
equation of electrodynamics is:
□A a µ = RA a µ (7.24)
where R is a well defined scalar curvature. The latter is missing completely
from the standard model but is a hitherto unconsidered source of electric power
from space-time. The generalization of the d’Alembert equation in ECE field
theory is obtained from the tensor notation of equations 7.7 and 7.10:
and
F aµν = ∂ µ A aν − ∂ ν A aµ + ω µa b Aνb − ω νa bA µb (7.25)
∂ µ F aµν = µ 0 ˜J aν . (7.26)
The indices of the spin connection have been raised with the appropriate metric,
that of ECE space-time with torsion and curvature:
So we obtain from Eqs.7.25 and 7.26:
ω µa b = gµσ ω a σb. (7.27)
□A aν = RA aν = µ 0 ˜J aν + ∂ µ
(
∂ ν A aµ − ω µa b Aνb + ω νa bA µb) . (7.28)
The current is therefore defined by:
˜J aν = 1 µ 0
(
RA aν − ∂ µ
(
∂ ν A aµ − ω µa b Aνb + ω νa bA µb)) (7.29)
and the scalar curvature by [1]– [33]:
Using the Ansatz in the form:
R = q λ a∂ µ ( Γ ν µλq a ν − ω a µbq b λ)
. (7.30)
A aν = A (0) q aν (7.31)
it is seen that ˜J aν is derived completely from geometry. In tensor notation:
˜J aν = A(0)
µ 0
(
Rq aν − ∂ µ
(
∂ ν q aµ − ω µa b qνb + ω νa ba µb)) (7.32)
This result can be compared with the current from Eq.7.11 in form notation.
These sources of current are not present in special relativity and so are absent
from the standard model. They are, however, the currents responsible for the
generation of electricity from space-time as observed experimentally [37]. In
these equations the tetrad with raised index is defined as usual through the
metric.
Similarly, a wave equation can be built up from Eqs.7.7 and 7.8 to show
that j a can also be built up from space-time in a manner analogous to Eq.7.32.
These equations give new insights into the process of absorption - which takes
electromagnetic energy from space-time, stores it in a material such as an atom,
and changes it into other forms of energy within the material. Such a change
of energy can take place within a superconductor or semiconductor, or a well
101

7.3. DISCUSSION
designed circuit. The end process is an output voltage - the production of
electricity from ECE space-time. The structure of the wave equation 7.28 is:
□A µ = −k 2 A µ = µ 0 J µ (7.33)
which is also the structure of the wave equation of superconductivity [35]. The
space part of Eq.7.33 is the London equation [35]:
If A is time-independent then:
Ohm’s law is
J = −k 2 A. (7.34)
E = −∂A/∂t = 0. (7.35)
E = RJ (7.36)
so E = 0, J = 0, meaning that R = 0. The resistance to a finite current J
from ECE space-time is zero under these circumstances and so the material
thus defined is a superconductor. This theory is similar to Cooper pair theory
[35] and also gives the Meissner effect (exclusion of magnetic flux). The extra
insight given by ECE theory is that A µ , k 2 and J µ are identified as space-time
properties. Therefore under well defined circumstances it is conceivable that
the absorption of ECE space-time produces a superconductor.
7.3 Discussion
The photon in ECE theory is a space-time property defined by Eq.7.24, which
is obtained from the ECE Lemma:
□q a µ = Rq a µ (7.37)
with the fundamental ansatz 7.2. Here q a µ is the fundamental tetrad field and
R is defined by the Einstein ansatz [37]:
R = −kT (7.38)
where k is the Einstein constant and where T is a well defined [37] canonical
energy-momentum magnitude formed by index contraction. In the linear limit
where the photon does not interact with the gravitational field of for example
an electron:
( mp c
) 2
kT −→
(7.39)
where m p is the mass of the photon. The photon is therefore defined entirely by
A (0) , k and Cartan geometry in an appropriate representation space - that of the
boson. Wave equations such as 7.24 or 7.28 in ECE field theory can be solved
simultaneously with the field equations 7.8 and 7.10, and indeed equations such
as Eq.7.28 are a re-statement of the field equations as deduced in Section 7.28.
For the free electromagnetic field [1]– [33]:
ω a b = ɛ a bcq c , (7.40)
R a b = ɛ a bcT c . (7.41)
102

CHAPTER 7. ECE GENERALIZATION OF THE D’ALEMBERT, . . .
In contrast the photon in the standard model is defined by the Proca equation
(or the free space d’Alembert equation if the photon mass is neglected) and
by the field equations of the standard model:
in which the field is related to the potential by:
d ∧ F = 0 (7.42)
d ∧ ˜F = µ 0 J (7.43)
F = d ∧ A. (7.44)
In the standard model there are therefore several fundamental problems, for
example the inability to describe the interaction of electromagnetism with gravitation,
the conflict between spin accelerations and special relativity (where
there are no accelerations); the conflict between photon mass theory and gauge
theory; the absence of the Evans spin field [1]– [33] because of the absence of
general relativity.
The electron in ECE theory is described by
(□ + kT ) q a µ (7.45)
where q a µ is the wave function of the electron. The latter is a fermion in an
SU(2) representation space that defines the tetrad as:
[ ]
q a q
R
µ = 1 q R 2
q L 1 q L (7.46)
2
The Dirac spinor is obtained straightforwardly from the tetrad. Eq.7.45 describes
the trajectory of an electron in a gravitational field. The superscripts
R and L denote left and right fermion spin and the subscripts 1 and 2 denote
components of the two dimensional complex SU(2) representation space. The
free electron is defined when the gravitational field is vanishingly weak. In this
limit:
( me c
) 2
kT −→
(7.47)
where m e is the mass of the electron. In this limit Eq.7.48 becomes the Dirac
equation ( ( me c
) ) 2
□ + φ = 0 (7.48)
where:
φ =
[
φ
R
φ L ]
(7.49)
is the Dirac spinor, and where φ R and φ L are the Pauli spinors.
The interaction between an electron and photon is described by solving
Eqs.7.24 and 7.45 simultaneously - the fermion boson interaction problem in
general relativity. The wave-function of the fermion is a tetrad in SU(2) representation
space and the wave-function of the boson is a tetrad in O(3) representation
space. In the standard model this problem is approached in the
semi-classical limit or by using quantum electrodynamics with artificial renormalization
of unphysical infinities which do not occur in general relativity. The
103

7.3. DISCUSSION
process of renormalization introduces adjustable parameters and therefore quantum
electrodynamics is an incomplete theory of special relativity, and not general
relativity as required. In ECE theory the interaction of a photon and
electron [1]– [33] is defined by solving two simultaneous equations:
Within a factor A (0) Eq.7.51 is
□q a µe = R e q a µe, (7.50)
□A a µ = R p A a µ. (7.51)
□q a µp = R p q a µp. (7.52)
Here R e is the scalar curvature of the electron and R p is the scalar curvature of
the photon. The tetrad in Eq.7.52 denotes the wave function of the photon, and
the tetrad in Eq.7.50 denotes the wave function of the electron. The effect of the
electron on the photon is described by Eqs.7.50 and 7.52, which are derived from
Cartan geometry. The influence of the electron wave function on the photon
wave function is to set up the currents j a and J a . The wave function of the
free photon will be changed by the electron and vice versa. Therefore the basic
problem is one of interaction of curvatures, total curvature being conserved.
Before interaction:
R = R ei + R pi . (7.53)
After interaction:
and:
R = R ef + R pf (7.54)
R ei + R pi = R ef + R pf (7.55)
Therefore the curvature of the electron after interaction is:
and:
R ef = R ei + (R pi − R pf ) (7.56)
R ef − R ei = R pi − R pf . (7.57)
In conventional language this means that the electron has absorbed energy and
momentum from the photon, but ECE theory shows also that the photon originates
in Cartan geometry, and in curvature and torsion. Therefore in absorbing
a photon, the electron has absorbed energy from space-time itself. Depending
on material and circuit design a large amount of energy may be absorbed by
the electron, producing a large j a and J a , and providing electric power from
space-time as observed experimentally [37]. The simplest possible type of theory
may be developed for atoms and molecules, absorption and emission theory,
the theory of conduction, semi-conductors and superconductors. Gravitational
theory may be incorporated into quantum electrodynamics, the theory of radio
activity, and quantum chromo-dynamics, and conversely electromagnetic theory
may be incorporated into graviton, gravitino and supersymmetry theory.
Acknowledgments The British Government is thanked for a Civil List pension
(2005) and the staff of AIAS and others for many interesting discussions.
104

Bibliography
[1] M. W. Evans, Found. Phys. Lett., 16, 367, 507 (2003).
[2] M. W. Evans, Found. Phys. Lett., 17, 25, 149, 267, 301, 393, 433, 535, 663
(2004).
[3] M. W. Evans, Found. Phys. Lett., 18, 139, 259, 519 (2005), and papers and
letters in Found. Phys. and Found. Phys. Lett., 1994 to 2005.
[4] M. W. Evans, Generally Covariant Unified Field Theory (Abramis, U.K.,
2005), volume one.
[5] M. W. Evans, Generally Covariant Unified Field Theory (Abramis, U.K,
2005 in press preprint on www.aias.us and www.atomicprecision.com), volume
two.
[6] L. Felker, The Evans Equations of Unified Field Theory (preprint on
www.aias.us and www.atomicprecision.com, 2005).
[7] M. W. Evans, The Objective Laws of Classical Electrodynamics, the Effect
of Gravitation on Electromagnetism, (2005, preprint on www.aias.us and
www.atomicprecision.com).
[8] M. W. Evans, First and Second Order Aharonov Bohm Effects in
the Evans Unified Field Theory, (2005, preprint on www.aias.us and
www.atomicprecision.com).
[9] M. W. Evans, The Spinning of Space-time as Seen in the Inverse Faraday
Effect, (2005, preprint on www.aias.us and www.atomicprecision.com).
[10] M. W. Evans, On the Origin of Polarization and Magnetization, (2005,
preprint on www.aias.us and www.atomicprecision.com).
[11] M. W. Evans, Explanation of the Eddington Experiment in the
Evans Unified Field Theory, (2005, preprint on www.aias.us and
www.atomicprecision.com).
[12] M. W. Evans, The Coulomb and Ampère Maxwell Laws in the
Schwarzschild Metric: A Classical Explanation of the Eddington Effect
from the Evans Field Theory (2005, preprint on www.aias.us and
www.atomicprecision.com).
[13] M. W. Evans, Generally Covariant Heisenberg Equation from the
Evans Unified Field Theory (2005, preprint on www.aias.us and
www.atomicprecision.com).
105

BIBLIOGRAPHY
[14] M. W. Evans, Metric Compatibility and the Tetrad Postulate (2005,
preprint on www.aias.us and www.atomicprecision.com).
[15] M. W. Evans, Derivation of the Evans Lemma and Wave Equation
from the Tetrad Postulate (2005, preprint on www.aias.us and
www.atomicprecision.com).
[16] M. W. Evans, Proof of the Evans Lemma from the Tetrad Postulate (2005,
preprint on www.aias.us and www.atomicprecision.com).
[17] M. W. Evans, Self-Consistent Derivation of the Evans Lemma and Application
to the Generally Covariant Dirac Equation (2005, preprint on
www.aias.us and www.atomicprecision.com).
[18] M. W. Evans, Quark-Gluon Model in the Evans Unified Field Theory (2005,
preprint on www.aias.us and www.atomicprecision.com).
[19] M. W. Evans, The Origin of Intrinsic Spin and the Pauli Exclusion Principle
in the Evans Unified Field Theory (2005, preprint on www.aias.us and
www.atomicprecision.com).
[20] M. W. Evans, General Covariance and Co-ordinate Transformation in Classical
and Quantum Electrodynamics (2005, preprint on www.aias.us and
www.atomicprecision.com).
[21] M. W. Evans, The Role of Gravitational Torsion: the S Tensor (2005,
preprint on www.aias.us and www.atomicprecision.com).
[22] M. W. Evans, Explanation of the Faraday Disc Generator in the
Evans Unified Field Theory (2005, preprint on www.aias.us and
www.atomicprecision.com).
[23] M. W. Evans et al. (A.I.A.S. author group), Experiments to Test the Evans
Unified Field Theory and General Relativity in Classical Electrodynamics
(2005, preprint on www.aias.us and www.atomicprecision.com).
[24] M. W. Evans et al., (A.I.A.S. author group), ECE Field Theory
of the Sagnac Effect (2005, preprint on www.aias.us and
www.atomicprecision.com).
[25] M. W. Evans et al., (A.I.A.S. author group), ECE Field Theory, the Influence
of Gravitation on the Sagnac Effect (2005, preprint on www.aias.us
and www.atomicprecision.com).
[26] M. W. Evans et al. (A.I.A.S. author group), Dielectric Theory of ECE
Space-time (2005, preprint on www.aias.us and www.atomicprecision.com).
[27] M. W. Evans et al. (A.I.A.S author group), Spectral Effects of Gravitation
(2005, preprint on www.aias.us and www.atomicprecision.com).
[28] M. W. Evans, Cosmological Anomalies: EH versus ECE Space-time (2005,
preprint on www.aias.us and www.atomicprecision.com).
[29] M. W. Evans, Solutions of the ECE Field Equations (2005, preprint on
www.aias.us and www.atomicprecision.com).
106

BIBLIOGRAPHY
[30] M. W. Evans (ed.), Modern Non-linear Optics, in I. Prigogine and S. A.
Rice (series eds.), Advances in Chemical Physics, (Wiley Interscience, New
York, 2001, 2nd ed.), vols. 119(1)-119(3), circa 2,500 pages.
[31] M. W. Evans and L. B. Crowell, Classical and Quantum Electrodynamics
and the B (3) Field (World Scientific, Singapore, 2001).
[32] M. W. Evans and J.-P. Vigier, The Enigmatic Photon (Kluwer, Dordrecht,
1994 to 2002, hardback and softback), vols. 1-5.
[33] M. W. Evans and S. Kielich (eds.), first edition of reference (30) (Wiley-
Interscience, New York, 1992, 1993, 1997 (softback)), vols 85(1)-85(3), circa
2,500 pages.
[34] S. P. Carroll, Lecture Notes in General Relativity (graduate course in the
public domain, Harvard, UCSB and U. Chicago, arXiv : gr - gc 973019 v1
1997).
[35] L. H. Ryder, Quantum Field Theory (Cambridge Univ. Press, 2nd ed.,
1996).
[36] J. D. Jackson, Classical Electrodynamics (Wiley, New York, 3rd. ed., 1998).
[37] A. Hill, Mexican Group, personal communications (www.aias.us and
www.atomicprecision.com, 2005).
107

BIBLIOGRAPHY
108

Chapter 8
Resonance Solutions Of The
ECE Field Equations
Abstract
by
M. W. Evans
Alpha Foundation’s Institute for Advance Study (A.I.A.S.).
(emyrone@aol.com, www.aias.us, www.atomicprecision.com)
Resonance solutions of the Einstein Cartan Evans (ECE) field equations are
obtained by developing them in terms of the electromagnetic potential to give
linear inhomogeneous differential equations whose solutions were first discovered
by the Jacobi’s and Euler (1739 - 1743). There are four such resonance
equations, and in a well defined approximation it is shown that resonance absorption
from ECE space-time occurs. The net result is that electric power from
space-time is available in copious quantities given the circuit or material design
to take resonant energy from ECE space-time.
Keywords: Einstein Cartan Evans (ECE) unified field theory; resonant absorption
from ECE space-time, energy from ECE space-time.
8.1 Introduction
The mathematical structure of Einstein Cartan Evans (ECE) unified field theory
is that of standard differential geometry [1] – [35] within a scalar valued factor
A (0) , a vector potential magnitude. Thus, for example, the relation between
the electromagnetic field form (F ) and electromagnetic potential form ((A)) is
given by the first Cartan structure equation, and the field equations for F and
its Hodge dual ˜F are given by the first Bianchi identity. The Cartan structure
equations and the Bianchi identities are standard equations of Cartan geometry.
We use for clarity of mathematical structure a ”barebones” or index suppressed
notation [1] – [35] to give these equations as follows:
F = d ∧ A + ω ∧ A, (8.1)
109

8.1. INTRODUCTION
d ∧ F = µ 0 j, (8.2)
d ∧ ˜F = µ 0 J. (8.3)
Here j is the homogeneous current and J the inhomogeneous current, and µ 0 is
the S.I. permeability in vacuo. The symbol ∧ is the wedge product, d∧ is the
exterior derivative and ω is the spin connection. These quantities and notation
are fully defined elsewhere [1]– [35]. The Hodge dual of Eq.8.1 is denoted:
From Eqs.8.1 and 8.2:
and from Eqs.8.3 and 8.5:
˜F = d˜∧A + ω˜∧A (8.4)
= d ∧ B + ω ∧ B. (8.5)
d ∧ (d ∧ A + ω ∧ A) = µ 0 j (8.6)
d ∧ (d ∧ B + ω ∧ B) = µ 0 J. (8.7)
Eq.8.6 is the fundamental resonance equation of ECE field theory and Eq.8.7
is its Hodge dual. Eq.8.6 is a development of the well known linear inhomogeneous
equation whose resonance solutions [36] were first given by the Bernoulli’s
and Euler (1739 - 1743). In general such equations give amplitude resonance,
potential and kinetic energy resonance, Q factors, transient and equilibrium solutions,
phase lags and other features of interest in many aspects of physics and
electrical engineering, notably circuit theory [36]. In Eq.8.6:
where
is the torsion form [1]– [35] and where
j = A(0)
µ 0
(R ∧ q − ω ∧ T ) (8.8)
T = d ∧ q + ω ∧ q (8.9)
R = d ∧ ω + ω ∧ ω. (8.10)
R is the Riemann form of standard differential geometry. Eqs.8.9 and 8.10 are
the first and second Cartan structure equations, sometimes known as the master
equations of differential geometry. Therefore Eq.8.6 is in general:
where
j = 1 µ 0
d ∧ (d ∧ A + ω ∧ A) (8.11)
A = A (0) q. (8.12)
Thus, the current j is a source of resonance absorption from ECE spacetime.
A similar conclusion can be reached for the Hodge dual resonance equation 8.7.
The potential A also obeys the ECE Lemma [1]– [35]:
□A = RA (8.13)
110

CHAPTER 8. RESONANCE SOLUTIONS OF THE ECE FIELD . . .
where
R = −kT (8.14)
is a well defined scalar curvature, T is the index contracted canonical energymomentum
tensor, and k is Einstein’s constant. Therefore the ECE Lemma is
the subsidiary proposition of the ECE wave equation [1]– [35]:
(□ + kT ) A = 0. (8.15)
Therefore the fundamental mathematical structure of standard differential geometry
gives three equations, 8.6, 8.7 and 8.15 with which to investigate resonant
absorption of energy from ECE space-time.
In the standard model:
F = d ∧ A, (8.16)
Eqs.8.16 and 8.17 give the Poincaré Lemma [37]:
d ∧ F = 0, (8.17)
d ∧ ˜F = µ 0 J. (8.18)
d ∧ (d ∧ A) = 0 (8.19)
and the current j is missing. The current J in the standard model is introduced
empirically and is not recognized to originate in Cartan geometry. Therefore
many key resonance features are missing from the standard model, notably the
ability of ECE theory to take electric power from space-time in the shape of the
currents j and J. Within the factor A (0) /µ 0 these currents are defined completely
by the structure or geometry of space-time itself. In the standard model
of classical electrodynamics (the Maxwell Heaviside field equations) space-time
has no structure, it is the flat or Minkowski space-time and in consequence classical
electrodynamics in the standard model cannot be unified with gravitation,
in which space-time is structured. Therefore electric power cannot be taken
from space-time in the standard model. This is contrary to the reproducible
and repeatable experiments [38] of the Mexican Group, which has observed amplification
of power levels in excess of one hundred thousand in given circuit
designs, and amplification that is due to resonant absorption from ECE spacetime.
This paper is the first to offer a detailed explanation of this important
phenomenon.
In Section 8.2 the fundamental resonance equation:
d ∧ (d ∧ A + ω ∧ A) = µ 0 j (8.20)
is developed into four resonance equations in the vector notation used in electrical
engineering and circuit theory. One of these vector equations is solved
analytically using appropriate approximations. The result is resonance from
a driven undamped inhomogeneous structure. This simple analytical exercise
achieves our aim of showing that resonant absorption is possible from ECE
space-time, as observed experimentally [38]. Driven undamped resonance produces
an infinite Q factor and infinite amplitude resonance at the fundamental
frequency [36]. More generally [36] the solutions of the linear inhomogeneous
equation give finite Q factors and phase factors, transient and steady state effects,
and various types of resonances. These are briefly reviewed in Section 8.3
111

8.2. THE RESONANCE EQUATIONS
for the simplest type of linear inhomogeneous second order differential equation
[36]. Eq.8.20 is expected to have all these features in general, and several
more, and numerical methods will reveal all of them straightforwardly given initial
and boundary conditions. Most generally resonance from ECE space-time
is described by solving Eqs.8.6, 8.7 and 8.15 simultaneously with given initial
and boundary conditions. However the simplest type of linear inhomogeneous
structure (Section 8.3) is sufficient to give the features expected, most importantly
the ability of a circuit or material of given design to absorb j and J from
ECE spacetime and amplify them greatly.
8.2 The Resonance Equations
The source of electric current from ECE space-time is its torsion. In barebones
notation the currents are given by:
j = A(0)
µ 0
d ∧ T (8.21)
J = A(0)
µ 0
d ∧ ˜T . (8.22)
The torsion is defined by the tetrad and spin connection in the first Cartan
structure equation of differential geometry and the tetrad in turn is defined by
the eigenvalues of the ECE Lemma, Eq.8.13. The tetrad is the fundamental
field in the Palatini variation of general relativity and is a wave of space-time.
The potential field is governed by resonance equations, and within a factor A (0)
is the tetrad. In this section the resonance equation 8.20 is developed into vector
notation for use in engineering. The spin connection is always defined by
the second Bianchi identity, and for the free electromagnetic field is the dual of
the tetrad in the tangent space-time [1]– [35]. The scalar curvature is defined
as eigenvalues of the ECE Lemma and is proportional to the index contracted
energy-momentum tensor through the Einstein Ansatz 8.14. Therefore energy
and momentum are transferred from R to j and J, and total energy and momentum
are conserved. Total charge-current density is also conserved.
In the standard notation of differential geometry [1]– [35] the relevant equations
are:
j a = A(0)
µ 0
d ∧ T a , (8.23)
J a = A(0)
µ 0
d ∧ ˜T a , (8.24)
□q a = Rq a (8.25)
T a = d ∧ q a + ω a b ∧ q b , (8.26)
D ∧ (D ∧ ω a b) = 0. (8.27)
In the standard notation the tangent space-time indices appear but the base
manifold indices are the same on both sides of a given equation and are not
112

CHAPTER 8. RESONANCE SOLUTIONS OF THE ECE FIELD . . .
written out [1]– [35]. If we restore these indices for the sake of illustration and
completeness Eqs.8.23 to 8.27 become:
j a µνρ = A(0)
µ 0
(d ∧ T a ) µνρ
, (8.28)
J a µνρ =
(d A(0)
∧
µ ˜T a) , (8.29)
0 µνρ
□q a µ = Rq a µ (8.30)
T a µν = (d ∧ q a ) µν
+ ω a µb ∧ q b ν , (8.31)
D ∧ ( D ∧ ω a µb)
= 0. (8.32)
Therefore the barebones and standard notations must always be interpreted as
implying the presence of the various indices that appear in Eqs.8.28 to 8.32.
The advantage of the barebones notation is that it gives the basic structure
with greatest clarity. These equations and notations are fully developed and
explained in the literature [1]– [35] in differential form, tensor and vector notation.
The vector notation is used in this section because it is the notation
universally used in engineering. However all three notations are equivalent and
contain the same mathematical information. The differential form notation is
the most concise and elegant.
In the standard model
j = 0, (8.33)
R = 0, (8.34)
and there can be no battery powered by space-time, even on a qualitative level.
The reason for this is that classical electrodynamics in the standard model is
still the Maxwell Heaviside theory, which is a nineteenth century theory of special
relativity in which the field is thought of as a separate entity superimposed
on a Minkowski frame in four dimensions. To Maxwell, space and time were
still separate concepts, and there could be no structure to space-time. At the
time when Heaviside developed Maxwell’s quaternion equations into vector notation
(late nineteenth century), space and time were still thought of as separate.
Only when Lorentz and Poincaré developed the tensor notation of the Maxwell-
Heaviside field equations did space and time become unified into space-time.
This occurred at the beginning of the twentieth century. Even then however,
the electromagnetic field was still through of as an entity superimposed on a
SEPARATE Minkowski frame with metric diag (-1, 1, 1, 1). The concept of a
curving space-time appeared only in 1916, in the Einstein Hilbert (EH) theory
of general relativity, but that theory was applied only to gravitation, and not
to electromagnetism. In EH theory a field was thought of for the first time as
the curving frame of reference ITSELF, not as something superimposed on a
separate frame of reference. ECE theory, developed from 2003 onwards [1]– [35]
is a rigorously objective theory of general relativity in which the electromagnetic
field is the torsion of space-time itself and in which currents can be generated
by the torsion of space-time itself through Eqs.8.21 and 8.22. These currents
are real, observable and physical, and can be used for engineering. In ECE
theory electromagnetism is unified with gravitation using differential geometry
and space-time currents are a new source of energy that conserves Noether’s
113

8.2. THE RESONANCE EQUATIONS
Theorem. This section is designed to show how the currents can be maximized
by resonance. In the standard model, again, there is no concept of spin connection,
because the latter is the mathematical description of a spinning and curving
frame. When a frame itself spins or curves (or both spins and curves) the spin
connection must be non-zero. In electromagnetism the non-zero spin connection
is observed through the Evans spin field [1] – [35] using the phenomenon of
magnetization by a circularly polarized electromagnetic field. This is known as
the inverse Faraday effect, and is rigorously reproducible and repeatable, occurring
in all materials and at all frequencies of the applied electromagnetic field.
The Evans spin field is therefore the definitive proof of general relativity in the
electromagnetic field. In the standard model the inverse Faraday effect must be
explained by assuming the existence of a cross product of complex conjugates
of the potential [1]– [35] or equivalently of the electric field or magnetic field.
Even this purely empirical description (occurring in non-linear optics [1]– [35])
did not appear until the mid fifties of the twentieth century and therefore was
not present in the original Maxwell theory and was not considered by Maxwell
or Heaviside. In summary one cannot describe the inverse Faraday effect selfconsistently
and objectively without general relativity, which asserts that ALL
of the equations of physics must be generally covariant. This means that all
must retain their structure under the general coordinate transformation, i.e. all
of physics must be geometrical in nature. This is the very essence of general
relativity, and until this is realized field unification cannot occur in an objective
manner. The Maxwell Heaviside equations do not obey this fundamental
requirement, because they retain their mathematical (tensorial) structure only
under the Lorentz transformation, as described in many texts [39]. In order
for the equations of electrodynamics to be generally covariant as required by
general relativity, the spin connection must be non-zero, and the Evans spin
field must be non-zero [1]– [35]. This is exactly what is shown experimentally
by the inverse Faraday effect.
The resonance equations developed in vector notation in this section originate
in the ”master” equation 8.20, which in standard notation is:
i.e.
where
In tensor notation Eq.8.37 is [1]– [35]:
d ∧ ( d ∧ A a + ω a b ∧ A b) = µ 0 j a , (8.35)
d ∧ F a = µ 0 j a (8.36)
F a = d ∧ A a + ω a b ∧ A b . (8.37)
F a µν = −F a νµ = ∂ µ A a ν − ∂ ν A a µ + ω a µbA b ν − ω a νbA b µ. (8.38)
This equation can be developed into the electric field components:
F a 01 = −F a 10 = ∂ 0 A a 1 − ∂ 1 A a 0 + ω a 0bA b 1 − ω a 1bA b 0, (8.39)
F a 02 = −F a 02 = ∂ 0 A a 2 − ∂ 2 A a 0 + ω a 0bA b 2 − ω a 2bA b 0, (8.40)
F a 03 = −F a 30 = ∂ 0 A a 3 − ∂ 3 A a 0 + ω a 0bA b 3 − ω a 3bA b 0, (8.41)
and the magnetic field components:
F a 12 = −F a 21 = ∂ 1 A a 2 − ∂ 2 A a 1 + ω a 1bA b 2 − ω a 2bA b 1, (8.42)
114

CHAPTER 8. RESONANCE SOLUTIONS OF THE ECE FIELD . . .
F a 13 = −F a 31 = ∂ 1 A a 3 − ∂ 3 A a 1 + ω a 1bA b 3 − ω a 3bA b 1, (8.43)
F a 23 = −F a 32 = ∂ 2 A a 3 − ∂ 3 A a 2 + ω a 2bA b 3 − ω a 3bA b 2. (8.44)
The vector description of the electric and magnetic fields follows by using
the following definitions in covariant/contra-variant notation [40]:
A a µ = (A a 0, −A a ) , ω a µb = (ω a ob, −ω a b) , (8.45)
A aµ = ( A a0 , A a) , ω aµ b = ( ω a0 b, ω a b)
, (8.46)
( ) 1 ∂
∂ µ =
c ∂t , ∇ , (8.47)
The contravariant electromagnetic tensor is:
⎡
F µν =
⎢
⎣
0 −E 1 /c −E 2 /c −E 3 /c
E 1 /c 0 −B 3 B 2
E 2 /c B 3 0 −B 1
E 3 /c −B 2 B 1 0
and the contravariant four-derivative [1]– [35] is:
Therefore there are electric field components such as:
⎤
⎥
⎦ (8.48)
∂ µ = g µν ∂ ν . (8.49)
F 01a = − 1 c E1a = ∂ 0 A 1a − ∂ 1 A 0a + ω 0a bA 1b − ω 1a bA 0b , (8.50)
i.e.
− 1 c Ea x = 1 ∂
c ∂t Aa x + ∂
∂x A0a + ω 0a bA b x − ω a xbA 0b , (8.51)
and it follows that the complete electric field vector is:
E a = − ∂Aa
∂t
− c∇A 0a − cω 0a bA b + cω a bA 0b . (8.52)
Similarly there are magnetic field components such as:
i.e.
F 12a = ∂ 1 A 2a − ∂ 2 A 1a + ω 1a bA 2b − ω 2a bA 1b = −B 3a , (8.53)
−B a z = − ∂Aa y
∂x
and the complete magnetic field vector is:
+ ∂Aa x
∂y + ωa xbA b y − ω a ybA b x (8.54)
B a = ∇ × A a − ω a b × A b . (8.55)
The classical electromagnetic field equations of ECE theory [1]– [35] in vector
notation are:
∇ · B a = µ 0˜j a0 , (8.56)
∇ × E a + ∂Ba
∂t
= µ 0˜ja , (8.57)
∇ · E a = cµ 0 ˜J 0a , (8.58)
115

8.2. THE RESONANCE EQUATIONS
∇ × B a − 1 ∂E a
c 2 = µ 0
∂t c ˜J a , (8.59)
in which the currents are defined by:
( ) 1
˜j aν =
c˜j a0 ,˜j a , (8.60)
˜J aν =
( 1
c ˜J a0 , ˜J a )
, (8.61)
Therefore the resonance equations are obtained by substituting Eqs.8.52 and
8.55 into each of Eqs.8.56 to 8.59.
The simplest equation is found by substituting Eq.8.55 into Eq.8.56 and
using the vector identity [41]:
to give
∇ · ∇ × A a = 0 (8.62)
∇ · (ω a b × A b) = −µ 0˜j a0 . (8.63)
In this equation summation is implied over repeated b indices as follows:
∇ · (ω a 0 × A 0 + · · · + ω a 3 × A 3) = −µ 0˜j a0 (8.64)
Therefore the charge density available from space-time is:
˜j a0 = − A(0)
µ 0
∇ · (ω a b × q b) (8.65)
where q b is the vector part of the tetrad.
A linear inhomogeneous [36] second order differential equation is found by
substituting Eq.8.52 into Eq.8.58 to give:
∇ · ∇A a0 + 1 c
∂
∂t (∇ · Aa ) + ∇ · (ω 0a bA b) − ∇ · (ω a bA 0b)
= −µ 0 ˜J 0a .
(8.66)
As discussed further in Section 8.3, the linear inhomogeneous structure gives
resonance solutions and resonances in the current ˜J 0a . This is the key to amplification
of currents from ECE space-time. These concepts and equations are also
used [36] in circuit theory for example, atomic absorption theory, or laser theory.
Before proceeding to derive the other two resonance equations of this section
the self-consistency of the mathematics being used is checked for Eqs.8.63 and
8.66 when the spin connection is dual to the tetrad [1]– [35]:
ω a µb = −κɛ a bcq c µ. (8.67)
Here κ has the units of wave-number (inverse metres) and the Levi-Civita symbol
is:
ɛ a bc = g ad ɛ dbc (8.68)
where g ad is the metric of the tangent space-time (a Minkowski metric). Therefore
there are components [1]– [35]:
ω 1 µ2 = −ω 2 µ1 = κq 3 µ, (8.69)
116

CHAPTER 8. RESONANCE SOLUTIONS OF THE ECE FIELD . . .
and so on. For a = 0 Eq.8.63 is:
ω 2 µ3 = −ω 3 µ2 = κq 1 µ, (8.70)
˜j 00 = − A(0)
µ 0
∇ · (ω a b × q b) (8.71)
where the relevant component of the spin connection is:
whose vector part is:
ω 0 µb = −κɛ 0 bcq c µ (8.72)
ω 0 b = −κɛ 0 bcq c . (8.73)
From the cyclically symmetric properties of ɛ a bc
, summation over b in Eq.8.73
would be over space-like indices, 1, 2 and 3. From Eq.8.73:
ω 0 1 = −κɛ 0 1cq c ,
= −κ ( ɛ 0 12q 2 + ɛ 0 12q 3) ,
(8.74)
Therefore in this approximation:
ω 0 2 = −κ ( ɛ 0 21q 1 + ɛ 0 23q 3) , (8.75)
ω 0 3 = −κ ( ɛ 0 31q 1 + ɛ 0 32q 2) . (8.76)
ω 0 b × q b = ω 0 1 × q 1 + ω 0 2 × q 2 + ω 0 3 × q 3
= −κ(ɛ 0 12q 2 × q 1 + ɛ 0 13q 3 × q 1
+ ɛ 0 21q 1 × q 2 + ɛ 0 23q 3 × q 3
+ ɛ 0 31q 1 × q 3 + ɛ 0 32q 2 × q 3 ).
(8.77)
Now use the properties:
ɛ 0 12 = −ɛ 0 21 = 1, (8.78)
ɛ 0 23 = −ɛ 0 32 = 1, (8.79)
ɛ 0 31 = −ɛ 0 13 = 1, (8.80)
and use the complex circular basis ((1), (2), (3)) [1]– [35] to obtain:
(
˜j 00 = 2κ A(0)
∇ · q (2) × q (1) + q (1) × q (3) + q (3) × q (2)) . (8.81)
µ 0
For plane waves:
and
Also [1]– [35]:
and
q (1) = q (2)∗ = 1 √
2
(i − ij) e iφ , (8.82)
∇ · q (2) × q (1) = 0. (8.83)
q (1) × q (3) = −iq (2)∗ = −iq (1) (8.84)
q (3) × q (2) = −iq (2) , (8.85)
117

8.2. THE RESONANCE EQUATIONS
so
Therefore it is found that:
(
˜j 00 = −2iκ A(0)
∇ · q (1) + q (2)) = 0. (8.86)
µ 0
which is self-consistent with the fact that:
˜j 00 = ˜j 01 = ˜j 02 = ˜j 03 = 0 (8.87)
˜j = 0 (8.88)
when Eq.8.67 applies, Q.E.D. Therefore the equation 8.63 is mathematically
self-consistent.
In order to check the consistency of Eq.8.66 recall that in the standard model
there is no spin connection, so Eq.8.66 reduces to:
1 ·∂Aa + ∇ · ∇A a 0 = 0. (8.89)
c ∂t
For each polarization index a, A a 0 is the electric scalar potential φ. Using the
Lorentz condition:
1 ∂φ
c ∂t + ∇ · A = 0 (8.90)
it is found that Eq.8.89 reduces to:
i.e.
1 ∂ 2 φ
c 2 ∂t 2 − ∇2 φ = 0 (8.91)
□φ = 0 (8.92)
which is the relativistic wave equation of the standard model for a scalar potential
φ. In order to obtain space-time resonance however, the complete Eq.8.66
is needed.
The third resonance equation is obtained by substituting Eq.8.52 and 8.55
into Eq.8.57. Using the vector properties [41]:
and
it is found that:
∂
∂t ∇ × Aa = ∇ × ∂Aa
∂t
(8.93)
∇ × ∇ a0 = 0 (8.94)
∂ (
ω
a
∂t b × A b) + c∇ × ( A 0b ω a b − ω a b 0 A b) = µ 0˜ja . (8.95)
This is a first order differential equation in the potential. The current ˜j a is nonzero
if and only if the spin connection is non-zero. So the current ˜j a is unique
to ECE theory and general relativity and does not occur in the standard model.
The self consistency of Eq.8.95 can be checked again by using Eq.8.67, in which
case we obtain:
ω a b × A b = ω 1 2 × A 2 + ω 1 3 × A 3
= κ
A (0) (
A 3 × A 2 + A 2 × A 3) = 0
118
(8.96)

CHAPTER 8. RESONANCE SOLUTIONS OF THE ECE FIELD . . .
and two more equations:
cA 0b ∇ × ω 1 b =
c
A (0) (
A 02 ∇ × A 3 + A 03 ∇ × A 2) (8.97)
and
−cω 10 c∇ × A b =
c (
A 03 ∇ × A 2 + A 02 ∇ × A 3) (8.98)
A (0)
which self-consistently sum to zero, Q.E.D.
The final resonance equation is obtained by substituting Eqs.8.52 and 8.55
into Eq.8.59 and is:
1 ∂ 2 A a
c 2 ∂t 2 + 1 (
∇A 0a − A b0 ω a b + ω a b 0 A b) +∇× ( ∇ × A a − ω a b × A b) = µ 0
c
c ˜J a .
(8.99)
This is a generalization of the linear inhomogeneous structure discussed further
in Section 8.3 in which an analytical solution is given of Eq.8.99 in a well defined
approximation. The simple type of linear inhomogeneous structure [36] is:
ẍ + 2βẋ + ω 2 0x = A cos ωt (8.100)
which is a driven damped oscillator equation of classical dynamics. It is seen
that Eq.8.99 is a generalization of Eq.8.100. Solutions of Eq.8.100 were first
discovered by the Bernoulli’s and Euler (1739-1743) and show resonance in the
amplitude A of Eq.8.100, resonance in the kinetic and potential energies, Q
factors, phase lags, transient and steady state effects. Therefore Eq.8.99 has
similar solutions and is also more richly structured.
8.3 Analytical Solution
Eq.8.100 is a development of the linear inhomogeneous [36] class of equations:
In the special case:
Eq.8.101 reduces to the linear homogeneous class
whose general solution is:
with the auxiliary equation
d 2 y
dx 2 + a dy + by = f(x). (8.101)
dx
f(x) = 0 (8.102)
d 2 y
dx 2 + a dy + by = 0 (8.103)
dx
y = c 1 e r1x + C 2 e r2x , r 1 ≠ r 2 , (8.104)
r 2 + ar + b = 0. (8.105)
Eq.8.104 holds when the roots of Eq.8.103 are real and unequal, i.e. r 1 ≠ r 2 . If
the roots of Eq.8.103 are imaginary (α ± iβ), then:
y = e αx (c 1 cos βx + c 2 sin βx)
= µe αx sin (βx + δ) .
(8.106)
119

8.3. ANALYTICAL SOLUTION
Now let:
be the general solution of
and let
be any solution of
then
y = u (8.107)
y ′′ + ay ′ + by = 0 (8.108)
y = v (8.109)
y ′′ + ay ′ + by = f(x) (8.110)
y = u + v (8.111)
is a solution of Eq.8.101. The function u is the complementary function and v is
the particular integral. One must find by inspection a function v that satisfies:
v ′′ + av ′ + bv = f(x). (8.112)
Eq.8.100 of Section 8.2 is a special case of the linear inhomogeneous class 8.100
and Eq. 8.100 can be rewritten as
mẍ + bẋ + kx = F 0 cos ωt. (8.113)
This is the equation of driven oscillation [37]. In Eq.8.113 the external driving
force varies harmonically with time, and is applied to the oscillator. The total
force on the particle is:
F = −kx − bẋ + F 0 cos ωt (8.114)
and consists of a linear restoring force, −kx, (Hooke’s Law), and a viscous
damping force −bẋ. Therefore the master equation 8.35 of ECE theory has all
these features and is also more richly structured. In this Section an analytical
solution of Eq.8.99 is found in a well defined approximation using the properties
of the linear inhomogeneous class of equations 8.101.
Resonance solutions of Eq.8.113 are found from the complementary function
x c (t) and the particular integral x p (t). The former is:
( ( (β
x c (t) = e −βt A 1 exp 2 − ω 2 ) 1/2
) (
0 t + A 2 exp − ( β 2 − ω 2 ) 1/2
))
0 t (8.115)
and the latter is [37]:
x p (t) = D cos (ωt − δ) . (8.116)
It follows that
x p (t) = A
( (ω
2
0 − ω 2) 2
+ 4ω 2 β 2) −1/2
cos (ωt − δ) (8.117)
where
( ) 2ωβ
δ = tan −1 ω0 2 − ω2
(8.118)
The general solution is:
x(t) = x c (t) + x p (t). (8.119)
120

CHAPTER 8. RESONANCE SOLUTIONS OF THE ECE FIELD . . .
The term x c (t) represents transient effects that depend on the initial conditions.
These damp out with time because of the factor e −βt . The term x p (t) represents
steady state effects which dominate for t >> 1/p. The quantity δ is the
phase difference between the driving force and the resultant motion, i.e. a delay
between the application of force and the response of the system. For a fixed ω 0 ,
as ω increases from 0, the phase increases from δ = 0 at ω = 0 to δ at π/2 and
to π as ω → ∞.
The amplitude resonance frequency ω R is that at which the amplitude D is
a maximum. It is defined by:
i.e.
dD
dω
∣ = 0 (8.120)
ω=ωR
ω R = ( ω 2 0 − 2β 2) 1/2
. (8.121)
We see that for an equation such as 8.92 in which ω 0 and β are both zero, there is
no resonance. In an equation in which ω 0 is zero but β is non-zero the resonance
frequency ω R is pure imaginary and unphysical. Therefore the requirement for
resonance is that ω 0 and D be non-zero. If the amplitude D is initially zero it
cannot be maximized from Eq.8.120. These conditions are very important for
the resonant acquisition of energy and for resonant counter-gravitation.
The degree of damping in an oscillatory system is described by the quality
factor:
Q = ω R
2β . (8.122)
In loudspeakers for example [36] the values of Q may be a few hundred, in quartz
crystal oscillators or tuning forks up to 10,000. Highly tuned electric circuits
(of interest to extracting resonance energy from ECE space-time) may have Q
up to 100,000 [36]. This is the order of magnitude of the amplification observed
by the Mexican Group. The oscillation of electrons in atoms leads to optical
radiation. The sharpness of the spectral lines is limited [36] by the damping
due to loss of energy by radiation (radiation damping). The minimum width of
a line is, classically, about:
δω = 2 × 10 −8 ω. (8.123)
The Q of such an oscillation is therefore of the order 10 7 . The largest known Q
occurs from radiation from a gas laser, about 10 14 . Therefore resonant energy
from ECE space-time and resonant counter-gravitation are also governed by
such features. A current j (barebones notation) is set up by Eq.8.21 and can
set electrons in a circuit or within a material into resonant motion, producing a
resonance current from space-time as observed experimentally [1]– [35]. Eq.8.21
shows that the current is generated by the geometry of space-time itself.
Resonance in kinetic energy (T ) is defined by the value of ω for which T is
a maximum, where [37]:
T = 1 2 mẋ2 (8.124)
It is found from:
d 〈T 〉
dω
∣ = 0 (8.125)
ω=ωE
121

8.3. ANALYTICAL SOLUTION
and is
where
ω E = ω 0 (8.126)
〈T 〉 = mA2
4 ω2 ( (
ω
2
0 − ω 2) 2
+ 4ω 2 β 2) −1/2
. (8.127)
The potential energy is proportional to the square of the amplitude, and occurs
at the same frequency as amplitude resonance. The kinetic and potential energies
resonate at different frequencies because the damped oscillator is not a
conservative system [36] of dynamics. Energy is continuously exchanged with
the driving system. In energy from ECE space-time energy is therefore continuously
exchanged between space-time and the circuit or material, total energy
being conserved by Noether’s Theorem.
Atomic systems within a material taking resonant energy from ECE spacetime
can be represented classically as linear oscillators. When light falls on
matter it causes the atoms and molecules to vibrate. Similarly ECE space-time
causes the atoms and molecules to vibrate, light being ECE space-time within
the factor A (0) of Eq.8.12. A resonant frequency occurs at one of the spectral
frequencies of the system. When light (i.e. ECE space-time) having one of the
resonant frequencies of the atomic or molecular system falls on the material,
electromagnetic energy (i.e. energy from ECE space-time) is absorbed, causing
the atom or molecule to oscillate with large amplitude. This is what happens
in a circuit or material such as that of the Mexican Group [1]– [35]. A large
amount of energy is resonantly absorbed from ECE space-time. This can be
released as electric current or power, the governing equation is equation 8.35.
Large electromagnetic fields (ECE space-time dynamics) are produced by the
oscillating electric charges. Electric circuits are non-mechanical oscillations.
Therefore resonance theory and electric circuit theory can be used to explain
energy from space-time. The mechanism is clear from Eq.8.35, i.e.:
j = A(0)
µ 0
(d ∧ (d ∧ q) + d ∧ (ω ∧ q)) . (8.128)
The current j is picked up from ECE space-time and is represented by q and
ω of Eq.8.128, a driven damped oscillator equation. Amplitude, kinetic energy
and potential energy resonances occur. The electrons in a well designed circuit
or material oscillate in constructive interference, producing a surge of current
and electric power. This is observed experimentally in the reproducible and
repeatable work of the Mexican group of AIAS [1]– [35].
These qualitative remarks are underlined as follows with an analytical solution
of Eq.8.99 with well defined approximations. First use
ω a µb = −κɛ a bcq c µ (8.129)
so
Then use
A b0 ω a b = ω a0 bA b , (8.130)
ω a b × A b = 0. (8.131)
∇ 2 A a = − ω2 0
c 2 Aa , (8.132)
122

CHAPTER 8. RESONANCE SOLUTIONS OF THE ECE FIELD . . .
with:
Eq.8.99 then simplifies to
∇ · A a = 0, (8.133)
∂A 0a /∂t = 0, (8.134)
∇ × (∇ × A) = −∇ 2 A + ∇ (∇ · A) . (8.135)
1 ∂ 2 A a
c 2 ∂t 2 + ω2 0
c 2 Aa = µ 0
c ˜J a . (8.136)
This is an undamped driven oscillator, it has the structure of Eq.8.100 with
From Eqs.8.132 and 8.133
β = 0. (8.137)
A a = A(0)
√
2
(i − ij) e −iω0Z/c (8.138)
is a possible solution. From the analytical solution of Eq.8.100 already discussed
in this Section:
A (0) = A (0)
c + A (0)
p (8.139)
where
assuming:
Resonance occurs at
with:
A (0)
c = A 1 e iω0t + A 2 e −iω0t (8.140)
A (0)
p = D, (8.141)
µ 0
c ˜J a = A a (i − ij) cos ωt (8.142)
ω R = ω 0 (8.143)
δ = 0, Q → ∞,
D → ∞.
In this case there is a surge of current of infinite amplitude:
(8.144)
˜J a → ∞ (8.145)
because there is no damping. This simple illustration, using well defined approximations,
shows how resonant energy from space-time occurs mathematically
within ECE theory. More realistic results with finite damping can be produced
numerically from Eq.8.99, and under certian conditions will reproduce the factor
of 100,000 amplification observed by the Mexican Group [1]– [35] and found
independently to be reproducible and repeatable.
Acknowledgments The British Government is thanked for the award of a
Civil List pension and the AIAS staff and environment for many interesting
discussions.
123

8.3. ANALYTICAL SOLUTION
124

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[29] M. W. Evans, Solutions of the ECE Field Equations (2005, preprint on
www.aias.us and www.atomicprecision.com).
126

BIBLIOGRAPHY
[30] M. W. Evans, ECE Generalization of the dAlembert, Proca and Superconductivity
Wave Equations: Electric Power from ECE Space-time, (2005,
preprint on www.aias.us and www.atomicprecision.com).
[31] M. W. Evans, (ed.), Modern Non-Linear Optics, in I. Prigogine and S. A.
Rice (series eds.), Advances in Chemical Physics, (Wiley Interscience, New
York, 2001, 2nd ed.) Vols. 119(1)-119(3), circa 2,500 pages.
[32] M. W. Evans and L. B. Crowell, Classical and Quantum Electrodynamics
and the B (3) Field (World Scientific, Singapore, 2001).
[33] M. W. Evans and J.-P. Vigier, The Enigmatic Photon, (Kluwer, Dordrecht,
1994 - 2002, hardback and softback) vols. 1-5.
[34] M. W. Evans and S. Kielich (eds.), first edition of ref. (31), (Wiley-
Interscience, New York, 1992, 1993 and 1997 (softback)), vols. 85(1)-85(3),
circa 3,000 pages.
[35] A. Hill, personal communications (www.aias.us and
www.atomicprecision.com 2005).
[36] J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and
Systems, (Harcourt Brace, New York, 1988, 3rd ed.).
[37] L. H. Ryder, Quantum Field Theory (Cambridge Univ. Press, 1996, 2nd
ed.).
[38] J. Shelburne, personal communication (www.aias.us and
www.atomicprecision.com 2005).
[39] J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998, 3rd ed.).
[40] S. P. Carroll, Lecture Notes in General Relativity (graduate course in the
public domain given at Harvard, UCSB and U. Chicago, arXiv: gr - gc
973019 v1 1997).
[41] E. G. Milewski (Chief Ed.), Vector Analysis Problem Solver (Research and
Education Association, New York, 1987, revised printing).
127

BIBLIOGRAPHY
128

Chapter 9
Resonant Counter
Gravitation
by
M. W. Evans
Alpha Foundation’s Institute for Advance Study (A.I.A.S.).
(emyrone@aol.com, www.aias.us, www.atomicprecision.com)
Abstract
Generally covariant unified field theory has been used to show that the equations
of classical electrodynamics are unified with those of gravitation using standard
Cartan geometry (Einstein Cartan Evans (ECE) field theory). By expressing
the ECE field equations in terms of the potential field, linear inhomogeneous
field equations are obtained for each of the fundamental laws of electrodynamics
unified with gravitation. These equations have resonant solutions, and in this
paper the possibility of resonant counter gravitation is demonstrated by showing
that the Riemann curvature can be affected by the electromagnetic field.
Examples are the Coulomb law and Ampère law respectively of electro-statics
and magneto-statics. At resonance the effect is greatly amplified (as for any
resonant phenomenon), so in theory, circuits can be built for effective resonant
counter gravitation and used in the aerospace industry.
Keywords: Resonant counter gravitation; Einstein Cartan Evans (ECE) field
theory; generally covariant unified field theory; linear inhomogeneous differential
equations; resonance.
9.1 Introduction
The principle of general relativity is the fundamental hallmark of objective
physics, a natural philosophy that is independent of the observer, independent
of subjective input. The principle means that every equation of physics
has to be generally covariant, meaning that it must retain its form under any
type of coordinate transformation. The principle must evidently be applied
to all equations of physics, including electrodynamics. Only in this way can
129

9.1. INTRODUCTION
an objective unified theory of physics emerge - a generally covariant unified
field theory [1]. It is well established [2] that the principle of general relativity
as applied to gravitational theory by Einstein and Hilbert [3] is very accurate
when compared with experimental data, but the principle of general relativity
is not applied to electrodynamics in the standard model. In the latter [4]
electrodynamics is a theory of special relativity in which the field is thought of
as an entity independent of the frame. The space-time of electrodynamics in
the standard model is the Minkowski (”flat”) space-time. As a result standard
model electrodynamics is not generally covariant, it is Lorentz covariant, and
as such cannot be unified with generally covariant gravitational theory in the
standard model. It is well known that Riemann geometry with the Christoffel
connection is the geometrical basis of gravitational general relativity. However
in this type of geometry the torsion tensor is zero [5]. It was first suggested by
Cartan [6] that the electromagnetic field be the torsion form of Cartan geometry.
In 2003 [7]– [40] a generally covariant unified field theory was developed
using this suggestion and using standard Cartan geometry. It has since been
developed in many directions [1, 7]– [40].
In Section 9.2 the field equations of ECE theory are expressed as linear
inhomogeneous equations with resonant solutions. The Riemann term is isolated
and it is shown that the electromagnetic part of the unified field can change
the Riemann curvature, i.e. change the gravitational field. At resonance this
effect is greatly amplified. In Section 9.3 this general conclusion is exemplified
using the Coulomb and Ampère laws unified with gravitation. This means that
a static electric or static magnetic configuration can change the gravitational
field. In order to maximize the effect numerical methods of solution are needed
to model a circuit which optimizes resonant counter gravitation. An assembly
of such circuits can be placed aboard a device such as an aircraft or spacecraft,
and is expected to be particularly effective in regions of near zero gravitation
in outer space. Under the usual laboratory conditions it is well known that the
electromagnetic and gravitational fields are essentially independent and have no
influence on each other. This is observed experimentally in the Coulomb and
Newton inverse square laws for example. If two charged masses are considered,
then changing the charge on one of them has no effect on the Newton inverse
square law. Similarly changing the mass of one of them has no effect on the
Coulomb inverse square law. However, it is known through the Eddington
effect that gravitation and electromagnetism interact and ECE theory was the
first to give a classical explanation of the Eddington effect [7]– [40]. Einstein’s
famous prediction was based on photon mass and a semi-classical treatment.
The Eddington effect is however tiny in magnitude, the enormous mass of the
sun bends grazing light by a few seconds of arc only. Therefore resonant counter
gravitation is the only practical method of counter gravitation. All claims to
have observed an effect of electromagnetism on gravitation without resonance
are almost certainly artifactual. Recently however the Mexican group of AIAS
have observed resonantly enhanced electric power from ECE spacetime, the
output power from a circuit was observed reproducibly [41] to exceed input
power by a factor of one hundred thousand. This has been explained using
ECE theory by the use of linear inhomogeneous differential equations of the
same type as used in this paper for counter gravitation. The two phenomena
are explained by a generally covariant unified field theory based on Cartan
geometry.
130

CHAPTER 9.
RESONANT COUNTER GRAVITATION
9.2 The Resonance Equations Of ECE Field Theory
The overall aim of this section is to develop the ECE field equations to define
the effect of electromagnetism on gravitation. In order to do this the Riemann
term is isolated on the right hand side of the following field equations:
d ∧ F + ω ∧ F = A (0) R ∧ q, (9.1)
d ∧ ˜F + ω ∧ ˜F = A (0) ˜R ∧ q, (9.2)
F = d ∧ A + ω ∧ A. (9.3)
In these equations we have used a notation [1] which suppresses the various
indices on the quantities on the left and right hand sides. This concise notation
is used to reveal the basic structure of the equations. Later they will be developed
into differential form, tensor and vector notation. Here F denotes the
electromagnetic field, ω the spin connection, R the Riemann curvature and q
the tetrad. The symbol ∧ denotes Cartan’s wedge product. The tilde denotes
the Hodge dual [1] and A the potential field. Finally A (0) is the proportionality
constant between F and the Cartan torsion:
F = A (0) T (9.4)
which is the ECE ansatz [1]. So Eqs. 9.1 and 9.2 balance electromagnetic terms
on the left hand side and on the right hand side a gravitational term R ∧ q
multiplied by A (0) . Using Eq. 9.3 in Eq. 9.1 gives a linear inhomogeneous
equation:
d ∧ (d ∧ A + ω ∧ A) + ω ∧ (d ∧ A + ω ∧ A) = A (0) R ∧ q, (9.5)
with resonance solutions [42]. Therefore resonance amplification of the effect
of electromagnetism on R ∧ q is possible in general relativity. In the standard
model gravitation is described by:
R ∧ q = 0 (9.6)
which is the Ricci cyclic equation [1] of Einstein Hilbert field theory. In tensorial
notation the Ricci cyclic equation is:
R σµνρ + R σρµν + R σνρµ = 0 (9.7)
where R σµνρ is the index lowered Riemann curvature tensor. Eq. 9.6 or equivalently
Eq. 9.7 are true if and only if the Christoffel connection is assumed:
Γ κ µν = Γ κ νµ. (9.8)
The assumption 9.8 implies that the torsion tensor is zero:
T κ µν = Γ κ µν − Γ κ νµ = 0. (9.9)
In the standard model the spin connection is missing because the Minkowski
frame is not spinning, and so in the standard model:
d ∧ F = 0, (9.10)
131

9.2. THE RESONANCE EQUATIONS OF ECE FIELD THEORY
Using Eq. 9.12 in Eq. 9.10 gives the Poincaré Lemma:
d ∧ ˜F = µ 0 J, (9.11)
F = d ∧ A. (9.12)
d ∧ (d ∧ A) = 0 (9.13)
which does not have resonance solutions. In order to understand the influence
of electromagnetism on gravitation numerical solutions of Eq. 9.5 are needed.
At resonance the effect is greatly amplified. In the standard model J is not
recognized as originating in elements of the Riemann tensor. The interaction
between electromagnetism and gravitation is defined by:
R ∧ q ≠ 0 (9.14)
and by a non-zero and asymmetric spin connection. Both conditions are needed.
It is important to note that for rotational motion, as for example in a free space
electromagnetic field in ECE theory [1], the spin connection is dual to the tetrad:
ω a b = − κ 2 ɛa bcq c (9.15)
where κ is a wave-number and ɛ a bc
is the index raised Levi-Civita tensor in the
tangent space-time. Eq. 9.15 implies that the Cartan torsion tensor is dual to
the Riemann tensor:
R a b = − κ 2 ɛa bcT c . (9.16)
So it must be clearly understood that there is a Riemann spin tensor for free
electromagnetism in ECE field theory. There is also a Riemann tensor for
gravitation, the well known curvature Riemann tensor. When electromagnetism
and gravitation are mutually influential R∧q is not zero, and Eqs. 9.15 and 9.16
no longer apply. This is the condition needed for resonant counter gravitation.
If R ∧ q is zero then electromagnetism does not influence gravitation. Similarly
if the spin connection is dual to the tetrad there is no mutual influence, and
when the Cartan torsion is dual to the Riemann spin tensor, there is no mutual
influence. These conclusions follow directly from Cartan geometry. For the free
electromagnetic field, the homogeneous field equation [1] reduces to:
d ∧ F a = 0 (9.17)
which for each polarization index a, and using vector notation, gives the Gauss
law applied to magnetism:
∇ · B a = 0 (9.18)
and the Faraday law of induction:
∇ × E a + ∂Ba
∂t
= 0 (9.19)
The inhomogeneous field equation of ECE theory [1] is:
d ∧ ˜F ( )
= A (0) ˜R ∧ q − ω ∧ ˜T . (9.20)
132

CHAPTER 9.
RESONANT COUNTER GRAVITATION
When the electromagnetic field is free of gravitation:
( ) (
˜R ∧ q = ω ∧ ˜T
)
e/m
e/m
(9.21)
and the inhomogeneous field equation 9.20 reduces to:
d ∧ ˜F ( )
= A (0) ˜R ∧ q := µ 0 J. (9.22)
This means that the inhomogeneous current is derived from the mass of an
electron in the Einstein Hilbert limit, i.e purely from the curving of space-time
as described by the Schwarzschild metric [1]. When the electromagnetic and
gravitational fields are mutually independent, there is no interaction between
the spinning and curving of space-time. In this limit Eq. 9.22 gives for each
index a the Coulomb Law:
∇ · D a = ρ a (9.23)
and the Ampère Maxwell law:
∇ · D a − ∂Da
∂t
grav
= J a . (9.24)
In the weak field limit of gravitation uninfluenced by electromagnetism the
Newton inverse square law is also recovered.
In standard differential form notation Eq. 9.1 is:
and this in tensor notation is:
d ∧ F a + ω a b ∧ F b = A (0) R a b ∧ q b (9.25)
∂ µ F a νρ + ∂ ν F a ρµ + ∂ ρ F a µν + ω a µbF b νρ + ω a νbF b ρµ + ω a ρbF b µν
= A (0) ( R a bνµq b ρ + R a bρνq b µ + R a bµρq b ) (9.26)
ν .
Now use:
and the right hand side of Eq. 9.26 becomes:
Eq. 9.27 is the same as:
Similarly Eq. 9.2 becomes:
In the standard model Eq. 9.29 is:
R ∧ q = −q ∧ R (9.27)
−A (0) ( q b µR a bνρ + q b ν R a bρµ + q b ρR a bµν)
. (9.28)
∂ µ ˜F aµν + ω a µb ˜F aµν = −A (0) q b µ
˜R
a µν
b
. (9.29)
∂ µ F aµν + ω a µbF aµν = −A (0) q b µR a µν
b
. (9.30)
∂ µ ˜F µν = 0 (9.31)
and Eq. 9.30 is:
∂ µ F µν = µ 0 J ν . (9.32)
133

9.2. THE RESONANCE EQUATIONS OF ECE FIELD THEORY
From Eq. 9.31 we obtain the Gauss law and Faraday law of induction of the
standard model, and from Eq. 9.32 we obtain the standard model’s Coulomb
Law and Ampère Maxwell Law. In ECE theory these laws must be obtained
from Eqs. 9.29 and 9.30. The details are given in Appendix K of ref. [1].
The Coulomb Law in vector notation in ECE theory will be derived and
explained in detail later in this section. The result is:
∇ · E a + ω a ′
b · E b = −cA b′ · R a b. (9.33)
Similarly the Ampère Maxwell Law in vector notation in ECE theory is:
∇ × B a + ω ′ a
b × B b − 1 ( ) ∂E
a
c 2 + ω ′ a
∂t ob E b = µ 0 J a′ . (9.34)
These laws are required to understand the effect of electromagnetism on gravitation,
and to design devices for resonant counter gravitation. Gravitation is
represented by the Riemann terms on the right hand sides of Eqs. 9.33 and
9.34, and electromagnetism by the terms on the left hand sides. The equations
therefore show that elements of the Riemann tensor can be affected by electric
and magnetic fields. The engineering challenge is to maximize the effect with
resonance amplification. The latter possibility is seen by writing out Eq. 9.3 in
vector notation [7]– [40]:
and
E a = − ∂Aa
∂t
− c∇A 0a − cω 0a bA b + cω a bA 0b (9.35)
B a = ∇ × A a − ω a b × A b . (9.36)
By substituting Eqs. 9.35 and 9.36 into 9.33 and 9.34 linear inhomogeneous
differential equations are obtained. The final step is to solve these numerically
to design circuits that give resonance amplification of the effect of electromagnetism
on gravitation. If the Riemann tensor is decreased, gravity is lessened,
and conversely. This is a highly non-trivial problem in general and shows why
previous attempts to understand this problem are naive. In the standard model
these linear inhomogeneous differential equations are replaced by the d’Alembert
wave equation [43] using the Lorentz gauge condition. The solutions are the
Liennard- Wiechert potentials, and these are electromagnetic waves without
resonance and without the information given by the interaction for gravitation
and electromagnetism of ECE field theory.
The Ampère law of magneto-statics is obtained when there is no electric field
present, only a magnetic field, so eq. 9.34 reduces to:
∇ × B a + ω ′ a
b × B b = µ 0 J a′ . (9.37)
As discussed in the introduction the standard model’s Coulomb, Ampère and
Newton inverse square laws hold to high precision [43]. Therefore no influence
of electromagnetism on gravitation has hitherto been detected in the laboratory.
The reason is that resonance amplification has not been used, and resonance
amplification occurs only in general relativity, not in special relativity. However
an influence of gravitation on electromagnetism has been detected in the
Eddington effect. ECE field theory is the first self-consistent explanation of the
134

CHAPTER 9.
RESONANT COUNTER GRAVITATION
Eddington effect both on the classical and quantum levels. Einstein’s famous
prediction which led to the Eddington experiment was semi-classical and was
based on Einstein Hilbert field theory, so used only gravitation and not the
required unified field theory. Light in Einstein’s prediction was a photon with
mass, the concomitant electric and magnetic fields were not considered. The
semi-classical theory happens to be very accurate for the solar system, [1], but
in general effects are expected due to the interaction of gravitation and electromagnetism.
These may occur not only in a cosmological context but also
in an atom or a circuit on the microscopic scale. In the vicinity of an electron
space-time curvature is large because of the small electron radius, and electric
fields are intense, giving plenty of scope for the interaction of space-time spin
and curvature.
The origin of Coulomb’s law in ECE field theory is the inhomogeneous field
equation [1]:
∂ µ F aµν = µ 0 J aν = A (0) ( R a µν
b
q b µ − ω a µbT bµν) (9.38)
and the law is obtained by using
in Eq. 9.38 to give:
ν = 0 (9.39)
∂ 1 F a10 + ∂ 2 F a20 + ∂ 3 F a30 + ω a 1bT b10 + ω a 2bT b20 + ω a 3bT b30
= A (0) ( R a b 10 q b 1 + R a b 20 q b 2 + R a b 30 q b ) (9.40)
3
which translates into the vector notation of Eq. 9.33. The primed quantities
arise because the metric g µν must be used to raise and lower indices, so:
A a ′
µ := g µν A aν , (9.41)
ω a ′
µb := g µν ω νa b, (9.42)
where the unprimed quantities are defined by convention as metric free. Other
conventions may be adopted, but in ECE theory the metric is not the Minkowski
metric in general, so contra-variant and covariant quantities must be defined
carefully. It is no longer sufficient just to switch the sign from positive (contravariant
space part of a four-vector) to negative (covariant space part of a fourvector).
These details must be programmed carefully in numerical applications.
The resonant version of Eq. 9.33 may now be developed from Eq. 9.35
substituted into Eq. 9.33 to give:
c∇ · ∇A 0a + ∇ · ∂Aa + c∇ · (ω 0a
∂t
bA b − ω a bA 0b)
( )
+ ω a ′ ∂A
b
b · + c∇A 0b + cω 0b
∂t
cA c − cω b cA 0c
= −cA b′ · R a b.
(9.43)
If we restrict consideration to a static electric field configuration Eq. 9.43 simplifies
as follows. In order to guide this simplification exercise consider first the
standard model’s static electric field:
∇ · E = ρ/ɛ 0 , (9.44)
135

9.2. THE RESONANCE EQUATIONS OF ECE FIELD THEORY
∇ × E = 0, (9.45)
∇ 2 Φ = −ρ/ɛ 0 , (9.46)
where Φ is the scalar potential of the Poisson equation [43]. Eq. 9.45 implies
that:
E = −∇Φ. (9.47)
For a time-dependent electric field:
so a static electric field means:
E = −∇Φ − ∂A
∂t , (9.48)
∂A
∂t
If we further assume for the sake of approximation that:
Eq. 9.43 simplifies to:
= 0. (9.49)
∇ · A b = 0 (9.50)
∇ · ∇A 0a − ∇ · (ω a bA 0b) + ω a′ b · ∇A 0b
+ ω a ′
b · ω 0b cA c − ω a ′
b · ω b cA 0c = −A b′ · R a (9.51)
b
:= cµ 0 J 0a
This is still a complicated equation so to simplify further we consider the limit
of weak interaction between the electromagnetic and gravitational fields [1]:
d ∧ ˜F ( )
= µ 0 J −→ A (0) ˜R ∧ q . (9.52)
In this limit: (
˜R ∧ q
)
e/m
The structure of Eq. 9.33 simplifies to:
∼
grav
(
ω ∧ ˜T
)
, ˜Tgrav ∼ 0. (9.53)
e/m
∇ · E a ∼ −cA b′ · R a b (9.54)
and the spin connection in Eq. 9.35 can be considered to be approximately dual
to the tetrad. So Eq. 9.43 simplifies further to
)
∇ ·
(− ∂Aa − c∇A 0a − cω 0a
∂t
bA b + cω a bA 0b ∼ −cA b′ · R a b (9.55)
with
ω a b ∼ − κ 2 qc ɛ a bc. (9.56)
If we use a static electric field and asume Eq. 9.50, Eq. 9.55 simplifies to:
∇ · ∇A 0a − ∇ · (ω a bA 0b) ∼ −A b′ · R a b. (9.57)
If we do not assume Eq. 9.50 we obtain:
∇ · ∇A 0a + ∇ · (ω 0a bA b) − ∇ · (ω a bA 0b) ∼ −A b′ · R a b. (9.58)
136

CHAPTER 9.
RESONANT COUNTER GRAVITATION
Eq. 9.57 is a Hooke’s Law type of resonance equation with a driving term on
the right hand side, Eq. 9.58 has an additional damping term. In both cases
gravitation is resonantly affected by a static electric field. In order for this to
occur the spin connection must be identically non-zero, meaning that κ in Eq.
9.56 must be identically non-zero. In the limit of an identically static electric
field, κ is identically zero, and we recover the standard model’s Coulomb law.
In so doing we lose the possibility of influencing gravitation with a static electric
field.
The Ampère Maxwell law can be expressed [1] in ECE theory as:
where
∇ × B a − 1 ∂E a
c 2 = µ 0 J a (9.59)
∂t
J a = J a x i + J a y j + J a z k. (9.60)
To isolate the Riemann term Eq. 9.61 is developed as:
∇ × B a + ω ′ a
b × B b − 1 ( ) ∂E
a
c 2 + ω ′ a
∂t 0b E b = µ 0 J a′ (9.61)
where [1]:
J a ′
x
= − A(0) (
R
a 10
0 + R a 2 12 + R a 13 )
3 , (9.62)
µ 0
J a ′
y
J a ′
z
= − A(0) (
R
a 20
0 + R a 1 21 + R a 23 )
3 , (9.63)
µ 0
= − A(0) (
R
a 30
0 + R a 1 31 + R a 32 )
2 , (9.64)
µ 0
We first check Eq. 9.61 for units. We obtain in S.I.:
A (0) = JsC −1 m −1 = voltsm −1 , (9.65)
R = m −2 , (9.66)
µ 0 = Js 2 C −2 m −1 , J = Am −2 = Cs −1 m −2 . (9.67)
When electromagnetism and gravitation are independent of each other the elements
of the Riemann tensor appearing in Eq. 9.61 are precisely those of the
Schwarzschild metric [1], elements which represent the curvature of space-time
due to the mass of an electron or ensemble of electrons. However when electromagnetism
and gravitation influence each other the elements of the Riemann
tensor are changed, and this gives rise to the possibility of resonant counter
gravitation.
The magneto-static Ampère law can be developed into a linear inhomogeneous
differential equation by using Eq. 9.36 in Eq. 9.37 to give:
∇ × (∇ × A a ) − ∇ × ( ω a b × A b)
+ω ′ a
b × (∇ × A a ) − ω ′ a
b × ( ω b c × A c)
= µ 0 J a′ .
(9.68)
This equation must be solved in general on a computer, but some simplifying
assumptions may be made as for the Coulomb Law. The gravitational term on
137

9.3. BASIC DEFINITIONS AND CONVENTIONS FOR . . .
the right hand side of Eq. 9.68 is balanced by the magnetic terms on the left
hand side. Eq. 9.68 reduces to the Ampère law of the standard model for each
index a when the spin connection vanishes.
In tensor notation the Coulomb and Ampère Maxwell laws are given in
resonant form by substituting
F aµν = ∂ µ A aν − ∂ ν A aµ + ω aµ b Abν − ω aν bA bµ (9.69)
into Eq. 9.30 to give the linear inhomogeneous tensorial equation:
□A aν − ∂ ν (∂ µ A aµ ) + ω aµ b ∂ µA bν − ω aν b∂ µ A bµ
+ ω a (
µb ∂ µ A bµ − ∂ ν A bµ) + (∂ µ ω aµ b ) Abν − (∂ µ ω aν b) A bµ
+ ω a µbω bµ cA cν − ω a µbω bν cA cµ
= −A (0) R a µν
b
q b µ = −A (0) R aµν µ
(9.70)
in which the gravitational term on the right hand side is balanced by the electromagnetic
terms on the left hand side. Eq. 9.70 is therefore the tensorial
equivalent of Eq. 9.5.
9.3 Basic Definitions And Conventions For Numerical
Solutions
The electric and magnetic fields in ECE theory are defined from Cartan geometry
by Eqs. 9.35 and 9.36. In these equations the tetrad is defined by:
q a ′
µ = g µν q νa . (9.71)
For each index a the contravariant tetrad is defined as the four-vector:
q νa = ( q 0a , q a)
Similarly the spin connection is defined by:
= ( q 0a , q 1a , q 2a , q 3a)
= ( (9.72)
q 0a , q a x, q a y, q a )
z .
ω a µb ′ = g µν ω νa b (9.73)
adopting the convention:
ω νa b = ( ω 0a b, ω a )
b
= ( ω 0a b, ω 1a b, ω 2a b, ω 3a )
b
= ( (9.74)
ω 0a b, ω a xb, ω a yb, ω a )
zb .
The role of the spin connection in ECE theory can be illustrated with reference
to the fundamentally important Evans spin field B (3) [1] observed in the inverse
Faraday effect. The spin connection means that electromagnetism is Cartan
torsion, so the frame is spinning. Similarly gravitation is Riemann or Cartan
curvature, so the frame is curving. A spinning or curving frame means that
there must be a connection present. If there is no connection the space-time
138

CHAPTER 9.
RESONANT COUNTER GRAVITATION
is Minkowski space-time, called flat” space-time, the space-time of special relativity.
The spin field is defined in the complex circular basis [7]– [40] with
polarization indices denoted by:
a = (1), (2), (3) (9.75)
so:
q (1) × q (2) = iq (3)∗ , (9.76)
The spin field is a special case of:
q (2) × q (3) = iq (1)∗ , (9.77)
q (3) × q (1) = iq (2)∗ . (9.78)
B a spin = −A (0) ω a b × q b (9.79)
and exists only in general relativity. Its existence therefore shows that classical
electrodynamics is a theory of general relativity, and not of special relativity.
This is a fundamentally important finding, because the spin field is an experimental
observable of the inverse Faraday effect, which therefore shows experimentally
that classical electrodynamics is a theory of general relativity. After
realizing this, the unification of electromagnetism with gravitation follows selfconsistently
from the rules of Cartan geometry, so the spin field is fundamentally
important for the development of a generally covariant unified field theory and
for the study of resonant counter gravitation. This point is emphasized here by
some technical details as follows.
If we consider a circularly polarized electromagnetic field independent of
gravitation, and use for example:
the spin field is:
a = 3 (9.80)
B 3 = −A (0) ( ω 3 1 × q 1 + ω 3 2 × q 2) (9.81)
where summation over repeated covariant contravariant indices has been used,
together with:
ω 3 3 = 0. (9.82)
Eq. 9.82 follows because for circular polarization [1]:
where:
ω a µb = − κ 2 ɛa bcq c µ (9.83)
ɛ a bc = η ad ɛ dbc (9.84)
and where η ad is the Minkowski metric of the tangent space-time of Cartan
geometry. Thus
⎡
⎤
−1 0 0 0
η ad = η ad = ⎢ 0 1 0 0
⎥
⎣ 0 0 1 0 ⎦ = diag (−1, 1, 1, 1) (9.85)
0 0 0 1
139

9.3. BASIC DEFINITIONS AND CONVENTIONS FOR . . .
and
ɛ dbc = 1, even permutation
(9.86)
= −1, odd permutation
i.e.
ɛ 123 = −ɛ 132 = 1 etc., ɛ 1 23 = ɛ123 = 1 etc. (9.87)
Therefore the spin connection elements are:
ω 1 2 = − κ 2 ɛ1 23q 3 = − κ 2 q3 , (9.88)
and the spin field is [1]:
B 3 = A (0) κ 2
ω 3 1 = − κ 2 ɛ3 12q 2 = − κ 2 q2 , (9.89)
ω 3 2 = − κ 2 ɛ3 21q 1 = − κ 2 q1 , (9.90)
(
q 2 × q 1 − q 1 × q 2) = −A (0) κq 1 × q 2 . (9.91)
Finally switch to the complex circular basis and use:
to find the original Evans spin field [1]:
A 1 = A (0) q 1 , A 2 = A (0) q 2 (9.92)
B (3) = B (3)∗ κ
= −i
A (0) A(1) × A (2) = −igA (1) × A (2) . (9.93)
Historically the spin field was proposed in 1992 [1] from the experimental existence
of the conjugate product A (1) × A (2) in the inverse Faraday effect and
developed in several ways [7]– [40] using gauge theory. The spin field was incorporated
into a generally covariant unified field theory from 2003 onwards [1].
The gauge theoretical methods were replaced completely by the fully self consistent
methods of Cartan geometry. In gauge theory the indices a are abstract
entities, in the final generally covariant unified field theory they are indices of
the tangent spacetime as in standard Cartan geometry and as such have a clearly
defined geometrical role which is rigorously self-consistent and consistent with
the principle of general relativity. Gauge theory on the other hand superimposes
an abstract a index on a flat Minkowski space-time, and so gauge theory cannot
lead to a generally covariant unified field theory. Gauge theory cannot be
used to investigate the mutual interaction of gravitation and electromagnetism,
and neither can string theory. Only a geometrically based theory can do this,
and for self-consistently the theory must be one that is in accordance with the
fundamental principle of general relativity.
In these equations the covariant derivative [1] is defined as usual by applying
a correction to the flat four-derivative:
( ) 1 ∂
∂ µ =
c ∂t , ∇ . (9.94)
The contravariant flat space derivative is:
( ) 1
∂ µ = η µν ∂
∂ ν =
c ∂t , −∇
(9.95)
140

CHAPTER 9.
RESONANT COUNTER GRAVITATION
and the flat space d’Alembertian operator is:
□ = ∂ µ ∂ µ . (9.96)
It follows that Eq. 9.95 must also be used to define ∂ µ in ECE space-time,
defined as the four-dimensional space-time with curvature and torsion both
present in general. The contravariant derivative in ECE space-time is:
but:
D µ = g µν D ν (9.97)
D µ = ∂ µ + · · · (9.98)
therefore the d’Alembertian operator remains the same in ECE space-time, and
is defined by:
D µ D µ = □ + · · · (9.99)
Therefore □ is the ”flat part” of D µ D µ . Similarly:
g µν = η µν + · · · (9.100)
and ν µν is the ”flat part” of g µν . As discussed by Carroll [7]– [40] in his chapter
3, the derivative operator ∂ µ in flat space-time is a map from (k, l) tensor fields to
(k, l+1) tensor fields, the derivative operator acts linearly on its arguments and
obeys the Leibnitz Theorem for tensor products. The derivative operator D µ of
ECE theory therefore performs the functions of ∂ µ in a way that is independent
of coordinates. This property is fundamentally required by general relativity.
Since D µ obeys the Leibnitz Theorem it may always be written as the ∂ µ plus
a linear transformation. In Riemann geometry and for a given vector V ν the
covariant derivative is therefore:
D µ V ν = ∂ µ V ν + Γ ν µλV λ (9.101)
where Γ ν µλ is the connection. Thus ∂µ in ECE theory is the same as Eq. 9.95,
and ∂ µ in ECE theory is the same as Eq. 9.94. It follows that the d’Alembertian
operator of ECE theory is defined by Eq. 9.96, i.e.:
□ = ∂ µ ∂ µ = 1 c 2 ∂ 2
∂t 2 − ∇2 . (9.102)
In order to produce a numerical solution of the ECE field equation the differential
operators must be defined as in flat space-time, i.e. as:
( ) 1 ∂
∂ µ =
c ∂t , ∇ = (∂ 0 , ∂ 1 , ∂ 2 , ∂ 3 )
( 1 ∂
=
c ∂t , ∂
∂x , ∂ ∂y , ∂ ) (9.103)
∂z
and
( 1
∂ µ =
c
( 1
=
c
)
∂
∂t , −∇ = ( ∂ 0 , ∂ 1 , ∂ 2 , ∂ 3)
∂
∂t , − ∂
∂x , − ∂ ∂y , − ∂ ∂z
141
)
.
(9.104)

9.3. BASIC DEFINITIONS AND CONVENTIONS FOR . . .
With these definitions and conventions the ECE electromagnetic field tensor is:
F aµν = −F aνµ = ∂ µ A νa − ∂ ν A νa + ω µa b Abν − ω νa bA bµ (9.105)
which compares with the standard model’s:
F µν = −F −νµ = ∂ µ A ν − ∂ ν A µ . (9.106)
The polarization index and spin connection are missing in the standard model.
In S.I. units the following convention is adopted for the field tensor:
⎡
⎤
0 −E a1 /c −E a2 /c −E a3 /c
F aµν = ⎢ E a1 /c 0 −B a3 B a2
⎥
⎣ E a2 /c B a3 0 −B a1 ⎦ . (9.107)
E a3 /c −B a2 B a1 0
In this convention the units of the field tensor are those of magnetic flux density
(tesla) or electric field strength (volt m −1 ) divided by c. Other conventions for
the field tensor may be used if preferred, provided that care is taken that all
S.I. units are balanced on the right and left hand sides of any equation. In Eq.
9.107:
E a1 = E a x, E a2 = E a y , E a3 = E a z ,
(9.108)
B a1 = B a x, B a2 = B a y , B a3 = B a z ,
thus:
and
Therefore:
F a01 = −E a1 /c = −F a10 , (9.109)
F a02 = −E a2 /c = −F a20 , (9.110)
F a03 = −E a3 /c = −F a30 , (9.111)
F a12 = −F a21 = −B a3 , (9.112)
F a13 = −F a31 = −B a2 , (9.113)
F a23 = −F a32 = −B a1 . (9.114)
F a01 = − 1 c Ea1 = ∂ 0 A a1 − ∂ 1 A a0 + ω a 0
b A b1 − ω a 1
b A b0
= − 1 c Ea x = 1 c
∂A a x
∂t
+ ∂Aa0
∂x + ωa 0
b A b x − ω a 1
xb A b0 (9.115)
from which we obtain Eq. 9.35. In the standard model (S. I. units):
Similarly:
E = − ∂A
∂t − c∇A0 := − ∂A − ∇φ. (9.116)
∂t
F a12 = −B a3 = −B a z = ∂ 1 A a2 − ∂ 2 A a1 + ω a 1
b A b2 − ω a 2
b A b1
= − ∂Aa y
∂x
+ ∂Aa x
∂y + ωa xbA b y − ω a ybA b x.
(9.117)
142

CHAPTER 9.
RESONANT COUNTER GRAVITATION
⎤
Now use the definition of the vector curl:
⎡
i j k
A x A y A z
∇ × A = ⎣ ∂/∂x ∂/∂y ∂/∂z ⎦ (9.118)
and vector cross product:
⎡
A × B = ⎣
i j k
⎤
A x A y A z
⎦ (9.119)
to obtain Eq. 9.36.
The following displays give a summary of the translation of notation. In
ECE theory:
F = d ∧ A + ω ∧ A →
In the standard model (S.I. units):
E a = − ∂Aa − c∇A a0
∂t
− cω a b 0 A b + cω a bA b0 ,
B a = ∇ × A a − ω a b × A b .
(9.120)
F = d ∧ A →
E = −∂A ∂t − ∇φ . (9.121)
B = ∇ × A
In order to develop the resonance formulation of the Faraday law of induction
in ECE field theory it is convenient to use the ECE Faraday law of induction
in its dielectric form [7]– [40]:
∇ × (ɛ r E a ) + ∂ ∂t
( B
a
µ r
)
= 0 (9.122)
where µ r and ɛ r are respectively the relative permeability and permittivity of
ECE space-time considered as a dielectric. The homogeneous current of Eq. 9.1
is re-defined in the dielectric formulation as:
˜ja := ∂Ma
∂t
− c 2 ∇ × P a (9.123)
where the magnetization is:
M a =
( 1
µ 0
− 1 µ)
B a (9.124)
and the polarization is:
P a = (ɛ − ɛ 0 ) E a . (9.125)
In Eqs. 9.123–9.125 and ɛ 0 respectively are the vacuum permeability and permittivity,
and µ and ɛ are the permeability and permittivity of ECE space-time
143

9.3. BASIC DEFINITIONS AND CONVENTIONS FOR . . .
regarded as a dielectric. Using Eqs. 9.35 and 9.36 in Eq. 9.122 gives a resonance
equation in the dielectric formulation. The magnetization is:
and the polarization is:
P a = A (0) (ɛ − ɛ 0 )
M a = A (0) ( 1
µ 0
− 1 µ) (∇
× q a − ω a b × q b) (9.126)
(− ∂qa
∂t − ∇q0a − cω 0a bq b + cω a bq 0b )
. (9.127)
The homogeneous current is:
(( 1
− 1 (∇
× q
µ 0 µ) a − ω a b × q b))
( ∂ ˜ja = A (0) ∂t
(
)))
−c 2 ∇ × (ɛ − ɛ 0 )
(− ∂qa
∂t − c∇q0a − cω 0a bq b + cω a bq 0b .
(9.128)
The numerical task is to find resonance solutions of Eq. 9.128. In general µ and
ɛ are functions of ct, X, Y and Z:
ɛ = ɛ (ct, X, Y, Z) , (9.129)
µ = µ (ct, X, Y, Z) , (9.130)
and in general both ɛ and µ are tensorial quantities (as for example in crystals).
They are scalars only in an isotropic homogeneous dielectric. We may
approximate Eq. 9.128 by considering ɛ and µ as scalars, so Eq. 9.128 simplifies
using:
((
∂ 1
− 1 ) ) ( 1
∇ × A a = − 1 ∇ ×
∂t µ 0 µ
µ 0 µ) ∂Aa
∂t , (9.131)
c∇ × ( (ɛ − ɛ 0 ) ∇A 0a) = 0, (9.132)
and which is a linear inhomogeneous differential equation with mixed derivatives.
Eq. 9.128 has only two input parameters ɛ and µ.
When the electromagnetic and gravitational fields are independent:
µ = µ 0 , ɛ = ɛ 0 ,˜j = 0, (9.133)
and the relative permittivity and permeability become:
ɛ r = 1, µ r = 1. (9.134)
In this limit of independent fields we obtain self consistently the Faraday law of
induction of ECE field theory with no homogeneous current:
∇ × E a + ∂Ba
∂t
= 0. (9.135)
In this limit the spin connection is dual to the tetrad as in Eq. 9.15 and the Riemann
spin form is dual to the torsion form as in Eq. 9.16. In this limit the Evans
spin field is obtained self consistently as in Eq. 9.93. Note carefully that Eq.
9.135 is a standard model Faraday law of induction for each index a [1,7]– [40].
144

CHAPTER 9.
RESONANT COUNTER GRAVITATION
In the standard model the Evans spin field is missing, because electrodynamics
in the standard model is a theory of special relativity (the Maxwell Heaviside
field theory). Therefore in the standard model classical electrodynamics is incompatible
with the principle of general relativity, and this is a major weak point
of the standard model because the Evans spin field is an experimental observable
in the inverse Faraday effect, indicating that classical electrodynamics must
originate in the torsion of space-time, torsion indicating the presence of a spin
connection that is, indeed, detected experimentally in the inverse Faraday effect.
Classical electrodynamics is not an entity superimposed on flat space-time (the
nineteenth century view) because in this view there is no spin connection and
no inverse Faraday effect, contrary to reproducible data. The historical origin
of this major weak point is well known but worth recounting briefly as follows.
Classical electrodynamics in its modern vector formulation was developed in
the late nineteenth century (from James Clerk Maxwell’s original quaternion
equations of the mid nineteenth century), by Oliver Heaviside, before special
relativity was developed. Heaviside’s vectorial equations for electrodynamics
were put in tensorial form by Lorentz and Poincarè at the turn of the twentieth
century and were assumed to be Lorentz covariant. Only later, in 1905, did
Einstein develop special relativity for the whole of physics. In 1915 Einstein
and Hilbert developed the generally covariant field equation of gravitation in
general relativity, but electrodynamics remained a Lorentz covariant theory of
special relativity. Therefore gravitation and electrodynamics could not be unified,
being conceptually (i.e fundamentally) different. Attempts at unification
have been made ever since and the first successful generally covariant unified
field theory is now generally recognised [7]– [40] as being ECE theory. This did
not begin to emerge until 2003. ECE theory now gives the basic understanding
needed to evaluate resonant counter gravitation and many other phenomena
new to physics [7]– [40].
If the Faraday law of induction is considered from Eq. 9.1 to be [7]– [40]:
∇ × E a + ∂Ba = µ
∂t 0˜ja , (9.136)
and if we restrict consideration to scalar, time-independent µ and ɛ, Eqs. 9.126
and 9.127 used in Eq. 9.123 give:
( 1 − 1 µ 0 µ + c2 (ɛ − ɛ 0 ))∇ × ∂Aa − ( 1 − 1 ∂t µ 0 µ ) ∂ ∂t (ωa b × A b )
(9.137)
c 3 ∇ × (ω 0a bA b − ω a bA 0b ) = ˜j a
Now use the approximation:
which is equivalent to:
ω µa b −→ −κ 2 qµc ɛ a bc (9.138)
˜j → 0. (9.139)
For index a = 1:
( 1
− 1 (
µ 0 µ + c2 (ɛ − ɛ 0 ))
∇ × ∂A1 1
− − 1 ) ∂ (
ω
1
∂t µ 0 µ ∂t 2 × A 2 + ω 1 3 × A 3)
− κ 2 c3 ∇ × ( q 03 A 2 − q 02 A 3 − q 3 A 02 + q 2 A 03) = ˜j 1 → 0
145
(9.140)

9.3. BASIC DEFINITIONS AND CONVENTIONS FOR . . .
i.e.:
(( 1
− 1 ) (
+ c
µ 0 µ)
2 (ɛ − ɛ 0 ) ∇ × ∂A1 1
+ κ − 1 ) ∂ (
q 2 × A 3)
∂t µ 0 µ ∂t
− κ 2 c3 ∇ × ( q 03 A 2 − q 02 A 3 − q 3 A 02 + q 2 A 03) = ˜j 1 → 0.
(9.141)
Now use:
and switch to the complex circular basis to obtain:
( 1
µ 0
− 1 µ + c2 (ɛ − ɛ 0 ))
∇ × ∂A(1)∗
The structure of this equation is:
q 03 = q 02 = A 03 = A 02 → 0 (9.142)
∂t
( 1
+ κ − 1 ∂A
(1)∗
=
µ 0 µ)
∂t
˜j (1)∗ → 0.
(9.143)
x∇ × ∂A(2)
∂t
where the scalars x and y are:
+ yκ ∂A(2)
∂t
= ˜j (2) → 0 (9.144)
x = 1 µ 0
− 1 µ + c2 (ɛ − ɛ 0 ) , (9.145)
y = 1 µ 0
− 1 µ . (9.146)
Eq. 9.144 is again a linear inhomogeneous differential equation with resonant
solutions. So in ECE theory the fundamental equations of classical electrodynamics
all develop a resonant structure never considered previously in the
history of physics and engineering because a generally covariant unified field
theory was not available.
Finally in this section the standard model’s Lorentz force law is developed
into a generally covariant equation of unified field theory. This shows how
gravitation is expected to affect the law. In the standard model the Lorentz
force law originates in the Lorentz transformation of the field tensor [1]:
where x µ is the four-coordinate:
F ′ µν = ∂x′ µ
∂x ρ ∂x ′ ν
∂x σ F ρσ (9.147)
x µ = (ct, X, Y, Z) . (9.148)
In the standard model the Lorentz transformation is used from K to a frame K ′
translating uniformly at v with respect to K . Using the Lorentz transformation
in Eq. 9.147 gives [7]– [40] in S.I. units:
E ′ = γ (E + v × B) −
γ2 v
( v
)
γ + 1 c c · E , (9.149)
B ′ = γ
(B − v )
c 2 × E − γ2 v
( v
)
γ + 1 c c · B , (9.150)
146

CHAPTER 9.
RESONANT COUNTER GRAVITATION
where:
γ =
(1 − v2
c 2 ) −1/2
. (9.151)
The Lorentz force law as usually given in the textbooks as:
and is an approximation to Eq. 9.149 when:
F = eE ′ = eγ (E + v × B) (9.152)
v ≪ c, γ ≠ 1. (9.153)
The non-relativistic limit of the Lorentz force law is obtained from the approximation
9.153 in the limit:
γ → 1 (9.154)
and is the familiar:
F = e (E + v × B) . (9.155)
In ECE theory [7]– [40] the Lorentz force law is obtained from the rules [7]– [40]
of general coordinate transformation of the torsion tensor in Cartan geometry,
i.e.:
T a′ µ ′ ν = ∂x µ ∂x ν
′ Λa′ a T a
∂x µ′ ∂x ν′ µν (9.156)
where only Λ a′ a is a Lorentz transformation matrix and where ∂x µ /∂x µ′ and
∂x ν /∂x ν′ are general coordinate transformation matrices. The electromagnetic
field tensor is [7]– [40]:
F a µν = A (0) T a µν (9.157)
so the Lorentz force in ECE field theory manifests itself in:
F a′ µ ′ ν ′ = A(0) T a′ µ ′ ν ′ (9.158)
multiplied by charge. Contained within the Cartan torsion is the spin connection,
which is related to the Riemann curvature and to gravitation.
Acknowledgments The British Government is thanked for a Civil List pension
and the AIAS environment for many interesting discussions.
147

9.3. BASIC DEFINITIONS AND CONVENTIONS FOR . . .
148

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[41] The Mexican Group of AIAS (www.aias.us and www.atomicprecision.com).
[42] J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and
Systems, (Harcourt Brace, New York, 1988, 3rd. Ed.).
[43] L. D. Ryder, Quantum Field Theory (Cambridge Univ. Press, 2nd. Ed.,
1996, softback).
151

BIBLIOGRAPHY
152

Chapter 10
Wave Mechanics And ECE
Theory
by
M. W. Evans
Alpha Foundation’s Institute for Advance Study (A.I.A.S.).
(emyrone@aol.com, www.aias.us, www.atomicprecision.com)
Abstract
Generally covariant wave mechanics is developed from Einstein Cartan Evans
(ECE) field theory. The ECE lagrangian density is identified and used in the
ECE Euler Lagrange equation to identify the origin of the Planck constant as
a minimized action of general relativity. It is shown that the Planck constant
as used in special relativity (standard model wave mechanics) is a special case
in which volume is fortuitously cancelled out. More generally the commutator
equation of Heisenberg must include volume. The Cartan structure equation,
Cartan torsion, and Bianchi identity are derived from the lagrangian density.
The Aspect experiment is explained using ECE wave mechanics, and quantum
entanglement is described using the spin connection term of ECE theory. The
Bohr Heisenberg indeterminacy is discarded in favor of a causal, objective and
unified wave mechanics. Phase velocity, v, in ECE wave mechanics can become
much greater than c (which remains the universal constant of relativity theory)
and the equations defining the condition v ≫ c are given.
Keywords: Einstein Cartan Evans (ECE) field theory; generally covariant wave
mechanics, origin of the Planck constant; ECE lagrangian density; derivation of
the Cartan structure equation, Cartan torsion and Bianchi identity; description
of the Aspect experiment and quantum entanglement, greater than c phase
velocity.
153

10.1. INTRODUCTION
10.1 Introduction
Recently [1]– [38] it has been shown that the origin of generally covariant wave
mechanics is the tetrad postulate of Cartan geometry [39, 40], the fundamental
requirement that a vector field be independent of the coordinate system used to
describe it. General covariance in physics means that its equations are covariant
under the general coordinate transformation. This means that they retain their
form, a tensor in one coordinate system must be a tensor in any other coordinate
system. The equations of physics are therefore objective to an observer in one
reference frame moving in an arbitrary way with respect to an observer in any
other reference frame. The requirement of objectivity in physics manifests itself
as this fundamental principle of general relativity and without this principle
there is no objective physics, nature would mean different things to different
observers. Special relativity is known to be accurate to one part in twenty seven
orders of magnitude and general relativity to one part in one hundred thousand
for the solar system. So the principle of objectivity is tested to high precision.
The other fundamental attribute of relativity theory is that the constant c be
universal. This is usually interpreted to mean that no information can travel
faster than c and other constants in physics are based on a fixed c in standards
laboratories worldwide. The constancy of c is needed to ensure causality, to
ensure that nothing happens without a cause.
Throughout the twentieth century, general relativity was thought to be incompatible
with the principle of indeterminacy developed mainly by Bohr and
Heisenberg. This principle states that pairs of variables such as position x and
momentum p behave in such a way that if one is known exactly (for example x),
the other (for example p) is unknowable. This assertion is based on a variation
inferred by Heisenberg of the Schrödinger equation of non-relativistic wave mechanics.
There is nothing, however, in the original Schrödinger equation which
implies indeterminacy, the Schrödinger equation is based [41] on the fact that action
is minimized in particles by the classical Hamilton principle of least action,
and that time interval is minimized in waves by the classical Fermat principle
of least time. The Heisenberg commutator equation is a restatement of the
Schrödinger equation. It has been shown [1]– [38] that the Schrödinger equation
is a well defined non-relativistic quantum limit of the Einstein Cartan Evans
(ECE) wave equation of general relativity. Therefore the Schrödinger equation
has been shown to be objective and causal and has been shown to be an equation
of relativity theory. It follows that the Heisenberg commutator equation
is also objective and causal. It cannot lead to Bohr Heisenberg indeterminacy
and cannot lead to anything that is unknowable. Recently [42]– [45] the Bohr
Heisenberg indeterminacy principle has indeed been refuted experimentally in
several independent ways, all of which are repeatable and reproducible. Indeterminacy
is therefore an intellectual aberration which worked itself uncritically
into thousands of textbooks of the twentieth century era.
In Section 10.2 the lagrangian density of generally covariant unified field theory
is deduced and used to derive the fundamental ECE wave equation. Therefore
from the outset the concept of volume is introduced into wave mechanics
because the lagrangian density has the units of energy divided by volume. It
has been shown [1]– [38] that the experiments of Croca et al. [42], experiments
which refute indeterminacy experimentally, can be explained by ECE theory
with the introduction of volume into the Heisenberg commutator equation. The
154

CHAPTER 10.
WAVE MECHANICS AND ECE THEORY
ECE lagrangian density inferred in this Section is the fundamental origin of
this volume. Key quantities in wave mechanics must therefore be densities, in
common with the rest of general relativity. This deduction is seen at work in
the fundamental ECE wave equation [1]– [38]:
(□ + kT ) q a µ (10.1)
Here k is Einstein’s constant, T is the index reduced canonical energy momentum,
a concept first introduced [46] by Einstein, and q a µ is the tetrad of Cartan
geometry [39,40], the fundamental unified field of ECE theory. In the rest frame:
T = m/V (10.2)
which is mass divided by volume. The lagrangian density in this limit is:
L = c 2 T = mc2
V
(10.3)
and is the rest energy divided by volume. All other wave equations of physics
are limits of the ECE wave equation [1]– [38], so volume is inherent in all of
them. In this section it is shown that the Cartan structure equation and the
Bianchi identity of Cartan geometry can be derived form the same lagrangian
density. It is thus inferred that all of physics (both classical and quantum)
derives from the tetrad postulate, the fundamental mathematical requirement
that a complete vector field is independent of the way it is written, independent
of the coordinate system used to define the vector field. This inference leads to
an unprecedented degree of simplicity and fundamental understanding.
In Section 10.2 the fundamental origin of the Planck constant is discussed
within ECE field theory using fact that action is:
S = 1 ∫
Ld 4 x (10.4)
c
an integral of the lagrangian density L over the four-volume d 4 x. Action has
the units of energy multiplied by time, and these are also the units of angular
momentum. Using these concepts the fundamental Planck Einstein and de
Broglie equations of quantum mechanics are derived within the concepts of
ECE field theory and thus of general relativity. This derivation is not possible
in the standard model because there, wave mechanics is not generally covariant.
The evolution of the tetrad in ECE theory is governed by:
( ) iS
q a µ (x µ ) = exp q a µ (0) (10.5)
where S is the action and a constant of proportionality introduced to make the
exponent dimensionless as required. This is the reduced Planck constant. The
fundamental origin of Eq. 10.5 is wave particle duality. In ECE theory there
is no distinction between wave and particle, both are manifestations of ECE
spacetime. The Dirac electron, for example, is defined by the limit [1]– [38]:
kT = m2 c 2
2 (10.6)
155

10.2. LAGRANGIAN FORMULATION OF GENERALLY . . .
of the ECE wave equation 10.1. This is not a point particle, because from Eqs.
10.3 and 10.6 emerges the rest volume of any particle:
V 0 = k2
mc 2 . (10.7)
The wave nature of the Dirac electron is governed by the same ECE wave
equation through the SU(2) representation of the tetrad [1]– [38]. In Section
10.3 it is shown that the Planck constant is a limit of Eq. 10.4, a limit in which
the volume V fortuitously cancels. There is a lot more to the Planck constant in
generally covariant unified wave mechanics than the standard model’s quantum
mechanics.
In Section 10.4 the Aspect experiment and quantum entanglement are discussed
within the context of generally covariant and causal wave mechanics, and
finally in Section 5 it is shown that under well defined circumstances, the phase
velocity, v, of a generally covariant wave can become much larger than c, and
indeed approach infinity. The phase velocity v ≫ c, combined with the spin
connection, lead to many new inferences and possible new technologies.
10.2 Lagrangian Formulation Of Generally Covariant
Wave Mechanics
It is seen by inspection that the generally covariant Euler Lagrange equation:
( )
∂L
= D µ ∂L
∂ (D µ q ν (10.8)
a)
with the lagrangian density:
gives:
∂q ν a
L = c 2 T + D µ q a µD µ q ν a (10.9)
D µ (D µ q a ν ) = 0. (10.10)
This is the ECE Lemma [1]– [38], which is obtained by covariant differentiation
of the tetrad postulate [39, 40] of Cartan geometry:
Using the fundamental definition:
the Leibnitz Theorem is applied to give:
Using Eq. 10.11 in Eq. 10.13 gives:
D µ q a ν = 0. (10.11)
q a µq µ a = 1 (10.12)
D ν
(
q
a
µ q µ a)
= q
a
µ (D ν q µ a) + ( D ν q a µ)
q
µ
a (10.13)
D ν q µ a = 0. (10.14)
Therefore Eq. 10.9 is:
L = c 2 T. (10.15)
156

CHAPTER 10.
WAVE MECHANICS AND ECE THEORY
In the rest frame limit this becomes Eq 10.3 as discussed in Section 10.1. Therefore
the second term in Eq. 10.9 is needed to give the lagrangian from which
the ECE Lemma and wave equation are derived by variational calculus and
minimization of action. Eq. 10.10 can be rewritten in the form [1]– [38]:
□q a µ = Rq a µ (10.16)
where R is a scalar curvature defined as follows. Using the Einstein Ansatz [1]–
[38]:
R = −kT (10.17)
the ECE Lemma becomes the ECE wave equation 10.1 of Section 10.1. This
is the fundamental and generally covariant wave equation of ECE field theory.
All the major wave equations of physics can be derived from Eq. 10.1 [1]– [38]
in various limits, for example the Dirac equation of special relativistic wave
mechanics and the Proca equation of electrodynamics (the d’Alembert equation
with photon mass). The ECE Lemma 10.16 follows [1]– [38] from the tetrad
postulate:
D µ q a λ = ∂ µ q a λ + ω a µbq b λ − Γ ν µλq a ν = 0. (10.18)
Here ω a µb is the spin connection and Γν µλ
is the gamma connection [1]– [40].
Therefore:
D µ (D µ q a λ) = ∂ µ (D µ q a λ) = 0. (10.19)
i.e.
or
∂ µ ( ∂ µ q a λ + ω a µbq b λ − Γ ν µλq a ν
)
= 0 (10.20)
□q a λ = ∂ µ ( Γ ν µλq a ν
)
− ∂
µ ( ω a µbq b λ)
. (10.21)
Now define the scalar curvature:
Rq a λ = ∂ µ ( Γ ν µλq a ν − ω a µbq b )
λ
(10.22)
and use Eq. 10.12 to obtain:
R = q λ a∂ µ ( Γ ν µλq a ν − ω a µbq b )
λ
(10.23)
and to deduce Eq. 10.16, Q.E.D. Therefore Eq. 10.23 is the fundamental
definition of scalar curvature in ECE wave mechanics:
R = −kT = q λ a∂ µ ( Γ ν µλq a ν − ω a µbq b λ)
. (10.24)
This lagrangian derivation of the ECE Lemma and wave equation is fully selfconsistent
and is based on Hamilton’s principle of least action for the particle
and Fermat’s principle of least time for the wave. Therefore wave and particle
are terms which become obsolete: in ECE theory they are both manifestations
of space-time. Wave and particle are simultaneously observable as indicated by
recent experiments [42]– [45]. In the now obsolete Bohr Heisenberg indeterminacy
the wave and particle are never simultaneously observable.
The lagrangian formulation of ECE theory is also the lagrangian formalism
of Cartan geometry itself. A powerful simplicity of understanding is achieved
through the tetrad postulate, which is the fundamental mathematical requirement
that a complete vector field V be independent of the system of coordinates
157

10.2. LAGRANGIAN FORMULATION OF GENERALLY . . .
used to define it. The key difference between Cartan and Riemann geometry
resides in the basis elements used to define the tangent spacetime at a point P
in the base manifold [1]– [40]. In Riemann geometry the basis elements always
form the set of partial derivatives. In Cartan geometry the basis elements are
more generally defined and labelled by a. The tetrad q a µ is the rank two mixed
index tensor defined by:
V a = q a µV µ (10.25)
where the vector elements V a are defined in the tangent space-time (Minkowski
space-time) and the vector elements V µ are defined in the base manifold (ECE
space-time). The tetrad postulate 10.18 follows from the fact that the complete
vector field V in the tangent space-time must be the same with basis set labeled
a and the Riemann basis set of partial derivatives. Eq. 10.18 is the rule for
the covariant differentiation of a mixed index rank two tensor [39], i.e. D − µ
acting on the rank two tensor q a ν ). Therefore the tetrad is always a rank two
tensor with a matrix structure. With these fundamentals clearly defined it is
seen that the lagrangian density of ECE theory and Cartan geometry, Eq. 10.9,
contains a product of two tetrad postulates and the ECE Lemma and wave
equation are obtained by covariant differentiation of the tetrad postulate. So
everything stems from the fact that a complete vector field V is independent
of the vector components and basis element used to describe it. For example a
vector field V in three dimensional Euclidean geometry may be represented by
the cartesian unit vectors, i, j and k (the basis elements), and by the cartesian
vector components V x , V y and V z :
V = V x i + V y j + V z k. (10.26)
The same vector field V in spherical polar coordinates will have different components
and different basis elements but is the same vector field. If we extend
this reasoning to Cartan geometry the tetrad postulate 10.18 is the inevitable
result [1]– [40]. All of physics stems from this property of vector field V via the
ECE field theory. So physics is fundamental geometry.
The well known Cartan torsion is also a direct consequence of the tetrad
postulate. This was first demonstrated in vol. 2 of ref. [1] and the proof is as
follows. Consider two tetrad postulates:
∂ µ q a λ + ω a µbq b λ = Γ ν µλq a ν , (10.27)
∂ λ q a µ + ω a λbq b µ = Γ ν λµq a ν . (10.28)
All that has been done is to change the index labeling, so we have written out
the tetrad postulate twice. Subtract Eq. 10.28 from Eq. 10.27 to obtain the
Cartan torsion:
T a µλ = T ν µλ q a ν = ( Γ ν µλ − Γ ν λµ)
q
a
ν
= ∂ µ q a λ − ∂ λ q a λ + ω a µbq b λ − ω a λbq b µ.
(10.29)
In differential form notation Eq. 10.29 is the first Cartan structure equation [39]:
T a µν = (d ∧ q a ) µν
+ ω a µb ∧ q b ν . (10.30)
In the standard notation of Cartan geometry the Greek indices of the base manifold
are omitted, because they are always the same on both sides of any equation
158

CHAPTER 10.
WAVE MECHANICS AND ECE THEORY
of Cartan or differential geometry, so Eq. 10.30 is written conventionally [39]
as:
T a = d ∧ q a + ω a b ∧ q b
:= D ∧ q a (10.31)
.
The electromagnetic tensor is then defined directly from the Cartan torsion
through the ECE Ansatz [1]– [38]:
F a = A (0) T a (10.32)
where A (0) is a fundamental and universal voltage within a factor of c.
So we have obtained the electromagnetic tensor and the lagrangian density
using the same tetrad postulate. The lagrangian density 10.9 is the same for
both the wave and field equations of ECE theory. In the standard model [47] the
lagrangian formulation is both incomplete and considerably more complicated.
The complexity of the standard model is lack of understanding, the converse of
the complete and simple ECE theory given here.
The field equations of ECE theory [1]– [38] are obtained from the first Bianchi
identity of Cartan geometry:
D ∧ T a = d ∧ T a + ω a b ∧ T b = R a b ∧ q b (10.33)
which states that the covariant derivative of the Cartan torsion is identically
equal to a cyclic sum of Riemann tensor elements [39, 40]. The first Bianchi
identity 10.33 again follows from the tetrad postulate, as demonstrated in full
detail in the appendices of chapter 17 of ref. [1]. Using the Ansatz 10.32 and
the equivalent ansatz:
A a µ = A (0) q a µ (10.34)
in Eq. 10.33 produces the homogeneous field equation of ECE theory:
d ∧ F a = µ 0 j a = A (0) ( R a b ∧ q b − ω a b ∧ T b) . (10.35)
The Hodge dual [1]– [40] of Eq. 10.35 is the inhomogeneous field equation of
ECE theory:
d ∧ ˜F a = µ 0 J a = A (0) ( ˜Ra b ∧ q b − ω a b ∧ ˜T b) . (10.36)
So we have linked all the fundamental equations of ECE theory with a lagrangian
formalism based on the minimization of action (Hamilton principle) and time
interval (Fermat principle).
Having achieved this unification of basic concepts it is now possible to develop
the fundamental equations of the incomplete standard model quantum
mechanics into principles of the completed theory sought for by Einstein and
Cartan: generally covariant wave mechanics. The minimization of action and
time interval are concepts which are central to wave mechanics and wave particle
dualism. It is now possible to develop wave particle dualism into the concept
of indistinguishability of wave and particle because wave mechanics has been
recognized as a property of space-time. The ECE Lemma asserts that scalar
curvature itself is quantized, space-time itself is quantized. The wave is the
tetrad eigenfunction of the ECE wave equation or Lemma, the particle is also
space-time, always occupying a finite volume. In the rest frame limit of the
Dirac equation this volume is given by Eq. 10.7 In the standard model the concept
of point particle is still used, and this concept is in conflict with relativity
because it introduces singularities and the complexity of renormalization.
159

10.3. PLANCK CONSTANT, PLANCK-EINSTEIN AND DE . . .
10.3 Planck Constant, Planck-Einstein And de
Broglie Equations, And The Schrödinger
Equation
In the rest frame the ECE lagrangian density is the rest energy divided by the
rest volume. Therefore the action in the rest frame is:
∫ mc
S 0 = d 4 x 0 . (10.37)
V 0
The four volume in the rest frame is:
Now identify the rest action with the Planck constant:
d 4 x 0 = V 0 cdt 0 . (10.38)
S 0 = (10.39)
and the integral over the time interval with the inverse rest frequency:
∫
dt 0 = 1 ω 0
(10.40)
to obtain the Planck Einstein equation in the rest frame:
E 0 = ω 0 . (10.41)
If applied to the photon rest mass this is also known [1]– [38] as the de Broglie
equation for photon rest mass. It is known experimentally that Eq. 10.41 also
holds for the photon when it travels for all practical purposes infinitesimally
close to the speed of light with respect to an observer in the rest frame. In
this case rest frequency is changed to ω. In special relativity this would be a
Lorentz transformation of angular frequency but in general relativity a general
coordinate transformation. However, the rest mass of the photon cannot be
identically zero in ECE theory, because the action would be identically zero.
The rest mass of the photon is very small but not zero. In ECE theory the
Planck constant is the rest frame limit of the action:
∫
S = c T d 4 x. (10.42)
In the rest frame:
∫ d 4 x 0
= mc
(10.43)
V 0
and if: ∫ d 4 x 0
V 0
we obtain:
= c
ω 0
(10.44)
E 0 = ω 0 = mc 2 . (10.45)
More generally, and for any particle, the action that gives the ECE wave equation
in any frame of reference is given by:
S = 1 ∫
Ld 4 x (10.46)
c
160

CHAPTER 10.
WAVE MECHANICS AND ECE THEORY
and is the generalization of the Planck constant to any frame of reference. The
Planck constant in the rest frame is:
S 0 = = 1 ∫
L 0 d 4 x 0 (10.47)
c
where V 0 is the rest volume defined by Eq. 10.7. Therefore ECE theory shows
that the Planck constant in the rest frame must have an internal structure
defined by the four volume d 4 x 0 and rest volume V 0 . The same is true in any
other frame, the rest volume being replaced by the volume in that frame of
reference. If the four volume is:
it is found that
d 4 x 0 = V 0 cdt 0 (10.48)
= mc 2 ∫
V0
V 0
dt 0 = mc 2 t 0 . (10.49)
The time interval t 0 must be a constant for a given mass m. The existence of the
Planck constant means that a particle is never quite at rest, it must have a rest
frequency defined by Eq. 10.40 So there must be zero point energy defined by
Eq. 10.45. Classically the particle in the rest frame does not move relative to the
observer in the same rest frame, and there is no rest energy in a classical theory.
The rest energy mc 2 is the result of special relativity theory as is well known.
ECE theory gives both the rest energy and the Planck energy ω 0 , showing
that it is a unified field theory . The fact that the Planck constant has an
internal structure that depends on volume is of key importance in modifying the
Heisenberg commutator equation in accordance with the experimental findings
[42] of Croca et al. This modification has been initiated in volume 2 of ref. [1].
The Fermat principle of least time [48] is the classical principle that governs
the propagation of light in optics. The path taken by the light through a medium
is such that the time of passage is a minimum. The amplitude of a light wave
at point P 1 is related to the amplitude at point P 2 by:
where the phase φ is defined by:
ψ (P 2 ) = e iφ ψ (P 1 ) (10.50)
φ = 2π x λ . (10.51)
Here x is the coordinate and λ the wavelength [48]. Eq. 10.50 is the fundamental
origin of the Schrödinger equation. Light takes paths such that the phase is
minimized. This is the precise statement of the Fermat principle. In the limit
of geometrical optics φ is infinite, the light appears to travel in straight lines.
There is no curvature, and this is a ”weak field limit” of ECE theory in which
the interval t 0 is minimized.
The propagation of particles is given classically by the Hamilton principle
of least action. Particles select paths between two points such that the action
associated with that path is a minimum. This classical statement is equivalent
to Newtonian dynamics in the weak field limit of ECE field theory. Particles
adopt a least path and waves a least time. The reason is the same, the phase
φ is minimized. So particles and waves become indistinguishable if the phase is
161

10.3. PLANCK CONSTANT, PLANCK-EINSTEIN AND DE . . .
made proportional to action and this is the fundamental idea of wave mechanics.
Thus φ is proportional to S and so the constant of proportionality must have
the units of inverse action because φ is unitless. In the classical limit φ is infinite
so the constant of proportionality approaches zero. Schrödinger’s equation is
recovered from this argument if:
φ = S . (10.52)
Eq. 10.50 describes a path from P 1 (x 1 , t 1 ) to P 2 (x 2 , t 2 ) [48]. Thus:
Differentiate [48] Eq. 10.53 with respect to t 2 :
ψ (x 2 , t 2 ) = e iS/ ψ (x 1 , t 1 ) . (10.53)
∂
ψ (x 2 , t 2 ) =
∂ (
)
e iS/ ψ (x 1 , t 1 ) . (10.54)
∂t 2 ∂t 2
Now use the Leibnitz Theorem:
∂
(
)
e iS/ ψ (x 1 , t 1 ) = ψ (x 1 , t 1 ) ∂ e iS/ iS/ ∂ψ
+ e (x 1 , t 1 ) . (10.55)
∂t 2 ∂t 2 ∂t 2
Since ψ(x 1 , t 1 ) is not a function of t 2 :
and
Thus:
∂ψ
∂t 2
ψ (x 1 , t 1 ) = 0 (10.56)
∂
e iS/ = i ∂S
· e iS/ . (10.57)
∂t 2 ∂t 2
∂
ψ (x 2 , t 2 ) = i ∂S
e iS/ ψ (x 1 , t 1 ) . (10.58)
∂t 2 ∂t 2
Finally use Eq. 10.53 to obtain the Schrödinger equation in time dependent
form [48]:
∂
∂t ψ = i ∂S
ψ. (10.59)
∂t
This is not strictly a wave equation because a wave equation in mathematics
contains second derivatives, but it is the famous equation of non-relativistic
quantum mechanics. The more familiar form of the Schrödinger equation is
obtained by using [48]:
E = − ∂S
(10.60)
∂t
where E is the total energy, the sum of kinetic and potential energy. So Eq.
10.59 becomes:
i ∂ψ = Eψ. (10.61)
∂t
Finally define the operator:
H = i ∂ ∂t
(10.62)
to obtain the familiar:
Hψ = Eψ (10.63)
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CHAPTER 10.
WAVE MECHANICS AND ECE THEORY
It is seen that the Schrödinger equation is a causal differential equation and
as such cannot be interpreted as an expression of something that is acausal or
unknowable. Using the operator 10.62 for energy it is seen that the Schrödinger
equation is mathematically the same as:
[H, t] ψ = iψ (10.64)
where the time t multiplies the function ψ. Eq. 10.64 is an example of a
Heisenberg commutator equation in the non-relativistic quantum limit. There is
no more meaning to Eq. 10.64 than Eq. 10.63 because Eq. 10.64 is a restatement
of Eq. 10.63 and thus contains the same mathematical information. This is the
causal deterministic view of Einstein, de Broglie, Schrödinger, Bohm, Vigier
and followers. The Copenhagen interpretation of Eq. 10.64 is that if t is known
exactly, E is unknowable, and vice versa. This is the view of Bohr, Heisenberg
and followers.
This non-relativistic analysis can be extended to ECE theory [1]– [38] by
using the equation for the propagation of the tetrad wave function:
where S(x µ ) is defined by Eqs. 10.8 and 10.9.
Differentiate Eq. 10.65 to obtain:
q a µ (x µ ) = e iS(xµ )/ q a µ (0) (10.65)
∂ ν q a µS (x µ ) = i ∂ν Se iS/ q a µ (0) . (10.66)
The second term disappears in analogy with the derivation of Eq. 10.59 from
Eq. 10.53. The following definitions are used:
( 1
∂ ν :=
c
∂
∂t 1
,
∂
∂X 1
,
∂
∂Y 1
,
)
∂
, (10.67)
∂Z 1
q a µ(0) := q a µ (ct 2 , X 2 , Y 2 , Z 2 ) , (10.68)
and:
∂ ν q a µ(0) = 0. (10.69)
Eq. 10.66 is the generally covariant Schrödinger equation in which the wave
function is the tetrad. Now differentiate Eq. 10.66 once more:
(
∂ ν ∂ ν q a ) i
µ =
∂ (
ν (∂ ν S) q a µ)
(10.70)
to obtain the following generally covariant wave equation:
□q a µ = i (
□S + i )
∂ν S∂ ν S q a µ = Rq a µ. (10.71)
The second equality in Eq. 10.71 follows from the ECE Lemma. Therefore we
obtain the following expression for the scalar curvature in terms of the action:
R = i (
□S + 1 )
∂ν S∂ ν S
(10.72)
163

10.4. THE ASPECT EXPERIMENT AND QUANTUM . . .
If Eq. 10.39 is not used, i.e. if it is not assumed a priori that S 0 is , then Eq.
10.6 becomes:
q a µ = exp( iS
S 0
)q a µ(0) (10.73)
and the scalar curvature can be expressed as:
R =
i (
□S + i )
∂ ν S∂ ν S . (10.74)
S 0 S 0
In the limit of the Dirac equation [1]– [38]:
R → −
( mc
) 2
(10.75)
and we obtain the wave form of the Dirac equation:
(□ + m2 c 2 )
2 q a µ = 0. (10.76)
This limit may be used to identify the Planck constant as:
m 2 c 2
2 = − i (
□S + i )
∂ ν S∂ ν S
S 0 S 0
(10.77)
where
S 0 → . (10.78)
The standard model does not consider the general covariance of the Planck
constant, because in the standard model quantum mechanics is not unified with
general relativity. The Hamilton and Fermat principles are classical, i.e. nonrelativistic.
In volume 2 of ref. (1) it was suggested that the key quantity
to consider is the density of action, not the action itself. So there may be
experimentally observable departures from quantum mechanics when general
covariance is properly considered. These may show up in hyperfine spectral
structure.
10.4 The Aspect Experiment And Quantum Entanglement
In this section the Aspect experiment [49] and quantum entanglement [50] are
developed as two examples of how ECE wave mechanics is applied to data.
In the Aspect experiment two photons are emitted at the same time and are
circularly polarized in opposite senses. The photons travel along different paths
and filters define their orientations a and b, subtending between them the angle
θ. Therefore a circularly polarized tetrad wave:
A (1) = A(0)
√
2
(i − ij) e iφ (10.79)
is split into
A (1) a = A(0)
√
2
ie iφ (10.80)
164

CHAPTER 10.
WAVE MECHANICS AND ECE THEORY
and
A (1) b = −iA(0) √ je iφ (10.81)
2
if a and b are at right angles. A photo-multiplier tube detects either A (1) a or
A (1) b , one detector for A(1) a and one for A (1) b
. In the detector, +1 is registered
for A (1) a and −1 for. The ±1 signals are collected on a coincidence counter.
This procedure occurs for both the right circularly polarized tetrad wave:
and the left circularly polarized tetrad wave:
A (1) R = A(0)
√
2
(i − ij) e iφ (10.82)
A (1) L = A(0)
√
2
(i + ij) e iφ . (10.83)
Therefore there are A (1) Ra , A(1) Rb , A(1) La and A(1) Lb . The components A(1) Ra
and A (1) La both register a +1 and A(1) Rb and A(1) Lb
both register a −1.
This coincidence counter only accepts results if the time delay between receiving
signals from the photo-multiplier tubes on sides A and B is less than
a certain interval t. The latter is half the time it takes for a signal c to travel
from one filter to the other. If an event occurs within the interval t the result
on side A (+1 or −1) is multiplied by the result from side B and the average
value found from repeated measurements. The average value is defined by the
expectation value, which is the sum of all the resulting values multiplied by the
probability for that value:
〈P 〉 = P ++ − P +− − P −+ + P −− (10.84)
The expectation value is a function of the filter orientation, or the angle θ
between a and b. Here P ++ is the probability that both detectors registered a+
and P −− that both detectors registered a−. These probabilities are measured
experimentally by recording the number of counts of a particular type and
dividing this record by the total number of counts recorded. For example, P −+
is the number of times the left detector registered −1 at the same time as the
right detector registered +1. The ”same time” means ”within the t interval”.
Within the factor A (0) the quantities A (1) Ra , A(1) Rb , A(1) La and A(1) Lb are
tetrads, or wave functions. So the Aspect experiment investigates the statistical
properties of tetrad waves. By statistical is meant statistical averaging of causal
wave-functions, which within A (0) are waves of spinning space-time. In generally
covariant wave mechanics (ECE theory) the Aspect experiment is considered as
follows. One filter detects linear polarization along the a direction, the other
along the b direction. The expectation value is [49]:
Ω = cos 2θ (10.85)
and this is what is measured experimentally in the Aspect experiment. Thus
ECE theory must be used to explain Eq. 10.85, the experimental result. From
Eq. 10.65 the basic equation to be used is:
q a µ (t 1 , r 1 ) = e iS/ q a µ (t 2 , r 2 ) . (10.86)
165

10.4. THE ASPECT EXPERIMENT AND QUANTUM . . .
Now use the identity:
and denote:
It is found that:
cos 2θ = Re ( e iθ e 2iθ e −iθ) (10.87)
ψ = e iθ , ψ ∗ = e −iθ , (10.88)
Ω = e 2iθ (10.89)
cos 2θ = ψΩψ ∗ (10.90)
This equation is similar to the definition [48] of expectation value in quantum
mechanics:
∫
∫
〈Ω〉 = ψ ∗ ΩψdV/ ψ ∗ ψdV (10.91)
where V is a volume. Usually in quantum mechanics [48] the denominator in
Eq. 10.91 is normalized to unity, so:
∫
ψ ∗ ψdV = 1. (10.92)
Therefore cos 2θ/V in Eq. 10.90 is the density of expectation value. As in
lagrangian dynamics and relativity theory it is the density that is the key quantity.
In Eq. 10.91 the density of expectation value is a weighted sum of the
eigenvalues of Ω [48]. The wave function is expanded as the sum:
ψ = ∑ n
C n ψ n (10.93)
where
Ωψ n = ω n ψ n . (10.94)
In the simple example of the identity 10.87 the wave function is:
ψ = e iθ (10.95)
and the eigen-operator Ω operates on the wave-function ψ:
Ω = e 2iθ (10.96)
So (cos 2θ) /V is the density of the expectation value of Ω.
described by Eq. 10.50:
A light wave is
Now identify the angle θ as:
to obtain Eq. 10.50 in the form:
ψ (P 2 ) = exp (2πi (x 2 − x 1 ) /λ) ψ (P 1 ) . (10.97)
θ = π (x 2 − x 1 ) /λ (10.98)
ψ (P 2 ) = e 2iθ ψ (P 1 ) . (10.99)
This is always true for any light wave. Quantization into photons occurs when
the angle in Eq. 10.9 is further identified as:
θ = S/. (10.100)
166

CHAPTER 10.
WAVE MECHANICS AND ECE THEORY
In the special case:
we recover:
ψ (P 1 ) = ψ ∗ (P 2 ) (10.101)
cos 2θ = Re ( ψe 2iθ ψ ∗) = Re (ψψ) . (10.102)
Finally apply this analysis to the tetrad propagation equation 10.86:
A a µ = e iS/ A a µ (0) (10.103)
which within A a µ describes the propagation of the electromagnetic potential
and photon simultaneously. The Planck Einstein and de Broglie equations are
recovered by identifying the electromagnetic phase with quantized action:
Eq. 10.104 gives:
S = (ωt − κZ) . (10.104)
En = ω, p = κ, (10.105)
which are the archetypical photon equations. In these equations is a universal
constant for the free electromagnetic field. However, as argued in Section 10.3,
when light interacts strongly with gravity, must be generally covariant. The
root cause of photons in the free electromagnetic field is the universal constancy
of the action . This is the minimum action or angular momentum of the
electromagnetic field free from gravity.
In the ECE description of the Aspect experiment the wave and particle
co-exist, they are parts of the ECE wave equation. In the Copenhagen interpretation
the wave and particle are not simultaneously knowable. The Aspect
experiment does not distinguish between these two points of view, the experiment
is meant to test the Bell inequalities [49] and hidden variable theory.
However, contemporary experiments [42]– [45] refute Bohr Heisenberg indeterminacy
while supporting relativity to very high precision as described in the
introduction. Young interferometry is an example of such experiments [50].
Indeterminacy is beginning to be supplanted [50] by the concept of quantum
entanglement. To end this section entanglement is briefly described with ECE
theory.
Quantum entanglement is the appellation originally given by Schrödinger
to the wave function of two interacting systems. In ECE theory these are two
different tetrads, q a µ and q b ν . These tetrads are governed by the ECE wave
equations:
□q a µ = R 1 q a µ (10.106)
and
□q b ν = R 2 q b ν . (10.107)
The entangled state is defined by the tensor product [1]:
which obeys the ECE wave equation:
g ab µν = q a µq b ν (10.108)
□ ( q a µq b ν
)
= R
(
q
a
µ q b ν
)
. (10.109)
The entangled state in ECE theory is therefore a tensor valued metric [1]– [38]:
g ab µν (entangled) = q a µq b ν . (10.110)
167

10.4. THE ASPECT EXPERIMENT AND QUANTUM . . .
An entangled quantum state is therefore a space-time property. The eigenfunction
g ab µν can always be written [1]– [38] as the sum of a symmetric and
antisymmetric components. The symmetric component is:
g µν = q a µq b ν η ab (10.111)
where η ab is the Minkowski metric of the tangent space-time of Cartan geometry,
and obeys the wave equation:
The antisymmetric component is:
and obeys the wave equation:
□g µν = R s g µν . (10.112)
g c µν = q a µ ∧ q b ν (10.113)
□g c µν = R A g c µν . (10.114)
For pure rotational motion [1]– [38] the tetrad is dual as follows to the spin
connection:
ω a b = − κ 2 ɛa bcg c (10.115)
where κ is a wave-number magnitude (inverse meters). Therefore g c µν is proportional
to the spin connection term ω a b ∧ qb of the Cartan torsion defined
by:
T a = d ∧ q a + ω a b ∧ q b (10.116)
If for example we consider the spin-spin interaction of two different spinning
particles, (e.g. two electrons in an atom), a net Cartan torsion is set up in
general. Since ω a b ∧ qb cannot exist without the d ∧ q a term the most general
spin-spin interaction is described by the wave equation:
□T a = V T a (10.117)
where V must have the units of volume. There is local spin-spin interaction,
defined by the d ∧ q a term, and non-local spin-spin interaction, described by
the ω a b ∧ qb term. Spin - spin interaction is observed [48] in fine and hyperfine
spectroscopy and in ESR, NMR and so forth. So these spectra are manifestations
of Cartan torsion. In optics and electrodynamics Eq. 10.116 becomes:
using the ansatzen:
and Eq. 10.117 becomes:
F a = d ∧ A a + ω a b ∧ A b (10.118)
A a = A (0) q a , (10.119)
F a = A (0) T a , (10.120)
□F a = V F a (10.121)
where the field F a has become a wave function. There may be entanglement
between different photons. One photon is described by the local term d∧A a and
the non-local term ω a b ∧ Ab . In analogy with the Aharonov Bohm effects [1]–
[38] the non-local part of the photon, ω a b ∧ Aa , can be observed experimentally
168

CHAPTER 10.
WAVE MECHANICS AND ECE THEORY
in regions where the local part of the photon, d ∧ A a , does not exist. So when
light (i.e. d ∧ A) travels through one aperture of a Young interferometer [50] it
is always accompanied in ECE theory by its non-local ω a b ∧ Ab , even on a one
photon level. This is precisely what is observed in contemporary experiments
[50], where one photon appears to ”interfere with itself”. The ”particle” is
not localized, it is always accompanied by the wave, and both are observed
simultaneously. Thus ECE theory and relativity are preferred experimentally
to indeterminacy. The extra ingredient given by ECE theory is the non local
term ω a b ∧ Ab due to the spin connection.
The archetypical entanglement effect is when one particle affects another
when they are separated by a large distance, for example two spins. In ECE
theory this is another experimental example of a non-local effect due to the spin
connection. The influence of one spin on another is due to the spinning of spacetime
itself, and such experiments prove that space-time spins. In ECE theory,
as in all theories of relativity, c is a universal constant, but as discussed in
Section 10.5, the phase velocity v of a tetrad may become much greater than c.
Entanglement proves that ”information” can be transmitted to a remote region.
In ECE theory this information is transmitted by the spin connection while c
remains constant. So the information is not being transmitted by the speed
of light c. It is transmitted by spinning space-time. The concept of spinning
space-time does not exist in the standard model, in which quantum mechanics
is almost always developed in a flat space-time without spin, the Minkowski
space-time of special relativity. So in the standard model effects such as single
photon interferometry, quantum entanglement and the Aharonov Bohm variety
are impossible to understand self-consistently. An understanding needs a unified
field theory which is generally covariant [1]– [38]. Also a single particle other
than a photon (for example an electron), can also exhibit Young interferometry.
In ECE theory this is understood in the same way, a particle and wave cannot
be separated, they are different aspects of ECE space-time. So we arrive at the
principle of wave particle indistinguishability, and introduce the terminology
”wave-particle”. In analogy, relativity unified space and time and introduced
the terminology ”space-time”.
10.5 Phase Velocity Of ECE Waves
To illustrate the ability of ECE theory to produce phase velocity v >> c consider
the homogeneous ECE field equation [1]– [38]:
∇ × E a + ∂Ba
∂t
= µ 0˜ja . (10.122)
This can be expressed as:
∇ × (ɛ r E a ) + ∂ ( ) 1
B a = 0. (10.123)
∂t µ r
Here E a is the electric field strength (volt m −1 ), B a is magnetic flux density
(tesla), µ 0 is the vacuum permeability and ˜j a is the homogeneous current. In
Eq. 10.123:
µ r = µ/µ 0 , ɛ = ɛ/ɛ 0 (10.124)
169

10.5. PHASE VELOCITY OF ECE WAVES
where µ r is the relative permeability of ECE space-time, µ is its absolute permeability,
ɛ r is its relative permittivity and ɛ is its absolute permittivity. Here
µ 0 and ɛ 0 are the vacuum permeability and permittivity respectively. The refractive
index of ECE space-time is
n 2 = µ r ɛ r . (10.125)
The phase velocity of a wave in ECE space-time is defined as:
v = c
n 2 = c . (10.126)
µ r ɛ r
The relative permeability and permittivity are complex quantities in general:
µ r = µ ′ r + iµ ′′
r (10.127)
ɛ r = ɛ ′ r + iɛ ′′
r (10.128)
so
v =
c
x + iy
(10.129)
where:
x = µ ′ rɛ ′ r − µ ′′
r ɛ ′′
r , (10.130)
y = µ ′ rɛ ′′
r + µ ′′
r ɛ ′ r. (10.131)
So the real-valued and physical part of the phase velocity is:
x
Re(v) =
(x 2 − y 2 c.
)
(10.132)
It is seen that for finite constant c:
when
Re(v) → ∞ (10.133)
x 2 = y 2 , x = ±y. (10.134)
If x = y then:
µ ′ rɛ ′ r − µ ′′
r ɛ ′′
r = µ ′ rɛ ′′
r + µ ′′
r ɛ ′ r. (10.135)
If x = −y then:
µ ′ rɛ ′ r − µ ′′
r ɛ ′′
r = − (µ ′ rɛ ′′
r + µ ′′
r ɛ ′ r) . (10.136)
The phase velocity v is that of the generally covariant unified field. The interaction
with gravitation occurs through the permittivity and permeability of
ECE space-time. It is this interaction that results in µ and ɛ of ECE space-time
being different from the vacuum values. This is confirmed experimentally in
the Eddington effect, the bending of light rays by the sun’s mass, and in other
cosmological effects of gravitational lensing. ECE theory is the first complete
theory of this famous effect. When electromagnetism is independent of gravitation
Eq. 10.122 becomes the Faraday law of induction for each polarization
index a. As we have seen, wave mechanics is unified with general relativity
through the ECE wave equation 10.1. The phase velocity of ECE waves can be
much greater than c as illustrated here. The constant c itself remains a universal
constant as required in any theory of relativity.
Acknowledgments Parliament and the British Head of State, Queen Elizabeth
the II, are thanked for a Civil List pension, and the AIAS environment
for funding and many interesting discussions. Franklin Amador is thanked for
meticulous typesetting.
170

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174