Electromagnetism And The Structure Of Matter:
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Preface
The theory of electromagnetism, in the form conceived by J.C. Maxwell,
can boast 130 years of honorable service. It has withstood the severest tests,
proving itself to be, for completeness and elegance, among the most solid
theories. Very few would doubt its validity, to the extent that they may
be more inclined to modify the point of view of other theories, rather than
question Maxwell’s equations. In fact, faith in the model has been strong
enough to obscure a certain number of “minor” incongruities, resulting in
a whole string of justifications and leading to the development of other
theories.
However, the truth is that although these time-honored equations excellently
solve complex problems, they are nevertheless unable to simulate
the simplest things. They are not capable, for instance, of describing what
a solitary signal-packet is, which is one of the most elementary electromagnetic
phenomena. Alternative models have been proposed with the aim of
including solitons in the theory, but they have been unsuccessful in acquiring
long-term credibility, because based on deliberate adjustments, which,
while accommodating specific aspects, cause the model to lose general properties.
The development of modern field theory, which was very prosperous in
the first half of the last century, has magnified the role of the equations,
giving them a universal validity in the relativistic framework. However, this
progress has come to a halt, despite the impression one has of being not
too far from the goal of unifying electromagnetism and gravitation theory.
We are going to make some statements that many readers will certainly
consider heretical. We think that the various anomalies in Maxwell’s model
are not incidental, but rather consequences of a still insufficient theoretical
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viii Electromagnetism and the structure of matter
description of electromagnetic phenomena. In fact, it is our opinion that
the flaws run deeper than might be expected, and therefore that this fundamental
building block of physics needs extensive revision. The process
of review we are facing is so radical that the entire conceptual framework
needs to be re-thought from the beginning. Then again, if it were just a
matter of small adaptations, this revision would have already been made a
long time ago.
We shall start by pointing out some facts, which may be considered
marginal at a practical level, in order to highlight contradictions. We solve
these problems by making appropriate adjustments to the Maxwell equations.
This will allow for the construction of a new model, whereby all the
inconsistencies will be solved and a better understanding of electromagnetic
phenomena will be achieved. The suitability of this approach will quickly
be made evident to the reader, by a sequence of remarkable coincidences,
which make the model as elegant as Maxwell’s, while providing greater
scope for development. Indeed, the new set of equations explains many
open questions and establishes links between electromagnetism and other
theories that have either been the subject of research for a long time, or
have been hitherto unimaginable.
None of the gracefulness that characterizes the Maxwell model will be
lost. The reader who has the patience to follow our arguments through to
the end will discover that all the pieces fit together in the global scheme
with due elegance and harmony. The model will be built up step by step,
up to its final form, so that the reader may appreciate the phases of its
maturation. The mathematical tools we have used are classical, possibly
outdated. However, our intention is to examine what would have happened
to the evolution of physics if our model had been applied instead of Maxwell
equations. We will elaborate and clarify many important concepts, pointing
the way to future developments in the investigation of nature’s most
intimate secrets.
This book is an improved and enlarged version of a preliminary
manuscript (see Funaro (2005)), which has never been submitted, since the
aim was to publish a definitive, comprehensive and self-contained version
that included some more persuasive material. In chapter 1, detailed arguments
are provided showing that the set of Maxwell equations in vacuum,
and the corresponding wave equations, do not properly describe the evolution
of electromagnetic wave-fronts, in the way it is commonly supposed.
Based on these indications, in chapter 2, a nonlinear corrected version, that
Preface ix
is proven to be far better suited to modelling electromagnetic phenomena,
is proposed. A velocity vector field, determining the direction of movement
of the fronts, explicitly appears in the set of partial differential equations.
The Lagrangian coincides with the one of the classical approach, but it is
minimized on a constrained space that enforces the wave-packets to follow
the rules of geometrical optics. The continuity equation and other classical
energy conservation laws are automatically implied. In this setting, requiring
the speed of light to be constant turns out to be equivalent to satisfy
the eikonal equation, governing the geometric development of wave-fronts.
Moreover, an extended range of soliton-like solutions with compact support
is explicitly found, as well as perfect spherical waves (not available in the
Maxwellian theory, despite common belief). This wide spectrum of solutions,
called free-waves, adds a new perspective to the study of light-wave
phenomena. As a matter of fact, the corrected model is proven to be invariant
under Lorentz transformations, unifying under a single statement
some of the axioms of special relativity. At this stage, it will be definitively
clear to the reader how a wave can be interpreted, at the same time, both
as a whole electromagnetic phenomenon and a bundle of photons.
The interaction of free-waves with matter is examined in chapter 3. This
qualitative study, based on well-known facts, allows for a further generalization
of the model. In fact, while the rays associated with free-waves
can only proceed along straight trajectories, new sets of solutions, called
constrained waves, are introduced in order to simulate those phenomena
where light, due to external perturbations, is forced to deviate from the
natural path. In this context, light rays are identified with the stream-lines
of a fluid evolving as prescribed by the non-viscous Euler equation, so that
the velocity vector field can now be subjected to transversal accelerations.
The additional equation is supplied with a forcing term, depending on the
electromagnetic fields, that turns out to be zero when there are no disturbances
acting on the wave (reproducing free-waves, in this special case).
Thus, a strong coupling, between the electromagnetic signals lying on the
front surface, and the path of the rays ruled by the laws of fluid mechanics,
is created. It is important to remark that the final set of model equations
only acts on vector fields in vacuum. Indeed, wave-packets moving at the
speed of light and reacting in accordance to deterministic rules, are the
main ingredients of such a universe.
In chapter 4, the equations are written, according to general relativity, in
covariant form. As far as the evolution of free-waves is concerned, requiring
the divergence of the classical electromagnetic stress tensor to be zero,
x Electromagnetism and the structure of matter
excellently combines with the new set of equations. For constrained waves,
the sum of the electromagnetic stress tensor with a suitable mass tensor
yields the whole set of model equations and provides the expected link
between electromagnetic and velocity fields. Successively, the combination
of the two energy tensors is put on the right-hand side of Einstein’s equation
and meaningful explicit solutions are found. Constrained waves follow the
geodesics of the resulting metric environment, ensuring the preservation of
the rules of geometrical optics, as well as the conservation of energy and
momenta. The study of the scattering of two or more interacting photons
can then be undertaken.
In chapter 5, the case of 2-D waves turning around an axis is studied.
Also in this situation explicit solutions are computed. They come
from an elliptic-type eigenvalue problem, derived from the model equations,
and display a quantized behavior. Therefore, even if quantum effects
are not directly included in the constitutive equations, they naturally come
out when handling particular solutions. This analysis, partially extended
in 3-D, leads to the construction of a non-singular deterministic model of
stable elementary particles, based on traditional electromagnetic and gravitational
fields. In this framework, the electron consists of rotating photons
in a toroid-shaped geometry, perfectly similar to a fluid dynamics vortex
ring. Thanks to Einstein’s equation, the space-time is modified, giving rise
to a situation of equilibrium, so that the electromagnetic fields are forced to
remain in the same gravitational environment generated by their own evolution.
Quantitative considerations demonstrate that the obtained structure
matches reality in all respects, opening the path to the understanding of the
structure of matter and its properties. Furthermore, the foundations for a
causal explanation of quantum phenomena are set forth. At atomic level, a
possible scenario of the consequences of this approach is investigated, using
heuristic arguments, in the concluding chapter 6.
D. Funaro
Contents
Preface vii
1. Something is wrong with classical electromagnetism 1
1.1 Maxwell equations and wave-fronts . . . . . . . . . . . . . 1
1.2 Wave-front propagation . . . . . . . . . . . . . . . . . . . 8
1.3 Fronts from an oscillating dipole . . . . . . . . . . . . . . 11
1.4 Preliminary conclusions . . . . . . . . . . . . . . . . . . . 21
2. First steps towards the new model 23
2.1 Modified Maxwell equations . . . . . . . . . . . . . . . . . 23
2.2 Perfect spherical waves . . . . . . . . . . . . . . . . . . . . 26
2.3 Travelling signal-packets . . . . . . . . . . . . . . . . . . . 30
2.4 Lagrangian formulation . . . . . . . . . . . . . . . . . . . 36
2.5 Free-waves and the eikonal equation . . . . . . . . . . . . 41
2.6 Lorentz invariance . . . . . . . . . . . . . . . . . . . . . . 46
3. Interaction of waves with matter 53
3.1 Wave bouncing off an obstacle . . . . . . . . . . . . . . . 53
3.2 Diffraction phenomena . . . . . . . . . . . . . . . . . . . . 59
3.3 Adding the mechanical terms . . . . . . . . . . . . . . . . 64
3.4 Properties of the new set of equations . . . . . . . . . . . 68
4. The equations in the framework of general relativity 77
4.1 Preliminary considerations . . . . . . . . . . . . . . . . . . 77
4.2 The energy tensor . . . . . . . . . . . . . . . . . . . . . . 84
4.3 Unified field equations . . . . . . . . . . . . . . . . . . . . 91
4.4 The divergence of the magnetic field . . . . . . . . . . . . 105
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xii Electromagnetism and the structure of matter
5. Building matter from fields 111
5.1 Adding the pressure tensor . . . . . . . . . . . . . . . . . 111
5.2 On the existence of particle-like solutions . . . . . . . . . 119
5.3 Looking for 2-D constrained waves . . . . . . . . . . . . . 129
5.4 Neutrinos, electrons and protons . . . . . . . . . . . . . . 146
5.5 Connections with a Dirac type equation . . . . . . . . . . 161
6. Final speculative considerations 169
6.1 Towards deterministic quantum mechanics . . . . . . . . . 169
6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Bibliography 185
Index 189
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Electromagnetism And The Structure Of Matter: Electromagnetism and the structure of matter by Daniele Funaro.JPG
