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Electromagnetism and
the Structure of matter
Daniele Funaro Università di Modena e Reggio Emilia, Italy
World Scientific NEW JERSEY
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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data Funaro, Daniele, 1958– Electromagnetism and the structure of matter / Daniele Funaro. p. cm. Includes bibliographical references and index. ISBN-13: 978-981-281-451-7 (hardcover : alk. paper) ISBN-10: 981-281-451-5 (hardcover : alk. paper) 1. Electromagnetism. 2. Matter. QC760.F88 2008 530.14'1--dc22 2008009652
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ELECTROMAGNETISM AND THE STRUCTURE OF MATTER Daniele Funaro Department of Mathematics University of Modena and Reggio Emilia Via Campi 213/B, 41100 Modena (Italy) E-mail:
[email protected] http://cdm.unimo.it/home/matematica/funaro.daniele/
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To those who trusted in me
v
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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 vii
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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
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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,
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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
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Contents
vii
Preface 1.
Something is wrong with classical electromagnetism 1.1 1.2 1.3 1.4
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Modified Maxwell equations . . . . . Perfect spherical waves . . . . . . . . Travelling signal-packets . . . . . . . Lagrangian formulation . . . . . . . Free-waves and the eikonal equation Lorentz invariance . . . . . . . . . .
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Wave bouncing off an obstacle . . . . Diffraction phenomena . . . . . . . . . Adding the mechanical terms . . . . . Properties of the new set of equations
Preliminary considerations . . . . . . The energy tensor . . . . . . . . . . Unified field equations . . . . . . . . The divergence of the magnetic field xi
23 26 30 36 41 46 53
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The equations in the framework of general relativity 4.1 4.2 4.3 4.4
1 8 11 21 23
Interaction of waves with matter 3.1 3.2 3.3 3.4
4.
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First steps towards the new model 2.1 2.2 2.3 2.4 2.5 2.6
3.
Maxwell equations and wave-fronts Wave-front propagation . . . . . . Fronts from an oscillating dipole . Preliminary conclusions . . . . . .
1
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53 59 64 68 77
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5.
Building matter from fields 5.1 5.2 5.3 5.4 5.5
6.
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Adding the pressure tensor . . . . . . . . On the existence of particle-like solutions Looking for 2-D constrained waves . . . . Neutrinos, electrons and protons . . . . . Connections with a Dirac type equation .
Final speculative considerations 6.1 6.2
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111 119 129 146 161 169
Towards deterministic quantum mechanics . . . . . . . . . 169 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Bibliography
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Index
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Chapter 1
Something is wrong with classical electromagnetism
1.1
Maxwell equations and wave-fronts
In the following sections, we make some assertions regarding the evolution of electromagnetic waves and the way they are modelled by the Maxwell equations. We start by pointing out the deficiencies mainly at the level of mathematical elegance. These in turn will reveal other more serious inconsistencies. In the end, even taking into account the correctness, up to a certain degree of approximation, of the physical approach, our assessment will be somewhat negative. As a matter of fact, in chapter 2, a substantial revision will be undertaken with the aim of finding a remedy to the problems that will have emerged. From now on, we assume that we are in void three-dimensional space. As usual, the constant c indicates the speed of light. In this case, the classical Maxwell equations are: ∂E = c2 curlB ∂t
(1.1.1)
divE = 0
(1.1.2)
∂B = − curlE ∂t
(1.1.3)
divB = 0
(1.1.4)
where the vector field E is dimensionally equivalent to an acceleration multiplied by a mass and divided by an electric charge, while B is a frequency 1
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multiplied by a mass and divided by a charge. Cross products (such as curlB = ∇ × B) are right-handed, in a right-handed coordinate system. This remark, which is for the moment irrelevant, will acquire greater significance as we proceed with our investigation. The above equations are supposed to be satisfied point-wise at any instant of time. Their solutions are assumed to be smooth enough to allow differential calculus. Therefore, discontinuous or singular solutions are not allowed. We refute the existence of point-wise concentrated charges, and the strategy we will therefore follow is to avoid for the moment singular solutions and arrive at the practical construction of elementary particles in chapter 5. Equations (1.1.2) and (1.1.4) may be considered unnecessary, since they are easily deduced from (1.1.1) and (1.1.3) respectively, after applying the divergence operator. Later on, for reasons that will be explained, we will question the validity of (1.1.2) and (1.1.4). As a consequence, the entire formulation will lose its credibility. As far as the evolution of an electromagnetic plane wave (with infinite extent and linearly polarized) is concerned, we have no objections to make. In cartesian coordinates, a monochromatic wave of this type, moving along the direction of the z-axis, is written as: E = c sin ω(t − z/c), 0, 0 B = 0, sin ω(t − z/c), 0 (1.1.5)
In this case, the Maxwell equations are all satisfied point-wise. The energy associated with this solution is infinite, hence the wave does not exist in nature, although it is very frequently employed in theoretical dissertations. Some confusion arises, however, when one tries to simulate the evolution of a transversal fragment of wave. We first examine the case of the plane wave given in (1.1.5). For any fixed z, we can cut out a region Ω in the plane determined by the variables x and y, and follow its evolution in time. For simplicity, Ω can be the square [0, 1] × [0, 1]. Before proceeding, it is important to define exactly what is a wavefront. A correct understanding of this concept is absolutely necessary for the discussion that follows. For the current purposes, a wave-front will be a piece of a two-dimensional surface, suitably parametrized along the directions determined by the two vector fields E and B, which are mutually orthogonal (see figure 1.1). We also assume that |E| = |cB| at each point, having denoted by | · | the standard vector norm in 3-D. A proof of the fact that E and B must be orthogonal is given at the beginning of section 2.4.
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Fig. 1.1 Wave-front parametrized by the electromagnetic fields E and cB. The light rays, obtained as the envelope of the velocity vector field V, are orthogonal to the front. Point-wise, the vector V indicates the direction followed by the front during its evolution.
Such a single wave-front is locally required to develop following the direction of the light rays, that, at any fixed instant, are orthogonal to the surface. Notwithstanding, the rays may be allowed to turn, as will be seen in chapter 3 (consider, for instance, figure 4.1). The energy is carried away along the rays at the speed of light. The energy intensity is proportional to the norm of the Poynting vector P = E × B, locally tangent to each ray. Finally, we denote by V a velocity vector field tangent to the rays, normalized in such a way that: |V| = c
(1.1.6)
The field V will be of paramount importance for constructing the new model we are going to propose in the coming chapters. Concerning electromagnetic waves, fronts are generated one after the other, producing a longitudinal continuum. The fields defined on successive fronts may vary in intensity and polarization. Once emitted by some source,
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these fronts usually develop without influencing each other, unless they find some obstacle (another set of wave-fronts for example). For the moment, we focus on free solitary waves (or free-waves). For this reason, in a preliminary analysis, we simply analyze the evolution of a single front, even if this could be part of a more complex apparatus. Basically, the information present on each single wave-front does not change its nature during the evolution. However, if the front is stretched, vector fields defined on it are accordingly modified in intensity. This is done in such a way that the global energy, integrated on the whole surface, remains constant. To grasp the idea, the reader may look at the first two pictures of figure 1.4. Let us go back to the case in which the support of the wave-front is the square Ω. We assume with Maxwell that, inside Ω, the electromagnetic fields are given by (1.1.5), where the variable ζ = t − z/c is fixed depending on the specific front we would like to follow. Outside Ω, the fields E and B are supposed to vanish. The problem is understanding what happens at the boundary ∂Ω of Ω. It is not difficult to realize that, on the vertical sides {0}×]0, 1[ and {1}×]0, 1[, curlB and divE become singular, producing concentrated distributions. Similarly, on the two horizontal sides ]0, 1[×{0} and ]0, 1[×{1}, the quantities curlE and divB present singularities. Therefore, we have no way of relying on the Maxwell equations. Some readers may object that discontinuities of the fields may not exist in nature. Commonly, the right way to proceed is to consider a thin layer around ∂Ω, where the solution given by (1.1.5) smoothly decays to zero. The situation is considered in figure 1.2. Let the width of the layer tend towards zero. This in general allows us to determine special relations to be satisfied on ∂Ω (in place of the Maxwell equations, which are meaningless there). Unfortunately, the procedure presents some drawbacks. Let us first assume that the wave-front shifts along the z-axis maintaining its squared shape. We also assume that the fields E and B are orthogonal and smoothly decaying to zero in a neighborhood of ∂Ω. TheR energy of the front is finite and, up to dimensional constant, given by: Ω (|E|2 + c2 |B|2 )dxdy. We claim that there exists at least one point where Maxwell equations are not all satisfied, because divE and divB cannot both be zero at the same time. In fact, examining for instance figure 1.2, we discover that there are infinite points where either divE 6= 0 or divB 6= 0. These points form a set which area is different from zero. We are free to try other configurations by modifying the orientation of the electromagnetic fields. An example is
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Non-Maxwellian wave-front defined on a square Ω. The fields decay to zero as they approach the boundary of Ω. The divergence of each of the two fields is different from zero almost everywhere. The vector field V is orthogonal to the page. Fig. 1.2
provided in figure 1.3, where the vector fields on the front are defined in order to satisfy divB = 0 at any point. Nevertheless, there are plenty of points where the divergence of E is nonzero (in particular we must have divE 6= 0 close to the center and near the boundary of Ω). Whatever the distribution and polarization of the fields, we always reach the same conclusion: somewhere some rule of physics breaks down because of the vanishing boundary conditions on ∂Ω. Now, the question is: if we do not know what the governing rules are in a layer around ∂Ω, how can we go to the limit for the size of the layer tending to zero? To prove what we claimed before, we show using very standard arguments that it is not possible to construct solutions to Maxwell equations, having finite energy and travelling unperturbed at constant speed along
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Another non-Maxwellian wave-front (now Ω is a circle). One of the fields (the rotating one) has zero divergence at any point, but this condition is not fulfilled by the other one. If, instead, discontinuities were allowed, the front of figure 4.2 would be acceptable. Fig. 1.3
parallel straight-lines. We assume that the speed is c and the straight-lines are oriented along the z-axis. Without loss of generality, such a signalpacket is supposed to be of the following type: E = E1 (x, y), E2 (x, y), E3 (x, y) g(t − z/c) B =
B1 (x, y), B2 (x, y), B3 (x, y) g(t − z/c)
(1.1.7)
where g is a bounded function and all the components E1 , E2 , E3 , B1 , B2 , B3 are zero outside a two-dimensional set Ω. It is not difficult to check that (1.1.1) and (1.1.3) only hold when E and B are identically zero. We can see that E3 and B3 must be zero (otherwise g would not be bounded).
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Then, one discovers that E1 , E2 , B1 , B2 must be harmonic functions in Ω. Since they have to vanish at the boundary, they must vanish everywhere. At this point, it is unavoidable to make a distinction between Maxwellian and non-Maxwellian fronts. In the first case, the vector fields E and cB are defined in order to have divE = 0 and divB = 0, at any point of the surface. The divergence can be measured in a system of 3-D coordinates having the first two components (say x and y) lined up with the two electromagnetic fields, and the third one lined up with V. This reference frame is orthogonal. Since E and B are tangent to the surface, they have no component along the third coordinate axis. Therefore, in this setting, the divergence does not contain the derivative with respect to z. This is why we were able to decide if a front was Maxwellian or not, just by examining its displacement on a 2-D surface. We discovered (but the fact is indeed well-known) that, excluding plane fronts of infinite extent (with E and cB constant), it is hard to give examples of plane Maxwellian fronts. In fact, we have just proved that, if Ω is flat and bounded, this is impossible. It turns out that there is no way to construct Maxwellian fronts, which are zero outside Ω, without introducing internal singularities or boundary discontinuities. Nevertheless, we would not like to discard the fronts analyzed up to now, since they will be exactly those giving rise to solitary waves. Such waves shift in the direction of the z-axis without modifying their structure during their evolution. Due to the above-mentioned reasons, solitonic solutions are not described by the classical theory of electromagnetism. Efforts have been made in the past to generalize the Maxwell model in a nonlinear way, in order to include solitons. To give an example, the Born-Infeld theory (see Born and Infeld (1934)) predicts the existence of finite-energy soliton-like solutions (that have been successively called BIons). Research in this field is still very active, and, as in the previously mentioned paper, it is mainly based on the detection of suitable Lagrangian functions. These models have no relation with the one we are going to develop later on (in section 2.4 we will discover that our Lagrangian is exactly the same as classical electromagnetism). However, they demonstrate the necessity of looking for nonlinear versions of the model. A few additional remarks. In many applications, a standard approach is to reconstruct the two-dimensional profile of the fields inside Ω with the help of a truncated Fourier series. This is accomplished by a complete or-
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thogonal set of plane waves, each one carrying a suitable eigenfunction in the variables x and y. We must pay attention, however, to the fact that these eigenfunctions are of a periodic type. Therefore, they reproduce the same profile, not only inside Ω, but in a lattice of infinite contiguous domains. In this way, the global solution in question turns out to have infinite energy. Considering only one of these profiles, thus forcing to zero the solution outside Ω, unavoidably brings us again to a violation of the Maxwell equations near the boundary of Ω. Some clarifying comments on this issue can be found in Goodman (1968), p. 42. We recognize that the techniques based on Fourier expansions provide excellent results in many practical circumstances, as for example the study of diffraction. Nevertheless, it is necessary to adapt the solutions, introducing some approximation, if we want them to correspond to the real phenomenon. Therefore, this is not the perfect mathematical tool for the in-depth analysis we have in mind.
1.2
Wave-front propagation
Concerning the cases treated in the previous section, another possibility is that the wave-fronts, due to the strong variation of the fields near the boundary of the square Ω, are forced to bend a little. The electromagnetic fields are no longer on a plane, so we could probably find a way to enforce all the Maxwell equations. However, this implies that the Poynting vectors are not parallel to the z-axis anymore. Thus, the shape of Ω is going to be further modified during the evolution. A little diffusion is bearable, yet our impression is that the wave-fronts would rapidly change their form. The more they bend, the faster they produce other distortion. This is in contrast, for instance, with the fact that neat electromagnetic signals, of arbitrary transversal shape, reach our instruments after travelling for years between galaxies. The only acceptable rule is that all the Poynting vectors must stay orthogonal to the fronts and parallel to the actual direction of movement; if this does not happen the wave quickly deteriorates, fading completely. The above considerations bring us to the crucial question: what is the correct physical development of a wave-front, and what kind of equations rule this behavior? Recalling that we are in vacuum, a partial answer is given by the following Huygens principle:
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Each point of a wave is the center of a new spherical disturbance (wavelet), expanding with constant velocity c. The actual wave, at a subsequent instant, is the envelope of all these secondary waves. Although the Huygens principle has a difficult mathematical formalization (see for instance Baker and Copson (1939)), the idea is simple and clear. The first two pictures of figure 1.4 show the sections of single wave-fronts at different times of their evolution. Between time instants differing by δt, the front uniformly moves by δx = cδt. The distance is measured along the normal V/c to the surface, i.e., in the direction of the rays. Note that, according to equation (1.1.6), the speed of light is constant everywhere. It is important to observe that the way the front develops is independent of the information it carries. As far as we are concerned, the Huygens principle is a purely geometrical assumption. It is also important to point out that the study of solitary waves carrying a scalar quantity (the reader is referred to Filippov (2000) for a review) is far more simple than the study of those carrying vector fields (like the ones we are studying here). Indeed, the hypothesis that the vectors must be tangent to the wave-front, severely limits its freedom. There are some rules that need fixing. As we mentioned above, the magnitude and displacement of the fields should not affect the velocity of propagation, so that the future shape is only recovered through geometrical properties. The fields can only decide the direction of motion, which is given by the Poynting vector P = E × B. This specification is important, since by the Huygens principle alone we are unable to deduce it. Situations like the one shown in the third picture of figure 1.4, where the Poynting vector changes sign on the same wave-front, are to be avoided. Considering that the speed of the front does not depend on the magnitude of P, the field V turns out to be discontinuous and the front will be forced to break into pieces. We consider this behavior to be inconsistent with physics, unless such an effect is due to external perturbations (not taken into account for the moment). The same example will be reconsidered in the next section. Another reasonable assumption is to suppose that the signal imprinted on a certain front originates as a whole, at an initial time. Once produced, this signal escapes from the source, carried by the front. In the absence of perturbations, no corrections can be made to the signal, with the exception of those imposed by the geometrical evolution of the front itself, such as the change in intensity due to the enlargement of the surface (see the second picture of figure 1.4).
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Cross-sections of evolving single wave-fronts: a plane one and a spherical one. In the second case there is a decay of the energy density. The last picture shows a wave-front moving without respecting the Huygens principle.
Fig. 1.4
Some mild analogy between the Maxwell equations and the eikonal equation governing the movement of the fronts was devised a long time ago. The equivalence is valid within the limits of geometrical optics (see Born and Wolf (1987), p. 110). In spite of this, from examining the geometrical behavior of the fronts of figures 1.2 and 1.3, our impression is that their natural evolution is in conflict with all restrictions imposed by the Maxwell equations. This statement will be clearer as we proceed with our study. To tell the truth, even some aspects of the Huygens principle are not very convincing (for example, how the wavelets behave when approaching the boundary of a front). We return to this subject in section 4.3 (see in particular figure 4.3), when our theory will be more developed. We previously discovered that fronts based on a plane bounded set Ω are always of non-Maxwellian type. Therefore, they cannot be used as initial data for the Maxwell equations. Moreover, we conclude that the Maxwell equations are not the right tool for advancing wave-fronts of this nature. We will see that this is also true for fronts having different shapes, such as spherical ones. From the mathematical viewpoint, the theoretical interest in Maxwell equations is very limited. Due to their linearity and their energy conservation properties, the analysis of existence and uniqueness of solutions is a trivial issue once initial conditions are assigned. Unfortu-
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nately, the real problem is the almost complete lack of compatible initial conditions.
1.3
Fronts from an oscillating dipole
The next step, which is far more delicate, is to examine the case of a spherical wave. The wave could be generated by an oscillating dipole of negligible size. However, the way the wave is produced and supplied is not of interest to us at the moment, since we are more concerned with analyzing the geometrical aspects of its evolution at a distance from the source. Let us denote by P = E × B the Poynting vector. It is customary to assume that E and B are orthogonal, and that the wave-front propagates at constant speed c, through spherical concentric surfaces. One may argue that perfect spherical waves do not exist in nature. Nevertheless, for the sake of simplicity, we maintain this hypothesis, which can be removed later, without modifying the essence of our reasoning. We are confronted with two possibilities. In the first one, the Poynting vector follows exactly the radial direction. This means that E and B locally belong to the plane tangent to the wave-front. In such a circumstance, as detailed below, we are able to show that (1.1.1) and (1.1.3) cannot both be satisfied everywhere. More precisely, it is known that (1.1.1) and (1.1.3) are true up to an error that decays quadratically with the distance from the source. Since the intensity of a spherical electromagnetic wave only decays linearly in amplitude, the above mentioned inaccuracy has no influence on practical applications. However, we thus make our first negative point. The second possibility is that, in order to satisfy the complete set of Maxwell equations, we lose the orthogonality of the Poynting vector with respect to the spherical surface. This is a more unpleasant situation, considering that the Poynting vector represents the direction of propagation of the energy flow. The lack of orthogonality between the tangent plane to the sphere and the direction of propagation violates the Huygens principle (recall what we stated in the previous section), leading to a deformation of the front itself. As we will check later, this results in significant defects in the development of the wave-shape (see figure 2.4). Let us study the problem in more detail, by taking into account the transformation to spherical coordinates: (x, y, z) = (r sin θ cos φ, r sin θ sin φ, r cos θ)
(1.3.1)
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with 0 ≤ φ < 2π, 0 ≤ θ ≤ π and r large enough. We look for vector fields having the following form: B = (0, 0, u)
E = (v, w, 0)
(1.3.2)
where u, v, w are functions of the variables t, r and θ (no dependency on φ is assumed). In (1.3.2), the first component of the vectors is referred to the variable r, the second one to θ, and the third one to φ. The unknowns in the system of Maxwell equations are thus reduced from six to three. Choosing a more general form for the fields only complicates the computations, without adding anything to the substance. We start by observing that equation (1.1.4) is immediately satisfied. Moreover (the subscript denotes partial differentiation): u cos θ u u θ + , − ( + ur ), 0 (1.3.3) curlB = r sin θ r r divE = vr +
w cos θ wθ 2v + + r r sin θ r
(1.3.4)
w vθ curlE = 0, 0, wr + − r r
(1.3.5)
w vθ ut = − wr + + r r
(1.3.6)
c2 cos θ u + uθ r sin θ
(1.3.7)
Therefore, the equations in spherical coordinates become:
vt =
wt = − c2
u
+ ur
(1.3.8) r To avoid discontinuities, we must introduce the following boundary constraints: u(t, r, 0) = u(t, r, π) = 0
w(t, r, 0) = w(t, r, π) = 0
∂v ∂v (t, r, 0) = (t, r, π) = 0 ∂θ ∂θ
(1.3.9) (1.3.10)
In the case of the pure radiation field of an oscillating dipole, when r is sufficiently large, one usually sets v = 0 and w = cu. Within this
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hypothesis, the two equations (1.3.6) and (1.3.8) are equivalent. They bring us to the general solution: c (1.3.11) w(t, r, θ) = c u(t, r, θ) = f (θ) g(t − r/c) r where f (with f (0) = f (π) = 0) and g are arbitrary functions (the only restrictions apply to their regularity). Among these solutions there is the one corresponding to f (θ) = sin θ, which is often present in classical texts (see for instance Bleaney and Bleaney (1965), p. 284), being the one with the greatest physical relevance. In this case, the fields corresponding to a monochromatic wave are: c sin θ E = 0, sin ω(t − r/c), 0 (1.3.12) r B =
0, 0,
sin θ sin ω(t − r/c) r
(1.3.13)
Note that the wave-fronts are perfectly spherical (see figure 1.5) and evolve as the Huygens principle prescribes. They move at speed c independently of the signal carried. Therefore, there should be no distinction between the far-field and the near-field. In fact, since the source is infinitesimal, such an alternative has no reason to exist. Of course, the situation would be different if the oscillating dipole were a physical device of some size. This is a case that we do not want to discuss here. It is worthwhile to recall that the spherical symmetry of the fronts does not allow us to understand the orientation of the dipole. This information is instead contained in the signal transmitted. Nevertheless, we unfortunately note that equation (1.3.7) is compatible with v = 0 only when: c g(t − r/c), 0 (1.3.14) E = 0, r sin θ 1 g(t − r/c) (1.3.15) r sin θ which manifest singularities at the points corresponding to θ = 0 and θ = π. B =
0, 0,
In general, we have the following statement: 1 cos θ 1 f (θ) + f ′ (θ) = 0 ⇔ f (θ) = (1.3.16) divE = 2 g(t − r/c) r sin θ sin θ Note that such a strong singularity at the poles cannot be removed simply by requiring the wave-front not to be perfectly spherical.
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We observe that f can be taken in such a way that divE vanishes at the poles (for example f (θ) = (sin θ)2 ), but not in the proximity of them. In addition, we observe that, if f is regular with f (0) = f (π) = 0, for any fixed r, the points in which the divergence of E does not vanish belong to a two-dimensional set whose measure is different from zero. For instance, if f (θ) = sin θ, we discover that divE is proportional to cos θ, so that this set consists of all points of the sphere of radius r, with the exception of the equator. It is certainly true that even if the divergence is not zero, it is negligible when designing, for instance, a device like an antenna. This argument, however, is not going to be valid here, since we would like to carry out an in-depth analysis of what is really happening in the evolution of an electromagnetic wave, compared to what the Maxwell theory is able to predict. In short, we have proved that spherical fronts also cannot be of the Maxwellian type. This situation is very similar to the one of figure 1.3. If the field B is tangent to the parallels, then divB = 0. However, being E tangent to the meridians, we cannot impose divE = 0, without asking E to be singular at the poles.
Spherical (non-Maxwellian) wave-front. The electromagnetic fields are locally tangent to the surface. The rays develop along the radial direction r. There is no way to construct couples of regular (non-singular) tangential fields, being orthogonal and having divergence equal to zero at all points. Fig. 1.5
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Let us now follow a different path and try to find other solutions, of the form given in (1.3.2), satisfying the set of all Maxwell equations (including divE = 0). If we do not want f to be singular anywhere, we have to accept that v is different from zero. This means that E has a radial component, so that the Poynting vector cannot be perfectly radial. In view of the assumptions in section 1.2, such a situation is not recommended. Hence, we must carefully check what happens in this last case. It is well-known that the Maxwell equations lead to: 1 ∂2B 1 ∂2E = ∆E and = ∆B (1.3.17) c2 ∂t2 c2 ∂t2 The above are usually called “wave equations”, but, shortly, we will see that this name is not appropriate. The terminology is correct only if the fields involved are scalar. By differentiating (1.3.6) with respect to time and using (1.3.7) and (1.3.8), we arrive at the equation: 1 2 1 1 1 (1.3.18) u = (r u ) + (u sin θ) tt r r θ c2 r2 r2 sin θ θ corresponding to the third component of the second equation in (1.3.17) in spherical coordinates. It is worthwhile noting that (1.3.18) is not the wave equation for the scalar field u in spherical coordinates, due to the fact that in this framework the Laplacian of a vector field is not the Laplacian of its coordinates (even if only one of them is different from zero). The wave equation for u reads as follows: 1 1 1 utt = 2 (r2 ur )r + 2 (uθ sin θ)θ = ∆u (1.3.19) 2 c r r sin θ This is not a trivial warning, since many texts in electromagnetism erroneously confuse (1.3.19) with (1.3.18). Implicitly, we made the same mistake before, when looking for d’Alembert-type solutions of the form (1.3.11), generating, for this reason, solutions not compatible with all Maxwell equations. By separation of variables, for any k ≥ 1 and any n ≥ 1, we discover that (1.3.18) admits the following basis of solutions: 1
r− 2 cos(ckt) Jn+ 21 (kr) sin θ Pn′ (cos θ) 1
r− 2 sin(ckt) Jn+ 21 (kr) sin θ Pn′ (cos θ)
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1
r− 2 cos(ckt) Yn+ 21 (kr) sin θ Pn′ (cos θ) 1
r− 2 sin(ckt) Yn+ 21 (kr) sin θ Pn′ (cos θ)
(1.3.20)
where Jn+ 12 and Yn+ 21 are Bessel functions of first and second kind respectively, while Pn is the n-th Legendre polynomial. A classical reference for Bessel functions is Watson (1944). It is important to note that the solutions given in Watson (1944) on page 127, for the scalar wave equation in spherical coordinates, differ from the ones shown in (1.3.20). The reason is that the functions in Watson (1944) (having Pn (cos θ) in place of sin θ Pn′ (cos θ)) are those solving (1.3.19), which is not the vector version of the wave equation, as we have mentioned already. For example, if n = 1 we have (see Watson (1944), p. 54): r 2 sin kr J 32 (kr) = − cos kr πkr kr Y 23 (kr) =
r
2 πkr
cos kr + sin kr kr
P1′ (cos θ) = 1
(1.3.21)
In order to understand how the solutions in (1.3.21) look, it is standard to introduce some approximation. Thus, for n = 1, we first take the combination: 1 u(t, r, θ) = √ sin(ckt)J 32 (kr) + cos(ckt)Y 23 (kr) sin θ (1.3.22) r Then, using (1.3.21), it is possible to get asymptotically the monochromatic solution r−1 sin θ sin k(ct − r) (compare to (1.3.12)), up to an error which decays quadratically with r. Once again, one ends up with something similar to a travelling wave, although some cheating has been necessary (that is equivalent, in the end, to replacing (1.3.19) with (1.3.18) again).
On the other hand, suppose that u is exactly a linear combination of the functions in (1.3.20). One recovers v and w by (1.3.7) and (1.3.8) through time integration. Thereafter, it is possible to compute the Poynting vector: Z Z cos θ u u u (1.3.23) + ur dt, + uθ dt, 0 P = − c2 u r r sin θ
which has, as expected, a non-radial component.
Now, let us fix r and study the behavior, by varying θ, of the two components of P. In particular, we are interested to see what happens near the poles (θ = 0 or θ = π). We start by noting that, for any n ≥ 1,
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A piece of Maxwellian wave-front, as the one supposed to be generated by the oscillations of an infinitesimal electric charge. The condition divE = 0 forces the front to bend when approaching the pole. Fig. 1.6
the term Pn′ (cos θ) tends towards a finite limit for θ → 0 or θ → π (recall that Pn′ (±1) = 12 (±1)n+1 n(n + 1)). Therefore, according to (1.3.20), the first component in (1.3.23) behaves as (sin θ)2 near the poles. It is a matter of using known properties of Legendre polynomials, in particular the differential equation: (sin θ)2 Pn′′ (cos θ) − 2 cos θ Pn′ (cos θ) + n(n + 1)Pn (cos θ) = 0
(1.3.24)
to check that the second component in (1.3.23) behaves as sin θ near the poles. Technically, there is no waste of energy, since the magnitude of the Poynting vectors tends toward zero, when approaching the vertical axis. We are ready to draw some conclusions. Let us note that finally divE = 0, hence all the Maxwell equations are satisfied. As already remarked, it has been necessary to keep the non-radial component of P different from zero. Surprisingly, for any fixed r, such a non-radial component prevails on the radial one, when approaching the poles. This implies that the shape of the wave-fronts does not resemble a sphere.
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18
θ=0 6
yX X Y H
θ=0 X X XH 6Y XH H
H AK
A K A
A
6 -
* θ=π
6 -
θ=0 6
I @
θ=π
@
:
*
6 -
R @
-
θ=π
Qualitative behavior of the field E as a function of the angle θ. Case 1: E = (0, r−1 (sin θ)g(t − r/c), 0), the wavefronts are perfect spheres, but divE 6= 0. Case 2: E = (0, (r sin θ)−1 g(t − r/c), 0) the condition divE = 0 is satisfied, but there are singularities at θ = 0 and θ = π. Case 3: the corresponding Poynting vector is given in (1.3.23), the divergence of the electric field vanishes, but the wave-fronts are far from being spherical surfaces.
Fig. 1.7
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Heuristically, in order not to generate singularities at the poles, a Maxwellian front tries to bend (see for instance figure 1.6). Moreover, the front surface should not have boundaries. In fact, to avoid discontinuities, in proximity of the borders, the fields must tend toward zero. As we noticed earlier, it is hard to combine this property with the fact that the divergence must vanish. Thus, the shape of the wave-front is going to be far more complicated. The different situations analyzed up to now are summarized in figure 1.7 which should clarify our point of view: either we keep the singularity at the poles (manifested by infinite amplitude of the fields or strong geometrical distortion), or we allow the divergence of the electric field to be different from zero. The exact behavior of the solutions depends on the combination of fundamental modes used to build them up. Let us study a specific example. From (1.3.22), using (1.3.7) and (1.3.8), we recover the electric field. Up to a multiplicative constant, we obtain: 2 cos θ sin k(ct − r) E = + k cos k(ct − r) , 2 r r sin θ k cos k(ct − r) 1 2 + 2 − k sin k(ct − r) , 0 r r r
(1.3.25)
that agrees with the celebrated one given by Hertz (see Hertz (1889), Hertz (1962) or Born and Wolf (1987), p. 81). At a given time, the lines of force of this field are shown in figure 1.8 (see also Joos (1986), p. 340). Recalling that B is lined up with the parallels (parametrized by the angle φ), it is easy to realize that the corresponding 3-D wave-fronts are topologically similar to toroidal surfaces. They are contained one inside the other forming various clusters, with the radial width measuring half a wave-length. As time passes, the clusters move away from the origin at the speed of light (see figure 2.4). The upper part of the central core has a shape similar to that shown in figure 1.6. At fixed points, located near the vertical axis (take for example sin θ ≈ 1/r), E rotates clockwise or anti-clockwise depending on the sign of cos θ (positive in the upper hemisphere, negative in the lower one). It has to be noted that, according to the direction of the Poynting vector, the normal to the surface is oriented toward the interior in some regions of the front, and toward the exterior in other regions of the same front. This bad behavior recalls the one displayed in figure 1.4 (last picture). We do not see a chance of recognizing any sort of Huygens principle here. This
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Fig. 1.8
Lines of force of the Hertz solution.
is not what we would call a travelling front. Moreover, there is a more regrettable fact making this phenomenon less clear from the physical point of view. Let us interpret the antenna radiating at r = 0 as an outflow point-wise boundary launching its signals far away in all directions. It is a little puzzling to discover that part of the surface of the front (the one closer to the source) contains a signal belonging to the past, if compared to that carried by the farthest part. This behavior breaks another rule established in section 1.2, stating that a front must be generated at once. The bizarre topology of the “fronts” of the Hertz solution does not vary when r grows. On the other hand, neglecting the terms of order r−2 and r−3 in (1.3.25), we obtain the field given by (1.3.12), corresponding to a completely different topology of the wave-fronts. As we said, these latter fronts are spherical and move in agreement with the Huygens principle. Each front is generated all together at a given time t. There is no mixture of past and future. In this circumstance, the lines of force must be replaced by circumferences.
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However, figure 1.8 is still valid because it qualitatively represents the contour lines of the scalar quantity |E| (recall that the shape of wave-fronts has no relation with the scalar information they carry, although the two concepts are commonly confused). Moving portions of spherical waves (as the one shown in the second picture of figure 1.4) are even more difficult to explain. If, in order to avoid discontinuities, the electromagnetic fields are supposed to decay near the boundaries, we experience the same troubles encountered in section 1.1 concerning flat fronts. Other solutions, related to so-called multi-pole radiation, are obtained by summing up more terms of the type: Jn+ 12 (kr)Pn′ (cos θ) sin θ
and
Yn+ 12 (kr)Pn′ (cos θ) sin θ
for n > 1
For instance, with the aim of satisfying certain boundary constraints, in Jackson (2005), a suitable expansion is found. There, it is claimed that the Poynting vectors are correctly oriented at the far-field, although they continue to show an anomalous behavior in a layer near the z-axis, as well as at the near-field (see also McDonald (2004) and the other related papers of the same author). Although convincing explanations of these results are given, all the weird little facts collected up to this moment lead us to be suspicious of the real potentiality of classical electromagnetism in predicting the evolution of the entities we called fronts.
1.4
Preliminary conclusions
Since all the solutions discussed here can be physically explained with the help of slight adjustments, yet remaining within the so-called limits of the model, we could just stop our analysis here, with the trivial (well-known) conclusion that the Maxwell model is not perfect. However, we believe that the discrepancies we have pointed out are not just imperfections, but symptoms of a more profound pathology affecting the theory of classical electromagnetism. As a matter of fact, although the question of the dislocation of the fields of an oscillating dipole is classical, there is still a lot of debate. Further discussion on the antenna model, starting from the solution of Maxwell equations, leads us into disputes that we prefer to avoid here. Sometimes, physics textbooks mystify (unconsciously or deliberately) the results. Many authors deduce relevant consequences from “solutions”
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that do not satisfy the entire set of Maxwell equations. They start, for instance, by introducing perfect plane waves, but later they damp down their profiles transversally, maybe with the help of some Gaussian functions, in order to produce narrow bundle of rays, without considering that this procedure is inconsistent with the equations. A standard mistake is to compute the field E directly from the first equation in (1.3.17), without taking into account that the solutions to the wave equation do not necessarily satisfy the condition divE = 0 (take for instance E = (e−ny sinnx sin(t−z/c), 0, 0) for some n). In spherical coordinates, solutions similar to (1.3.12), which are not in agreement with the whole set of equations, are also widely used. This way to proceed is routine. Exact solutions are rarely used, and when this happens conclusions are often controversial. So, if “fake” solutions have more physical relevance, would it not be wiser to adapt the model and let them become real solutions? What we learn in these pages is that there are plenty of simple and interesting phenomena which are inadequately explained by the Maxwell model, because the equations impose too many restrictions. Consequently, our aim in the next chapter is to weaken the equations, thereby widening the range of solutions. In this way we can keep the fronts of figure 1.8 (even if we think they are not very meaningful), without disturbing consolidated theory. At the same time, we will be able to include in the new theory every kind of solitary wave, and allow the corresponding fronts to evolve in the right manner. We devote this last paragraph to the rehabilitation of the role of the Maxwell equations. In sections 5.3 and 5.4 they will be absolutely necessary in building the most basic elementary particle: the neutrino. In section 4.3, it will be discovered that, as long as we are in a flat metric space (from the point of view of general relativity), the way the Maxwell equations are usually written is not correct; however, in a suitable geometry (the one generated by the evolving wave as a result of the solution of Einstein’s equation), their appearance is the one we are used to. Although the above sentences are provocative and, at present, without foundation, we shall nevertheless endeavor to make everything clear in due course. Finally, note that the set of equations commonly attributed to J. C. Maxwell is not exactly the same as the one of his original papers (see Maxwell (1861) and Maxwell (1865)), but a revised version due to other scientists, such as W. Gibbs and O. Heaviside (see for instance Heaviside (1893)).
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Chapter 2
First steps towards the new model
2.1
Modified Maxwell equations
The demolition process started in the previous chapter is finished, so now it is time to rebuild. To begin, we propose the following model: ∂E E×B = c2 curlB − c (divE) ∂t |E × B|
(2.1.1)
∂B = − curlE ∂t
(2.1.2)
divB = 0
(2.1.3)
that will be further adjusted in the subsequent chapters. The normp| · | is the usual one in R3 , i.e., for Cartesian coordinates: |(x, y, z)| = x2 + y 2 + z 2 . We also define: J = V/c = P/|P| = (E × B)/|E × B|. Note that, when the Poynting vector P is zero, the direction of J is not determined (see also the comments at the end of section 3.1). The vector J is supposed to be adimensional (or, equivalently, P/|P| is multiplied by a constant, equal to 1, whose dimension is the inverse of the dimension of P). Consequently, cJ = V is a velocity vector (see also (1.1.6)). As the reader may notice, the “awkward” relation divE = 0 has been eliminated. It is also evident that in all the points in which divE = 0, we find again the classical Maxwell system. This states that the solutions of (1.1.1)-(1.1.4) are also solutions of (2.1.1)-(2.1.3). Therefore, the replacement of (1.1.1)-(1.1.2) by (2.1.1) yields the property we wanted, the enlargement of the range of solutions. 23
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We then have to understand and justify what is happening from the point of view of physics. Let us recall that we are in empty space, and that there are no electric charges or masses anywhere. We are only examining the behavior of waves. In spite of this, we assume that there may be regions where divE 6= 0. The situation is not alarming, since we checked that the condition divE 6= 0 is quite frequent in the geometrical study of waves. In any case, such a hypothesis is acceptable, as long as it is coherent with the basic laws of physics. In our theory, singular point-wise stationary charges will not be admitted. In section 5.4, we will learn that charged particles are extremely complicated dynamical structures, occupying a very small (but positive) volume. The nature of these objects is peculiar, at the point that the simple relation divE 6= 0 is not sufficiently accurate to describe them. Thus, it is not a crime to assume that divE 6= 0 in a region of space, without recognizing there the existence of a real solid particle. Even without the presence of matter, the laws we aim to develop here, and in the coming chapters, will be valid for fields as if the fields themselves were primitive sketches of what, later, will become real matter. Indeed, one of the merits of the Maxwell theory was the innovative achievement of translating into the “language” of fields properties that were experimentally evident for physical macroscopic devices. However, we now raise the doubt that the Maxwell model in vacuum is not actually the result of a passage to the limit, from the macroscopic world to a total lack of matter. We claim that a piece is missing, and this is exactly the nonlinear term that has been added to (1.1.1) in order to get (2.1.1). First of all, we observe that the new term in (2.1.1) has a strong analogy with the corresponding one in classical electromagnetism, that should appear on the right-hand side of (1.1.1) as a consequence of the Amp`ere law, and due to the presence of moving charges. In fact, by setting ρ = divE, the vector ρcJ can be assimilated, up to a dimensional multiplicative constant, to an electric current density. Thus, even if in our case there are no real charged particles, we have to deal with a continuous time-varying medium consisting of infinitesimal electrical sources living with the wave during its evolution. For a plane wave, we clearly have ρ = 0, because the area of the wave-fronts is infinite. Moreover, the added term does not compromise the theoretical study of the far-field of a macroscopic device like an antenna, since, at a certain distance, the quantity divE is negligible.
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By taking the divergence of (2.1.1), we get a very important relation: ∂ρ = − c div(ρJ) (2.1.4) ∂t which is, actually, the continuity equation for the density ρ = divE. The equation (2.1.4) testifies to a shift at the speed of light along the direction determined by J. Hence, something is flowing together with the electromagnetic fields; something that later, in sections 3.3 and 3.4, will be compared to a true mechanical fluid. On the other hand, this was also the interpretation at the end of the 19th century, before the theory of fields was rigorously developed. The fluid may change in density, but preserves its quantity, as stated by the continuity equation. It is extremely significant to remark that this property comes directly from (2.1.1), so it is not an additional hypothesis. In section 4.2, starting from the density ρ, we will construct a mass tensor that, due to (2.1.1), can be perfectly combined with the standard electromagnetic energy tensor. The perceptive reader will have already understood that this allows us to find the link between the world of electromagnetism and the one of mechanics. We are now going to collect other properties about the new set of equations. Considering that E × B is orthogonal to both E and B, a classical result is obtained: 1 ∂ (|E|2 + c2 |B|2 ) = c2 curlB · E − curlE · B = − c2 divP (2.1.5) 2 ∂t where the quantity |E|2 + c2 |B|2 , up to a multiplicative dimensional constant, is related to the energy of the electromagnetic field. Thus, the nonlinear term in (2.1.1) is not disturbing at this level, and the Poynting vector P preserves its meaning. By noting that J · J = 1 and that E · J = 0, we get another interesting relation: ∂(E · J) ∂E ∂J = − · J = − c2 (curlB) · J + c divE (2.1.6) E· ∂t ∂t ∂t Finally, one has: ∂2B ∂E = − curl = − c2 curl(curlB) + c curl(ρJ) ∂t2 ∂t = − c2 ∇(divB) + c2 ∆B + c curl(ρJ)
(2.1.7)
from which we deduce the following second-order vector equation with a nonlinear forcing term: ∂2B = c2 ∆B + c curl(ρJ) ∂t2
(2.1.8)
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that generalizes the second equation in (1.3.17). Thus, the “wave” equations for the fields E and B no longer hold true. On the other hand, it has emerged in section 1.3 that, in vector form, they are only a source of a lot of trouble. The nonlinear evolutive equation (2.1.8) opens the road to the physics of quantum phenomena (see section 6.1). According to the results of sections 5.3 and 5.4, solutions localized in a region of space are associated with eigenfunctions of (2.1.8), and the imposition of suitable boundary conditions produces quantized eigenvalues. From our point of view, energy quantization is not a direct consequence of the differential model, but a prerogative of some special solutions under peculiar circumstances. In the classical Maxwell equations the role of the field E can be interchanged with that of field cB. This is not true for the new formulation. We will later see, in section 3.3, how to solve this problem. For the moment, we keep working with (2.1.1)-(2.1.3), just because the theory will be more easy. In the coming sections 2.2 and 2.3, we will see how elegantly it is possible to solve the problems raised in chapter 1.
2.2
Perfect spherical waves
In the case of a plane wave of infinite extension, for both the Maxwell model and the new one we are able to enforce the condition divE = 0 and realize the orthogonality of the Poynting vectors with respect to the propagation fronts. Concerning a “fragment” of plane wave, the classical method runs into problems. However, with the new approach the situation radically improves. Let us see why. With the same assumptions as in section 1.1, let Ω be the square [0, 1] × [0, 1]. We have already noted that, on the vertical sides {0}×]0, 1[ and {1}×]0, 1[, the quantities divE and curlB become infinite. Nevertheless, when substituted into equation (2.1.1), they lead to a difference of the type +∞ − ∞. The two singular terms reciprocally cancel out, leaving a finite quantity, so that the equation has a chance of being satisfied. To show this, we can create a layer around the boundary of Ω. Then, without developing singularities, we go to the limit for the width of the layer tending to zero. The trick now works, because, in contrast to the classical Maxwell case, equation (2.1.1) can be satisfied exactly at all points, since it is compatible with the condition divE 6= 0. In the limit process we can also guarantee
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that the Poynting vectors remain parallel to the z-axis. Therefore, the fragment does not change its transversal shape. Explicit computations will be carried out in section 2.3, in the case in which Ω is a circle (see figure 1.3, having divB = 0). For example, the situation represented in figure 1.2 is perfectly compatible with our equations, except near the lower and upper sides. Actually, on the sides ]0, 1[×{0} and ]0, 1[×{1}, given that divB and curlE are singular, we still have problems (clearly equations (2.1.2) and (2.1.3) are not true). Similar problems are encountered by modifying the polarization of the wave. These questions will be solved in section 3.3 by unifying (2.1.2) and (2.1.3) in a single equation similar to (2.1.1), in such a way that the roles of E and cB are interchangeable. On the other hand, in section 4.4, we discover that configurations like the one in figure 1.2, although not forbidden, are not encountered in nature, where condition divB = 0 is usually taken for granted. The case of a spherical wave is very interesting. Let us consider the transformation of coordinates given in (1.3.1). Let us also suppose that the fields are given as in (1.3.2), with u, v, w not depending on φ. We have: E × B = (uw, − uv, 0) J =
E×B s(u) = √ w, − v, 0 2 2 |E × B| v +w
(2.2.1)
(2.2.2)
where s(u) = u/|u| is the sign of u.
The new equations in spherical coordinates become: vθ w + ut = − wr + r r
(2.2.3)
2 1 cos θ vr + v + wθ + w c2 cos θ r r r sin θ w (2.2.4) √ u + uθ − c s(u) vt = r sin θ v 2 + w2 wt = − c2
u r
+ ur
1 cos θ 2 w vr + v + wθ + r √r r sin θ v + c s(u) v 2 + w2
(2.2.5)
To avoid discontinuities, we also impose the boundary conditions (1.3.9) and (1.3.10). Now, by choosing v = 0 and w = cu, one obtains:
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cu cos θ c E×B divE = + uθ , 0, 0 |E × B| r sin θ r
(2.2.6)
w ut = − wr + r
(2.2.7)
vt = 0
(2.2.8)
Therefore, one gets:
wt = − c2 ur +
u r
(2.2.9)
Once again, the first and the last equations are equivalent, providing the general solution (1.3.11). In any case, this time, thanks to the nonlinear corrective term of (2.1.1), the second equation is compatible with v = 0. We are not obliged to choose f (θ) = 1/ sin θ, in order to enforce the condition divE = 0, because this constraint is no longer required. The conclusion is that perfect spherical wave-fronts are admissible with the new model. The functions f and g may be truly arbitrary (the only restriction being f (0) = f (π) = 0). The fields (1.3.12) and (1.3.13) are now acceptable solutions and a plot of |E|, for a fixed φ, is similar to that of figure 1.8, although, in this case, the lines of force are distributed on spherical surfaces. Later, we will construct infinitely many other solutions which are unobtainable with the classical Maxwell model. In the perfect spherical case, the Poynting vector P = (cu2 , 0, 0) only has the radial component different from zero. As expected, this component has constant sign (even if u and w oscillate). Since the set of equations is of a hyperbolic type, we can introduce the characteristic curves. In the example of the spherical wave, such curves are semi-straight lines emanating from the point r = 0, and the vector J = P/|P| = (1, 0, 0) is aligned with them. In other words, the light rays, having V as tangent vector at any point, are straight. It is worthwhile to note that such a property is not true in the standard Maxwell model. Actually, we already pointed out that the solutions, obtained as a combination of the functions in (1.3.20) (from which we get in particular the Hertz solution (1.3.25)), bring us to a very complicated distribution of the vector field P. In this case, it is very hard (probably impossible in an explicit way) to reconstruct the path of the light rays (see also figure 2.4). The nonlinear term introduced in (2.1.1) does not adversely affect the behavior of the wave, because, with v = 0 and w = cu, the corresponding
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equations (2.2.7) and (2.2.9) are linear. Therefore, the superposition principle is still valid. Any piece of information, present at the boundary of the sphere of radius r > 0, propagates radially at the speed of light, without being disturbed (except by the natural decay in intensity). The nonlinear effects of the model are latent. They show up when we try to force, with some external solicitation, the Poynting vector not to follow the characteristic lines. This circumstance will be taken into account in sections 3.1 and 3.2. In a very mild form, we can state that the divergence vanishes, by observing that, for any T : Z T +2π/ω divE dt = 0 (2.2.10) T
that is divE has zero average when integrated over a period of time. Nevertheless, in section 4.3, we will get an astonishing result. We will see that an electromagnetic wave produces, during its passage, a modification of the space-time. In the new geometry, the 4-divergence of the electric field is zero. This could make it difficult, or even impossible, to set up an experiment that emphasizes the condition divE 6= 0 at some point. The measure could be affected by the modified space-time metric in such a way that the condition cannot be revealed.
Among the stationary solutions we find: K2 K3 K1 v(t, r, θ) = 2 w(t, r, θ) = (2.2.11) u(t, r, θ) = r sin θ r r sin θ as well as: K1 v(t, r, θ) = K2 cos θ w(t, r, θ) = −K2 sin θ (2.2.12) u(t, r, θ) = r sin θ with K1 , K2 , K3 arbitrary constants. In particular, we recognize the classical stationary electric field: E = K2 r−2 , 0, 0
whose divergence is zero for any r > 0. Unfortunately, most of these solutions show singularities. As we said in the previous sections, stationary solutions will not be part of our theory. Therefore, from now on, they will be completely abandoned. Due to the nonlinearity of the equations, the study of the interference of waves looks quite complicated. As long as the waves are such that divB = 0 and divE = 0 (as in the plane case) there are no problems, since the nonlinear terms do not affect the outcome. For waves of different shape,
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the situation may be truly intricate. Although we do not wish to discuss it here, the subject is of crucial relevance and deserves to be studied in more detail. To a first approximation, the nonlinear effects should attenuate faster than the amplitude of the waves. Thus, at a certain distance, these anomalies would not normally be observed. Indeed, concerning several specific situations, we expect the nonlinearities to be negligible. In a similar fashion, at certain regimes, acoustic waves described by the compressible Euler’s equation sum up linearly, even if originally the model is strongly nonlinear (see for instance Landau and Lifshitz (1987), chapter VIII). In chapter 3, we will discover that the motion of an electromagnetic wave is actually ruled by the equations of fluid dynamics.
2.3
Travelling signal-packets
For the sake of convenience, in this section we express our new set of equations in cylindrical coordinates. After taking (x, y, z) = (r cos φ, r sin φ, z), we assume that the fields are of the form B = (0, 0, u), E = (v, w, 0), where the first component is referred to the variable r, the second to the variable z and the third to the variable φ. Moreover, for simplicity, the functions u, v and w will not depend on φ. In cylindrical coordinates, the counterparts of equations (2.2.3)-(2.2.5) are: ut = − wr + vz vt = c2 uz
wt = − c2
u r
v vr + + wz w − c s(u) √ r v 2 + w2
+ ur
v vr + + wz + c s(u) √ r v v 2 + w2
(2.3.1)
(2.3.2)
(2.3.3)
By setting v = 0, we can easily find solutions when u and w do not depend on z. In this case one has divB = 0 and divE = 0. From (2.3.1) and (2.3.3) it is easy to get the equations: w u and wtt = c2 (2.3.4) + ur + wr utt = c2 r r r r whose solutions are related to Bessel functions. Nevertheless, of particular interest are the following solutions in cylindrical coordinates:
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u(t, r, z) = g(t ± z/c)f (r)
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v(t, r, z) = ±c g(t ± z/c)f (r)
w(t, r, z) = 0
(2.3.5)
Note that the divergence of E is equal to ρ = vr + v/r, so that it is different from zero, unless f is proportional to 1/r. The functions f and g can be arbitrary (we establish that f (0) = 0 only to guarantee the continuity of the vector fields). The relations in (2.3.5) give rise to electromagnetic waves shifting at the speed of light along the z-axis. If f and g vanish outside a finite measure interval, for any fixed time t the wave is constrained inside a bounded cylinder. The packet travels unperturbed for an indefinite amount of time. The corresponding field E is perfectly radial and the vector J is parallel to the z-axis (see figure 2.1). Note that u and v are solutions of the scalar wave equations utt = c2 uzz and vtt = c2 vzz . Given r0 > 0, suppose that f is continuous and zero for r > r0 . Suppose also that f in a neighborhood of r0 has a sharp gradient. It is evident that the vector c2 curlB − c(divE)J = (c2 uz , 0, 0) = (±cg ′ f, 0, 0) remains bounded even if we let the derivative of f tend to ∞ at r0 . Therefore, as we anticipated at the beginning of section 2.2, we can give a meaning to equation (2.1.1), even if f is a discontinuous function in r0 . Further remarks on this issue will follow in section 4.3. We can get a transport equation for the unknown ρ = divE by using the equation (2.1.4), i.e.: ∂ρ = − c div(ρJ) = − cρ divJ − c∇ρ · J (2.3.6) ∂t which, thanks to the fact that J is a constant field, takes the simplified form of: ∂ρ ∂ρ = ±c (2.3.7) ∂t ∂z with the sign depending on the orientation of J. If we now follow the evolution of the front of figure 1.3, where divB = 0, we discover that it moves in the direction of V exactly as the Huygens principle prescribes (see also the first picture of figure 1.4). We recall that Maxwell’s equations do not allow for the existence of solitary waves like the ones we have just introduced. This is a very important result. According to the caption of figure 2.1, a solid macroscopic wire is now replaced by a sort of impalpable version, where only fields are present. As stated in section 2.1, the essence of the phenomenon is preserved, even if matter is absent.
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Displacement of the fields for a solitary wave satisfying divB = 0 (see also figure 1.3). Although there is no matter involved, the setting recalls that of a wire of infinite length, inside which a current of density ρ is flowing at the speed of light. There is no component of E along the z-axis (perfect conductivity), and B circulates around the axis as it would in the case of a real wire. Similarly, the sign of curlB depends on the orientation of the current flow (sign of V) and on the sign of the density ρ (directly deducible from the sign of E). Fig. 2.1
The energy E of these solitary waves is obtained by integrating the energy density, given by: 12 ǫ0 (|E|2 + c2 |B|2 ). Thus, one gets: Z +∞ Z +∞ g 2 (ξ)dξ (2.3.8) f 2 (r)rdr E = 2πǫ0 c2 0
−∞
Suppose that, at an initial time t0 , the electromagnetic fields are assigned compatibly with (2.3.5). The vector J is automatically determined and the wave is forced to move in the direction of J at speed c. In fact, each wave-front, having the intensity of its fields modulated by the function g, evolves independently, without interfering with those nearby. There are no stationary solutions, unless g is constant. However, in this last case, due to (2.3.8), the energy is not going to be finite. Wave-packets take their energy far away, without dissipation, until they react with other waves or more complicated structures (such as, for instance, particles).
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Let us study more closely the expressions given in (2.3.5). Assume that, for r ≥ 0, the function f is non-negative, and that f (0) = 0. If the function g has a constant sign, we distinguish between two cases, according to whether the sign is positive or negative (see figure 2.2). The sign determines the “orientation” of the vector curlB (note, however, that curlB is not exactly parallel to the z-axis, despite what is shown in figure 2.2). There are also subcases, depending on whether E is directed toward the z-axis or not. In conclusion, two possible types of solitary waves can occur, depending on the orientation of the electric field (external or internal). These will be denoted by γ + and γ − , respectively. In figure 2.2, J indicates the direction of motion. Of course, g could also have a non-constant sign. In this case, the corresponding wave can be seen as a sequence of waves of type γ + and γ − , shifting one after the other. In vacuum, electromagnetic fields at rest are assumed to be identically zero. During the passage of a soliton, the calm is momentarily broken only at the points “touched” by the wave. The information shifts, but does not irradiate. The fact that the wave-packet displays a negative or positive sign, depending on the orientation of E, has no bearing on what is usually called electric charge. Hence, as long as the cylinders containing two different solitons do not collide, they can pass very close to each other without influencing each other. On the contrary, we can expect some scattering phenomena by means of a mechanism that will be introduced in section 3.2. If we place a mirror parallel to the z-axis, at some distance from it, the reflected image of the travelling wave-packet will be the same wavepacket shifting in the opposite direction (because curlB changes sign, while E maintains its orientation), unless we also modify the sign of the vector product × or that of the reference frame (see also section 3.3). The same is true for the Maxwell equations. In both cases, we have no elements to decide the correct sign of the vector product × (left-hand or right-hand). As a matter of fact, without modifying the equations, a change of the sign of × can be compensated for by a change of the sign of the electric (or the magnetic) field. To learn more about this problem we need to study the interactions between waves and matter. Hence, for the moment, we have insufficient information to determine which part of the mirror our universe is from. An answer to these crucial questions will be given in section 5.2. Particles and anti-particles will be studied in section 5.4. In cylindrical coordinates, we can find many other solitonic solutions.
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g>0
curlB
g>0 J ????????? z 666666666
g<0
curlB
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6 66
curlB
6 ??
? ?
6
?
6 ?
66
6 ? ??
J
z
g<0 J ????????? z 666666666
6 66
curlB
6 ??
? ?
6
?
6 ?
J 66 6 ? z ? ?
Fig. 2.2 Behavior of the electric field for the two possible wave-packets (section for a fixed angle φ): γ − shifting from left to right, γ + shifting from right to left, γ − shifting from right to left, γ + shifting from left to right.
Here is another example: E = (±cu, 0, ∓ cv)
B = (v, 0, u)
(2.3.9)
with u = f1 (r, φ)g(t ± z/c) and v = f2 (r, φ)g(t ± z/c). In order to fulfill the condition divB = 0, it is necessary to impose: ∂f1 ∂(rf2 ) + = 0 ∂φ ∂r
(2.3.10)
Note that E · B = 0 and J = (0, ∓1, 0). Except for the condition (2.3.10), the functions f1 and f2 are arbitrary, so that the new solutions are very general. Actually, they include the previous ones (take f1 = f and f2 = 0). In section 3.3, we will remove the condition divB = 0, thus allowing for the existence of even more solutions. We can force f1 and f2 to vanish outside
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a two-dimensional domain Ω. Again, this train of flat fronts, modulated by the function g, travels along the z-axis at the speed of light. We may reasonably expect these solitary solutions to be modified or even destroyed when they encounter another external electromagnetic field. In fact, the equations being nonlinear, the superposition principle does not hold, in particular if the motion is disturbed in a way that is in contrast to the natural evolution along the characteristic curves (see also the comments at the end of section 2.2). We are not aware of documented experiments evidencing these facts. In section 4.1, we examine the behavior of solitary waves under the action of gravitational fields. Concerning the space variables, with a process corresponding to the socalled “partition of unity”, the signal on a wave-front can be decomposed into a sum of a finite number of terms having compact support. Each piece of front obtained in this way advances independently. Nevertheless, since the development is only governed by geometrical rules, the various pieces seem to be glued together, and march as a single entity (see figure 2.3). Therefore, electromagnetic radiation can be suitably considered as a cluster of solitary waves, travelling in the same direction. If, transversally, these solitons are of infinitesimal size, they can be compared to “light rays”. By giving a satisfactory answer to old questions, this observation clarifies how a wave can be viewed at the same time entirely as an electromagnetic phenomenon and simultaneously as a bundle of infinite microscopic rays. We hope that this brief investigation into the secrets of light proves, to use an appropriate term, “illuminating”. A better formalization of these concepts will be given in section 4.3. As for photons, they too are pure electromagnetic manifestations, yet, unfortunately, they are not modelled by the Maxwell equations. Modern atomic and subatomic physics would not exist without photons, yet they find no place in the classical theory of electromagnetism. This is a serious shortcoming. Although physicists are acquainted with this dualism, the general framework remains blurred (see for instance de la Pe˜ na and Cetto (1996), p. 56). From our new standpoint, we contend that the photons observed in nature have every chance of being successfully modelled by the equations introduced here. As a matter of fact, we have enough freedom to build solutions (no matter how complicated) resembling real photons. We can assign a “frequency”, by modulating them longitudinally or transversally. Then, we know that they must move at the speed of light and can have finite energy, given by the energy of their electromagnetic vector fields.
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Fig. 2.3 A front evolves according to geometrical rules. When the front is broken, each of the two disjoint pieces proceeds independently, albeit, along the same path as the whole front.
They can be “positive” or “negative” without being electrical charges. Even with no mass at rest, their motion can be distorted by gravitational fields (see section 4.1). If what we are proposing here is a new functioning model for electromagnetism (and we will collect other evidence supporting this hypothesis), then it explains why photons can be self-contained elementary entities and electromagnetic emissions at the same time. In this case, although our equations do not contain (for the moment) any visible source of “quantization”, a first link between classical and quantum physics is set forth. In section 2.5, we examine the impact of our new set of equations in the framework of Quantum Electro-Dynamics. The invariance of (2.1.1) with respect to Lorentz transformations will be shown in section 2.6. In order to understand the role of photons in our model of the universe, the reader must wait until section 6.1.
2.4
Lagrangian formulation
In order to recover the equations (2.1.1)-(2.1.3) from the principle of least action, we follow the path that leads to the classical Maxwell equations. Thus, we introduce the scalar potential Φ and the vector potential A =
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(A1 , A2 , A3 ), such that: B =
1 curlA c
E = −
1 ∂A − ∇Φ c ∂t
(2.4.1)
From the above definitions we easily get the equations: divB = 0 and ∂ ∂t B = −curlE. The third equation (2.1.1) is going to be deduced from the differentiation of a suitable action function. The potentials are not unique, but they are usually related through the so-called gauge condition. Let us first note that, by taking the potential Φ equal to zero (corresponding to a suitable choice of gauge) and setting ξ = t − (xJ1 + yJ2 + zJ3 )/c, one obtains: E = −
1 ∂A c ∂ξ
cB = ∇ξ ×
∂A 1 ∂A = −J× ∂ξ c ∂ξ
(2.4.2)
provided J is constant. This allows us to infer that B is orthogonal to E and that |E| = |cB| (see also Landau and Lifshitz (1962), p. 126). Otherwise, if J is not constant, the last two properties are not guaranteed. Subsequently, for i and k between 0 and 3, we introduce the electromagnetic tensor: Fik =
∂Ai ∂Ak − ∂xi ∂xk
(2.4.3)
where A0 = Φ and (x0 , x1 , x2 , x3 ) = (ct, −x, −y, −z). Explicitly put, we have: 0 −E1 −E2 −E3 E1 0 −cB3 cB2 (2.4.4) Fik = E2 cB3 0 −cB1 E3 −cB2 cB1 0
with E = (E1 , E2 , E3 ) and B = (B1 , B2 , B3 ). Replacing E by −E, one gets instead the contravariant tensor F ik : 0 E1 E2 E3 −E1 0 −cB3 cB2 (2.4.5) F ik = −E2 cB3 0 −cB1 −E3 −cB2 cB1 0
Therefore, up to multiplicative constants, the action turns out to be (see for instance Jackson (1975), p. 596): Z Z S = − Fik F ik dx0 dx1 dx2 dx3 = c Fik F ik dxdydzdt (2.4.6)
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where, summing-up over repeated indices, the Lagrangian is L = Fik F ik = 2(c2 |B|2 − |E|2 ). As usual, the variations are functions δAi having compact support both in space and time (between two fixed instants). With the help of well-known results, one obtains: Z ∂F ik δAi dx0 dx1 dx2 dx3 (2.4.7) δS = − 4 ∂xk Imposing δS = 0, we get exactly the standard Maxwell equations. As a matter of fact, due to the arbitrariness of the variations δAi , one recovers: − ∂x∂ k F ik = 0 (for i = 0, 1, 2, 3), which is equivalent to divE = 0 and ∂ 2 ∂t E = c curlB, where the operators curl and div are associated with the classical coordinates (x, y, z). Let us now introduce a novelty. If we require that the variations δAi are subjected to a certain constraint, the conclusions are different. Thus, we set the following condition: δΦ − J · δA = δA0 − J1 δA1 − J2 δA2 − J3 δA3 = 0
(2.4.8)
where J is the vector (E × B)/|E × B|, as defined in section 2.1. The relation (2.4.8) says, for instance, that, if the vector variation (δA1 , δA2 , δA3 ) locally belongs to the tangent plane generated by E and B, then the variation δA0 is zero. Although for the moment we can only provide vague explanations, we assert that (2.4.8) is the germ of the Huygens principle. The picture will become more focused as we proceed with our analysis. Consequently, we discover that the 4-vector − ∂x∂ k F ik is not identically vanishing, but, due to (2.4.8), it must have a component along the 4-vector (1, −J). This leads to: divE = λ
(2.4.9)
1 ∂E − c curlB = − λJ c ∂t
(2.4.10)
where the parameter λ is a Lagrange multiplier. By eliminating λ we easily arrive at equation (2.1.1). Thanks to (2.4.8), the set of possible variations is smaller than the set in which we impose no restrictions at all. Therefore, the equation δS = 0 is now less stringent. As we already know, this shows that (2.1.1) admits a space of solutions which is larger than the one corresponding to (1.1.1) and (1.1.2) together. Using the electromagnetic tensor, equation (2.1.1) can be written as: ik ∂F 0k ∂F for i = 1, 2, 3 (2.4.11) + Ji = 0 −c ∂xk ∂xk
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By defining J0 = −1, the above relation is also trivially satisfied for i = 0. In section 4.1, within the framework of general relativity, we will be able to write (2.4.11) in invariant tensor form. Let us try to understand the reason for the constraint (2.4.8). As far as the evolution of a plane or a spherical wave is concerned (and, surely, in more general cases), it is easy to check that: A = ΦJ
(2.4.12)
A straightforward way to get (2.4.12) is from explicit solutions. For example, in spherical coordinates, we can use (1.3.11) in order to find: 1 1 B = 0, 0, f (θ)g(t−r/c) J = (1, 0, 0) E = 0, f (θ)g(t−r/c), 0 r cr A = − F (θ)g(t − r/c), 0, 0 Φ = −F (θ)g(t − r/c) (2.4.13)
where F is such that F ′ = f . The relation (2.4.12) is also true in the case of solitons. In fact, in cylindrical coordinates, thanks to (2.3.5) one has: E = ±cf (r)g(t±z/c), 0, 0 B = 0, 0, f (r)g(t±z/c) J = (0, ∓1, 0) A = 0, cF (r)g(t ± z/c), 0
Φ = ∓cF (r)g(t ± z/c)
(2.4.14)
′
where F is such that F = f and f (0) = 0. Note that, in general, Φ and A are not uniquely determined. However, there exists at least one choice of Φ and A such that (2.4.12) is satisfied. Now, the equation A = ΦJ implies |Φ| = |A|, or equivalently: Φ2 − |A|2 = 0. Taking the variation of the last relation brings us to the constraint: 2 Φ δΦ − A · δA = 2Φ δΦ − J · δA = 0 (2.4.15) which is the same as in (2.4.8). Another way to get (2.4.8) is by directly evaluating the variation of (2.4.12): δA = δ(ΦJ) = δΦ J + Φ δJ ⇒ J · δA = δΦ
(2.4.16)
where we considered that |J| = 1 and δJ · J = 0 (a normalized vector is orthogonal to its variation). Obviously, the vector relation A = ΦJ implies the scalar relation: Φ = J·A
(2.4.17)
obtainable after scalar multiplication by J and by observing that |J| = 1. Another way of expressing (2.4.17) is to require that the scalar product between the 4-vectors (Φ, A) and (1, −J) is zero.
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In conclusion, the equation (2.1.1) can be recovered from the constrained variation of the action function associated with the classical Lagrangian. A similar approach was taken by Turakulov and Muminov (2006). In our case, the constraint originates from (2.4.12) which says, in particular, that A is lined up with J. As will be better explained in section 2.5, such a condition characterizes the propagation of waves, whose evolution is ruled
The field J (normalized Poynting vectors) at different stages of evolution, in the case of the Hertz solution (1.3.25). We recall that the space of solutions of Maxwell’s equations is very rare, and the above is one of the few representatives. The situation is different for a perfect spherical wave, where J is a radial and stationary field.
Fig. 2.4
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by the Huygens principle. On the other hand, this is not valid in the case of the Hertz solution (1.3.25). In fact, the corresponding potential A is: cos k(ct − r) cos θ, − sin θ, 0 (2.4.18) A = r which is lined up with the vertical axis, while J follows the errant behavior of the Poynting vector (see figure 2.4). Note also that |E| = 6 |cB|.
2.5
Free-waves and the eikonal equation
From now on, the waves satisfying (2.4.12) will be called “free-waves”. In sections 3.1 and 3.2, we will see that not all waves are of this type. An interesting relation, that is directly obtained by checking (2.4.13) or (2.4.14), is the following one (see figure 2.5): E + c J×B = 0
(2.5.1)
The above equation is extremely important, since it represents another characterization of free-waves, perhaps simpler than (2.4.12). Indeed, it is analogous to the Lorentz law for moving electric charges (recall that cJ is a velocity), even if here there are no particles. As will be explained in the coming sections, equation (2.5.1) tells us that the mechanical forces acting on a free-wave are zero. Therefore, the wave actually moves without external disturbances and in agreement with the Huygens principle. We will prove all these statements in section 3.4. The fact that (2.4.12) implies (2.5.1) can be proven with the same argument as in (3.4.14). An equivalent version of (2.5.1) is: − cB + J × E = 0
(2.5.2)
obtained by vector multiplication of (2.5.1) by J. Both (2.5.1) and (2.5.2) can be trivially deduced from the orthogonality of E with respect to B, and by the equality |E| = |cB| (see also the beginning of section 2.4). Therefore, they have quite a general validity. Solving the entire set of Maxwell equations may not lead to solutions satisfying (2.5.1). This is the case for example with the Hertz solution. By comparing the norm of E given in (1.3.25) with the one of the corresponding cB (obtainable from u in (1.3.22)), we get a discrepancy. Note that we cannot use the relation (2.4.2), since now J is not constant. Thus, in our opinion, the classical dipole radiation, resulting from the imposition of all
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Perfect balance between the vector E and V × B, where V = cJ. Such a geometrical property characterizes the so-called free-waves. When |E| 6= |cB|, this configuration will not be valid and, according to the results of chapter 3, V is subjected to transverse acceleration.
Fig. 2.5
the Maxwell constraints, does not generate free wave-fronts. We came to the same conclusion at the end of the previous section. In truth, we were skeptical about these solutions right from the beginning. Clearly, for V = cJ, (2.5.1) can be written in tensor form in the following way (see (2.4.5)): 0 −E1 −E2 −E3
E1 0 cB3 −cB2
E2 −cB3 0 cB1
c E3 V1 cB2 = 0 −cB1 V2 V3 0
(2.5.3)
which also includes the relation E · V = 0. The above expression opens the path to the analysis of our equations in the framework of the theory of relativity, starting in the next section. It will be better not to use the expression V = c(E × B)/|E × B|, since it cannot be translated in covariant form. We must interpret V as an independent field, implicitly defined through relation (2.5.1) (or (2.5.3)).
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We collect a few more properties. We begin by considering the following expression: 1 ∂A 2 2 2 − ∇Φ − c B · curlA |E| − c |B| = E · − c ∂t = −
1 ∂ 1 ∂E (E · A) + · A − E · ∇Φ c ∂t c ∂t
− c A · curlB − c div(A × B) 1 ∂ 1 ∂E 2 = − (E · A) + − c curlB · A c ∂t c ∂t − div(ΦE) + Φ divE − c div(A × B)
(2.5.4)
where we used the definitions in (2.4.1) and the known formulas of vector calculus. Introducing the constraint A = ΦJ (hence also Φ = J · A) we have that A is orthogonal to E, because J is also. Thus, (2.5.4) can be simplified: |E|2 − c2 |B|2 = − div Φ(E + c J × B) 1 + c
∂E 2 − c curlB + c(divE)J · A ∂t
(2.5.5)
Interestingly, when both (2.5.1) and (2.1.1) are satisfied, the Lagrangian vanishes. Note that, under the same assumptions, also the differential of the Lagrangian is zero, hence we have a saddle point. On the contrary, when E is orthogonal to B and |E| = |cB| (so that (2.5.1) holds), the relation (2.5.5) reveals that imposing equation (2.1.1) is a natural requisite. Finally, due to (2.4.1), equation (2.1.1) entails: ∂Φ ∂A 1 1 ∂2A − ∇ = c curl(curlA) + c div + ∆Φ J − c ∂t2 ∂t c ∂t ∂(divA) = −c ∆A + c∇(divA) + + c ∆Φ J (2.5.6) ∂t We also assume the following Lorenz condition: 1 ∂Φ = 0 (2.5.7) divA + c ∂t which is known to be an invariant expression in general relativity. With the help of (2.5.7), from (2.5.6) we get: 2 ∂2A ∂ Φ 2 2 − c ∆A = − c ∆Φ J (2.5.8) ∂t2 ∂t2
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which is in perfect agreement with the continuity equation (2.1.4), once we ∂2 define ρ = c−2 ∂t 2 Φ−∆Φ. Note, however, that (2.1.4) is true independently of (2.5.7) and (2.5.8). In the classical Maxwell case, the right and the left terms of (2.5.8) both vanish, providing, together with (1.3.17), additional “wave” equations. This is not necessarily true in our case. Finally, we observe that condition (2.5.7), combined with (2.4.12), provides the following continuity equation for the scalar potential Φ: ∂Φ = − c div(ΦJ) (2.5.9) ∂t Putting together the results we have just obtained, we end up with the system: 1 ∂2A − ∆A = ρJ c2 ∂t2 (2.5.10) 2 1 ∂ Φ − ∆Φ = ρ c2 ∂t2
which is perfectly compatible with the basic assumptions of Quantum Electro-Dynamics (see for instance Feynman (1962), p. 27). It is enough to combine (2.5.10) with some model equations relative to energy quantization (such as Klein-Gordon, Pauli or Dirac equations), in order to recover well-known results. Paradoxically, in the past, such results were correctly anticipated without having an exact notion of photon, which was roughly deduced after suitable normalization of standard plane waves. Our theory instead provides a formal description of solitary waves with compact support. Therefore, it fits the context of QED better, without conflicting with the theory itself. We examine possible connections with Dirac’s equation in section 5.5. For further information regarding our interpretation of quantum phenomena, see section 6.1. We end this section with another important observation. Let us suppose that V is a gradient, i.e.: V = ∇Ψ, where Ψ is a scalar potential not depending on time. Therefore, we have an irrotational vector field. This can be verified, for instance, in the case of plane or spherical solitons, but it can also be checked for free-waves in general (we will be more precise in section 3.4). Thanks to (1.1.6), we trivially have: |∇Ψ| = c
(2.5.11)
which is exactly the eikonal equation. This analytic property, obtained without approximation, goes beyond the famous limits of geometrical optics. This fact emphasizes the strategic presence of the field V in equation
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(2.1.1). According to (1.1.6) and (2.5.11), V evolves in agreement with the Huygens principle and, through (2.1.1) and (2.1.2), contributes to generating the two fields E and B. Correspondingly, the wave-fronts, obtained as the envelope of tangent electromagnetic fields, precisely move as prescribed by V, independently of the actual information they carry. The preservation of the energy, distributed on the global surface of each wave-front, is guaranteed by the Poynting theorem (see (2.1.5)) and by the continuity equation (2.1.4), both deducible from (2.1.1)-(2.1.3). As we said in section 1.2, this is the right way of propagating wave-fronts. We will generalize these statements in the next chapters. A preliminary version of the set of equations (2.1.1)-(2.1.3), together with (2.5.1), appeared in Levi-Civita (1907). The paper deals with “movements of pure electricity” by assuming a continuum flux of charges. Therefore, the hypothesis ρ = divE 6= 0 was seriously taken into consideration. In contrast to our approach, the intensity of the velocity field V was arbitrary and the system was successively integrated in specific situations depending on the geometry of the flowing fluid. It was also observed that the relation (2.5.1) actually tells us that no mechanical forces are acting on massless pure electromagnetic fields. Hence, the paper anticipates many of the conclusions we are getting here with a different formalism. A similar setting was also considered in Poincar´e (1906). In that celebrated paper, the electron itself was viewed as a continuum of charges having density ρ 6= 0. We will return to the Poincar´e electron model at a later stage in the development of our theory (see chapter 5). In the Extended Electromagnetic Theory (see for instance Lehnert, Roy and Deb (2000) and Lehnert (2006)), the nonzero divergence of the electric field in vacuum is also assumed. The model equations coincide with (2.1.1)(2.1.3), where V = cJ has been replaced by a vector C (satisfying |C| = c), in such a way that the system turns out to be Lorentz invariant. Properties of photons with nonzero mass are then studied with the help of a Proca type equation. Finally, the same set of solitonic type solutions given in section 2.3 is obtained in Donev and Tashkova (1995), where the equations being studied are a combination of those considered here. The model is derived from conservation properties of the electromagnetic stress tensor. As we will see in section 4.2, as far as the space of solutions is concerned, there is an equivalence between our approach and the one based on the differentiation of the energy tensor, as also proposed in Donev and Tashkova (1995).
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2.6
Lorentz invariance
Since we stated that our new set of equations is the right candidate to explain luminous phenomena, we check its consistency with the rules of special relativity. In other words, we need to prove that equation (2.1.1) is invariant under Lorentz transformations. Later, in chapter 4, we will examine the situation within the framework of general relativity. Using standard notation (see for instance Jackson (1975), section 11.3, or Atwater (1994), p. 12), we assume a linear change of variables, relating the set of coordinates (x0 , x1 , x2 , x3 ) to the set of coordinates (˜ x0 , x ˜1 , x ˜2 , x ˜3 ). The corresponding Jacobians are given by the matrices: γ βγ 0 0 βγ γ 0 0 ∂xi = aij = 0 0 1 0 ∂x ˜j 0 0 0 1 aij =
∂x ˜i ∂xj
with β =
v c
γ −βγ = 0 0
−βγ γ 0 0
and
γ = p
0 0 1 0
0 0 0 1
1 1 − β2
(2.6.1)
(2.6.2)
The constant v, with |v| < c, is the velocity of an observer with respect to some absolute reference system, considered to be at rest. Let us set (x0 , x1 , x2 , x3 ) = (ct, −x, −y, −z) and
(˜ x0 , x ˜1 , x ˜2 , x ˜3 ) = (ct˜, −˜ x, −˜ y , −˜ z ) = (γct + βγx, −βγct − γx, −y, −z) In this way we get: 1 ∂ ∂ γ ∂ − βγ = c ∂ t˜ c ∂t ∂x
−
∂ βγ ∂ ∂ = − γ ∂x ˜ c ∂t ∂x
∂ ∂ = − ∂ y˜ ∂y
−
∂ ∂ = − ∂ z˜ ∂z
−
(2.6.3)
Having switched the sign of x, based on the classical set of coordinates (x, y, z) the observer moves at velocity −v.
The next step is to take the contravariant tensor F ik in (2.4.5) and rewrite it in the reference frame of the observer in motion:
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F˜ ik = aim akj F mj =
0
E1
γ(E2 + vB3 )
γ(E3 − vB2 )
γ γ 2 2 −E1 0 − (vE2 + c B3 ) − (vE3 − c B2 ) c c γ 2 −γ(E2 + vB3 ) (vE2 + c B3 ) 0 −cB1 c γ (vE3 − c2 B2 ) cB1 0 −γ(E3 − vB2 ) c where we sum up on repeated indices. Then, we compute the divergence of the first row, namely: ! ∂ F˜ 01 ∂ F˜ 02 ∂ F˜ 03 ∂ F˜ 00 ∂ F˜ 0k = − + + + D0 = − ∂x ˜k ∂x ˜0 ∂x ˜1 ∂x ˜2 ∂x ˜3 = − = −
∂ F˜ 01 1 ∂ F˜ 00 ∂ F˜ 02 ∂ F˜ 03 + + + c ∂ t˜ ∂x ˜ ∂ y˜ ∂ z˜
∂E1 ∂ ∂ βγ ∂E1 + γ + γ (E2 + vB3 ) + γ (E3 − vB2 ) c ∂t ∂x ∂y ∂z = −
βγ c
∂E − c2 curlB ∂t
+ γ divE
(2.6.4)
1
where ( )1 denotes the first component of the vector and the operators curl and div are associated with the coordinates (x, y, z) of the reference frame at rest. In the same way, one can evaluate the divergence of the other rows: γ ∂E ∂ F˜ 1k 2 = − c curlB − βγ divE (2.6.5) D1 = − ∂x ˜k c ∂t 1 D2 = −
∂ F˜ 2k 1 = ∂x ˜k c
∂E − c2 curlB ∂t
(2.6.6)
D3 = −
1 ∂ F˜ 3k = ∂x ˜k c
∂E − c2 curlB ∂t
(2.6.7)
2
3
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It is obvious that, if the Maxwell equations (1.1.1) and (1.1.2) are true, then D0 , D1 , D2 , D3 are identically zero. This means that, if the relation − ∂x∂ k F ik = 0 holds, then we also obtain − ∂∂x˜k F˜ ik = 0, for i = 0, 1, 2, 3. This shows the Lorentz invariance of the equations of classical electromagnetism. In order to handle the modified equations, more work needs to be done. First of all, we have to deal with the vector V = (V1 , V2 , V3 ) = cJ. One has: V2 V3 1 (2.6.8) V + v, , (V˜1 , V˜2 , V˜3 ) = 1 1 + vV1 /c2 γ γ The above is the usual way to sum up velocities in special relativity (see Mould (1994), section 2.3). More correctly, in (2.6.8) we have the subtraction V1 + v = V1 − (−v), since x was replaced by −x. We can easily check that: V12 + V22 + V32 = c2
⇒
V˜12 + V˜22 + V˜32 = c2
(2.6.9)
which states that light moves at the same speed for both the observers, the one at rest and the one in motion. Note that, for i = 1, 2, 3 and V0 = c, we can also write (2.6.8) as follows: c ˜ Vi = aik Vk (2.6.10) a0j Vj For i = 0, we get V˜0 = c. This also follows from the relation: V˜0 = a0m aim Vi
(2.6.11)
We would like to have: −
V˜i ∂ F˜ 0k ∂ F˜ ik + ∂x ˜k c ∂x ˜k
!
= 0
for i = 1, 2, 3
(2.6.12)
which is the counterpart of (2.4.11) for J = V/c. The formula will be further adjusted in chapter 4 in order to conform with the rules of tensor theory. In particular, in Minkowski space, V˜i will be replaced by the contravariant version −V i , i = 1, 2, 3 (see (4.1.27)). In this way, by defining V 0 = c, we are able to satisfy (2.6.12) also for i = 0. We anticipate that (2.1.1) is just a first step towards a more elaborated theory that will later find its correct place in the context of relativity. Before arriving at the final model, we must study the interaction of waves with matter. This analysis will be developed in the next chapter.
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Afterwards, we note that D0 , computed in (2.6.4), also takes the following form: βγ ∂E − c2 curlB + V divE D0 = − c ∂t 1
vV1 + γ 1 + 2 c
divE
(2.6.13)
Assuming that (2.1.1) is satisfied, then the first term on the right-hand side of (2.6.13) is zero. Therefore, we recover: vV1 (2.6.14) D0 = γ 1 + 2 divE c For the same reason we deduce: γ ∂E 2 D1 = − c curlB + V divE c ∂t 1 −γ
V1 γ + β divE = − (V1 + v) divE c c
Then, from (2.6.14) and (2.6.15), for i = 1, we get: ! V˜1 ∂ F˜ 0k V1 + v ∂ F˜ 1k + = D1 + D0 = 0 − ∂x ˜k c ∂x ˜k c(1 + vV1 /c2 )
(2.6.15)
(2.6.16)
On the other hand, for i = 2 and i = 3, we have: ! V˜i ∂ F˜ 0k Vi ∂ F˜ ik + = Di + D0 − ∂x ˜k c ∂x ˜k cγ(1 + vV1 /c2 ) =
=
1 c
1 c
∂E − c2 curlB ∂t
i
+
Vi divE c
∂E − c2 curlB + V divE ∂t i
(2.6.17)
which is also zero, provided (2.1.1) holds. This concludes our analysis. It is interesting to observe that now the Lorentz invariance is explicitly and tightly related to the addition formula for velocities (2.6.9), combining the first and the second postulates of relativity. The invariance of the speed of light relies on the possibility of assuming the existence of luminous signals originating from a point-wise source and travelling in all directions in an anisotropic way. Such spherical wave-fronts are certainly included in our
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theory (see section 2.2). On the other hand, they are not appropriately modelled by the classical theory, as we argued in section 1.3. As a matter of fact, as shown in the analysis, it is difficult to admit the presence of straight rays in the evolution of a Hertzian wave (see figure 2.4). We remark that, from the viewpoint of the moving observer, the electromagnetic wave behaves as a free-wave. In fact, by studying the product F˜ ik V˜k (see (2.5.3)), one gets: 1 F˜ 0k V˜k = v(E + V × B)1 + E · V 2 1 + vV1 /c −1 c(E + V × B) + β E · V 1 1 + vV1 /c2
F˜ 1k V˜k =
F˜ 2k V˜k =
−c (E + V × B)2 γ(1 + vV1 /c2 )
F˜ 3k V˜k =
−c (E + V × B)3 γ(1 + vV1 /c2 )
where we defined V˜0 = c. Considering that E is orthogonal to V and recalling (2.5.1), all the above expressions vanish. Thus, we arrive at: F˜ ik V˜k = 0
for i = 0, 1, 2, 3.
(2.6.18)
As a final result we show the counterpart of the continuity equation (2.1.4) in the new framework: ∂ ∂ ρ˜ ∂ ˜ ∂ ˜ + (˜ ρV˜1 ) + (˜ ρV2 ) + (˜ ρV3 ) = 0 (2.6.19) ∂x ˜ ∂ y˜ ∂ z˜ ∂ t˜ where ρ˜ = D0 = γ ρ+β(ρV1 )/c , with ρ = divE, is given in (2.6.14). Note ˜ = ρ(γ(V1 +v), V2 , V3 ). Thus, by substituting that, from (2.6.8), one has: ρ˜V in (2.6.19) and computing the derivatives as prescribed in (2.6.3), we obtain: ∂ ∂ ∂ ∂D0 + (D0 V˜1 ) + (D0 V˜2 ) + (D0 V˜3 ) ∂x ˜ ∂ y˜ ∂ z˜ ∂ t˜ ∂ = γ ∂t 2
−
β β 2 ∂ ρ + (ρV1 ) − βcγ ρ + (ρV1 ) c ∂x c
βγ 2 ∂ ∂ ∂ ∂ ρ(V1 + v) + γ 2 ρ(V1 + v) + (ρV2 ) + (ρV3 ) c ∂t ∂x ∂y ∂z
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= γ 2 (1 − β 2 )
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∂ρ ∂ ∂ ∂ ∂ − β 2 γ 2 (ρV1 ) + γ 2 (ρV1 ) + (ρV2 ) + (ρV3 ) ∂t ∂x ∂x ∂y ∂z =
∂ρ + div(ρV) = 0 ∂t
Before going ahead with the development of our model, let us deal with a potential criticism. At first sight, the set of solutions admitted by the modified equation (2.1.1) may seem outrageously abundant. Photons with any shape, carrying almost any type of signal, are essentially entitled to be solutions. Nevertheless, if one thinks about the enormous variety of electromagnetic phenomena that surrounds our everyday life, this occurrence should not surprise us at all. It is enough to take into consideration our vision and the impressive polychromy of the various optic stimulations reaching our eyes, to realize that a huge space of solutions is actually a necessity. The infinite nonlinear combinations of the observed phenomena, resulting from their dynamic interference, make the multiplicity even more relevant. We may admit that there are solutions of our model equations not corresponding to real light manifestations (we will make this statement more precise in chapter 6), however, our common experience shows us an extremely variegated universe, not achievable with Maxwell’s original model.
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Chapter 3
Interaction of waves with matter
3.1
Wave bouncing off an obstacle
In this section and in the following one, we study what happens to a freewave when it meets an obstacle that we define as of “mechanical type”. The phenomenon can be extremely complicated, therefore the analysis will be conducted on very simple cases. For the moment, we will discuss some basic facts and try to elucidate the underlying ideas. In section 3.3, we offer a better formalization of the problem by writing down the final equations. Only in chapter 5 will we have the necessary tools to face the entire problem in all its complexity. There, we will also disclose the innermost secrets of matter. We first take into account an example concerning the reflection of electromagnetic radiation. We assume a monochromatic plane wave, linearly polarized, which is totally reflected by a perfectly-conducting metallic wall (no refraction at all). In Cartesian coordinates, the wall corresponds to the plane y = 0. Initially, each wave-front propagates forming an angle ζ 6= 0 with the y-axis. Referring to figure 3.1, the incident wave may be described by the fields: E(i) = 0, − c sin ζ, c cos ζ sin ω(t − (y cos ζ + z sin ζ)/c) B(i) =
1, 0, 0 sin ω(t − (y cos ζ + z sin ζ)/c)
(3.1.1)
The reflected electric field E(r) is such that, for y = 0, the component parallel to the obstacle of the vector E(r) + E(i) vanishes (see for instance Bleaney and Bleaney (1965), p. 270). Concerning the magnetic field at 53
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z 6
I J @ J(r) @
(r) E
ζ ζ
-
y
E(i) @ I @ J J(i) @ Reflection of a plane wave when the magnetic field is normal to the plane of incidence x = 0. The vectors B(i) and B(r) are therefore orthogonal to the page, pointing in the same direction. Fig. 3.1
y = 0, we have: B(r) = B(i) . One easily gets: E(r) = 0, − c sin ζ, − c cos ζ sin ω(t + (y cos ζ − z sin ζ)/c) B(r) =
1, 0, 0 sin ω(t + (y cos ζ − z sin ζ)/c)
(3.1.2)
To justify the imposition of the boundary conditions on y = 0, it is customary to assume the existence of instantaneous electric currents on the conducting surface, which forces the tangential electric field to be zero (see Born and Wolf (1987), p. 558, and Jackson (1975), p. 335). The simulation of this phenomenon is more realistic if we consider that the bouncing wave-fronts do not have an infinite extension. We know that this case can be easily handled by the model developed in the previous chapter. Classical texts dealing with optics cannot apply this localization process, since it is not allowed by Maxwell equations. Therefore, this problem is usually neglected without justification. For simplicity, we continue to work with plane wave-fronts, despite not being obliged to. After reflection, the wave is very similar to the incident one, with the exception that J(i) = (0, cos ζ, sin ζ) has changed to J(r) =
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(0, − cos ζ, sin ζ). Note that relation (2.5.1) is valid both for the incident and the reflected waves. Nevertheless, during impact, in which an instantaneous flip of the signs occurs, the wave-front at y = 0 does not show a behavior consistent with the one corresponding to a free-wave. Each ray, when reaching the wall, evolves for a single moment without respecting the eikonal equation. Of course, what we are considering here is just a mathematical idealization. More realistically, the wall is made of matter, so it would be more correct to check what happens to the wave when it interacts with the atoms of the wall (the powerful influence that matter exerts on photons is one of the subjects of discussion in section 6.1). Anyway, we do not think it is useful to study more complicated situations, as in figure 3.2, since they only modify the form but not the substance of facts. We believe that the main idea has already been outlined: when encountering an obstacle a free-wave may lose its likeness and become, for a small amount of time, a “constrained wave”. The reaction of the obstacle can be so strong that, as in the case of the reflecting wall, the wave-fronts are forced to modify abruptly the direction of their movement. What we would like to do in the following pages is to understand what happens at those instants. We recall that |E(i) | = |E(r) |. We also recall that, for y = 0, the vector E + E(i) does not have the same length of E(i) before the impact (or E(r) after the impact). Therefore, some electromagnetic energy turns out to be missing. We conjecture that it has been spent, with the help of the mechanical constraint, to allow the variation of the direction of the wavefront of an angle π − 2ζ. For a moment, this energy has gone elsewhere, compensated by something not of an electromagnetic kind (see (3.4.12)). We would like to find out what this is. To this end, let us introduce a new vector field: 0 if y 6= 0 G = (3.1.3) J(r) − J(i) 2 c lim if y = 0 y→0 y (r)
Note that, dimensionally, the vector G corresponds to an acceleration. At y = 0, G is oriented as the vector (0, −1, 0), displaying however an infinite magnitude. If we imagine the wave as a bundle of rays (we saw in section 2.3 that the two things are strictly related), then G provides a measure of their curvature. When we are dealing with a free-wave, the rays proceed along straight lines. Correspondingly, we have G = 0. When the rays hit the wall, then G becomes different from zero. In the particular case we are
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Smooth reflection of a wave-front in proximity of a perfectly conducting wall. A suitable vector field G, orthogonal to the rays, can be introduced with the purpose of providing a way to measure their curvature. Fig. 3.2
examining, G is a singularly distributed field, but if we allow the rays to change their trajectories smoothly, then G is going to be finite (see figure 3.2). This would give the idea of a centripetal time-varying acceleration, responsible for the rotation of the rays and the corresponding wave-fronts. In the coming sections, we will show that a non-vanishing vector G appears each time a wave evolves without following the Huygens principle, as a consequence of external perturbations. Some suitable way of estimating the magnitude G should allow us to compensate the missing electromagnetic energy. The theory is not going to be easy. In chapter 4, we will discover that, in order to change the trajectories of the rays, it is necessary to pass through a modification of the space-time metric. Thus, we cannot be more precise until we are ready to carry out our analysis in the framework of general relativity. Before that, we have to work a little more on the definition (3.1.3). This will be done in section 3.3. For the moment, let us put together other basic facts. We now vary the polarization of the incident wave of 90 degrees (see figure 3.3). For y = 0, the component, orthogonal to the reflecting wall, of the magnetic field must be zero. At the same time, the whole electric field vanishes (since, for y approaching zero, the field E(r) is opposite to E(i) ). Therefore, more electromagnetic energy is missing when the wave
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J
I @
(r)
57
z 6
B(r) J @ ζ ζ
-
y
B(i) @ I @ L J(i) @ Fig. 3.3 Reflection of a plane wave when the electric field is normal to the plane of incidence x = 0. The vectors E(i) and E(r) are therefore orthogonal to the page, pointing in different directions.
hits the wall. Actually, in this case, the effects of the obstacle are stronger: together with the deviation of the wave-front, we also observe a torsion that modifies the polarization by 180 degrees. Such a torsion process is instantaneous, but qualitatively similar to that generating the wave-fronts of a circularly-polarized plane wave, like the one for example expressed by the following fields: E = c cos ω(t − y/c), 0, c sin ω(t − y/c) B =
sin ω(t − y/c), 0, − cos ω(t − y/c)
(3.1.4)
where the polarization constantly changes at finite speed. We note that this wave is also a solution to Maxwell equations and is more “energetic” than the corresponding one with constant polarization. The bouncing of a single wave-front is qualitatively shown in figures 3.2 and 3.4. Two different possibilities are actually allowed. We may surmise that they depend on the angle of incidence being above or below the Brewster angle. In the second case, the reflected front interacts with other
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incoming fronts. By some complicated mechanism, the various signals carried by the fronts sum up linearly (to a first approximation), even if the model equations for a free-wave are nonlinear (see also the comments at the end of section 2.3). In order to achieve a clearer picture of what may really be happening, we need to introduce the concept of pressure (see section 5.1). If, together with reflection, we also have refraction within a medium of different density, the investigation becomes more involved. Upon reaching the plane of reflection, the rays bifurcate. In our opinion, this is due to the arbitrariness of the vector J at the time of the impact (for example because P is zero). The incident wave splits into two solutions (the reflected and
Wave-fronts are suddenly reflected back from a conductive wall. We can explain this phenomenon if light rays are identified with the stream-lines of a kind of compressible fluid. The wave-fronts slow down in proximity to the obstacle, each pressing against the other, until they stop and then reversing the direction of motion. In fact, we know that light exerts pressure on illuminated objects. Nevertheless, the fronts continue to travel at the speed of light. This paradox becomes clear if we assume a strong alteration of the space-time metric, due to the interaction of the incoming wave with suitable electromagnetic layers produced by the atoms of the wall (see chapters 5 and 6). Fig. 3.4
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the refracted waves), both compatible with the same boundary conditions on the plane y = 0, brought along by the incoming solution. We do not necessarily have bifurcation each time that P is zero (this actually happens very frequently), but only when, in addition to this, suitable uncommon conditions are satisfied. Based on the above observations, we conclude this section with a brief philosophical dissertation. The equations (2.1.1)-(2.1.3) are of local and deterministic type. Hence, for the given initial data, the solution is uniquely determined. Nevertheless, there may be circumstances in which, following the evolution of a certain solution, several other branches of solutions to equation (2.1.1) may be admissible. As we noticed, this could be true because the evolution of vector J turns out to be compatible with different options. Thus, the original solution splits, and this event is also deterministic. As a matter of fact, when the right conditions are fulfilled, an incident wave bifurcates, giving rise to a reflected wave and a refracted wave. There is no uncertainty: both solutions are systematically observed. However, this situation becomes extremely unstable when reversing time. We are indeed allowed to think that, marching backward in time, the inverse of a bifurcation phenomenon could occur. In this case, as a film runs in reverse, two waves would meet in perfect phase and melt, producing a single wave. Nevertheless, this event has zero probability of happening. A little perturbation is sufficient to modify completely the evolution of the two waves, with no chance of their fusion occurring. In conclusion, our equations are hyperbolic, deterministic and energy preserving. Nevertheless, the particular nature of the nonlinear term can be a source of instabilities when reversing the sign of time. The consequence is that some original information can be lost, and there is no practical way to recover it by following the reverse path. There is a question whether this could be an explanation (at least in part) for the second law of thermodynamics. Some related issues will be taken into consideration in section 6.1.
3.2
Diffraction phenomena
We continue our qualitative analysis on the interaction of waves with matter. Our second example is the development of diffraction, where an electromagnetic wave encounters an aperture. Once again, for simplicity, the
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device used to generate the phenomenon is a pure mathematical abstraction. We just put together some well-known facts, trying to predict the mechanism of their origin. Let us suppose that a plane monochromatic wave propagates in the direction of the y-axis (with y increasing) and hits a perfectly conducting wall at y = 0. This time, however, there is a passage through the strip 0 ≤ z ≤ a, for some positive width a. For y < 0, the incident wave is polarized as in (3.1.1) with ζ = 0. Consequently, we have: E = 0, 0, c sin ω(t − y/c) , B = sin ω(t − y/c), 0, 0 E×B = P = J =
P = (0, 1, 0), |P|
0, c[sin ω(t − y/c)]2 , 0 divE = 0,
divB = 0
(3.2.1)
A specific wave-front is obtained by fixing t − y/c. We will observe the phases of its evolution by varying y. At y = 0, for z < 0 and z > a, the wave is reflected. In actual fact, it is not a perfect reflection, since it is affected by some perturbations originating at the boundaries of the aperture. However, we neglect this aspect and focus our attention on the study of the sources of the disturbances. We assume that along the two straight-lines y = 0, z = 0 and y = 0, z = a, instantaneous currents push the electric field to zero, so that the condition E = 0 is enforced (see Born and Wolf (1987), p. 559). At the instant in which the wave reaches the obstacle, the electric field develops discontinuities. Its divergence is therefore a concentrated Dirac distribution along the above mentioned straight-lines. Thus, for y = 0, it is easy to see that: divE = c [δ0 (z) − δa (z)] sin ωt
(3.2.2)
On the other hand, for 0 < z < a, we can expect that: lim P = (0, c(sin ωt)2 , 0)
y→0+
(3.2.3)
because these are points in which the wave does not hit the obstacle.
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Diffraction of a wave-front passing through an aperture. The rules of geometrical optics hold before and after the obstacle. When the front is aligned with the wall, the rays instantaneously deviate. The action of the obstacle is described by a suitable vector field G, that is formally defined in section 3.3. Fig. 3.5
Next, let us examine in greater detail what happens on the straight-lines y = 0, z = 0 and y = 0, z = a. There, considering that E is zero, the wave undergoes a sudden energy collapse. Some quantitative information could be recovered by examining equation (2.1.5). We know that the Poynting vector P presents a natural pulsating variation along the direction of propagation of the wave. But, since diffraction is a kind of diffusive phenomenon, during the encounter with the obstacle, we have to add another transversal variation, due to the instantaneous change to the flux of energy. As we can see from figures 3.5 and 3.6, during impact, vector field P shows some momentary dispersion of the vertical component, and its divergence suddenly grows. Thus, for just a moment, the quantity − c2 divP registers a negative peek.
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As mentioned in the previous section, the change of curvature of the rays is accompanied by the creation of a new vector field G, concentrated on the obstacle. A suitable pressure also needs to be introduced. The term divP will also depend on the variation of pressure (see (5.1.13)). At this stage, we cannot be more precise. In fact, the correct law of energy balance, taking into account the “reaction” of the obstacle, is very complicated and involves Einstein’s equation. The result is a modification of the space-time geometry, where the rules of geometrical optics are still verified and the light rays follow suitable geodesics. We formalize these ideas more accurately in the coming sections. Concerning diffraction, we will be able to provide some additional qualitative explanations in section 5.1. As in figure 3.6, the rotation of the fields turns out to be clockwise at the points (x, 0, 0). It is anti-clockwise at the points (x, 0, a). This is true whatever the sign of the electric field (note that E and B change sign together and J always maintains the same orientation). Even if the rotation is at infinite speed, the rotation angle is finite. In a more realistic situation, the change of direction of the Poynting vector field is not instantaneous, but develops smoothly. A remarkable achievement is that our equations are appropriate to handling the condition divE 6= 0, which is verified near the boundaries (see for instance (3.2.2)). Such a peculiarity is very helpful, especially in numerical simulations. To a first approximation, the successive evolution of the wave after the obstacle follows the Huygens principle. In fact, for y > 0, the wave is free. It displays a slight spreading due to the rearrangement of the electromagnetic fields, described earlier. It is well known that the behavior depends on the ratio between the width a and the frequency ω/2π. We do not investigate this aspect, since it has been intensively studied in the past. Here, we were only concerned with detecting the mechanism that leads to the deflection of the rays, when they meet the border of the aperture. The same mechanism takes place in the scattering of two solitons, when, with reciprocal influence, their electromagnetic fields collide. We have reasons to believe that the dispersion process here outlined is actually more complex. In section 2.5, we said that a free-wave is characterized by the relation E + cJ × B = 0 (see figure 2.5). Since we imposed that E vanishes at the border of the aperture, there the above equation does not hold. In section 3.3 we will describe the light rays as stream-lines of a suitable fluid governed by the inviscid Euler equation, where a forcing term proportional to E + cJ × B is also present. Hence, for E = 0, the
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z
63
6
as
0s
y
JE? ? ? ? ?
J 6 J R @ ?J J @ R @ 6 J
Plane wave encountering a thin and perfectly reflecting wall with an aperture. Qualitative behavior of the fields E and J before the impact and right after the impact. The vector B is orthogonal to the page. Fig. 3.6
acceleration conferred to the rays depends on the magnitude of B, varying with time. Consequently, during the passage through the aperture (y = 0), together with a general dispersion, the evolution of the fronts is modulated by transversal oscillations originating at z = 0 and z = a, similar to those produced by two small dipoles. This might be equivalent to imposing artificial boundary conditions at the points (x, 0, 0) and (x, 0, a), as also supposed by other theories (see, e.g., Born and Wolf (1987), chapter XI). In this way, due to mutual interference of the fronts originating from the boundaries, one can justify the classical fringes, visible on a screen placed beyond the aperture. Before starting our ambitious program, there are still some small practical problems to fix. Suppose that the incident wave is polarized in a different way, for instance by applying a rotation of 90 degrees: E = c sin ω(t − y/c), 0, 0 B = 0, 0, − sin ω(t − y/c) (3.2.4)
On the contact with the straight-lines y = 0, z = 0 and y = 0, z = a, we should now observe a prompt change of the magnetic field B, along the
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direction z. This event is not modelled by our equations, since we need to suppose that divB can be different from zero. In order to proceed, it is necessary to correct the model in such a way that the fields E and cB have the same role, as in the classical Maxwell equations. We also discuss this in the coming section.
3.3
Adding the mechanical terms
On the basis of some problems that emerged in the previous sections, we make a first adaptation of the set of equations (2.1.1)-(2.1.3), with the aim of bringing the two fields E and cB to the same level. Thus, the new formulation is: E×B ∂E = c2 curlB − c divE (3.3.1) ∂t |E × B| ∂B E×B = − curlE − c divB ∂t |E × B|
(3.3.2)
Now, (3.3.1)-(3.3.2) do not change if we replace E by cB and cB by −E, as in the Maxwell equations. This makes the model invariant under change of polarization. Such a rearrangement recalls the one proposed in M´ unera and Guzm´an (1997). Clearly, if the divergence of B vanishes, we come back to equations (2.1.2)-(2.1.3). The possibility for divB to be different from zero, does not imply the existence of magnetic monopoles, as the condition divE 6= 0 does not imply the existence of electrical charges. However, the issue is delicate, and will be discussed further at the end of chapter 4 and in section 5.2. The modified version (3.3.1)-(3.3.2) allows for an even greater space of solutions. The spherical waves analyzed in section 2.2 can now be constructed with the electric field following the parallels, and the magnetic field following the meridians, which is not, however, a natural configuration (see section 4.4). In this case, we have divE = 0. Other different intermediate polarizations, which may also vary in time, can be considered as well. Finally, we can also get solutions as the one shown in figure 1.2. It is sufficient to set: E = cf (x, y)g(t − z/c), 0, 0
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B = 0, f (x, y)g(t − z/c), 0 ,
J = 0, 0, 1
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(3.3.3)
where f is an arbitrary function, decaying to zero near the boundary of a two-dimensional domain Ω. The function g can also be arbitrary. If g vanishes outside a finite interval, then the solution in (3.3.3) is a travelling soliton, like the ones considered in section 2.3. The difference with respect to section 2.3 is that those solutions satisfy the condition divB = 0. Now, in cylindrical coordinates, the solutions given in (2.3.9) satisfy (3.3.1)-(3.3.2), without the need to enforce relation (2.3.10). However, we are not fully satisfied yet. There is too much symmetry now, while we know that, in most natural phenomena, the roles of fields E and cB are clearly differentiated. In fact, the difference is detectable when a wave interacts with matter. We can take, for example, the case of the wire-grid polarizers, where an incident wave hits a grate of parallel metallic wires. If the wave is polarized with the electric field orthogonal to the wires, then it passes the obstacle almost undisturbed (if its wavelength is much smaller than the distance between two wires of the grate). If the electric field has a component along the direction of the wire, then the wave changes polarization by a certain angle. Insisting on a similar example, we can go back to the end of section 3.2. We are now able to study the diffraction of the wave given in (3.2.4), where the electric field is parallel to the x-axis. We reach the same conclusions obtained for the wave given in (3.2.1), only differently polarized. However, this result is incomplete, because in the case of wave (3.2.4), together with the diffusion of the rays, there should be a change of polarization after passing the obstacle, which is not modelled by the equations (3.3.1)-(3.3.2), and which is not present in the case of the wave polarized as in (3.2.1). In addition to the above observations, in the reflection-refraction phenomenon, we also note that the way the incident wave is polarized affects the final result. Thus, it is necessary to further improve the model. For this purpose the material collected in sections 2.5, 3.1 and 3.2, will be useful. We need to introduce new vector fields (not of electromagnetic type), which are activated each time a free wave becomes a constrained wave. Let us begin by taking a velocity vector field V. We assume all vectors to be of constant norm, in particular: |V| = c, where c is the speed of light. As already stated in the previous chapters, the idea is that V is the tangent vector field to a bundle of light rays. Subsequently, we propose the following system of time-dependent partial differential equations, with three unknown vector fields:
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∂E = c2 curlB − (divE)V ∂t
(3.3.4)
∂B = − curlE − (divB)V ∂t
(3.3.5)
∂V = − (V · ∇)V − µ E + V × B ∂t |V| = c
(3.3.6)
(3.3.7)
The constant µ > 0 is dimensionally equivalent to an electric charge divided by a mass. An estimate of µ is provided at the end of section 5.4. The last condition (3.3.7) has been previously anticipated by (1.1.6). It is customary, in fluid mechanics, to introduce the material (or substantial) derivative: ∂V DV = + (V · ∇)V (3.3.8) Dt ∂t where G is an acceleration field. Hence, the equation (3.3.6) is equivalently written as: DV = −µ E + V×B (3.3.9) Dt G =
D Geometrically, the vector Dt V provides a measure of the curvature of the stream-lines, which in this case are identified with the rays (recall (3.1.3)). As will become clearer, the knowledge of the vector field V is secondary with respect to the determination of its variation G. Note that V is a discontinuous field (for example, it is arbitrarily defined when there is no electromagnetic signal). As a consequence, the writing (V · ∇)V only holds in some prescribed regions.
In our theory, everything will evolve at the speed of light. Therefore, what really matters are accelerations, that automatically introduce the concept of force. The idea that the derivative of a velocity, not the velocity itself, is the primary agent in our universe dates back to Galileo. Indeed, in later times, the theory of relativity provided the right formalism and clarified many philosophical questions. Assuming once again that we are in vacuum, with no particles of any kind around, equation (3.3.9) nevertheless bears a strong similarity with
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the Lorentz law for a density of charge moving at the speed of light. All the ingredients are there. Multiplying by a mass, the left-hand side in (3.3.9) is a force: its component along the direction of motion turns out to be proportional to the electric force field, while the orthogonal component is proportional to the magnetic force field. As we can see, the symmetry is broken and the electric and magnetic fields cannot be interchanged. However, this only happens in the case of constrained waves (G 6= 0). For free-waves, we recall that the relation (2.5.1), corresponding to V = cJ and G = 0, is true. Moreover, replacing E by cB and cB by −E, we obtain the relation (2.5.2), which is also true. Therefore, all the free waves, no matter what kind of polarization they have, are included in the new model. The interesting part is to study the behavior of constrained waves. We will discuss some general properties in section 3.4. We need to say something about the orientation of V. For this purpose, assuming a right-handed coordinate system, we define:
s =
+1 −1
if the triplet (E, B, V) is right-handed (3.3.10) if the triplet (E, B, V) is left-handed
Then, we generalize (3.3.4) and (3.3.5) by setting: ∂E = sc2 curlB − (divE)V ∂t
(3.3.11)
∂B = − s curlE − (divB)V (3.3.12) ∂t In this way, cross-products and curls can remain right-handed, but the equations self-adapt to the prescribed orientation of the frame (E, B, V). Note that changing sign to V is equivalent to reverse time. Since right and left are questionable concepts, a better definition of s is as follows: if E × B is oriented as V +1 s = (3.3.13) −1 if E × B is opposite to V whatever the sign of × is.
The reader has certainly noticed that the positive constant µ in (3.3.9) is preceded by a minus. At first glance, this may sound a bit strange since the Lorentz law is usually written as: F = (E + v × B)e, where e is a
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charge and v its velocity. Nevertheless, if we accept the idea that we are in a left-handed universe (which means that the triplet (E, B, V) is lefthanded in a right-handed reference system), we also have to admit that an electron has positive charge (motivations are provided in section 5.2). In this new perspective, we must switch the sign of the Lorentz law and set: F = −(E + v × B)e, which is now in agreement with (3.3.9). Undoubtedly, there will be some readers who will dislike such a revolution. We would invite them to be patient and wait for developments. We can use the definition (3.3.13) to provide a formulation which is valid for any signature of the framework. In this way (3.3.9) becomes: DV = − µ E + sV × B (3.3.14) Dt where × is right-handed. At the moment, we do not have other physical explanations for the equation (3.3.9). After adding another (absolutely essential) term containing pressure (see (3.4.11) and section 5.1), the general picture should be clearly delineated.
Before proceeding, we feel that some clarification is necessary concerning the meaning of the word “mass”, which has been used, perhaps improperly, several times in this paper. In our discussion, there are no masses in the classical sense, since there are no elementary particles of any sort. Nevertheless, we needed to make a distinction, in terms of dimensionality, between electromagnetic and mechanical (later they will be called gravitational) fields. This task has been assigned to the constant µ, which provides the dimensional link between the two “flavors”. Although other names could have been appropriate to this purpose, the choice of the term “mass” is not incidental, since, as we proceed with our arguments, it will prove to be consistent with the standard setting.
3.4
Properties of the new set of equations
The new system of equations (3.3.4)-(3.3.6) is able to describe electromagnetic phenomena where the wave-front, locally evolving in the direction determined by V, could be subjected to transversal perturbations modifying the trajectories of the rays. The propagation of the wave is governed by the first two equations. Through a feedback process, the third equation, from the current knowledge of the local electromagnetic fields, allows for
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D V, setting up the new direction of motion. This the determination of Dt coupling is possible because we have been able to include the vector V in the description of the electromagnetic part (the term cJ was added to (2.1.1) in the same way). We thus obtained a remarkable result: a link between electromagnetic and mechanical forces. Using the standard Maxwell equations such a connection could never be established. These statements will be properly formalized in chapter 4.
Let us continue with our analysis. From known formulas of vector calculus, we first deduce that: ∇|V|2 DV ∂V ·V = ·V + − V × curlV · V = 0 (3.4.1) Dt ∂t 2 where we took into account that ∇|V|2 = 0 and that V, since V has constant norm.
∂ ∂t V
is orthogonal to
We recall that the definition of J in (2.1.1) implies: E · J = 0. Similarly, by (3.4.1) and by scalar multiplication of (3.3.9) by V, one easily gets: E·V = 0
(3.4.2)
Although one has B · J = 0, nothing can be deduced in general for the scalar product B · V. By vector multiplication of (3.3.9) by V, we get: DV (3.4.3) = − µ V × E − c2 B + (V · B)V V× Dt that generalizes (2.5.2). Finally, by scalar multiplication of (3.3.4) by E and (3.3.5) by B, one obtains: 1 ∂ (|E|2 + c2 |B|2 ) = − c2 div(E × B) − c2 (V · B) divB 2 ∂t which is the counterpart of (2.1.5).
(3.4.4)
Referring to figure 3.7, let J(t) be the normalized Poynting vector at time t and J(t+δt) the one at time t + δt. Let V be the vector at time t obtained by backward parallel transport, along the stream-lines, of the vector J(t+δt) . Then, we have: DV V − cJ(t) = lim (3.4.5) δt→0 Dt δt We recall that the same was done in section 3.1 in order to define the vector G (see (3.1.3)). Therefore, for small time variations δt, we can write: V ≈ cJ + Gδt = cJ − µ(E + cJ × B)δt
(3.4.6)
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t t − δt
t + δt (t)
J HH
Hj G V H
E
(t+δt) HJ H HH j
?
Fig. 3.7 Typical behavior of different fields, when a wave-front is forced to turn. The magnetic field is orthogonal to the page.
with J = (E × B)/|E × B|. If E · B = 0, the relation (3.4.6) can be rewritten as: E V ≈ cJ − µ |E| − |cB| δt (3.4.7) |E| after noting that: (E × B) × B = (E · B)B − |B|2 E = −|B|2 E and |E × B| = |E||B|. This shows that it is sufficient to have |E| = 6 |cB| in order to activate the transversal field G (see figure 2.5). Thanks to (3.4.2), at time t + δt, E is adjusted in order to remain orthogonal to J(t+δt) .
We can compare the evolution of an electromagnetic phenomenon to that of an inviscid fluid, whose mass density, up to dimensional constant, is given by ρ = divE. Note, however, that a real “mass” does not exist. Note also that ρ can also attain negative values. We do not define the density ρ = divB for reasons that will be detailed in sections 4.3 and 4.4. The following continuity equation holds (see also (2.1.4)): ∂ρ = − div(ρV) ∂t
(3.4.8)
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obtainable by taking the divergence of (3.3.4). Equation (3.4.8) can also be written as: Dρ = − ρ divV (3.4.9) Dt For a plane wave (or soliton) having ρ 6= 0, we obtain divV = 0 as well as G = 0. Then, (3.4.9) tells us that the fluid shifts, without modifications, along the direction determined by V. The fluid travels at constant speed c, showing rarefaction and compression. More properly, it evolves like an incompressible fluid, but with density not equally distributed in space. Regarding a spherical wave, one has divV > 0 and G = 0. As expected, this implies that the mass density diminishes (in absolute value) while time passes, since it spreads on spheres of growing area. In both examples (the plane and the spherical), we have curlV = 0. In other words, the fluid is irrotational. Then, as in section 2.5, let us suppose that V = ∇Ψ is a gradient. Due to (2.5.11), we know that the eikonal equation holds true. ∂ Ψ = 0 implies: We then observe that relation ∂t ∂Ψ DV |∇Ψ|2 = 0 (3.4.10) = ∇ + Dt ∂t 2 This confirms a remarkable result: the eikonal equation (hence, the evolution of the wave-fronts based on the Huygens principle) is perfectly compatible with the condition G = 0. In equation (3.3.9), the term V × B = −B × V = T (V) can be viewed as a suitable strain tensor T applied to the vector normal to the front of propagation (see for instance Batchelor (1967), p. 10). A full relation with the inviscid incompressible fluid dynamics will be set up in section 5.1, where (3.3.9) is corrected in the following way: DV ∇p = −µ E + V×B − (3.4.11) Dt ρ after suitably defining a pressure p. Correspondingly, an equation of state will be introduced. However, we are not yet ready to discuss this extension. From numerical tests, still in progress, one observes that the passage through a narrow aperture of a solitary wave actually displays the dynamic behavior of a non-viscous flow in motion. As we already know, forced variations of the electric field produce changes in the motion of the fronts. If these are combined with forced variations of the magnetic field, a torsion is also introduced, which modifies the polarization of the wave. We guess, that, when the external perturbations end, the electromagnetic fields return to their natural equilibrium in
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which |E| = |cB| and (2.5.1) is satisfied, so that the fluid again takes an irrotational motion. From the examples discussed in sections 3.1 and 3.2, this behavior corresponds to what is commonly observed in nature, and certainly comes from the minimization of some Lagrangian. At the present time, however, we do not have a theoretical explanation for this conjecture. An energy balance involving the vector G is hard to achieve. Only in section 4.3 will we be able to say something in tensor form with the help of Einstein’s equation. More or less, the scheme will be of this type:
potential energy
+
electromagnetic energy
=
kinetic energy
(3.4.12)
In this way, if we want to preserve kinetic energy, any decay in the electromagnetic energy should be compensated by variation of potential energy. This potential energy is somehow related to |G| and, in the framework of general relativity, to the space-time curvature. However, an explicit expression is not available at the moment. Furthermore, we note that stationary electric fields, for example with B = 0, are no longer solutions. We can check this by examining relations (3.3.9) and (3.4.2). These two equations force the velocity field V to turn itself around (V deviates in the direction of E, but E remains orthogonal to it). More generally, equation (3.3.6) requires the solutions to be in continuous evolution. We contend that the new system of equations does not admit stationary solutions having finite energy. We made the same consideration in section 2.3, in the particular case of solitons. Nevertheless, there could be non-stationary solutions localized in space. We can imagine for instance the case of two (or more) solitons, in such a situation that they are constrained, by influencing each other’s electromagnetic fields, to revolving around a common center. We still do not have all the elements to study these phenomena, which, as we will see in the following pages, need the environment of general relativity to be stated properly. In particular, the case of rotating solitons will be examined in chapter 5. We are unable at the moment to obtain the equations (3.3.4)-(3.3.6) from the stationary points of a suitable action function as we did in section 2.4 (concerning (3.3.6) alone, something in this direction will be obtained in the next chapter). One may consider the usual Lagrangian L = 2(c2 |B|2 − |E|2 ) and the generalization of the relation (2.4.12), i.e.: cA = ΦV
(3.4.13)
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We could then differentiate the same action function given in (2.4.6) using constraint (3.4.13). Nevertheless, we would not obtain the desired result, since (3.4.13) is too restrictive. In this way, we only get a set of equations describing free-waves. As a matter of fact, we can prove that, if V has the same direction as A, then one automatically has G = 0. Assuming divB = 0, this check can be done as follows. Considering (2.4.1) and (3.4.13), we have: µ DV µ ∂A µ D c A − A = − − (V · ∇)A Dt Φ c Dt c ∂t c DV µ + µE + µ∇Φ − (A · ∇)A Dt Φ ∇Φ2 DV µ − = + µE − + (A · ∇)A Dt Φ 2 =
=
µ DV + µE − Dt Φ =
−
∇|A|2 + (A · ∇)A 2
A DV + µ E + × curlA Dt Φ
DV + µ(E + V × B) = 0 (3.4.14) Dt where we used that Φ2 = |A|2 . The last equality is true thanks to (3.3.9). Then, noting that (c/Φ−µ/c)A = (1−µΦ/c2 )V, the relation (3.4.14) leads to: D µΦ µ DΦ µΦ DV 0 = 1− 2 V = − 2 V + 1− 2 (3.4.15) Dt c c Dt c Dt =
By scalar multiplication by V, due to (3.4.1), we recover: µΦ DV DΦ µ DΦ |V|2 + 1 − 2 ·V = −µ = 0 − 2 c Dt c Dt Dt
(3.4.16)
D Thus, Φ turns out to be constant along the stream-lines ( Dt Φ = 0). For this D A = 0). reason, from (3.4.14), A is also constant along the stream-lines ( Dt D Therefore, from V = cA/Φ, one has Dt V = G = 0. We also conclude that, when the rays bend (G 6= 0), then the vector A cannot be aligned to the direction of motion.
We mentioned in the previous sections that the mechanical effects are implicitly included in the term c2 divP, where it is necessary to distinguish
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between the contribution due to the variation of the Poynting vector along the actual direction of propagation of the front, and the transversal contribution (which is zero when G = 0). Differentiating with respect to time the expression J = P/|P|, we get: ! ∂ P · ∂t P ∂P 1 P (3.4.17) − G = |P| ∂t |P|2 In particular, by scalar multiplication of G by P, (3.4.17) shows the orthogonality relation G · P = 0. Furthermore, from (2.1.5), the energy can be described as a work by integrating − 2c2 divP with respect to time. This yields: Z Z t2 2 divP dt = − 2 divP V · ds −2c (3.4.18) t1
Γ
where we set ds = Vdt, which implies V · ds = |V|2 dt = c2 dt. The last integration is made along the curve Γ representing the path of the light ray in the time interval [t1 , t2 ]. We end this section by illustrating another interesting relation. Let us define as usual ρ = divE. Let us then assume that ρ 6= 0 and define ω ¯ = F/ρ, where F = curlV − µB. Then, along the stream-lines we have:
Dω ¯ = (¯ ω · ∇)V (3.4.19) Dt Note that ω ¯ is dimensionally equivalent to a time multiplied by a charge and divided by a mass. Equation (3.4.19) recalls the analogous one for isentropic flows, which is introduced in fluid dynamics by defining ω ¯ as the curl of the velocity field divided by the mass density (see Chorin and Marsden (1990), p. 24). Using (2.4.1), the field ω ¯ also takes the following form: ω ¯ =
curl(V − µA/c) 1 ∂ − divA − ∆Φ c ∂t
(3.4.20)
The equation (3.4.19) can be proven as follows:
Dω ¯ − (¯ ω · ∇)V ρ Dt = curl
∂V ∂t
− µ
1 Dρ 1 DF = ρ − 2 F − (F · ∇)V ρ Dt ρ Dt
∂B + (V · ∇)F + F divV − (F · ∇)V ∂t
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= curl − (V · ∇)V − µ(E + V × B) + µ(curlE + V divB) + (V · ∇)F + F divV − (F · ∇)V =
h
curl(V × F) − V divF + (V · ∇)F + F divV − (F · ∇)V −
h
i
i curl[(V · ∇)V] + curl(V × curlV)
1 = − curl (V · ∇)V + (V × curlV) = − curl ∇|V|2 = 0 (3.4.21) 2 where we used (3.4.9), (3.3.6), (3.3.5) in that order, some well-known calculus properties and the fact that ∇|V|2 = 0.
In the case of plane solitary waves (of any transversal shape), we have curlV = 0, hence ω ¯ = −µB/ρ (when ρ 6= 0). Therefore, ω ¯ remains D ω ¯ = 0. Then, orthogonal to V, so that the relation (3.4.19) becomes Dt the quantity ω ¯ shifts, remaining constant along the stream-lines determined by the velocity field V (which are straight-lines in this case).
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Chapter 4
The equations in the framework of general relativity
4.1
Preliminary considerations
Our first step, in this section, is to recover equation (3.3.9) through the stationary points of a suitable Lagrangian. To this end we work in space-time using 4-vectors. Let us start by defining (x0 , x1 , x2 , x3 ) = (ct, −x, −y, −z) and (e0 , e1 , e2 , e3 ) = (1, −1, −1, −1). Then, for the vector (V0 , V1 , V2 , V3 ) = (V0 , V), one has: 3 X i=0
ei Vi2 = V02 − |V|2
(4.1.1)
As in section 2.4, we assume that divB = 0 and introduce the potentials Φ and A by (2.4.1). Also let (A0 , A1 , A2 , A3 ) = (Φ, A). Up to multiplicative constants, we can define a Lagrangian in the following way (see also Landau and Lifshitz (1962), p. 50): q µ L = c V02 − |V|2 + (A · V − ΦV0 ) (4.1.2) c
The quantities Vi , i = 0, 1, 2, 3, are the independent variables, while the potentials depend on xi , i = 0, 1, 2, 3. By setting V0 = c, the term in P3 parentheses of (4.1.2) can be written as: c−1 i=0 ei Ai Vi . For the moment, we do not impose the condition (3.3.7), implying that the sum in (4.1.1) is zero.
Suppose that we are moving along a stream-line (or curved light ray), between two instants of time t1 and t2 , then the action function takes the form: Z t2 L dt (4.1.3) S = − t1
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The search for stationary points leads us to the Euler-Lagrange equation (see Jackson (1975), p. 577): ∂L ∂L ∂L d ∂L d ∂L = =c =− i = 1, 2, 3 (4.1.4) dt ∂V0 ∂t ∂x0 dt ∂Vi ∂xi
∂ ∂ , ∂ , ∂ , ∂ ). In particular, for , −∇) = ( ∂x where we observed that ( 1c ∂t 0 ∂x1 ∂x2 ∂x3 i = 1, 2, 3, we have: ! d ∂L µ −cVi d p + = Ai dt ∂Vi dt c V02 − |V|2 3
= − and
DVi µ ∂Ai µ X ∂Ai dxk + + Dt c ∂t c ∂xk dt
(4.1.5)
k=1
µ ∂L = ∂xi c
V·
∂A ∂Φ − V0 ∂xi xi
(4.1.6)
D where, in (4.1.5), the substantial derivative Dt Vi gives the variation, along the stream-lines, of the coordinatesRof p the velocity field, parametrized with t respect to the arclength: s = |c|−1 t1 V02 − |V|2 dξ, t ∈ [t1 , t2 ].
If we now define (2.4.5), we get:
d dt xm
= Vm , for m = 1, 2, 3, thanks to (2.4.3) and
µ DVi = Dt c
∂Ai ∂Φ − ∂t ∂xi
3 ∂A µ X ik µ V · ∇Ai + = F Vk + c ∂xi c
(4.1.7)
k=0
When V0 = c, the last term in (4.1.7) is equal to the i-th component of the vector −µ(E + V × B). This implies the equation (3.3.9). Concerning k = 0, we have: d ∂L DV0 dΦ ∂Φ = − µ = −µ − µ V · ∇Φ dt ∂V0 Dt dt ∂t ∂L ∂A µ ∂Φ and V· = − V0 ∂t c ∂t ∂t
(4.1.8) (4.1.9)
D If we fix V0 to be constantly equal to c, one obtains Dt V0 = 0. Therefore, due to (4.1.4), by equating the two last expressions, we recover: 3 X 1 ∂A F 0k Vk = − µ E · V (4.1.10) ·V = −µ 0 = µ ∇Φ + c ∂t k=0
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which corresponds to (3.4.2). Considering (4.1.10), by scalar multiplication of (3.3.9) by V, we deduce that the velocity fields, stationary for the action, D V = 0. Hence, the norm |V| is constant. If such a must satisfy: V · Dt constant is c, we finally obtain the relation (3.3.7), which says that the solutions evolve on the light-cone. At this point, it should be noted that, by choosing |V|2 = c2 , the first part of the Lagrangian in (4.1.2) vanishes. This does not mean that it vanishes identically, but only in correspondence to the stationary point. Instead, the second part of the Lagrangian is zero when A·V = cΦ, which is very similar to the condition (2.4.17), obtained from the constraint (2.4.12) (see also (3.4.13)). This coincidence is quite significant. In a more general setting, we will return to discuss Lagrangians in section 5.1. By multiplying equation (4.1.7) by Vi , i = 1, 2, 3, and equation (4.1.10) by V0 , we get: F ik Vk Vi = 0
(4.1.11)
where we sum up on repeated indices from 0 to 3. This also trivially follows from the anti-symmetry of the tensor F ik . Equation (2.4.11) is also written as: ik 0k ∂F ∂F V0 − ei Vi = 0 for i = 0, 1, 2, 3 (4.1.12) ∂xk ∂xk Otherwise, equations (1.1.3) and (1.1.4), can be recovered from the expression (see for instance Fock (1959), p. 150): Fikj =
∂Fik ∂Fkj ∂Fji + + = 0 ∂xj ∂xi ∂xk
(4.1.13)
where here there is no sum on repeated indices. The rank-three tensor Fikj is anti-symmetric and called the cyclic derivative of Fik . On the other hand, equation (3.3.5) follows from the expression: ∂Fik ∂F23 ∂Fkj ∂Fji ∂F31 ∂F12 V0 + + = ± em Vm + + (4.1.14) ∂xj ∂xi ∂xk ∂x1 ∂x2 ∂x3 where the indices m, j, i, k (taken in this order) are all different. The sign ± depends on the permutation (even or odd) of these indices (the sign is plus if m = 0, j = 1, i = 2, k = 3). In (4.1.14) the term in parentheses on the right-hand side is equal to c divB. In a more contracted form, the last equation reads as follows: V0 Fjik = ± em Vm F123
(4.1.15)
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In the results obtained above, we basically considered V as the velocity field of an infinitesimal entity moving at the speed of light. On the other hand, in a wave there are infinite contiguous trajectories. As a matter of fact, the evolution of a wave is a global phenomenon that should be taken as a whole, not studied independently along each stream-line (see figure 4.1). For such in-depth analysis, we need to work in the context of general relativity. We are going to show that the passage of a wave modifies the space-time structure. For a free-wave this does not affect the evolution of the wave itself (see section 4.3), but for constrained waves the change of the geometry influences their entire behavior. The analysis will allow us to find the coupling between the fields describing the wave and its corresponding geometry, hence the link between electromagnetic and gravitational phenomena. In addition, we will make a rather strong assertion, namely: gravitation is at the service of electromagnetism. The distortion of the space-time geometry exists for the purpose of permitting photons (the most basic entities) to follow curved trajectories, when interacting between them in agreement with the usual laws of dynamics, including conservation of momenta. Indeed, a universe where photons can only go straight in a flat space, would be quite uninteresting. Gravity is then a necessary by-product designed to guarantee energy preservation, according to the rough outline given in (3.4.12). We will do our best to convince the reader of these facts. Before discussing these serious issues, we first need to introduce some classical definitions. Mainly, we adopt the notation used in Fock (1959). We use Roman alphabet instead of Greek letters, and the summations will go from 0 to 3, unless otherwise stated. The space-time geometry is locally determined by a symmetric bilinear form (the metric tensor), whose coefficients are denoted by gij . From now on, the signature of the tensor will be (+, −, −, −). Then, the Christoffel symbols are defined in the following way: g im ∂gmk ∂gmj ∂gkj i Γkj = (4.1.16) + − 2 ∂xj ∂xk ∂xm where we sum over the index m. The coefficients g ij are such that: gim g mj = δij
(4.1.17)
The coefficients gij are adimensional, while the Christoffel symbols are the inverse of a distance. For example, in Cartesian coordinates, if the space is “flat” (Euclidean or Minkowski space), one can set g ik = gik = ei δik . As a consequence all the Christoffel symbols vanish.
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In turning wave-fronts, the external rays follow larger trajectories if compared to those of internal rays. If we want to preserve the condition |V| = c, one has to alter the space-time metric. This should result in a front behaving as a single solid entity and, at the same time, representing a bundle of independent infinitesimal solitons. Fig. 4.1
As usual, we denote by g the determinant (which is negative) of the tensor gik . A lemma due to Ricci (see Fock (1959), p. 129) claims that the covariant differentiation of the metric tensor is zero. For any j = 0, 1, 2, 3, one has: ∇j g ik =
∂g ik + Γimj g mk + Γkmj g im = 0 ∂xj
(4.1.18)
where ∇j is the covariant differentiation operator. The same is true for the coefficients gik . In particular, for the 4-divergence, we have: √ 1 ∂( −g g ik ) ∇k g ik = √ + Γijm g jm = 0 (4.1.19) −g ∂xk
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Moreover, the coefficients g ik are said to be harmonic when: √ 1 ∂( −g g ik ) √ = 0 −g ∂xk
(4.1.20)
Next, we define V i = g im Vm . The values Vm are the entries of a velocity vector expressed in the coordinates system (x0 , x1 , x2 , x3 ). For the sake of convenience, we will set V 0 = c. This limitation is going to be easily removed later, since we want to handle covariant entities. Then, the condition (3.3.7) is generalized in the following way: V i Vi = g ik Vk Vi = gim V i V m = 0
(4.1.21)
The above can still be considered to be the eikonal equation (see (2.5.11)). As a matter of fact, according to Fock (1959), section 36, (4.1.21) is related to the invariant wave equation: |∇ω|2 = g ik
∂ω ∂ω = 0 ∂xi ∂xk
(4.1.22)
and the condition ω(x0 , x1 , x2 , x3 ) = 0 describes the geometrical motion of a surface. This means for example that, in the appropriate metric space, the fronts of figure 4.1 are moving by following the rules of geometrical optics, while this is certainly not true in a flat space. In this more general framework, equations (4.1.7) and (4.1.10) are rewritten as: DV i µ + Γijk V j V k = F im Vm for i = 0, 1, 2, 3 (4.1.23) Dτ c where τ is the so-called proper time. For i = 0, 1, 2, 3, we also define (see Fock (1959), p. 217): DV i + Γijk V j V k = V m ∇m V i (4.1.24) Dτ From (4.1.21) and (4.1.23) we easily recover the orthogonality relations: Gi =
Gi V i = Gi Vi = 0
(4.1.25)
where Gi = gim Gm . Finally, let us define G = (G1 , G2 , G3 ), which is dimensionally equivalent to an acceleration. With the new notations, equation (2.4.12), valid for free-waves, takes the form: Am V m = Ak Vk = 0
(4.1.26)
In general relativity, the gravitational field is somehow identified with the tensor gij . Of course, the vector G may vanish without the space being
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flat. Although G does not fully characterize the properties of space-time, it will be called the vector gravitational field. Equation (4.1.23) enables us to understand how the trajectory of a “thin” solitary wave can be distorted when immersed in a given gravitational field. The soliton being a free-wave, the right-hand side of (4.1.23) vanishes (see (2.5.1)). Thus, its path follows a suitable geodesic in spacetime, the shape of which is determined by the external gravitational field. From the viewpoint of an observer, this should correspond to some transversal bending in the direction locally individuated by a vector G. With this reasoning, we have to neglect a couple of facts, both due to the change of direction: the modification of the electromagnetic fields and the “gravitational reaction” of the soliton (a curving wave produces new gravitational field). As we argued in section 3.4, these should be minor effects, since the wave, for some principle of least action, tries to compensate the electromagnetic fields in order to enforce (2.5.1). From the point of view of the soliton, the path followed is straight, even if it actually travels on a curved geodesic. To get more reliable quantitative results, we must solve a quite complicated system of equations. Further, if some theoretical passages may be formally similar, the important clue is that there is no need to suppose that a soliton is a particle (obtained as a limit when mass tends to zero and velocity tends to c) to justify that it is attracted by a gravitational field. After recalling that F ik is an anti-symmetric tensor, in a general metric space equation (4.1.12) becomes: √ √ 1 ∂( −g F ik ) 0 ∂( −g F 0k ) i √ =0 i = 0, 1, 2, 3 (4.1.27) V − V −g ∂xk ∂xk or, in more contracted form:
(∇k F ik )V 0 − (∇k F 0k )V i = 0
i = 0, 1, 2, 3
(4.1.28)
The above expression is metric independent as one can easily check by taking into account formula (4.1.18) and considering that V 0 = g 0m gmi V i (see also (2.6.11)). Note that (4.1.28) states that the two vectors V i and ∇k F ik are parallel. An appropriate covariant form is obtained by writing: ∇k F ik = j i
i = 0, 1, 2, 3
(4.1.29)
where j = (j 0 , j 1 , j 2 , j 3 ) is a 4-vector (one then deduces: V i = V 0 j i /j 0 ). Equation (4.1.13) remains unchanged. However, it can also be written in the following way (see Fock (1959), p. 133): Fikj = ∇j Fik + ∇i Fkj + ∇k Fji = 0
(4.1.30)
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Besides, equation (4.1.15) becomes: V 0 Fjik = ± V m F123 (4.1.31) By taking the 4-divergence of the contravariant vector in (4.1.27) and considering once again that the tensor F ik is anti-symmetric, we arrive at the continuity equation: √ √ 1 ∂( −g F 0k ) 1 ∂( −g ρE V i ) √ =0 with ρE = √ (4.1.32) −g ∂xi −g ∂xk We got an analogous result in section 2.1, by taking the standard divergence of the vector equation (2.1.1). The time derivative came from the ∂ term div ∂t E and the term div(curlB) was zero. All the pieces here combine in a completely different manner. Nevertheless, the final result is extraordinarily similar. By examining the right-hand side of (4.1.29) one gets j 0 = ρE and j i = c−1 ρE V i .
4.2
The energy tensor
Let us first work in Cartesian coordinates. We will define the symmetric electromagnetic stress tensor in the classical way (see Fock (1959), p. 96), i.e.: 3 X 1 2 2 (4.2.1) c |B| − |E|2 ei δik ej Fij Fkj − Uik = − 2 j=0 P3 We have U00 = 12 (|E|2 + c2 |B|2 ) and i=0 ei Uii = 0. Its contravariant version is given by U ik = ei ek Uik and of course we P3 have i=0 ei U ii = 0. The explicit expression of the contravariant tensor is the following: 1
2 2 (|E|
+ c2 |B|2 )
cB2 E3−cE2 B3 cE B −cB E 1 3 1 3 cB1 E2−cE1 B2
cB2 E3−cE2 B3
cE1 B3−cB1 E3
−E12 +c2 B22 +c2 B32 − 21 (c2 |B|2 −|E|2 )
−E1 E2−c2 B1 B2
−E1 E2−c2 B1 B2
−E22 +c2 B12 +c2 B32 − 12 (c2 |B|2 −|E|2 )
−E1 E3−c2 B1 B3 −E2 E3−c2 B2 B3
cB1 E2−cE1 B2
−E1 E3−c2 B1 B3 2 −E2 E3−c B2 B3 2 2 2 2 2 −E3 +c B1 +c B2 − 12 (c2 |B|2 −|E|2 )
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If (3.3.4) and (3.3.5) are satisfied, then an important property of this last tensor is that, in the case of free-waves (hence in the absence of mechanical terms), its 4-divergence vanishes. Indeed, we have for i = 0, 1, 2, 3: ∂U ik = 0 ∂xk
(4.2.2)
provided (2.5.1) (or (2.5.2)) is satisfied. We recall that for a free-wave one has E · B = 0 and |E| = |cB|. These hypotheses also imply that |E × B| = |E||B| and V · B = 0. Let us prove (4.2.2) starting from i = 0. Thanks to (2.1.5), one has: 1 ∂ ∂U 0k = |E|2 + c2 |B|2 + c div(E × B) = 0 ∂xk 2c ∂t
(4.2.3)
As far as the other values of i are concerned, let us begin to define: N = (N1 , N2 , N3 ) =
∂E − c2 curlB + (divE)V ∂t
∂B + curlE + (divB)V (4.2.4) ∂t Thus, if equations (3.3.4) and (3.3.5) are true, then we get N = 0 and M = 0. We are ready to check (4.2.2) for i = 1 (the other cases are treated in a very similar way). We have: M = (M1 , M2 , M3 ) =
∂ ∂ ∂U 1k = (B2 E3 − E2 B3 ) − (−E12 + c2 B22 + c2 B32 ) ∂xk ∂t ∂x +
∂ ∂ 1 ∂ 2 2 (−E1 E2 − c2 B1 B2 ) − (−E1 E3 − c2 B1 B3 ) c |B| − |E|2 − 2 ∂x ∂y ∂z = (M2 E3 − M3 E2 + N3 B2 − N2 B3 ) + (E1 + V2 B3 − V3 B2 ) divE + (c2 B1 + V3 E2 − V2 E3 ) divB = 0
(4.2.5)
The last three terms in (4.2.5) are actually zero for the following reasons. In the first one we recognize the second and the third components of N and M, which vanish, if we assume that equations (3.3.4) and (3.3.5) are satisfied. The second one contains the first component of the vector E+V×B, which vanishes in the case of a free-wave. Concerning the last term, the part in parentheses is the first component of the vector c2 B − V × E, which also vanishes (see (2.5.2)).
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The property (4.2.2) is reported in many texts (see for instance Fock (1959), p. 97). But, it is extremely important to observe that, in the case of Maxwell’s equations, the last two terms are zero because divE = 0 and divB = 0. This is the reason why we decided to double check equation (4.2.2), which turns out to be fulfilled even when the divergence of the fields E and B is not zero (the assumption we are supporting throughout this book). Therefore (4.2.2) holds under weaker hypotheses. As expected, a converse statement also holds: assuming that (4.2.2) is true, then we can recover equations (3.3.4) and (3.3.5). This amounts to differentiating the equation of energy conservation, in order to obtain the corresponding Euler equations. Arguing as we did to get (4.2.5), we arrive at: 1k ∂U ∂U 2k ∂U 3k , , = (M × E − N × B) ∂xk ∂xk ∂xk + (E + V × B) divE + (c2 B − V × E) divB
(4.2.6)
Assuming, as previously, that we are dealing with a free-wave, after eliminating in (4.2.6) the vanishing terms, we are left with (M×E−N×B). Since, by hypothesis, equation (4.2.2) is true, if the vector N is zero, then M must also be zero (likewise, if M is zero, then N is zero). Therefore, (3.3.4) is satisfied if and only if (3.3.5) is satisfied. This is the same situation ∂ B + curlE and encountered in the classical Maxwell equations, where ∂t ∂ 2 divB both vanish if and only if ∂t E−c curlB and divE are both zero. In the standard approach, the first pair of equations are satisfied by choosing the potentials A and Φ as in (2.4.1). The second pair is obtained by means of variational type arguments. The idea of describing solitary-wave solutions by imposing the equation M × E − N × B = 0, was firstly taken into account in Donev and Tashkova (1995) (and related papers), where a theory under the name of Extended Electrodynamics was introduced. Effectively, in this way, we reobtain all the solutions examined in section 2.3. That paper was also inspired by the vivid perception that Maxwell equations are totally unsatisfactory in explaining the propagations of waves in vacuum. Of course, we can find “intermediate” situations, by suitably redefining the two potentials. Let us take for example: B = curlA
E = −
1 ∂A − ∇Φ c ∂t
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with
λE + (1 − λ)cB E = p λ2 + (1 − λ)2
and
λcB − (1 − λ)E B = p λ2 + (1 − λ)2
87
(4.2.7)
where λ is a real parameter. From the relations (4.2.7) we can explicitly compute the fields E and B in terms of E and B. These also imply:
∂B = − c curlE (4.2.8) ∂t Then, it is a matter of finding the stationary points of the usual Lagrangian. Introducing the constraint A = ΦV/c, one gets the equation: divB = 0
∂E = c curlB − V divE (4.2.9) ∂t that, for λ = 1, is equivalent to equation (3.3.4). The equations in (4.2.8) are equivalent to require (λ − 1)N + λM = 0, while the one in (4.2.9) brings to λN + (1 − λ)M = 0.
It is to be noted that V has the same direction of E × B, which is also like that of E × B. So, from the energy tensor it is not possible to figure out what the parameter λ is, as well as the polarization of the wave. This information has to be provided with the initial conditions. For instance, the wave in (3.1.4), circularly polarized, produces the same tensor U ik of a linearly polarized wave, moving in the same direction with twice the intensity. As a further consequence, we finally observe that U ik does not change if E takes the place of −cB and cB takes the place of E. Such a permutation corresponds to the choice λ = 0. We can now argue in a general framework. For a given metric tensor gik , the electromagnetic stress tensors must be modified in the following way (see Fock (1959), p. 151): Uik = − g mj Fim Fkj − 41 gik Fmj F mj U ik = − gmj F im F kj −
mj 1 ik 4 g Fmj F
(4.2.10)
where Fik is given in (2.4.4), while F ik comes from the relation: F ik = g im g kl Fml
(4.2.11)
Assuming a situation similar to that of a free-wave, equation (4.2.2) has to be replaced by the following one: √ 1 ∂( −g U ik ) ik + Γimj U mj = 0 (4.2.12) ∇k U = √ −g ∂xk
The proof of (4.2.12) is given for instance in Jackson (1975), p. 606, for the classical Maxwell equations. This is also true in the case of our new set
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of equations (hence under weaker hypotheses). At the end of this section, we will evaluate the 4-divergence of the tensor U ik in general coordinates. Such generalizations are unavoidable since, even the simple case of a plane wave provokes a modification of the space-time geometry, requiring work with tensors of the form (4.2.10). These aspects will be better studied in the next section. When the electromagnetic phenomenon is not a free-wave, we cannot expect that (4.2.2) and its generalization (4.2.12) are verified. This means that the system constituted by only the electromagnetic part is not energy preserving. We know that, in this case, the energy balance has to take care of the mechanical effects. Thus, we study how to introduce them. We start by assuming that divB = 0, leaving the discussion of the more general case to section 4.4. Then, let us define a mass tensor as follows: Mik = Vi Vk divE
(4.2.13)
The contravariant version is given by M ik = ei ek Mik , which is explicitly written as (see also (4.1.32)): −V0 V1 −V0 V2 −V0 V3 V02 −V V V12 V1 V2 V1 V3 0 1 M ik = ρE (4.2.14) −V0 V2 V1 V2 V22 V2 V3 2 −V0 V3 V1 V3 V2 V3 V3
where V0 = c and ρE = divE is a kind of mass density (dimensionally this is not correct, but this aspect will be altered later). We recall that ρE can also be negative. Let us check what happens to ∂x∂ k M ik . For i = 0 we have: ∂M 0k ∂ρE ∂(ρE V1 ) ∂(ρE V2 ) ∂(ρE V3 ) = c c − − − ∂xk ∂x0 ∂x1 ∂x2 ∂x3 = c
∂ρE + div(ρE V) = 0 ∂t
(4.2.15)
This is true because of the continuity equation (3.4.8) with ρ = ρE . For the other indices i = 1, 2, 3, we have: ∂(ρE Vi ) ∂(ρE V1 Vi ) ∂(ρE V2 Vi ) ∂(ρE V3 Vi ) ∂M ik = − c − − − ∂xk ∂x0 ∂x1 ∂x2 ∂x3
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= −
89
∂ρE ∂(ρE Vi ) + div(ρE Vi V) = − Vi + div(ρE V) ∂t ∂t
−ρE
∂Vi + (V · ∇)Vi ∂t
= −
DVi divE Dt
(4.2.16)
where we again used the continuity equation. We conclude for instance D V = 0), then one that, if the light rays are straight-lines (that is: G = Dt ∂ ik gets ∂xk M = 0, for i = 0, 1, 2, 3. In non-Euclidean geometry, it is necessary to generalize the mass tensors in the following way: Mik = ρE Vi Vk
M ik = ρE V i V k
(4.2.17)
where now ρE is a suitable scalar. We may deduce from equation (4.1.29) √ that ρE = j 0 = (−g)−1/2 ∂x∂ k ( −g F 0k ).
With the help of the continuity equation (4.1.32) and the definition (4.1.24), it is easy to get, for i = 0, 1, 2, 3: √ 1 ∂( −g M ik ) + Γimj M mj ∇k M ik = √ −g ∂xk = ρE V k
µ ∂V i + ρE Γimj V m V j = ρE Gi = ρE F ik Vk ∂xk c
(4.2.18)
where the last equality is a consequence of (4.1.23) (see also (2.5.3)). Hence, for unperturbed trajectories (i.e.: Gi = 0), the 4-divergence of the mass tensor vanishes. Moreover, we observe that the mass tensor does not contain the pressure term. This will be added in section 5.1, where a suitable equation of state is also defined. In order to combine electromagnetic and mechanical effects, we sum up the corresponding tensors, by defining: µ (4.2.19) Tik = 4 − µUik + Mik c where we recall that the constant µ is dimensionally equivalent to a charge divided by a mass. It follows that Tik has the same dimension of a curvature, that is the inverse of the square of a distance. A minus has appeared in (4.2.19). It is not a mistake. We justify this in the next section. Now, in a flat space-time, even if we are not dealing with a free-wave, we may write: ∂T ik = 0 ∂xk
(4.2.20)
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As a matter of fact, due to (3.3.9), if the term µ(E + V × B)divE of the electromagnetic part does not vanish (see (4.2.6)), it is nevertheless D V of the mechanical part compensated by the corresponding term divE Dt (see (4.2.16)). In the general case, the relation (4.2.20) is substituted by: √ 1 ∂( −g T ik ) ∇k T ik = √ + Γimj T mj = 0 −g ∂xk
(4.2.21)
Before ending this section, we would like to verify that (4.2.21) actually corresponds to the Euler equations. As a matter of fact, (4.2.21) is satisfied when (4.1.28), (4.1.31) and (4.1.23) are true. We recall that these last three equations are the generalizations of (3.3.4), (3.3.5), (3.3.6), respectively. For the moment, we will only treat the case in which divB = 0, leaving the general discussion to section 4.4. Let us start by computing the 4-divergence of the tensor U ik . First of all, we have: ∇k U ik = − ∇k (gmj F im F kj ) + = g im ∇k (Fmj F jk ) +
ik mj 1 ) 4 ∇k (g Fmj F
mj 1 ik ) 4 g ∇k (Fmj F
(4.2.22)
where we notice that gmj F im = g im Fmj (thanks to (4.2.11)), that F kj = −F jk and that, due to (4.1.18), it is allowed to exchange the metric tensor with the covariant derivative (see also Fock (1959), p. 129). It follows that: ∇k U ik = g im (∇k F jk )Fmj + g im (∇k Fmj )F jk + = c−1 ρE Fmj g im V j +
mj 1 ik 2 g (∇k Fmj )F
1 im (∇k Fmj )F jk 2g
+ 21 g im (∇k Fmj + ∇m Fjk )F jk
(4.2.23)
where we used (4.1.29) (which is equivalent to (4.1.28) for V 0 = c and j 0 = ρE = ∇k F 0k ). The other passages were obtained by suitably renaming the indices. Recalling the definition of Fikj given in (4.1.30), we have: ρE Fmj g im V j + 21 g im (∇k Fmj )F jk ∇k U ik = c + 21 g im Fmjk F jk − =
ρE im F Vm + c
1 im (∇j Fkm )F jk 2g 1 im Fmjk F jk 2g
(4.2.24)
In the last passage two terms have been deleted, since, after renaming the indices, they resulted in being equal and with opposite signs. The last term
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in (4.2.24) is zero because of (4.1.30) (remember that we are studying the case divB = 0, thus F123 = 0). Of course, the final result is zero when, for instance, ρE = 0, as in the classical Maxwell case. But it is also zero when F im Vm = 0, which corresponds to the case of a free electromagnetic wave (recall (2.5.3)). On the contrary, we need to consider the contribution of the mass tensor. If Fmjk = 0, taking into account the relations (4.2.19) and (4.2.18), one finally obtains: µ ∇k T ik = 4 − µ∇k U ik + ∇k M ik c µ µ µ = 4 − ρE F im Vm + ρE F ik Vk = 0 c c c
4.3
Unified field equations
In the previous section, we built the symmetric tensor Tik that includes both the energy contribution of an electromagnetic wave and that of mechanical type. The latter takes into account possible deviations from the natural propagation path of the wave. The properties of Tik insure the preservation of energy and momentum. Hence, we can put Tik on the right-hand side of the Einstein equation: Rik −
1 2 gik R
= − χ Tik
(4.3.1)
in which we recognize the Ricci tensor: Rik =
∂Γm ∂Γm j im m ik − + Γjik Γm jm − Γim Γkj ∂xm ∂xk
(4.3.2)
the scalar curvature: R = g ik Rik
(4.3.3)
and an adimensional positive constant χ, that, at the moment, it is not directly related to the gravitational constant. Being the signature of the space (+, −, −, −), the minus sign on the right-hand side of (4.3.1) is correct. We recall that the Christoffel symbols are defined in (4.1.16). Let us note that, by (4.2.10) and (4.2.17), the metric tensor, which is now our unknown, also appears on the right-hand side of (4.3.1). By taking the trace of (4.3.1), one obtains: g ik (Rik −
1 2 gik R)
= R − 2R = − R = − χg ik Tik
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χµρE ik χµ ik µg Uik − g ik Mik = − g Vi Vk = 0 (4.3.4) 4 c c4 where we considered that the trace of the electromagnetic tensor Uik is zero. Moreover, we took into account (4.1.21). In the end, we obtain: =
R = 0
(4.3.5)
For the moment, let us work with the above condition, since it will be enough for studying free-waves. Later, in section 5.1, we will examine the case R 6= 0, after adding a suitable pressure term to the mass tensor.
Since we are in the mood for radical changes, it is time to deliver another blow to the foundations of physics. Let us discuss the sign of the right-hand side of (4.3.1). If we examine the nature of the curvature tensor, there should be no reason to express a preference. Einstein’s equation is of a hyperbolic type (see for instance Bona and Mass´o (1992)), therefore there is nothing implicitly defined as positive or negative. The choice of a specific sign, depending on the signature of the metric tensor, was made in order to conform to what is actually observed in astronomy. Of course, we must adhere to this decision. In fact, the mass tensor, which is responsible for the inertial properties of matter, appears with the correct sign. In this way, when the electromagnetic phenomena are negligible, we have an opportunity to rediscover well-known results. Once a polarity was decided, scientists wisely established that the electromagnetic stress tensor had to follow suit. This was far from being a good idea, and we are now going to show why with several examples. The fact that U00 is positive should not influence the decision on the sign of Uik , since, as we said, the curvature tensor does not have a polarity. According to (3.4.12), the correct place for the electromagnetic tensor should be beside the curvature tensor, on the left-hand side of the Einstein equation. A recent paper (see Lo (2006)) points out that a suitable adaptation of the right-hand side of the Einstein’s equation should be invoked, in order to justify the coexistence of photons and gravitational waves, taking into account both physical and mathematical aspects. The answers proposed in Lo (2006) are different from ours, but the questions raised are pertinent and with similar significance. We know that, at this level, our arguments may be totally vague. The reader will have more elements to judge the correctness of this viewpoint in chapter 5. In any case, the most important consequence of taking the opposite sign for the electromagnetic stress tensor is the astonishing possibility of building meaningful and extremely simple explicit solutions (in contrast to the very poor material generally obtainable by working with the standard sign). We discuss these examples in full.
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Let us examine the response of equation (4.3.1) to the passage of the most elementary plane wave. We take for instance the expression given in (1.1.5), where we have E1 = cB2 = c sin ω(t−z/c), divE = 0 and divB = 0. We will verify that, even in this simple case, the space-time geometry that comes from the solution of (4.3.1) is not Euclidean. In fact, we look for a metric tensor gik of the following type: 1 0 0 0 1 0 0 0 0 −1/σ 2 0 0 −σ 2 0 0 0 (4.3.6) g ik = gik = 0 0 0 −1 0 0 −1 0 0 0 0 −1 0 0 0 −1 where σ is a function, to be determined, of the variable ξ = t − z/c. Somehow, we are expressing a preference for the direction of the x-axis, which is oriented with the electric field. The determinant g of gik is equal to −σ 2 . The corresponding Christoffel symbols are:
−σσ ′ σ′ σσ ′ Γ311 = Γ101 = Γ110 = Γ113 = Γ131 = (4.3.7) c c cσ where the prime denotes the derivative with respect to ξ. All the other symbols vanish. The non-zero coefficients of the Ricci tensor are instead: Γ011 =
R00 = R03 = R30 = R33 = −
σ ′′ c2 σ
(4.3.8)
The scalar curvature R is zero. As the divergence of E is zero, the mass tensors Mik and M ik vanish. In fact, one should check that ρE = 0, where ρE is the divergence of the electric field evaluated in the new metric defined by gij . This is also true, as the comments at the end of this section illustrate. The tensors Uik and U ik have to be computed through (4.2.10). First of all, one has: 0 −u 0 0 u 0 0 u (4.3.9) Fik = c 0 0 0 0 0 −u 0 0 F ik
0 −u/σ 2 = c 0 0
where u = B2 = E1 /c.
u/σ 2 0 0 −u/σ 2
0 0 0 0
0 u/σ 2 0 0
(4.3.10)
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Note that (V0 , V) = (c, 0, 0, c) and (V 0 , V 1 , V 2 , V 3 ) = (c, 0, 0, −c). Hence, one gets F ik Vk = 0, from which we deduce that the wave is free, as is already known (see also (2.5.3)). According to (2.4.3), a potential corresponding to (4.3.9) is (Φ, A) = (A0 , A1 , A2 , A3 ) = (−xu, 0, 0, −xu), which satisfies the Lorenz gauge condition (2.5.7). Note that, in agreement with the relation (3.4.13), we have Ak V k = 0 and V0 Ak = A0 Vk , for k = 0, 1, 2, 3. Subsequently, we have: (u/σ)2 0 0 (u/σ)2 µ2 0 0 0 0 (4.3.11) Tik = − 2 0 0 0 0 c (u/σ)2 0 0 (u/σ)2 (u/σ)2 0 0 −(u/σ)2 2 µ 0 0 0 0 (4.3.12) T ik = − 2 0 0 0 0 c −(u/σ)2
0
0
(u/σ)2
Thus, (4.3.1) and (4.3.12) bring us to the equation: −σ ′′ σ = µ2 χu2
(4.3.13) √ For u = sin ω(t − z/c), we finally obtain σ = µ χ/ω sin ω(t − z/c), which is the solution we are looking for. This also confirms that χ must be positive (if the equation to be solved had been σ ′′ σ = µ2 χu2 , we would have had little chance of getting any interesting information out of it). The case when u is a non-zero constant function in time should not be allowed by the theory. In fact, equation (4.3.13) is solvable for a constant right-hand side only given certain assumptions. The solution σ, however, cannot be constant. Actually, if σ in (4.3.6) were constant, then the Ricci tensor would be zero, while, for u 6= 0, the electromagnetic tensor cannot vanish. It is also worthwhile to observe that the magnitude of the wave σ is inversely proportional to the frequency of oscillation. There are surely other geometries compatible with the same plane wave. Note that the one presented here satisfies the relation (4.1.20). Another one will be shown later in this section. We also observe that there are points in which the metric becomes singular, that is, the determinant g is zero. Another equivalent possibility is to exchange g11 and g22 in (4.3.6), and make σ oscillate with the magnetic field. Comments about this option will be given in the next section. We point out the following relation: ∂Ei − Γjik Ej Vk = 0 ∂t
(4.3.14)
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where E0 = 0. The above equation can be deduced, with the help of (4.3.7), by direct verification. Thus, the electric field is parallel transported along the flux lines (see Fock (1959), section 39). The solution just obtained can be interpreted as a transversal (perfectly plane) gravitational wave, travelling in phase with the electromagnetic one. It must also be noted that, even if the space is officially non-Euclidean, the geodesics involved in the motion of the wave are straight-lines. The field G, defined by (4.1.24), is identically zero. This is in agreement with our viewpoint: the geometry may be deformed, but there is no creation of a real gravitational vector field. Pure gravitational solutions resembling plane waves, were formally derived in Bondi, Pirani and Robinson (1959) (see also Misner, Thorne and Wheeler (1973), section 35.9). We obtain the explicit (and very simple) solution above because we put the “wrong” sign before the electromagnetic stress tensor. In addition, we were resolute enough to assume the dependence from the metric tensor of the right-hand side of the Einstein equation. As far as we can deduce from the literature, in contrast to our general approach to the problem, it is customary to construct the electromagnetic energy tensor in vacuum (thus, in Minkowski space-time), also because such an assumption is supposed (erroneously) to simplify the computations. One reaches a set of solutions, but, as proved, this is not the correct setting. Note also that, commonly, gravitational waves are sought among the solutions of the linearized homogeneous Einstein equation, obtained after perturbation of the flat space-time. Our solution is far from being a small perturbation of the flat metric, hence, there is little chance of getting it through standard arguments. We can recover the laws of motion by evaluating the 4-divergence of the tensor T ik in (4.3.12). The geometry is non-Euclidean, therefore, the √ relation (4.2.20) has to be replaced by (4.2.21), where −g = |σ|. For i = 0, one has: √ −µ2 1 ∂(u2 /σ) ∂(u2 /σ) 1 ∂( −g T 0k ) 0 mj √ + Γmj T = 2 + −g ∂xk c σ c ∂t ∂z u2 1 ∂σ ∂u2 ∂σ −µ2 1 1 ∂u2 − 2 (4.3.15) + + = 2 c σ σ c ∂t ∂z σ c ∂t ∂z
The situation is exactly the same for i = 3. The last term in (4.3.15) is zero, when for instance: 1 ∂u ∂u 1 ∂σ ∂σ + = 0 + = 0 (4.3.16) c ∂t ∂z c ∂t ∂z
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Our plane electromagnetic-gravitational wave is actually the solution to both the above equations, once the proper initial conditions have been assigned. Let us now discuss the example of a circularly-polarized plane wave: E = c cos ω(t − z/c), c sin ω(t − z/c), 0 B =
− sin ω(t − z/c), cos ω(t − z/c), 0
(4.3.17)
The classical divergence of the electric field vanishes, as well as that of the magnetic field. Let us then take the following metric tensor: 1 0 0 0 µ2 χ 0 −[cos ω(t − z/c)]2 0 0 (4.3.18) gik = 2 2 0 0 −[sin ω(t − z/c)] 0 ω 0 0 0 −1
in such a way that the coordinates x and y are synchronized with the electric field (the reasons for this choice will be explained at the end of this section). In this case, the non-vanishing coefficients of the Ricci tensor are: R00 = R03 = R30 = R33 = 2ω 2 /c2 . They coincide with the respective coefficients of the stress tensor: T00 = T03 = T30 = T33 = −2ω 2 /χc2 . Therefore, once again, the Einstein equation is verified. The wave is free and we have R = 0 and G = 0. Slightly more complicated is the case of a plane wave where divE is nonzero. This happens for instance when u = B2 = E1 /c = f (x) sin ω(t − z/c). As we know, the solution satisfies equations (2.1.1)-(2.1.3), but not the classical Maxwell equations. We suggest looking for a metric tensor of the form: 1 0 0 0 0 −σ 2 f 2 0 0 gik = (4.3.19) 0 0 −1 0 0
0
0
−1
√ where σ is a function of the variable ξ = t − z/c. One has: −g = |f σ|. The tensor Fik is the same as in (4.3.9). Regarding the other tensors, we get: 0 u/(σf )2 0 0 −u/(σf )2 0 0 u/(σf )2 (4.3.20) F ik = c 0 0 0 0 0 −u/(σf )2 0 0
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(u/σf )2 µ2 0 =− 2 0 c (u/σf )2
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(4.3.21)
We must point out an extraordinary fact: in the new geometry, the 4divergence of the electric field turns out to be zero. As a matter of fact, by noting that u/f and σ do not depend on x, one has:
ρE
=
√ 1 ∂( −g F 0k ) = √ −g ∂xk
−1 ∂(|f σ| F 01 ) −c ∂(u/f ) = = 0 |f σ| ∂x f σ2 ∂x
(4.3.22)
Thus, the mass tensor still vanishes. The Christoffel symbols are slightly different from the ones in (4.3.7) (in particular Γ111 is not zero), but the coefficients of the Ricci tensor are exactly equal to those given in (4.3.8). Therefore, equation (4.3.13) must be modified as follows: 2 u ′′ 2 (4.3.23) −σ σ = µ χ f thereby admitting the same solution σ obtained in the case of the plane wave at uniform density. The laws of motion are the same as in (4.3.16). It is important to observe that they do not contain derivatives with respect to the variables x and y. They tell us that u and σ shift at the speed of light along the z-axis. They do not specify however the function f , which must be assigned through the initial conditions. We do not discuss the case when f depends on y, since it implies divB 6= 0. However, we can easily study, in cylindrical coordinates, the situations examined in section 2.3 (see also figures 1.3 and 2.1), where divB = 0. We are ready to illustrate the case of a spherical wave. With the same notation of sections 1.2 and 2.2, we set the coordinates to obtain: (x0 , x1 , x2 , x3 ) = (ct, −r, −θ, −φ). Let us assume that E = (0, cu, 0) and B = (0, 0, u), where u = 1r f (θ) sin ω(t − r/c). We recall that, to avoid singularities at the poles, the function f is not allowed to be constant and f (0) = f (π) = 0. This is similar to the case of a variable-density plane wave. We also have that (V 0 , V 1 , V 2 , V 3 ) = (c, −c, 0, 0). Then, let us begin by giving the metric tensor:
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gik
1 0 = 0 0
0 −1 0 0
0 0 −σ 2 f 2 0
0 0 0 −1
(4.3.24)
where σ is a function of the variable ξ = t − r/c. In (4.3.24), we have that g22 is zero at the poles. For the electromagnetic tensors we get: 0 0 −ru 0 0 0 −ru 0 Fik = c ru ru 0 0 0 0 0 0 F ik
0 0 = c −ru/(σf )2 0
0 0 ru/(σf )2 0
ru/(σf )2 −ru/(σf )2 0 0
0 0 0 0
In order to evaluate Fik , we started from (2.4.3), recalling that, by (2.4.13), one has (Φ, A) = (−F (θ) sin ω(t − r/c), −F (θ) sin ω(t − r/c), 0, 0), where F is a primitive of f . The metric in (4.3.24) is the same as the one we would have obtained if we had worked with a plane electromagnetic wave. The fact that we are in spherical coordinates is actually contained in the electromagnetic tensors, where we find ru in place of u (see for instance Atwater (1994), p. 124). Note that (V0 , V1 , V2 , V3 ) = (c, c, 0, 0), from which one obtains the relation F ik Vk = 0, confirming that the wave is free. As far as energy is concerned, we get: (ru/σf )2 (ru/σf )2 0 0 µ2 (ru/σf )2 (ru/σf )2 0 0 Tik = − 2 0 0 0 0 c 0
0
Therefore, from the Einstein equation, we get: 2 ru ′′ 2 −σ σ = µ χ f
0
0
(4.3.25)
where we observe that the right-hand side only depends on the variable ξ = t − r/c. Equation (4.3.25) once again gives the solution: √ σ = µ χ/ω sin ω(t − z/c). Let us also note that there is no relation between (4.3.24) and the metric expressing the standard inertial spherical system of coordinates, given by gij = diag{1, −1, −r2 , −r2 sin2 θ}, and having the Ricci curvature tensor equal to zero.
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Finally, by differentiating T ik (for i = 0 and i = 1), we find the Euler equation in spherical coordinates: 1 ∂(ru) 1 ∂u + = 0 (4.3.26) c ∂t r ∂r The function u is the solution to (4.3.26), after assuming the appropriate initial conditions. We return for a moment to the case of the plane wave (although all the other situations previously discussed can be dealt with in the same manner). A metric tensor alternative to the one given in (4.3.6) is: 0 0 0 1 2 0 0 1 0 −1/σ 2 0 −σ 2 0 0 0 0 ik (4.3.27) g = gik = 0 0 0 −1 0 0 −1 0 1 0 0 −2 1 0 0 0
where, in order to satisfy Einstein’s equation, σ is a solution to (4.3.13). This way of writing the metric tensor is a particular case of the more general expression: 2J0 J1 J2 J3 J1 gik = J2 γik + Ji Jk /J0 J3
g ik
0 −J 1 /J0 = −J 2 /J0 −J 3 /J0
−J 1 /J0
−J 2 /J0
γ
ik
i
−J 3 /J0
k
− J J /J0
(4.3.28)
depending on the vector J = V/c = (J1 , J2 , J3 ) and the 3 × 3 symmetric invertible tensor γik . We define J 0 = V 0 /c = 1 and, as usual, J i = g ik Jk , Ji = gik J k . Note that one has: gim g mj = δij ⇐⇒
⇐⇒
J k Jk = 0
For i = 1, 2, 3, one also has: Ji =
3 X
m=1
J0 · 1 + J1 J 1 + J2 J 2 + J3 J 3 = 0
γ im Jm
⇐⇒ Ji =
V k Vk = 0 3 X
γim J m
(4.3.29)
(4.3.30)
m=1
In addition, for J0 > 0, one recovers: 2 Jk (ds)2 = J0 dx0 + + J0 (dx0 )2 + γik dxi dxk dxk J0
(4.3.31)
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where x0 = ct and the indices i and k vary from 1 to 3. We notice that the special case the choice of tensors: −σ 2 0 J = (0, 0, 1) γik = 0 −1 0 0
(4.3.27) is associated with J0 = 1 and 0 0 −1
γ ik
−1/σ 2 0 0 = 0 −1 0 0 0 −1
In this setting two major implications are obtained. First of all, the vector V directly appears among the unknowns of the metric tensor and automatically satisfies relation (4.1.21). Secondly, due to (4.3.31), the solution simulates a gravitational wave moving along the flow field J. This is exactly what we got in the examples discussed here. This way of organizing the tensor gik is very similar to the one corresponding to the 3+1 splitting form (see Arnowitt, Deser and Misner (2004)), where space and time are treated separately. By suitably projecting the curvature tensors on the three-dimensional subspace one can substitute the Einstein equation by an evolutive first-order nonlinear system (see also Misner, Thorne and Wheeler (1973), chapter 21). As far as numerical simulations are concerned, such simplifications are of primary interest (the reader can find numerous algorithms in the vast literature). The splitting procedure borrows various technicalities from differential geometry (such as the Gauss-Codazzi equation) and its description is a bit involved. Therefore, for the sake of simplicity, we do not examine this issue in detail. The results of this section, although only restricted to the analysis of free-waves, focus our attention on some important issues. Up to now, we have claimed that a good theory of electromagnetism was meaningful only by allowing divE to be different from zero. Here instead we find that ρE = 0. In our opinion, what is happening can be explained as follows. The spacetime “reacts” to the passage of a wave, by varying itself synchronously, in order to make the 4-divergence of the electric field vanish. The perturbation of the geometry is however weak enough to maintain G equal to zero. The classical divergence divE may instead attain arbitrary values. We ask ourselves if it is possible to set up an experiment showing that, at some point and at a certain time, one has divE 6= 0. Perhaps, this is not possible since, due to the modification of the metric, the instruments are unavoidably affected by the deformation of time and distances (with respect to the Euclidean reference frame). Therefore, in place of divE, we could end up measuring ρE . But the last quantity is always zero (at least for free-waves). As a consequence, we conclude that some divergence
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vanishes, albeit the relativistic one, not the classical one. According to this new interpretation of the facts, in some sense the Maxwell theory was correct. Thus, we come to the following conclusions. The equations of physics manifest themselves in various fashions, depending on the reference system and the given metric, but, if properly stated, they have a meaning independent of the context. The equations modelling electromagnetism (in which we include the velocity vector V) can actually be written in covariant form. They look like the usual Maxwell equations, when the metric is the one obtained by solving the Einstein equation associated with the phenomenon under investigation. In other metric spaces, they display a different form. In particular, in a flat space, one ends up with the equations analyzed in chapter 2. A natural reference frame is then associated with the triplet of vectors (E, B, V), and the metric itself is built depending on the orientation and the magnitude of these three vectors, continuously changing with time. Of course, one can assume the existence of other suitable reference systems and metrics, but the one represented by (E, B, V) is very special since it gives a plausible way, perhaps the simplest, to directly combine electromagnetic vector fields with properties of the metric. Note that, without the introduction of the vector field V, this step was certainly not possible. Let us examine more closely the case of a photon. By computing σ from (4.3.13) and considering (4.3.21), for a monochromatic wave we trivially get: 1 0 0 1 µ2 0 0 0 0 (4.3.32) Tik = − 2 χc 0 0 0 0 1 0 0 1
This is true for any f and ω. The same can be said for the curvature tensor Rik . Correspondingly, one always has ρE = 0.
Concerning other types of photons, like, for instance, those described by (2.3.9) and (2.3.10) in cylindrical coordinates, similar results hold. Suppose that the wave-fronts of the photon are taken as in figure 1.3. Then, in the modified geometry, the same fronts appear like the one shown in figure 4.2. All the vectors have the same size, because the local unit of measure varies exactly in a proportional manner. The fields are zero at the central point P and outside the domain Ω of definition of the front. Therefore, the front terminates abruptly at the border of Ω. Nevertheless, we cannot claim that there are discontinuities. For two-dimensional beings living inside the front
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The wave-front of figure 1.3 (relative to a flat metric space) is here expressed in its own metric space, generated through the Einstein equation. The discontinuities at the border and at the center are only apparent. As for the equations of non-viscous fluid dynamics, the only variable actually involved in the evolution of the front is the one related to the direction of V, which is orthogonal to the page. Fig. 4.2
the world is limited to Ω − {P }. Trying to reach the boundary is equivalent to walking on a path of infinite length, since they get smaller as they approach the external rim. We can recognize an analogy with the Poincar´e disk model (see for instance Anderson (1999)), where arcs of circumferences (corresponding to geodesics of infinite length) intersect perpendicularly the boundary of the disk, producing a peculiar fractal structure. A probe, capable to examining (without contamination) the world at the level of the Planck constant, would be heavily influenced by the local geometry. At such extremely small scale, tensors assume a trivial form (see
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for instance (4.3.32)). The probe would be completely lost, transmitting no valuable piece of information. This is an elegant way to get around the indefinitely small paradox. In the local metric there is no trace of the original intensity of the fields (the one we observe from a flat space, as in figure 1.3). The photon exists inside Ω − {P }; it does not exist outside. Note that the domain Ω − {P } is not a simply-connected region. Thus, we expect the entire 3-D photon to be topologically equivalent to a toroid. As we shall see in section 5.4, this is also the shape of elementary particles. The reason for the difference with the classical case is that we do not need to compute derivatives in a direction transversal to the motion of the photon (as we do when evaluating for example divE or curlB). It is the same situation that we encounter in non-viscous fluid dynamics, when we can follow a single point along a stream-line. Since there is no diffusion, each infinitesimal fluid particle evolves without worrying about affecting its neighbors, so that transversal discontinuities are theoretically allowed. At this stage, we can better understand how the fronts geometrically propagate according to the Huygens principle (see section 1.2). In fact, the construction of the so-called wavelets is problematic when, in flat Minkowski space, one approaches the rim of each front. As explained in the caption of figure 4.3, a deformation of the shape of the wavelets, based on the indications provided by the local metric, must occur. There is another point that needs to be clarified. The problem is why the geometry changes depending on the electric field, and not the magnetic field, especially after we said that for free-waves the two fields have the same role. Firstly, we note that, in all examples studied in this section, the condition divB = 0 was always fulfilled. In addition, if similarly to ρE , we define ρB , we discover that this new quantity is also zero (see section 4.4). If we imagine the wave like a fluid in motion, then this condition says that there is no flow of some “magnetic density”. In truth, it is reasonable to assume that a single electromagnetic fluid exists (not two, a separate electrical one and a magnetic one). As will become clear in the next section, where we analyze the case divB 6= 0, such a fluid turns out to be associated with a pulsation along a specific transversal direction (in principle, not necessarily corresponding to that of the electric field). Exchanging cause with effect, in section 4.4 we will support the following statement: from the behavior of natural events, we are inclined to name the direction of the electric field as being that identified by the privileged transversal deviations of the fluid in motion. The situation is schematically described in figure 4.4.
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Far more complicated phenomena show up when we suppose that the waves are no longer free (thus, G 6= 0). In this context, real gravitational fields come to life. We will face this difficult problem in chapter 5.
Treatises on classical optics systematically forget to mention that there are problems in constructing spherical-shaped wavelets near the border of a given front, as depicted in the upper picture. The difficulty is comparable to that of extending Maxwell’s equations up to the boundary of a wave-packet (the subject is often avoided by stating the equations in integral form). We have seen that a full comprehension of electromagnetic wave phenomena cannot neglect this important aspect. Now, we know exactly how the passage of a wave deforms the space-time. Hence, “spherical” wavelets are not rounded, but they look distorted, as shown in the lower picture. In this way, they never touch the border of the front.
Fig. 4.3
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The divergence of the magnetic field
In the previous sections, some situations were discussed under the hypothesis divB = 0. Although our equations now have a general validity, the assumption is necessary, for instance when introducing the potentials A and Φ. Regarding this condition, we would like to add further comments in this section. It is standard to introduce a transformation that exchanges the role of the electric and magnetic fields. This can be done through the pseudo-tensor: 0 when at least two indices are equal ǫmjik = (4.4.1) 1 if the indices form an even permutation −1 if the indices form an odd permutation
The parity of the permutations is counted starting from the set: {0, 1, 2, 3}. We then define: ǫmjik = em ej ei ek ǫmjik = − ǫmjik
(4.4.2)
We may now introduce the dual of the tensors (2.4.4) and (2.4.5) in the following way: Fˆmj = 12 ǫmjik F ik
Fˆ mj = em ej Fˆmj = − 21 ǫmjik Fik
(4.4.3)
Therefore, we obtain for example: Fˆ01 = F 23
Fˆ02 = F 31
Fˆ03 = F 12
Fˆ23 = F 01
Fˆ31 = F 02
Fˆ12 = F 03
The original tensors and their dual have the same structure, with the difference that E replaces −cB and cB replaces E.
In a similar way, the dual of the anti-symmetric rank-three tensor Fjik (defined in (4.1.13)) is given by: Fˆ m = − 16 ǫmjik Fjik
(4.4.4)
Hence, up to even permutations of the lower indices, one has: Fˆ 0 = F123
Fˆ 1 = −F023
Fˆ 2 = F013
Fˆ 3 = −F012
In general coordinates, it is customary to define: √ ′ ′ ′ ′ 1 ∈mjik = −g g mm g jj g ii g kk ǫm′ j ′ i′ k′ = − √ ǫmjik −g √ ∈mjik = −g ǫmjik (4.4.5) So that the dual in (4.4.3) and in (4.4.4) are generalized as follows: Fˆmj =
1 2
∈mjik F ik
Fˆ mj = − 12 ∈mjik Fik
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Fˆ m = − 16 ∈mjik Fjik
(4.4.6)
where Fik is provided in (2.4.4) and F ik can be found in (4.2.11). From Fock (1959), p. 134, we know that: √ 1 ∂( −g Fˆ mj ) √ = Fˆ m −g ∂xj
(4.4.7)
where we supposed that Fˆ m is the dual of the cyclic derivative Fjik of the tensor Fik (of which Fˆ mj is the dual). Passing to the dual, the equation (4.1.31) becomes: V 0 Fˆ m = V m Fˆ 0 . Therefore, by (4.4.7) we get for m = 0, 1, 2, 3: ! √ √ ∂( −g Fˆ mj ) 0 1 ∂( −g Fˆ 0j ) m √ V − V =0 −g ∂xj ∂xj
(4.4.8)
which is the exact counterpart of (4.1.27). Equation (4.4.8) represents, in a general coordinates system, the equation (3.3.5), that is equivalent to (3.3.4), after taking E in place of − cB and cB in place of E. From (4.4.8), we can recover the continuity equation: √ √ 1 ∂( −g Fˆ 0k ) 1 ∂( −g ρB V i ) √ =0 with ρB = √ (4.4.9) −g ∂xi −g ∂xk It is worth noting that ρB has the same dimensions as ρE . For example, according to (4.4.6), the dual of Fik in (4.3.9) is: 0 0 −u 0 c 0 0 0 0 Fˆ mj = √ (4.4.10) 0 0 −u −g u 0 0 u 0
Since we supposed that u does not depend on y, we obtain ρB = 0. Based √ on the metric given by (4.3.19) (where −g = |f σ|), we have the proof that, together with the condition divB = 0, the 4-divergence of the magnetic field also vanishes. Going back to equation (4.2.24), this time we cannot assume that Fmjk = 0. On the other hand, we can use (4.4.7) and (4.4.8) with V 0 = c, to get: √ ∂( −g Fˆ lj ) ˆ jk im ˆ l ˆ im 1 1 im Flm Fmjk F = g F Flm = g √ 2g −g ∂xj =
ρB im ˆ ρB ˆ im ρB ˆ im F glm V l = F Vm g Flm V l = c c c
(4.4.11)
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The first passage follows on from a direct counting of the permutations of the indices, thanks to the definitions provided in (4.4.6). Substituting in (4.2.24), one finally gets: 1 (4.4.12) ρE F im Vm + ρB Fˆ im Vm ∇k U ik = c Let us observe that, for i = 0, we have F 0m Vm = 0 (due to (4.1.10)), while ρB Fˆ 0m Vm recalls the product −c2 (B · V)divB. Thus, the first line of (4.4.12) turns out to be equivalent to (3.4.4). The equation (4.4.12) is the generalization of (4.2.6) with N = 0 and M = 0. In spite of its elegance, it is not very convincing, since it involves two mass densities (see also the comments at the end of section 4.3). Let us try to understand what is happening. Without going into technical detail, ˆ we may make some remarks. We first note that Fˆik = −Fik , which means that, after applying the dual twice, one gets the opposite of the original tensor. Then, for any real λ, we consider the two tensors: h i 1 λFik + (1 − λ)Fˆ ik Fik = p λ2 + (1 − λ)2 i h 1 (4.4.13) λFˆ ik − (1 − λ)Fik Fˆ ik = p λ2 + (1 − λ)2
where the second one is the dual of the first one. As in (4.2.11) we have: F ik = g im g kl Fml . Moreover, we can check that the tensor Uik in (4.2.10) does not change if in place of Fik and F ik we take Fik and F ik , respectively. Therefore, the electromagnetic stress tensor does not depend on λ, even if this parameter varies in space and time. Actually, we have already observed in section 4.2 that the energy tensor does not recognize the polarization of the electromagnetic field. At this point, we can introduce the two new densities (see p also (4.2.7)): 0k 0k 2 2 ˆ ρE = ∇k F and ρB = ∇k F p, where E = λE+(1−λ)cB / λ + (1 − λ) 2 2 and B = λcB − (1 − λ)E / λ + (1 − λ) .
So, another equivalent way of writing equation (4.4.12) is: 1 (4.4.14) ρE F im Vm + ρB Fˆ im Vm ∇k U ik = c For λ = 1 the two versions are actually the same. Now, by allowing λ to vary, suppose that it is possible (at least in simple situations) to modify the polarization of the fields E and B at each point, in order to get ρB = 0. In this way, we are left with a single density ρE , which is the one to be used in constructing the mass tensor Mik = ρE Vi Vk .
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Under perturbations, an electromagnetic “fluid” may change trajectory while evolving. The direction of variation determines the orientation of the electric field. Before the disturbance, the original displacement of the fields was determined by preceding natural causes. Fig. 4.4
Let us restate the situation in brief. Every non-trivial electromagnetic wave presents regions where the classical divergence of any of the two fields is non-zero. The electromagnetic energy tensor does not distinguish between the two types of fields (electric or magnetic). In the end, what matters is the intensity of the wave and the modality of propagation of its fronts, without paying attention to the way each front has been parametrized. We can associate a fluid in motion at the speed of light with the wave. Synchronized with the frequency of the wave, we assume that the density ρ of the fluid oscillates. Independently of the actual orientation of the fields E and B, we can locally build two other fields E and B, so that the intensity of the first one oscillates together with ρ and the second one satisfies divB = 0. This fictitious change of polarization has no influence on the electromagnetic energy tensor. The 4-divergence of E, when different from zero, is used to construct the mass tensor.
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In principle, the fields E and B are not directly associated with E and B. However, in the natural evolution of electromagnetic phenomena, the two entities usually coincide. As a matter of fact, all the examples analyzed in the previous section satisfy ρB = 0 and ρB = 0, hence they were already well suited to the case λ = 1, corresponding to E = E and B = cB. In particular, the case in spherical coordinates simulates the real behavior of a wave generated by an infinitesimal electric dipole oscillating in a vertical direction. Somehow, the dipole imparts mechanical oscillations to the fluid, in the same direction as the electric field. Formally, we can now exchange the role of the fields E and cB, by polarizing the spherical wave by 90 degrees. In this new situation, we have divE = 0, ρE = 0 and divB 6= 0. By choosing λ = 0, we realize the condition ρB = 0 and the fictitious fields E and B turn out to be anti-rotated by 90 degrees. Therefore, there is no longer a coincidence of E and B with the corresponding E and cB. Nevertheless, a spherical wave having the second kind of polarization is difficult to observe in nature, since it should correspond to the one generated by an infinitesimal magnetic monopole (see for instance Jackson (1975), p. 251). It is certainly true that our equations are not capable of recognizing the polarization of free-waves. This is a specification that comes with the initial conditions. However, free-waves are created by some causes inherent to natural events, which have a strong influence in determining polarization. The problem resides at the origin, for example in the non-existence of magnetic monopoles (we will have a short discussion about this in section 5.2). Recall that, in equation (3.3.6), the electric and magnetic fields cannot be interchanged. Certainly, this equation influences the creation of a spherical wave through the mechanical oscillations of an electric charge. The conclusion is that, at least for free-waves, we can expect λ = 1, which implies that the direction of a possible transversal perturbation of the fluid is in accordance with that of the electric field (see figure 4.4). More precisely, this can be taken as a definition of an electric field. Suppose that an external mechanical perturbation is applied to a free-wave having E = E, in a direction not aligned with that of field E, in such a way that the direction of E changes. We may then think that the wave reacts by varying its polarization (see sections 3.1-3.3) in order to correct its posture, bringing field E to coincide once more with field E. This behavior, which matches reality, actually distinguishes the electric field. In this way, a different perception of electric and magnetic phenomena clearly emerges, because of the asymmetry of (3.3.6).
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Chapter 5
Building matter from fields
5.1
Adding the pressure tensor
We propose the last improvement to our set of equations by correcting the mass tensor in (4.2.17) as follows: Mik = ρE Vi Vk + gik p
(5.1.1)
0m
where ρE = ∇m F . In (5.1.1), the scalar p is dimensionally equivalent to mass×(acceleration)2 /charge. Although the dimensions of p do not correspond to those of a classical pressure, its role will be practically equivalent. Therefore, we will deliberately confuse p with a real pressure. Thus, by taking the 4-divergence, we get (see also (4.2.18)): ∇k M ik = ρE Gi + ∇k (g ik p) = ρE Gi + g ik
µ ∂p = ρE F ik Vk ∂xk c
(5.1.2)
where we used that ∇k g ik = 0 (see (4.1.19)). Therefore, (4.1.23) can now be replaced by: DV i µ im g im ∂p + Γijk V j V k = F Vm + (5.1.3) Dτ c ρE ∂xm For i = 1, 2, 3, in Minkowski space, the above set of equations is equivalent to (3.4.11), corresponding to Euler’s equation for compressible fluids. For i = 0, considering that DV 0 /DT = 0 for V 0 = c, one gets: ∂p = µ(divE)(E · V) (5.1.4) ∂t This says that we have to drop the condition E · V = 0 (see (4.1.10)), which has hitherto been taken for granted. Therefore, the non-orthogonality of the two vectors E and V is responsible for the development of pressure. 111
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In this chapter we will also drop the constraint |V| = c (see (3.3.7)). This does not mean that the length of V can be arbitrary. In fact, in the appropriate metric space, we need to fulfill condition (4.1.21), stating that the wave-fronts must develop by following the eikonal equation (see (4.1.22)). Note that we can include equation (4.1.21) within the metric tensor, as was done in (4.3.28). For the solutions examined in section 4.3, although the space-time modified by the waves was not perfectly flat, in the end we always found |V| = c. We are going to examine the behavior of turning fronts (see figure 4.1). Therefore, from the reference viewpoint of a flat metric, V cannot have constant norm. The property that the rays all propagate at the same speed c will be true however in the deformed metric, specified by Einstein’s equation. By scalar multiplication of the equation in (3.4.11) by V, one gets: −µρE · V = ρG · V + ∇p · V (we cannot use that G · V = 0 since |V| is not constant along the stream-lines). Hence, substituting in (5.1.4), we obtain: ∂p = − ∇p · V − ρG · V = − div(pV) + p divV − ρG · V (5.1.5) ∂t which yields: Dp ρ D|V|2 = − ρG · V = − Dt 2 Dt
(5.1.6)
∂p = − div(pV) (5.1.7) ∂t that is the continuity equation for pressure, provided suitable assumptions are made. p divV = ρG · V
⇐⇒
Equation (5.1.6) is our electromagnetic version of the Bernoulli’s principle (see for instance Batchelor (1967) or Landau and Lifshitz (1987)). Recalling the continuity equation for ρ and using (3.4.9), we easily rewrite (5.1.6) as: ρ 1 ρ Dρ D p + |V|2 = |V|2 = − |V|2 divV (5.1.8) Dt 2 2 Dt 2 In the cases in which the right-hand side is equal to zero, the above is exactly the Bernoulli’s equation (after adjusting the dimensionality). The choice in (5.1.1) is equivalent to modifying the right-hand side of the Einstein equation (4.3.1) as follows: µ Tik = 4 (− µUik + Mik ) c µ = 4 µg mj Fim Fkj − 14 µgik Fmj F mj + ρE Vi Vk + gik p (5.1.9) c
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An important relation is obtained by taking the trace of (4.3.1) (compare to (4.3.4)). This yields: 4χµ χµ (5.1.10) −R = − 4 g ik gik p = − 4 p c c bringing us to the following “equation of state”: p =
c4 R 4χµ
(5.1.11)
that replaces (4.3.5) and has to be coupled with (5.1.3). It is worthwhile to observe that, if ρE = 0, we cannot pass from (5.1.2) to (5.1.3). Therefore, when ρE = 0 in a region of space, we will also assume p = 0 and R = 0. We can give a physical meaning to the scalar relation (5.1.11) by arguing that the existence of a non-vanishing pressure p implies a curvature of the space-time. As far as free-waves are concerned, we have R = 0. The sequence of wave-fronts evolves in such a way that they do not influence each other during the motion, so that we expect no pressure to develop. On the contrary, the fronts of a constrained wave, as a result of their bending and twisting, tend to develop internal mutual compressions, and the scalar curvature is going to be different from zero. An example is the pressure exerted by light when hitting an obstacle (see also the caption of figure 3.4). Of course, the explicit determination of the path of a constrained wave turns out to be a more difficult problem, as we have to deal with a new unknown p and the extra equation (5.1.11). Note that (5.1.11) is not just a simple equivalence, but involves the second derivatives of the coefficients of the metric tensor. Theoretically, considering that in (5.1.3) one finds ∂x∂m p, the whole differential problem becomes of the third order. A relation between density and pressure is usually recovered by requiring ∇k M ik to be zero (see Fock (1959), section 32). This assumption is too strong. Instead we imposed ∇k T ik = 0, where Tik in (5.1.9) contains the sum of both the electromagnetic and the kinematic contributions. Equation (4.3.1) now represents an energy balance where we read on the left-hand side the curvature tensor that can be assimilated to a potential energy (although we are aware of the comments in Misner, Thorne and Wheeler (1973), section 20.4); the right-hand side takes into account the electromagnetic sources and the related kinematic energy (see the scheme (3.4.12)). Thanks to (5.1.11), the trace is the same on both sides, and by computing the 4divergence of T ik , we end up with the entire set of equations. This is a wonderful and very consistent result.
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We recall that A. Einstein himself made an adjustment of the trace by proposing the variant: Rij − 12 gij R+Λgij = −κTij , Λ being the cosmological constant (note instead that our pressure p is not constant). The correction was made to contrast the contraction properties of gravity in a stationary universe, by creating a negative pressure counter-balance. As we shall see in the next sections, a similar idea keeps an elementary particle in equilibrium. It is interesting to observe that (5.1.3), for i = 0, is also deducible from the equations in (4.1.28) and energy conservation. We check this in Minkowski space. It is a matter of multiplying (3.3.4) by E and (3.3.5) by B (with divB = 0). Since we cannot use the orthogonality of E with respect to V, equation (2.1.5) now becomes: 1 ∂ (|E|2 + c2 |B|2 ) + c2 div(E × B) = − (divE)(E · V) 2 ∂t
(5.1.12)
From the above identity, by the relation ∇k T ik = 0 for i = 0 (recall the expression of ∂U 0k /∂xk provided in (4.2.3)), it is easy to arrive at (5.1.4). Finally, another way of writing (5.1.12) is as follows: p ∂ 1 2 |E| + c2 |B|2 + = − c2 div(E × B) (5.1.13) ∂t 2 µ With the introduction of the pressure term we are now able to include in our theory situations in which the electric field has a component along the direction of propagation of the wave. In other words, this means that the pair of electromagnetic vectors does not lie on the plane locally tangent to the wave-front surface. An extreme case is given by the following shearwave, with the electric field perfectly orthogonal to the wave-front (see figure 5.1). In Cartesian coordinates, we choose: E = 0, 0, c sin ω(t − z/c) , B = 0, V = (0, 0, c)
so that E is lined up with V. We have: divB = 0, curlE = 0, curlB = 0, ∂ ∂t E = −(divE)V = (0, 0, cω cos ω(t − z/c)). Hence (3.3.4) and (3.3.5) are satisfied. Moreover, after defining p = − 21 µc2 [sin ω(t − z/c)]2 , equations (3.4.11) and (5.1.4) are also satisfied. It is very important to observe that, although E and B could be generated by potentials A and Φ satisfying the Lorenz condition, it is easier to enforce the Coulomb gauge by assuming A = 0 and Φ = −(c2 cos ω(t − z/c))/ω. Note also that, at any point, the integral of the electric field is zero in a period of time of width 2π/ω. The pressure turns out to be negative; is there any relation with the Casimir effect?
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In this particular front the magnetic field is zero and the electric field is lined up with the velocity field V. In this way, we can build global spherical solutions that, although non-stationary, simulate the radial field emitted by a point-wise charge.
Fig. 5.1
In the relativistic framework, it is sufficient to define the metric tensor as: 1 0 gij = 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −σ 2
(5.1.14)
√ with σ = ( χµ/cω) sin ω(t − z/c). This brings to R = 2ω 2 and p = 2 4 ω c /2χµ. In this metric space we have ρE = 0. Therefore, including the pressure term, the tensor in (5.1.9) becomes:
Tik
0 ω2 0 =− χ 0 0
0 1 0 0
0 0 1 0
0 0 0 0
(5.1.15)
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One can actually check that the Einstein equation is satisfied by gij given in (5.1.14). Surprisingly, the energy tensor does not contain information about the oscillating behavior of the electric field. This is implicitly imposed through (5.1.11), corresponding to a second order differential equation, where the ratio σ ′′ /σ is required to be constant. Note that there is no need, in the new metric, to define the velocity vector, since equation (5.1.3) is not meaningful. In fact, (5.1.3) should be derived from the relation ∇k T ik = 0, after collecting the terms multiplied by ρE (see (4.2.18) and (4.2.24)). But, in the metric space defined by (5.1.14), one has ρE = 0, so that (5.1.3) does not necessarily hold in this framework. According to (4.3.28), after defining Vk = c 12 (1+σ)2 , 0, 0, σ , V k = c 1, 0, 0, − 12 (1+σ)2 /σ and γ33 = −2(σ/(1+σ))2 , an alternative metric is given by: 0 0 0 1/σ (1+σ)2 0 0 σ 0 −1 0 0 −1 0 0 g ij = 0 gij = 0 0 0 −1 0 0 −1 0 2 1/σ 0 0 −(1+1/σ) σ 0 0 0 Translating into spherical coordinates, we can model the evolution of a genuine radial electric pulse. For a given function g, we introduce the fields E = g(ct − r)/r2 , 0, 0 , B = 0 and V = (c, 0, 0). Now, the variable r has to be greater than some r0 > 0. In this way (3.3.4) is true, since we have: ∂E1 cg ′ 2E1 ∂E1 = − ρV1 = 2 = c + ∂t r ∂r r
Moreover, we can suitably define a pressure p such that: DV/Dt = 0 = −µE − (∇p)/ρ, where the differential operators are taken in spherical coordinates. Note that pure stationary electric fields, such as E = (±1/r2 , 0, 0), are not allowed by the model (being ρ = 0, we are unable to determine p). However, we can add to the first component of a time-dependent electric field a stationary term with zero divergence, i.e.: E = ([q+g(ct−r)]/r2 , 0, 0), where q is a constant. Stationary fields could then be conveniently modelled by slow-varying fields. In the example just considered, the longitudinal signal starts from the surface of a given sphere and propagates on concentric spheres of growing radius r. The amplitude of the signal decays as r−2 , so that the global energy is preserved (note that, since B = 0, the Poynting vector field is not defined). As a matter of fact, at speed c, the signal can reach any point of
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the space, but the quantity |E|2 , integrated on the whole space, turns out to be bounded with respect to time. The situation is completely different from the one analyzed in the first chapter, where the fields radiated by a sinusoidal infinitesimal dipole were studied. In that case, there was a continuous production of energy via the central source. Consequently, the decay of the signal was only proportional to r−1 . By integrating the energy field on the whole space, we get an unbounded function of time. At time t = +∞, the wave fills the entire space and the produced energy is infinite. It is also interesting to remark that, among the equations ruling the dynamics of the fields, the only one actually displaying the constant c is: ∂E/∂t = c2 curlB − (divE)V. Thus, when B = 0, such a constant disappears. Hence, we could in principle assume the existence of pure radial electric waves travelling at any velocity V, not necessarily equal in intensity to the speed of light. As we shall see later in sections 5.3 and 5.4, the new radial waves here introduced, are akin to those emanated by neutrinos. We will explain there what kind of mechanism allows for the production of such longitudinal-type waves. We can use the results of this section with the aim of explaining the initial dispersion of the light fronts in the diffraction phenomenon (the situation of section 3.2). When hitting the border of the perfectly conducting wall, the electric field is suddenly set to zero and the divergence |ρ| has a vertical peak. At the same instant, it is reasonable to assume that the velocity V goes to zero, which is equivalent to imposing a no slip condition at the obstacle. Simultaneously, |p| passes from 0 to infinity. Heuristically, this can be done compatibly with the Bernoulli’s equation (5.1.8). In truth, before the impact, one has p = 0, |V| = c and divV = 0, so that D 1 2 2 Dt ( 2 ρc ) = 0. For y = 0, we assume ρ|V| = 0. If we want to preserve the energy of the fluid along the stream-lines, p should grow to a value proportional to the kinetic energy just before the impact. Together with the modification of p, the geometrical environment changes (R 6= 0) and the light rays must modify their trajectories accordingly. In fact, if we set V = 0 in (2.1.1) we get an instantaneous rotation of the electric field. As a matter of fact, with the notation in (3.2.1), we examine the variation of the second component of E, that was initially set to zero. One gets: ∂E3 ∂B1 ∂E2 = c2 {curlB}2 − (divE)V2 = c2 = c ∂t ∂z ∂z
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+∞ = c divE = sign(E3 ) × 0 −∞
if z = 0 if 0 < z < a
(5.1.16)
if z = a
where we recalled (3.2.2). This leads to an instantaneous clockwise rotation at the points (x, 0, 0) and an anti-clockwise rotation at the points (x, 0, a). When the influence of the obstacle is over, the magnitude of |V| is restored to the value c and the fronts proceed unperturbed. These argumentations are a bit speculative. In order to validate the theory and collect quantitative information, numerical simulations are necessary. As a final result, we are able to propose a global Lagrangian function: χµ (5.1.17) L = R + 4 12 µFik F ik + ρE Vk V k − 2p c
Thus, let us see what happens by looking for the stationary points of the action function: Z √ S = − L −g dx0 dx1 dx2 dx3 (5.1.18) We vary the coefficients of the metric tensor (see for instance Dirac (1996), sections 26-28). First of all, we have (see Landau and Lifshitz (1962), section 95, and Dirac (1996), p. 54): Z Z √ √ (Rik − 12 gik R) δg ik −g dx0 dx1 dx2 dx3 δ R −g dx0 dx1 dx2 dx3 = Z Z √ √ δ Fik F ik −g dx0 dx1 dx2 dx3 = − 2 Uik δg ik −g dx0 dx1 dx2 dx3 where we considered that δg ik = −g im g kj δgmj (see Fock (1959), p. 205). Then, using the continuity equation (see Hawking and Ellis (1973), p. 67): Z Z √ √ ρE Vi Vk δg ik −g dx0 dx1 dx2 dx3 δ ρE Vk V k −g dx0 dx1 dx2 dx3 = Then, one has (see Fock (1959), p. 205): Z Z √ √ δ p −g dx0 dx1 dx2 dx3 = − 12 p gik δg ik −g dx0 dx1 dx2 dx3 Finally, putting it all together, we get: Z √ δS = Rik − 21 gik R + χTik δg ik −g dx0 dx1 dx2 dx3 = 0 (5.1.19)
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from which, in view of the arbitrariness of δg ik , one obtains (4.3.1), with Tik given in (5.1.9). Considering that Vk V k = 0 at the stationary point, the Lagrangian takes the value: L = R +
2χµp χµ2 Fik F ik − 4 2c c4
χµ2 2 2 χµ2 R 1 Fik F ik = + R + c |B| − |E|2 (5.1.20) 4 4 2 c 2 c where we used (5.1.11). Recall that for free-waves one finds L = 0. =
Correctly, the relation Vk V k = 0 is not directly obtained from the energy balance. The equation of state could be weakened by imposing: R = χµ(ρE V k Vk + 4p)/c4 , that results from taking the trace in (4.3.1). It is a fact to be proven that, if the condition Vk V k = 0 holds for initial data, it is maintained in the successive evolution.
5.2
On the existence of particle-like solutions
We begin by recalling the most fundamental results obtained so far. In Minkowski space, we started by introducing the following equations: ∂E = c2 curlB − (divE)V ∂t
(5.2.1)
∂B = − curlE − (divB)V ∂t
(5.2.2)
DV ∇p = −µ E + V×B − Dt ρ
(5.2.3)
where E is the electric field, B is the magnetic field and V is a velocity field that, for free-waves, satisfies: |V| = c
(5.2.4)
The constant µ is a charge divided by a mass, and c is the speed of light. The pressure p is only defined when ρ = divE 6= 0.
In the above equations the cross-products are right-handed and the triplet (E, B, V) is also right-handed (s = 1, according to (3.3.10)). Similar versions of the equations are available when dealing with left-handed triplets (see (3.3.11), (3.3.12) and (3.3.14) for s = −1). Since there exists anti-matter, which is the mirror image of matter, we have to be prepared
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to handle both situations (using, for example, only right-handed crossproducts). Then, in chapter 4, we wrote the equations in covariant form. Equation (5.2.2) is verified (with divB = 0) by computing the electromagnetic tensor Fik as ∇i Ak − ∇k Ai (see also (2.4.3)). Moreover, we have for i = 0, 1, 2, 3:
ρE i V c µ im − F Vm = ∇i p c
∇k F ik = ρE V j ∇j V i
(5.2.5) (5.2.6)
where ρE turns out to satisfy the continuity equation and ∇i p = g im ∂x∂m p. The normalizing condition (5.2.4) takes the form: V i Vi = 0
(5.2.7)
The scalar p is coupled with the scalar curvature through the following equation of state: p =
c4 R 4χµ
(5.2.8)
where χ is an adimensional constant. Note that ρE = 0 ⇒ p = 0 and R = 0. The fields and the metric tensor appear together as unknowns in the equation: Rik −
1 4 gik R
χµ mj mj 1 (5.2.9) µg F F − ρ V V − µg F F + ik mj E i k im kj 4 c4 which differs from (4.3.1) since the curvature tensor on the left-hand side is traceless (thus, according to Hawking and Ellis (1973), p. 41, it is related to the Weyl tensor associated with gij ). The 4-divergence of Rik − 14 gik R is not zero, but, with the help of (5.2.8), leads to the gradient of the pressure term appearing in (5.2.6). To build up the right-hand side of (5.2.9) one has to consider the rules provided in section 4.2 (in the construction of ρE = ∇m F 0m remember to take into account the warnings at the end of section 4.4). Such a coupling corresponds to quite a complex system that is able to describe space-time geometry in conjunction with electromagnetic phenomena. There may not be uniqueness of solution, since the symmetric tensor gij is only determined up to 6 degrees of freedom out of 10 (see Hawking and Ellis (1973), section 7). The set of equations (5.2.5)-(5.2.6), =
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which are true independently of the metric tensor, follow by taking the 4divergence on both sides of equation (5.2.9). Therefore, one can first solve these equations in the flat space and then compute the associated geometry via (5.2.9). Stable configurations, with R = 0, were actually found in section 4.3. Of course, the cases in which R 6= 0 are of substantial interest. In fact, nonzero curvature is also found in the absence of masses (see for instance the metric (5.1.14)). It is enough to have an electromagnetic phenomenon evolving in a complicated way (as, for example, the impact of two photons), in order to activate the pressure term. Due to the strong link we have established between electromagnetism and the geometrical properties of space, we should be allowed to talk about “empty space” only when there is no electromagnetic signal at all. Our set of equations contains the embryo of some of the main laws of physics. In (5.2.1) and (5.2.2), we recognize the equations of electromagnetism, more or less with the same structure as the Maxwell equations. Furthermore, (5.2.4) is the eikonal equation, so that the Huygens principle is also latent. On the right-hand side of (5.2.1) we partly recognize the Amp`ere law. The equation (5.2.3) expresses the Lorentz law, anticipating the Newtonian law in the form of momentum equation for the dynamics of fluids. In fact, we claimed that the light rays can be assimilated to streamlines of a certain fluid of density ρE . Moreover, we know that a continuity equation holds for ρE . Later, we will check that quantum phenomena are also contained in the model. Throughout the paper, we assumed we were in a universe that we could call “pre-Coulombian”. As a matter of fact, we developed a theory of electromagnetism without introducing any charges, and we spoke about fluids without having any masses. The only elements at our disposal were the fields. Here comes the big question: can we now build matter from these fields respecting the rules presented? In other words: can an elementary particle be a “solution” to our set of equations? A particle is quite a complicated thing. It has charge, magnetic momentum, spin, mass. It evolves and interacts with other particles according to the rules of quantum mechanics. Can we encapsulate all these factors in a solution localized in space? Although the problem is extremely complicated, we start by collecting some pieces of evidence, whose details are shortly to be discussed, in support of the possibility of creating particles from fields. Technical details and quantitative results will be examined in
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the next two sections, where we come out with a credible solution. Possible links with quantum like effects are discussed in chapter 6. We recall that other authors, through a qualitative analysis, have followed a similar idea of building electrons from photons (see for example William and van der Mark (1997) and the references therein). Experimentally, this could be achieved with the help of an extraordinary amount of energy (see for instance Burke et al. (1997)). We can give a rough idea of how a “particle solution” looks by examining figure 5.2 (see also figure 3.7), that shows, projected on a plane, the rotation of the fronts around an axis. From a qualitative viewpoint, field E oscillates radially, but, on average, mainly pointing either inward or outward. This creates the polarity of the electrical charge. Field B is parallel to the rotation axis. The rays form closed orbits and their vector curvature G points towards the center, producing a non-vanishing gravitational vector field. If the sign of E is changed, then G again points toward the inside. The displacement of field V matches the idea that something is “spinning”, and the associated electromagnetic fluid corresponds to a kind of vortex. We will discover in section 5.4 that this is not exactly what happens. Figure 5.2 is well-suited to studying the 2-D case, but it is inappropriate for the 3-D case. However, let us remain at this level of simplicity for the time being and obtain some qualitative conclusions. In reference to figure 5.2, let us suppose that the particle is an electron. The field E should then be directed toward the center and, using the standard vector product ×, B should point downwards. Nevertheless, a negative charge rotating clockwise produces a spin angular momentum pointing downward and a magnetic field pointing upward, which is in contrast to what we found previously. This happens because we are not using the suitable vector product ×. In fact, the correct one is left-handed. Since the magnetic dipole moment is independent of the sign of ×, the change of parity now confirms that B points upward (as actually illustrated in figure 5.2). If we want to maintain the same set of equations (s = 1) and righthanded cross products, we can solve the problem by just changing the sign of the electric field, so that the electron has a chance of existing only if the electric field vectors point outward. This will comply very well with the rule stating that currents flow from a positive pole to a negative one. Answering the question raised in section 2.3, the correct side of the mirror is the one where × and (E, B, V) have different orientations (chirality). This means that in equations (3.3.11), (3.3.12) and (3.3.14) we must set
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Fig. 5.2 Qualitative displacement of left-handed fields in the pure electromagnetic model of a 2-D electron.
s = −1. As we said, if we do not wish to modify the model equations, we need to switch the polarity of the electric fields. In such a way, an electron turns out to be positive and a proton negative. The asymmetry of our universe and, consequently, the determination of its parity, is a problem that emerged about 50 years ago. The effects of this dichotomy were predicted by T. D. Lee and C. N. Yang (see for instance Wehr, Richards and Adair (1984), p. 534), but the reasons for preferring left or right have still to be found. If the above arguments are free from errors, then here may lie the solution to the problem. More information will come in section 5.4, where we deal with particles and anti-particles (see for instance figure 5.11). Nevertheless, one can see that such a situation is not compatible with equation (5.2.4). The reason is that the outer orbits of the light rays are longer than the inner orbits, while (5.2.4) tells us that the information
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propagates at constant speed. In order to have a chance of finding solutions of the form described above, the use of the general relativistic framework is unavoidable. The modification of space-time geometry allows for the preservation of the momentum of inertia (a typical mechanical concept) and provides the “glue” that keeps the particle together. The rotating wave follows the geodesics of the new metric. At the same time, the curvature of such geodesics has to be compatible, through (5.2.6), with the electromagnetic setting. The geometry alters the relation between space and time in such a way that the rays, always travelling at speed c, can accomplish paths of different length in the same amount of time. This recalls the problem of the rigid rotating disk in general relativity. It is clear that the particle solution involves the use of the whole set of equations. Therefore, its determination, even from the point of view of numerical computations, is a demanding problem. Finally, by heuristic arguments, one can recognize that solutions similar to that of figure 5.2, where the magnetic field is exchanged with the electric one, should be forbidden by equation (5.2.6). This would imply the impossibility of building magnetic monopoles (see figure 5.3). In fact, we mentioned in section 4.4 that stable situations in which divB is different from zero seem to be forbidden at the origin. Nevertheless, we can imagine that a momentary formation of regions where divB 6= 0 may occasionally occur. As we proceed with our investigation, we will discover that figure 5.2 is oversimplified. When we try to carry out computations in the 3-D case, we realize that the spherical form is unrealistic. Alternatively, we will embrace the idea of a toroid shaped particle. The dream of explaining the structure of matter from the electromagnetic fields has a long history. In Poincar´e (1906), the electron was modelled as a moving continuum of charges. It was observed that, without some bonding forces (different in nature from those of the electromagnetic type), the particle was unable to keep its shape, tending to diffuse. The hypothesis that the bonds were a consequence of classical gravitational fields was taken into consideration, but was soon discarded because of their weakness. In 1906, general relativity was still to be developed. As far as we are concerned, the stability of the electron is only a matter of the geometrical properties of the space-time. The physical vector quantity G may actually be very small. Nonetheless it is the path of the geodesics, and not the intensity of G, that keeps the particle together. The gravitational vector field G is just a by-product of the shape of the geometry and does not
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Qualitative displacement of the fields in a magnetic monopole. The structure is not going to be stable, since the orientation of the vector G is not compatible with the curvature of the rays. Fig. 5.3
take an active part in the stabilization process. Similar observations are reported in Burinskii (2000), where the author makes a distinction between the topological properties of certain geometries and the effective mass that, in the case of the electron, is practically negligible. One positive aspect is that particle solutions are expected to be extremely stable (an electron is quite a difficult object to destroy). Another aspect is that they are in some sense “unique” (there is only one type of electron or proton), and this property raises other questions. The equations (3.3.1) and (3.3.2) are “scalable”, meaning that we can multiply the fields of a free-wave by a constant, once more obtaining a solution. Thus, free-waves may be of any size and intensity. But, if we take into account constrained waves, then this property is no longer true, since (5.2.4) is not
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a scalable equation. This result does not penalize our theory. In fact, it may give more strength to it. As a matter of fact, we cannot have electrons of any size! Nevertheless, we have no basis for quantifying the values of the various parameters, unless we find the particle solution explicitly. Before ending this section, let us examine a possible candidate for the geometrical framework of a particle. This is given by the famous KerrNewman solution (see Kerr (1963) or Hawking and Ellis (1973), section 5.6). Let η ≥ 0 be a constant. One then considers the Boyer-Lindquist ˆ φ) (see Boyer and Lindquist (1967)), which coordinate reference frame (ˆ r, θ, is obtained by parametrizing the ellipsoid: z2 x2 + y 2 + = 1 rˆ2 + η 2 rˆ2
(5.2.10)
in the same way a sphere is parametrized in classical spherical coordinates (see also Henrici (1974), p. 294). As a matter of fact one has: p p y = rˆ2 + η 2 sin θˆ sin φ z = rˆ cos θˆ x = rˆ2 + η 2 sin θˆ cos φ
The variable rˆ is now related to the distance from the disk x2 + y 2 ≤ η 2 with z = 0. Next, for certain dimensional constants m and q (to be specified later in section 5.4), we define: Ξ =
2mˆ r − q2
(5.2.11)
rˆ2 + η 2 cos2 θˆ
If the signature of the metric is (+, −, −, −), the Kerr-Newman geometry gij turns out to be associated with the following tensor:
1−Ξ 0
0 −
2
rˆ + η rˆ2
2
+ η 2 cos2 θˆ
0
0
Ξη sin2 θˆ
0
We have:
Ξη sin2 θˆ
0 −1 −Ξ
0 0 2 2 2 ˆ 0 −(ˆ r + η cos θ) 2 ˆ 2 2 −(ˆ r + η ) sin θ 0 −Ξη 2 sin4 θˆ
√ ˆ sin θ| ˆ −g = (ˆ r2 + η 2 cos2 θ)|
and
R=0
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The constant q is usually related to the presence of charge. Then, by defining the potentials: ! qη rˆ sin2 θˆ q rˆ (5.2.12) , 0, 0, (A0 , A1 , A2 , A3 ) = rˆ2 + η 2 cos2 θˆ rˆ2 + η 2 cos2 θˆ we get the equation: Rik −
1 2 gik R
= Rik = 2Uik
(5.2.13)
where Uik is the standard electromagnetic stress tensor (see section 4.2). There is no mass tensor since we have ρE = 0. Note that, when q = 0, then: Rik = 0. For η = 0, r = rˆ, θ = θˆ and q = 0, the Kerr-Newman solution reduces to the well-known Schwarzschild solution, corresponding to the metric tensor:
gij =
1 − 2m/r 0
0
0 −1
−(1 − 2m/r)
0
0
0
0
−r
0
0
0
0 2
0 −r2 sin2 θ
(5.2.14)
The Kerr-Newman geometry represents a model for the rotating (stationary) charged black-hole. The spin axis is oriented along the z-axis. The momentum is proportional to the constant η. The setting is actually similar to the one we are looking for. Perplexities about the sign of the right-hand side of (5.2.13) were raised in section 4.3. Some further comments are now necessary. First of all, we can adjust the potentials with a multiplicative constant in the following way: ! r qˆ r qη rˆ sin2 θˆ c2 2 (5.2.15) , 0, 0, (A0 , A1 , A2 , A3 ) = µ χ rˆ2 + η 2 cos2 θˆ rˆ2 + η 2 cos2 θˆ Such a trivial modification leads us to the equation: Rik −
1 2 gik R
=
χµ2 Uik c4
(5.2.16)
which is consistent with (5.2.9). There is a problem. If the signature of the metric was (−, +, +, +), the Einstein equation (5.2.16) would correctly describe, at a certain distance,
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the gravitational behavior of an object of positive mass (proportional to m). Nevertheless, we decided to work with the signature (+, −, −, −). Thus, some correction has to be made in order to avoid negative masses. The answer is surprisingly simple, and, as far as we know, seems not to have been reported in the literature. We modify Ξ in (5.2.11) by setting: Ξ =
2mˆ r + q2 rˆ2 + η 2 cos2 θˆ
(5.2.17)
It is astounding to note that, by replacing −q 2 with q 2 , the curvature of the space changes sign, while this is not true for the tensor Uik . The new setting is now in agreement with positive masses and a signature of the type (+, −, −, −). In other words, with the expression given by (5.2.17), the equation (5.2.16) still holds, even assuming m > 0 and a metric with signature (+, −, −, −). The replacement of −q 2 with q 2 leads to another important implication. For θˆ = 0, the term g00 = 1−Ξ is actually positive if: rˆ2 − 2mˆ r + η2 − q2 > 0 which is true for rˆ large enough. A minimum value rˆh > 0, defining the so-called horizon, exists provided the following relation holds: m2 + q 2 ≥ η 2
(5.2.18)
The whole mass is then supposed to be concentrated in a domain of radius rˆh , from where no light can escape. The inequality (5.2.18) is crucial for the study of the properties of elementary particles. If Ξ was taken as in (5.2.11), the counterpart of (5.2.18) would be m2 ≥ q 2 + η 2 (see Misner, Thorne and Wheeler (1973), p. 879), which is completely useless when we try to describe electrical objects with a very small mass. Instead, in (5.2.18), m would be negligible. Such an observation adds more value to the theory, already introduced in section 4.3, stating that the sign of the right-hand side of the Einstein equation must depend on the context. Indeed, for astronomical purposes, the standard choice of the signs is valid, but for phenomena dominated by electromagnetism, the corresponding stress tensor Uik prevails and must be accompanied by the opposite sign. With the formula provided in (5.2.17) we are able to handle both situations. Note however that when we refer to a supposed presence of mass, we mean its gravitational effects according to general relativity. In our opinion, we actually feel reactive inertial masses only when an appropriate mass tensor turns out to be added on the right-hand side of (5.2.16). This should occur if an acceleration is imparted to our particle.
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In view of equation (5.2.13) we can recover, after differentiation, our set of equations. However, although (5.2.5) can be directly checked, equation (5.2.3) is not valid. In fact, one can start by defining (V 0 , V 1 , V 2 , V 3 ) = (c, 0, 0, V 3 ) and then determine the rotational speed V 3 through the relation V k Vk = 0. Considering that R = 0, p = 0 and ρE = 0, it is a straightD V i + Γijk V j V k = forward computation to discover that the relation Dτ −1 im µc F Vm cannot hold true. As mentioned in the previous section, by differentiating the energy tensor, the terms involved in (5.2.6) are all multiplied by ρE = 0. This explains why the equation of fluid motion is not valid here. The equation ρE = 0 actually replaces (5.2.6) in this case. Incidentally, we do not accept this solution as a mathematical model of a particle. Firstly, because the electric field presents a singularity when approaching the circle x2 + y 2 = η 2 with z = 0. This can be checked by examining the denominator of (5.2.12) for θˆ = 21 π and rˆ tending to zero. In addition, a black-hole somehow corresponds to an attractive mass. Photons wandering around in its vicinity tend to approach a domain of radius rˆh built around the circle x2 + y 2 = η 2 . There is no balance between the gradient of electromagnetic pressure and the centripetal attraction exerted by gravitational forces. Finally, the geometry is stationary, while we would like it to be produced by spinning solitons that cannot escape from their own trap. Other explicit solutions, having Ricci curvature zero, also exist (see Tomimatsu and Sato (1973)), but we do not believe they are interesting to us. We must think in terms of time-varying electromagnetic waves. The metric tensor itself must be time-dependent. Due to the equation of state (5.2.8), in order to produce an effective pressure it is necessary to assume R 6= 0. Therefore, we have to look for completely different solutions. We could adopt however the Kerr-Newman solution as support for more complicated time-dependent geometries. Some hints will be given in section 5.4, although a complete analysis seems to be too difficult without the use of heavy numerical computations.
5.3
Looking for 2-D constrained waves
In a classical context we couple the equations of electromagnetism with those of fluids, to find out the behavior of rotating wave-fronts. The computations will be carried out in Minkowski space. The information collected here will be useful in the subsequent section, with the aim of building 3-D stable solutions localized in space.
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The variables are expressed in the cylindrical coordinate system (r, z, φ), where no dependency of z will be assumed. Such a reference system is left-handed. Hence, we are allowed to operate with left-handed triplets (E, B, V), without modifying the constitutive equations (s = 1). We begin by providing the potentials A = (A1 , A2 , A3 ) and Φ. Namely, for some ω, we define: ! ∞ ∞ X X A = γk ak (r) cos(cωt − kφ) γk ak (r) sin(cωt − kφ), 0, a0 (r) + k=1
k=1
∞ k+1 1X ′ ak (r) cos(cωt − kφ) γk ak (r) + Φ = ψ(r) + ω r k=1
(5.3.1)
where γk , k ≥ 1 are suitable constants. For ω > 0 the motion is anticlockwise (seen from a point with z > 0). The functions ak , k ≥ 1, are of Bessel type. More precisely, we set: (k + 1)2 a′k ak + ω 2 ak = 0 (5.3.2) − r r2 from which we obtain that any ak is a linear combination of Jk+1 (|ω|r) and Yk+1 (|ω|r), i.e. the Bessel functions of the first and the second kind, respectively. The two functions a0 and ψ are not specified yet. a′′k +
With this choice, the potentials satisfy the Lorenz condition (2.5.7). Actually, we have: A1 1 ∂A3 ∂A1 ∂Φ = − c divA = − c + + ∂t ∂r r r ∂φ = −c
∞ X
k=1
γk
a′k
k+1 ak sin(cωt − kφ) + r
It is easy to check that (see (2.4.1)): 1 ∂A 1 ∂Φ 1 ∂A3 ∂Φ 1 ∂A1 E = −∇Φ − =− + , 0, + c ∂t ∂r c ∂t r ∂φ c ∂t = −
ψ′ +
∞ 1X k+1 k a′k + ak cos(cωt − kφ), 0, γk ω r r k=1
! ∞ 1X k+1 k ′ 2 ak + ak − ω ak sin(cωt − kφ) γk ω r r k=1
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B = 1 = − c
0,
1 1 curlA = c c a′0
0,
1 ∂A1 ∂A3 A3 − − , 0 r ∂φ ∂r r
131
! ∞ k+1 a0 X ′ + γk ak + + ak cos(cωt − kφ), 0 r r k=1
where we used the equation: ′ k+1 k+1 k ′ ′ a + a + a a − ω2 a = r r r
(5.3.3)
We now introduce the velocity vector V = (0, 0, v(r)). Whatever the constants γk are, the following relations can be checked by direct calculation: div B = 0 ψ′ div E = ρ = − ψ ′′ + r ∂B + curlE = 0 ∂t
∂E a′0 a0 2 ′′ = − ρ (0, 0, v) − c curlB = c 0, 0, a0 + − 2 ∂t r r
In this way we enforced equations (5.2.1) and (5.2.2). The last condition sets a relation between ψ and a0 : v a0 ′ = (rψ ′ )′ (5.3.4) c a′0 + r r The potentials also satisfy equation (2.5.8) with J = (0, 0, v/c). In addition, we note that A is not parallel to V. According to what we claimed in section 3.4, this testifies that we are not dealing with a free-wave. We finally observe that, by choosing J proportional to (0, 0, r), the general expression of B is compatible with the nonlinear wave equation (2.1.8), although we do not have that |J| = 1.
Let us now consider (5.2.3). The substantial derivative of V along the stream-lines is: DV/Dt = (−v 2 /r, 0, 0). Hence, for some pressure p, we end up with the two equations: −
1 ∂p v2 = − µ(E1 − vB2 ) − r ρ ∂r 0 = − µE3 −
1 ∂p rρ ∂φ
(5.3.5) (5.3.6)
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Therefore, by integrating with respect to φ, we can recover the pressure from the second equation: ∞ ω 2 rak k+1 µρ X ′ cos(cωt − kφ) (5.3.7) ak − γk ak + p = p0 + ω r k k=1
where p0 is a function only depending on r. By differentiating with respect to r, we get: ∞ k+1 µX ω 2 rak ∂p ′ ′ ′ = p0 + ak − γk ρ ak + ∂r ω r k k=1
+ρ
k ω2 r − r k
a′k
k+1 + ak r
cos(cωt − kφ)
where we used (5.3.3). Hence, by substituting the explicit expressions of p, E1 , B2 and ρ in (5.3.5), we obtain a rather complicated expansion. For the terms not depending on φ, we deduce the relation: h a0 i ρv 2 v ′ a0 + + (5.3.8) p′0 = − µρ −ψ ′ + c r r On the other hand, if we consider, for any k ≥ 1, the component in (5.3.5) associated with the term cos(cωt − kφ), one must have: 0 =
−
µ ρ′ ω ρ
a′k +
µ ω
k+1 ak r
k+1 ω 2 rak ak − r k ⇐⇒
+
a′k +
k+1 r
−
µ ω
k vω − r c
k+1 k ω2 r a′k + ak − r r k
vω ω 2 r ρ′ − + c k ρ
a′k
ω 2 r2 vω ω 2 r ρ′ 1− ak = 0 − + c k ρ k(k + 1)
It turns out that the above expression is actually zero if ρ is constant and v(r) = cωr/k. It follows that the fluid is incompressible. Note that, under ∂ this hypothesis, (5.3.6) is equivalent to (5.1.4), considering that ∂t p = ∂ −(cω/k) ∂φ p. It seems anyway that there are no other interesting choices for ρ and v. Thus, for any fixed k ≥ 1, the velocity grows linearly with
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the radius r, matching the idea of a rigid rotating body. Consequently, there is an increase of wavelength when moving from an inner to an outer circumference, giving rise to red-shift (see for instance Mould (1994), p. 277). In addition, the higher the mode k associated with the angle φ, the slower the angular velocity of rotation. This means that there is no way to equate all the terms of the series with zero at the same time. Therefore, in order to satisfy (5.3.5) we have to deal with a single value of k. We discard the case k = 1, because it is the only one which is not symmetric with respect to the z-axis, giving the idea of an unbalanced rotating object. Therefore, let us study the case k = 2, which should correspond to the lowest energy level. According to the previous passages, solutions of (5.2.1)-(5.2.3) are obtained by choosing V = (0, 0, 12 cωr) and introducing the following potentials: A = a(r) sin(cωt − 2φ), 0 , a0 (r) + a(r) cos(cωt − 2φ) 3a(r) 1 ′ a (r) + cos(cωt − 2φ) ω r where the function a satisfies the differential equation: Φ = ψ(r) +
a′ 9a − 2 + ω2 a = 0 (5.3.9) r r We eliminated the constant γ1 , since it will be included in the definition of a, which is actually determined up to a multiplicative constant. When r > 0, in agreement with (5.3.4), the two functions ψ and a0 can be taken as follows: 1 1 2 ψ(r) = γ r + γ ˆ log r a0 (r) = 81 γ0 r3 + 14 γˆ0 r (5.3.10) 0 0 ω 2 for some arbitrary constants γ0 and γˆ0 . The function log r could also be added to a0 . We will use this option later on. After defining ζ = cωt − 2φ, the corresponding electromagnetic fields are: a′′ +
E =−
1 ω
γ0 r +
B =−
1 2c
γˆ0 2 ′ 3a ω 2 ra 2 ′ 3a a + cos ζ, 0 , a + sin ζ + − r r r r r 2
3a 0 , γ0 r2 + γˆ0 + 2 a′ + cos ζ, 0 r
(5.3.11)
We have divE = −2γ0 /ω. Let us also note that 2cB2 = ωrE1 , implying the relation c2 B = E × V. This last expression is analogous to the one
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allowing the determination of the magnetic field generated by a rotating charge. Another property worth mentioning is the following: ω 2 r2 ∂E1 ∂ ∂(rE3 ) = 1− = (E1 − vB2 ) (5.3.12) ∂r 4 ∂φ ∂φ For ρ constant, the above formula ensures that p is a gradient and can actually be recovered from relations (5.3.5) and (5.3.6). Concerning pressure (determined up to an additive constant), we have: 2µγ0 3a ω 2 ra ′ p = p0 − cos(cωt − 2φ) a + − ω2 r 2 with p′0 = −
γ0 c2 ωr γˆ0 µγ0 (4 − ω 2 r2 ) − γ0 r + 2 2ω r 2
(5.3.13)
Now, for r = 2/|ω|, we get V = (0, 0, ±c). Thus, for the radius equal to 2/|ω|, we obtain the only circular orbit travelling at the speed of light. We will assume that the support of our fields is an annular region having the inner circumference of radius rm = 2/|ω| and the outer circumference of radius rM = δ/|ω| (δ > 1 to be determined). The behavior of the fields outside such a ring is not interesting to us at the moment. Within the ring, the rays move at a speed greater than c. This produces a distortion of the signal carried by the rotating wave, resulting in the generation of a component of the electric field in the direction of motion (see the example of section 5.1). We suspect that this behavior is related to the Thomas precession phenomenon (see for instance Eisberg and Resnick (1985), appendix O). Thanks to (5.1.4), an azimuthal oscillating pressure is produced. Although the fluid is incompressible (ρ constant), the changes of pressure are related to the curvature of the space-time (recall the equation of state introduced in section 5.1). The qualitative analysis carried out above can be formalized by imposing appropriate boundary conditions. Let us first recall some properties of Bessel functions (see Watson (1944)): ν ν Yν′ (r) + Yν (r) = Yν−1 (r) (5.3.14) Jν′ (r) + Jν (r) = Jν−1 (r) r r 2ν 2ν Jν−1 (r)−Jν−2 (r) Yν (r) = Yν−1 (r)−Yν−2 (r) (5.3.15) r r We then write the general solution of (5.3.9) as a(r) = σJ3 (|ω|r) + τ Y3 (|ω|r). We note that Y3 is singular for r = 0. For the moment, we shall Jν (r) =
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0.6
0.5
0.4
0.3
0.2
0.1
0
2
2.5
3
3.5
4
4.5
5
–0.1
Plot of f (r) = J2 (r) + (τ /σ)Y2 (r), for τ /σ = −0.4386 and 2 ≤ r ≤ 5. For this choice we have f (δ) = 0 and f ′ (2) = 0, where δ ≈ 4.6884. According to (5.3.11), the two time-dependent components of E1 and B2 are proportional to f (|ω|r), while E3 is proportional to f ′ (|ω|r). This implies that the first two are zero in δ/|ω|, and the third one is zero in 2/|ω|. Fig. 5.4
stay off that point. A relation between the two constants σ and τ can be established by imposing: 3a ′ = 0 for r = rm = 2/|ω| (5.3.16) a′ + r Consequently, using (5.3.3) and (5.3.14) for ν = 3, one gets the equation: [σJ3 (2) + τ Y3 (2)] − [σJ2 (2) + τ Y2 (2)] = 0
(5.3.17)
Under this hypothesis, we obtain the result that the third component E3 of the electric field is zero at the inner boundary. Thus, the wave restricted to the circumference r = rm , has no longitudinal components, travels at speed c and cB2 = E1 6= 0. Therefore, although the path is circular, the behavior is like that of a thin piece of standard free-wave. If E and B are supposed to be zero for r < rm , there is a discontinuity. At any rate, as remarked in section 4.3, this should not be a problem, due to the hyperbolic nature of
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the equations involved and the fact that the circle r = rm is a stream-line. We will return to this issue later. On the other boundary, we impose: a′ +
3a = 0 r
for r = rM = δ/|ω|
(5.3.18)
which amounts to setting to zero the time-dependent components of E1 and B2 . Equation (5.3.18) is equivalent to requiring: σJ2 (|ω|rM ) + τ Y2 (|ω|rM ) = σJ2 (δ) + τ Y2 (δ) = 0
(5.3.19)
The differential equation (5.3.9) corresponds to an eigenvalue problem, hence not all the values of δ are admissible (but only a discrete number of them). We define δ to be the minimum value (estimated around 4.6884), for which both conditions (5.3.17) and (5.3.19) are simultaneously satisfied, determining a up to a multiplicative constant. The ratio τ /σ can be computed, yielding the approximated value of − 0.4386. A plot of a′ + 3a/r is shown in figure 5.4. The solutions we have obtained are extremely meaningful. In fact, we explicitly built a thick rotating photon occupying an annular region. The size of such a region is inversely proportional to the angular velocity. By freezing the variable t, we can assign a frequency ν = k/2π to the photon, depending on the number of oscillations of the function cos kφ in the interval [0, 2π]. Then, for k > 2, we generalize (5.3.17) and (5.3.19) by considering the two equations: [σJk+1 (k) + τ Yk+1 (k)] − [σJk (k) + τ Yk (k)] = 0 σJk (δ) + τ Yk (δ) = 0
(5.3.20)
The first one leads to the determination of the ratio τ /σ. We can then recover from the second one the minimum allowed value δ > k. On the stream-line r = k/|ω|, the velocity is c. It is greater than c for k/|ω| < r < δ/|ω|. Thus, the average radius grows with ν. Depending on the zeros, maxima and minima of Bessel functions, for the same k we can also find infinitely many other annular regions. For k = 3, 4, 5, 6, the situation is schematically depicted in figure 5.5, and it is similar to the ones that are found in quantum mechanics books, when introducing the Bohr atom (see for instance Eisberg and Resnick (1985), p. 113). In the dynamic version, the closed lines of figure 5.5 turn with different angular velocities (decreasing with the radius), since the “bumps” shift along each circle at
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90 120
60
30
150
180
0
210
330
240
300 270
Schematic representation of photons following circular orbits and carrying different frequencies. Fig. 5.5
constant velocity c (compare with figure 6.1). If in (5.3.2) we make the √ substitutions ak = rψ and l = k/2, we arrive at: ψ 2ψ ′ − l(l + 1) 2 + ω 2 ψ = 0 (5.3.21) ψ ′′ + r r Therefore, a relation can easily be established with the spherical Schr¨odinger equation with a centrifugal potential (see Eisberg and Resnick (1985), p. 536), after separating the variables. The first traces of quantization are emerging, and, as anticipated at the end of section 2.1, they are not artificially enforced by the model’s equations, but they belong to specific solutions, such as those studied here. The big innovation in our approach is that we know exactly what a photon is and how the corresponding electromagnetic fields are organized. We return on the problem of quantization in section 5.5. In section 6.1, we will learn to associate a wavelength to a particle in motion.
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0.4
0.3
0.2
0.1
0
1
2
3
4
5
Plot of the Bessel function J2 . Thanks to (5.3.24), the components E1 and B2 are zero for δ0 /|ω|, where δ0 is given in (5.3.23). The component E3 is proportional to J2′ (|ω|r). Fig. 5.6
Let us now analyze the pressure term. As we said, we could ask, for instance, to have zero radial pressure at r = rM . The time-dependent part is actually zero for both r = rm and r = rM . By examining (5.3.13), it is enough to impose: p′0 (δ/|ω|) = 0
⇐⇒
γˆ0 = −
c2 δ 2 ω γ0 δ 2 + 2 ω µ(δ 2 − 4)
(5.3.22)
For r = rm , we are left with the Euler equation ρDV/Dt + ∇p = 0, where no other external forces are present (i.e.: µ(E + V × B) = 0). Hence, the stream-line is free from constraints and the gradient of the flow pressure is perfectly balanced by the centripetal force. Moreover, let us note that p does not depend on time when r = rm . However, this is not the situation we expect, since we are still neglecting gravitational effects. There is another case worth considering. We first observe that, for r close to zero, the Bessel function Jk+1 behaves as rk+1 . Since we are dealing with k = 2, one can easily see that the fields E and B can be prolonged
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up to r = 0 in a continuous manner. This would not be possible for k = 1. Therefore, we can also construct solutions, defined on a disk Ω, centered at the origin and having a certain radius δ0 . Let us suppose that γ0 = γˆ0 = 0. We also eliminate the part of the solution containing Y3 , because of the singularity of this function at r = 0. We can then determine δ0 in such a way that E1 and B2 vanish at the boundary of Ω. It is simply a matter of seeking the first root of J3′ + 3J3 /r (equivalent to J2 , if we consider (5.3.14)). It is not difficult to recover the estimate: δ0 ≈ 5.135622
(5.3.23)
A plot of J2 can be found in figure 5.6. Let us summarize (see also figure 5.7). Up to a multiplicative constant σ, from (5.3.11), we have the fields: h J (ωr) ωJ (ωr) i 2σ J2 (ωr) 3 2 sin ζ cos ζ, 0, − E = − ω r r 2 σ 0, J2 (ωr) cos ζ, 0 (5.3.24) c for 0 ≤ φ < 2π and 0 ≤ r ≤ δ0 /|ω|. They solve the set of Maxwell equations, so that divE = 0. In such a circumstance, we do not need to define the velocity vector field V, that can be, in principle, arbitrary. However, as we noticed, a natural choice for V is (0, 0, 12 cωr). Therefore, our two-dimensional particle is spinning. The speed at the boundary of Ω is V3 = cδ0 /2, more than twice the speed of light. Anyway, such a velocity is relative since it depends on the reference frame. Thus, we are unable to define a consistent intrinsic spin, just by reasoning from the hypothesis that something is turning around. Actually, we have to work on a more elaborate definition. We will face this issue later in section 5.4. Of course, other interesting solutions may be obtained for k > 2. Finally, for any given time T , let us note that (see also (2.2.10)): Z T +2π/cω Z T +2π/cω B dt = 0 (5.3.25) E dt = 0 B = −
T
T
showing that the agitated jerks of the fields have no significant impact during an extended period of time. We may also try to get ρ 6= 0 without using the stationary terms in (5.3.10). We can do this by choosing potentials of the form: f (r) cos(cωt − 2φ) A = a(r) sin(cωt − 2φ), 0, b(r) cos(cωt − 2φ) , Φ = ω
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with a different from b and f = a′ + a/r + 2b/r. For V = (0, 0, 12 cωr), by taking into account (2.5.8), we enforce the equation: c(Att −c2 ∆A) = V(Φtt −c2 ∆Φ). Then, we come to the relations: a′ 5a 4b a′′ + − 2 − 2 + ω2 a = 0 (5.3.26) r r r b′ f′ 5b 4a r 4f b′′ + f ′′ + − 2 − 2 + ω2 b = − 2 + ω 2 f (5.3.27) r r r 2 r r Now, it is easy to check that any a and b satisfying (5.3.26) also satisfy (5.3.27) (differentiate (5.3.26) and remember the definition of f ). Consequently, we have enough freedom to construct the fields E and B with ∂ ∂t E = curlB − ρV and ρ 6= 0. In addition, we have 2cB2 = ωrE1 together with (5.3.12). Unfortunately, this approach leads to a dead end, because, when we try to impose (5.3.5) and (5.3.6), we discover that admissible solutions are only obtained for a = b. In fact, being Φ time-dependent, we can have ρ constant only if ρ = 0. Thanks to (5.3.9), this brings us back to the original setting. Then, it seems that, in order to get ρ 6= 0, one must add a stationary component to the fields in (5.3.24) (see figure 5.8). In particular, as prescribed in (5.3.11), let us sum − γ0 r/ω to E1 and − 12 γ0 r2 /c to B2 . In this way, one gets: ρ = −2γ0 /ω. The radial stationary electric field is the one we would obtain inside a cylindrical condenser of infinite length, when a difference of potential is applied between the central line and the outer surface. At this point, we need to recover more information about the magnitude of γ0 . We have already used all the means at our disposal, deducible from the equations of motion. To know more, it is necessary to work on the laws of energy conservation (note that (5.1.8) is also satisfied). Together with the electromagnetic energy E = 12 (|E|2 + c2 |B|2 ) of the wave, we also have the mechanical energy of the rotating fluid. A centripetal field G = DV/Dt is associated with the curvature of the trajectories. We can propose a potential U for the field ρG. In fact, considering that ρ is constant, one has: v2 (5.3.28) ∇U = ∇(ρG) = ρ − , 0, 0 = 12 γ0 c2 ωr, 0, 0 r Therefore, we can define: U = 14 γ0 c2 ωr2 . By integrating over the set Ω, we impose the following equivalence: Z Z U ≈ E (5.3.29) Ω
Ω
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Fig. 5.7 Displacement of the electric field in (5.3.24) for a fixed t. The magnetic field is orthogonal to the page. By varying t, the picture rotates. Note that, at the boundary of the disk, E is tangential.
Note that the two terms in the above expression are dimensionally compatible. In practice, one has: Z Z δ0 /|ω| πγ0 c2 δ04 r3 dr = U = 21 πγ0 c2 ω (5.3.30) 8ω 3 0 Ω
On the other hand, for the pure stationary part of the electromagnetic wave, one has: Z Z E = 21 (E12 + c2 B22 ) Ω
= π
Z
0
δ0 /|ω|
γ02 r2 γ 2 r4 + 0 2 ω 4
Ω
rdr =
πγ02 δ04 4ω 6
1+
δ02 6
(5.3.31)
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By comparing these results, we realize that relation (5.3.29) is satisfied if γ0 ≈ ω 3 . In E we could also include the time-dependent part given by (5.3.24). To do this, we freeze the rotating photon at a certain time (say t = 0) and integrate over Ω (see figure 5.7). Now the computation is a bit more involved. However, it is not difficult to discover that (5.3.29) holds whenever σ ≈ ω and γ0 ≈ ω 3 . In the end, since Jk (r) ≈ rk for small r, we discover that the behavior of the radial electric field near the origin is E1 ≈ ω 2 r, while for the magnetic field we have B2 ≈ ω 3 r2 . At first glance, these estimates look rough, but, as checked at the end of section 5.4, they provide substantial information. Based on the above considerations, as far as the stationary part is concerned, for a given constant q, the following case is taken into consideration: 2 qω r for 0 ≤ r ≤ δ0 /|ω| δ2 0 E1 = (5.3.32) q for r ≥ δ0 /|ω| r
The above definition will be suitably reviewed at the end of section 5.4 in a three-dimensional context, where, outside Ω, the field is required to decay quadratically. Correspondingly, the stationary part of the magnetic field will be given by:
B2 =
3 qω 2 2cδ 2 r 0 qδ0 2cr
for 0 ≤ r ≤ δ0 /|ω| (5.3.33) for r ≥ δ0 /|ω|
In this way (5.3.4) is satisfied. For the density we have:
ρ =
2 2qω 2 δ0 0
for 0 ≤ r < δ0 /|ω| (5.3.34) for r > δ0 /|ω|
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A radial stationary field has been added to the field of figure 5.7. When time passes, this new field oscillates, but the arrows continue to point outwards. Fig. 5.8
The electric and magnetic fields are continuous functions, although not differentiable. We evaluate the integral of ρ over the disk Ω: Z δ0 /|ω| Z ρr dr = 2πq (5.3.35) ρ = 2π Ω
0
It is very interesting to observe that the last expression does not depend on the parameter ω, which also determines the size of Ω. This is also true for the magnitudes of the fields outside the domain Ω. The idea here is to simulate charged particles. Hence, the size of a particle has no influence on its charge. Is this property a partial indirect proof of the uniqueness of the elementary charge? Using (5.3.13), we can evaluate the radial pressure gradient at the
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boundary of Ω, obtaining: q|ω|ω 2 δ02 − 4 ∂p δ0 ′ 2 = p0 (δ0 /|ω|) = c + qµ 2 ∂r |ω| 2δ0 δ0
(5.3.36)
We conclude that the acceleration ρ−1 ∇p , present at the boundary of Ω, may differ depending on the sign of q. Therefore, if we want to contrast this effect with gravitational attraction, point-wise masses of different magnitude should be placed accordingly at the center of the disk. As a matter of fact, for q > 0, we will simulate an electron, for q < 0, a proton. Note that, in the 3-D version, the field E1 in (5.3.32) is going to be equal to q/r2 outside Ω. Thus, formula (5.3.36) must be reviewed accordingly. We can stop at this point and recapitulate. We evaluated the electromagnetic part of our solutions by solving the equations of the curved motion of a wave in flat space-time. This, of course, is not enough to guarantee the energy balance, since all the gravitational part is still missing. First of all, the relation (5.2.4) is not verified. This means that, in the unknown modified geometry, we have to enforce condition (5.2.7). In fact, only through an appropriate metric can we ensure that the rays proceed at the speed of light, independently of the distance r. The second, and more important, issue concerns the possibility of creating a central time-dependent gravitational environment. The attractive force exerted by this field should be able to compensate the radial gradient of pressure, resulting in the system remaining stable. We expect this to happen only for specific values of ω. We soon admit that, due to its complexity, we will not be able to solve Einstein’s equation in this case. In order to operate in the relativistic framework, we start by introducing the coordinate system: (x0 , x1 , x2 , x3 ) = (ct, −r, −z, −φ). Then we need suitable potentials. Drawing on (5.3.1), we define: (A0 , A1 , A2 , A3 ) = (Φ, A1 , A2 , rA3 ) (note that the last entry has been multiplied by r). As usual, we define the electromagnetic tensor in cylindrical coordinates (see also Atwater (1994), p. 124):
Fik =
∂Ai ∂Ak − ∂xi ∂xk
0
E 1 = 0 rE3
−E1
0
0
0
0
0
− 12 ωr2 E1
0
−rE3
1 2 2 ωr E1 0 0
(5.3.37)
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where we recall that A2 = 0 and ωrE1 = 2cB2 . In the space induced by the metric: 1 0 0 −1 gij = 0 0 0 0
0 0 −1 0
0 0 0 −r2
we recover the equations in classical cylindrical coordinates (recall that √ −g = r). In particular, using (5.3.10), one obtains: 1 ψ′ 2γ0 1 ∂(E1 r) ∂E3 = ψ ′′ + = + ρE = ∇k F 0k = − r ∂r ∂φ r r ωr Furthermore, equation (5.2.5) for i = 3, yields: V 3 = −c. Consequently, we deduce: (V0 , V1 , V2 , V3 ) = c (1, 0, 0, r2 ). Thus, the condition Vk V k = 0 is not satisfied. This is obvious, because the geometry gij is the one automatically generated by the change of variables in cylindrical coordinates, and it is too poor for our purposes (basically, we are still in a flat space). Concerning the situation pertaining to (5.3.24), an interesting metric is given by the following tensor: 1 2 1 0 0 2 ωr 0 2 −E1 0 −E1 E3 (5.3.38) gij = 0 0 −1 0 1 1 2 4 2 2 2 −E1 E3 0 −E3 + 4 ω r − β 2 ωr
where β is an arbitrary constant. In the new corresponding framework, the contravariant electromagnetic tensor takes the form: 0 1 0 0 |β| −1 0 0 0 F ik = √ (5.3.39) 0 0 0 0 −g 0 0 0 0
exhibiting only the radial component of the electric field. This is true independently of the choice of E1 and E3 . We have g = −(βE1 )2 . By defining: 2 −2 2E3 , 0, , 0, 0, 1 V k = c 1, V = cβ k E1 (ωr2 + 2β) ωr2 + 2β ωr2 + 2β
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we also get V k Vk = 0. The equations in (5.2.5) are satisfied and ρE = 0. Therefore, because of (5.2.6), we must impose p = 0. Unfortunately, the metric gij is not the solution to the Einstein equation. This can be checked for instance by noting that R 6= 0, which is in contrast with the equation of state (5.2.8). Finding the right geometry is not an easy task, even from the numerical viewpoint. Note that we are looking for time-periodic solutions, therefore we cannot rely on the existence results available for the initial-value problem. In fact, we are a bit suspicious about the possibility that such solutions exist. We must not forget that we are dealing with the 2-D case, and that our rotating photons have infinite energy, since they are unbounded in the z direction. The 3-D case is indeed more realistic, although the computations become far more complex. We leave the problem to the experts. Nevertheless, we are free to imagine what is expected to happen in the modified geometry. First of all, from the experience developed in the previous chapters, we infer that stationary solutions alone are not sufficient to produce an adequate space-time curvature. This is why a dynamical component has always been added to the stationary fields examined in this section. Subsequently, the space-time deformation forces all the circular rays belonging to Ω to travel at speed c. However, this correction will be possible up to a certain limit (the boundary ∂Ω of Ω). A break-down then occurs. Some of the entries of the metric tensor tend to zero when approaching ∂Ω. We encountered a similar situation in section 4.3. Due to the hyperbolicity of the equations involved, the motion in the new geometry follows the geodesics and no transversal derivatives are involved. The rays move independently and cease to exist after ∂Ω. We cannot claim that there is a discontinuity on ∂Ω because there is no exchange of information between inside and outside Ω. Inside we have “matter”, outside the void. We discuss these crucial arguments further in section 6.1.
5.4
Neutrinos, electrons and protons
It is now time to face the problem of building examples of three-dimensional solutions, sharing their properties with elementary particles. Our first step is to look for spherical generalizations of the rotating fronts analyzed in section 5.3. This research will turn out to be unsuccessful, so we have to resort to other geometric settings.
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Working in spherical coordinates (r, θ, φ), we can define the potentials: A = a(r) sin θ sin(cωt − φ), − a(r) cos θ sin(cωt − φ),
1 Φ = ω
3a a + r ′
a(r) cos2 θ cos(cωt − φ)
sin θ cos(cωt − φ)
(5.4.1)
corresponding to the 3-D extensions of those in (5.3.1) for the parameter k = 1. The function a is obtained from the differential equation: 6a 2a′ (5.4.2) − 2 + ω2 a = 0 r r Therefore, a is a linear combination of the two Bessel type functions: r−1/2 J5/2 (|ω|r) and r−1/2 Y5/2 (|ω|r). We leave the reader the tedious exercise of checking that the above potentials satisfy the wave equations: Att = c2 ∆A and Φtt = c2 ∆Φ. a′′ +
By evaluating electric and magnetic fields, we arrive at: h1 i 3a 1 1 ′ 3a a + sin θ cos ζ, a′ + − ω 2 a cos θ cos ζ, E = − ω r r r r i h1 3a a′ + − ω 2 a cos2 θ sin ζ r r 1 3a sin θ cos θ cos ζ, B = − c r i ha 2a a 2 ′ ′ cos θ cos ζ, a + cos θ sin ζ + a + r r r
where ζ = cωt − φ.
We do not like these solutions very much for reasons we shall now spell out. It is known that the function r−1/2 J5/2 (r) behaves as r2 for r tending to zero. Therefore, by examining the term r−1 (a′ + 3a/r), one deduces that the fields cannot be continuously prolonged up to the origin (there is no compatible definition of E at the point r = 0). Thus, it is disappointing to observe that our spherical structure must have an internal hole, where it is not clear what boundary constraints have to be imposed. In the 2-D case, we found a way to overcome this trouble by replacing (cωt − φ) with (cωt − 2φ). In fact, Bessel functions associated to the higher mode k = 2 decay faster near the origin. Unfortunately, in 3-D, we could not
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find any explicit solution by taking 2φ instead of φ. After trying numerous possibilities, we gained enough experience to realize that it is not just a question of luck or perseverance. Something new had to be tried. Thus, we started by looking for solutions displaying a toroid shape. As a matter of fact, we can take a vertical segment of the unbounded cylindrical solution considered in section 5.3 and bend it to form a closed loop (with average radius equal to a given constant η > 0). Let us now study this option. In order to do so, we must go back to the cylindrical coordinates (r, z, φ) and consider the potentials: A = c a(t, r, z), b(t, r, z), 0
Φ = −
∂A A ∂B + + ∂r r ∂z
(5.4.3)
where a and b are arbitrary functions, while A and B are their primitives with respect to the time variable. It is easy to check that the Lorenz condition is satisfied. The corresponding electromagnetic fields are: E = − Φr − at , − Φz − bt , 0 B = (0, 0, br − az )
(5.4.4)
If we now ask E and B to satisfy Maxwell equations (or, equivalently: Att = c2 ∆A and Φtt = c2 ∆Φ), we come to the following system of hyperbolic differential equations: 1 ∂2a ∂2a ∂ = + 2 2 2 c ∂t ∂z ∂r
∂a a + ∂r r
∂2b ∂2b 1 ∂b 1 ∂2b = + + 2 2 2 c ∂t ∂z ∂r2 r ∂r which are coupled through the boundary condition: ∂b ∂a = ∂z ∂r
(5.4.5)
(5.4.6)
(5.4.7)
requiring B = c−1 curlA to be zero at the contour of a certain domain Ω in the plane (r, z), which has yet to be determined. On the same boundary, field E will only have a tangential component, because it is parallel to the ∂ ∂ E1 − ∂r E2 = 0). The functions a and b are going to gradient (we have ∂z describe the time evolution in the plane (r, z) of a rotating wave similar to the one studied in section 5.3. The set Ω is now “centered” around a point (η, 0), for some η > 0. Note however that Ω is not a perfect circle. Bearing
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in mind that we are not in polar coordinates, we shall rename the variable r by setting y = r − η, in order to avoid misunderstanding.
∂ a To study (5.4.5) and (5.4.6), we first perform the substitutions: u = ∂z ∂ ∂ and v = ∂r b = ∂y b. Differentiating the first equation with respect to z and the second one with respect to y, yields:
1 ∂2u ∂2u ∂ ∂u u = + + ∂z 2 ∂y ∂y y+η c2 ∂t2
(5.4.8)
1 ∂2v ∂2v ∂ ∂v v = + + c2 ∂t2 ∂z 2 ∂y ∂y y + η
√ with u − v = 0 on ∂Ω. Moreover, upon substitution u ˜ = u y + η, √ v˜ = v y + η, one gets: 1 ∂2u u ˜ ˜ ∂2u ˜ ∂2u ˜ 3 = + − 2 2 2 c ∂t ∂z ∂y 2 4 (y + η)2
(5.4.9)
v˜ ∂ 2 v˜ ∂ 2 v˜ 3 1 ∂ 2 v˜ = + − 2 2 2 c ∂t ∂z ∂y 2 4 (y + η)2
to be solved in a suitable bounded domain Ω, on the boundary of which the two unknowns are related through the condition u ˜ − v˜ = 0.
We are concerned with revolving solutions associated to the second mode with respect to the angle of rotation. The functions u ˜ and v˜ will have a phase difference of 90 degrees. The idea is that, for η tending to infinity, the set Ω converges to a circle and the electromagnetic fields are exactly those given in (5.3.24) (see figure 5.7). Our scope is to find the shape of Ω for finite values of η. We then introduce the new unknown w = u ˜ − v˜ and take the difference of the two equations in (5.4.9). Assuming that the angular velocity of rotation is cω/2, we freeze the time variable and pass to the stationary boundary-value problem: w ∂2w 3 ω2 w ∂2w − + = − ∂z 2 ∂y 2 4 (y + η)2 4 w = 0 on ∂Ω
in Ω (5.4.10)
that generalizes the eigenvalue problem (5.3.9). If we require w not to be identically zero, Ω cannot be arbitrary, but has to allow λ = ω 2 /4 to
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Sketch of the set Ω. This is going to be the section of a toroid Σ, simulating the shape of an elementary particle. Fig. 5.9
be a suitable eigenvalue of the linear positive-definite differential operator: ∂2 3 ∂2 −2 . In addition, since we would like to build, for L = − ∂z 2 − ∂y 2 + 4 (y + η) the same λ, independent eigenfunctions (having different phases), the multiplicity of λ must be equal to 2. Thus, going back to the evolutive problem, we should be able to create a continuum of eigenfunctions parametrized by t, simulating the rotation. The constant ω must not be too large in order to guarantee that the singularity point y = −η does not belong to Ω. From now on, the mathematical analysis requires skill and we would prefer avoid having to face this challenging problem here. Presumably, the characterization of Ω involves the study of the shape of convex domains with optimal properties. This is the case, for example, of domains minimizing the area, when the lowest eigenvalue of a positive-definite operator is fixed. For the Laplacian with homogeneous boundary conditions the circle is the set yielding the least lowest eigenvalue (see Faber (1923), Krahn (1925)), and, due to symmetry arguments, the eigenvalues related to the
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angular modes have multiplicity 2. In our case, we can assume that there are two positive constants such that: c1 (η) ≤ 34 (y + η)−2 ≤ c2 (η). This means that Ω is required not to touch the straight-line y = −η in the (y, z) plane. Let us suppose that the minimum eigenvalue is given; then, for the ∂2 ∂2 constant-coefficient operators Lm = − ∂z 2 − ∂y 2 + cm (η) (m = 1, 2), the corresponding optimal sets Ωm are circles with different radii. Therefore, we expect Ω to be comprised between Ω1 and Ω2 (see figure 5.9). Note that, in the time-dependent version, the solutions on both circles rotate with the same angular speed. The solution on Ω should turn around in accordance. For the moment, the theoretical existence of a stationary set Ω with the above prescribed properties is only a conjecture. Since there is no dependence on the variable φ, the electromagnetic fields are automatically defined on a doughnut Σ = Ω × [0, 2π], and ruled by the 3-D Maxwell equations (including divE = 0). The fluid dynamics analog is a vortex ring (see Batchelor (1967), section 7.2 and plates 20 and 21). At the boundary surface ∂Σ, the field E is tangent, while B is zero. At every point, the average in time, during a period of oscillation, of any of the two electromagnetic fields is zero (see (5.3.25)). Outside Σ, the oscillating electric field can be extended to fill the whole space, as we outlined in section 5.1. In this case, the vectors are orthogonal to the propagation surfaces (see figure 5.1). The rays emanating from ∂Σ carry a signal decaying in intensity as the square of the distance from the source, and having zero average over a time period. Although it is not necessary to define the field V (since divE = 0 everywhere), we can introduce velocity vectors as in section 5.3. Inside Σ, for any φ, V lies on the plane of the domain Ω and the corresponding stream-lines are concentric curves (see figure 5.10). Associated with V, a sort of spinning is implicitly determined, although, this time, there is nothing actually turning around an axis. This kind of spin cannot be reduced or eliminated by a trivial change of reference, as in the two-dimensional examples of section 5.3. A spinning frequency ν = c|ω|/2π may be introduced. Oriented as the z-axis, a spin vector can be officially introduced as suggested in figure 5.10. Although its magnitude is not specified, a coherent choice should involve the Planck constant. Here, with the standard notation, we just say that the spin is 12 . We will not insist further on this concept, but, of course, a formal definition would be necessary (see also the end of section 5.5). Instead, there is no spin associated with pure photons when they behave as free-waves. We can get an “anti-spin” by switching the sign of V (see also sections 3.2 and 5.2), so that particles and anti-particles (defined as
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mirror image) can be identified according to whether the triplet (E, B, V) is left-handed or right-handed. However, in the example we are dealing with, a rotation of 90 degrees of the z-axis produces the same effect as the change of signature of V. Hence, our solution coincides with its antisolution. We point out that this construction requires at least three spatial dimensions. We also observe that it is possible to comb a hairy doughnut, while one cannot comb the hair on a sphere (remember the problems we had in section 1.3 when studying spherical waves). To see if what we got is a stable structure, we should resolve the corresponding Einstein equation to compute the metric. From the energy balance we wish to discover that the gravitational space-time deformation is sufficient to keep the object in equilibrium. Much effort was required to find explicit solutions by symbolic manipulation, but the problem seems to be unsolvable without introducing numerical approximation. However, the exercise is quite absorbing, so we shall skip this part. Our verification is therefore incomplete. Nevertheless, we proceed by providing plausible conjectures. Our guess is that, in the metric naturally induced by taking the electromagnetic tensor on the right-hand side of Einstein’s equation, the scalar ρE will remain zero. According to this, the mass tensor is still not active. Reasonably, we can state that the newly found particle has no inertial mass in the classical sense. Nonetheless, by integrating on Σ the energy of the fields and dividing by c2 , a nonzero localized relativistic mass is shown to exist. In addition, the deformation of the metric indicates the existence of a gravitational mass, or, at least, a severe perturbation of the flat space. Radiation, in the form of longitudinal (shear) waves, are also expelled by the particle, carrying gravitational information far away. In other words, the particle is not going to be an isolated system and will act like a fermion (we will develop this issue further in section 6.1). It is also interesting to observe that the 3-D curvature of a torus is negative on the side facing the spin axis and positive on the opposite side. This might provide an illustration of the competition that exists between various different forces. Let us put all these facts together. We constructed an electromagnetic solution occupying an annular core Σ in a bounded region of space. The fields defined in Σ oscillate in all directions with zero average over time. This makes the particle neutral. Any effort to impart positive acceleration, by exposing the particle to external fields, is not going to produce observable effects. As a matter of fact, we assumed that no inertial mass should be
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Displacement of the velocity field around a particle.
attributed to the particle. The mirror image of the particle agrees with the particle itself. Therefore, particle and anti-particle are the same entity. What has already been created has a name in physics: it is the neutrino. Of course, we know how mysterious this particle is, and how much controversy surrounds its properties, especially regarding mass. We fully expect to be heavily criticized for our statements. Being at the moment unable to solve the Einstein equation completely, we still do not know how big our neutrinos are. In other words, we do not know how large ω is and what the intensity of the electromagnetic fields defined in Σ is. Some hints may come from section 5.3, where we used heuristic considerations based on energy conservation principles, to recover information about the magnitude of |E| in terms of ω. This restricts the problem to the determination of the parameters η and ω, which will reasonably be related to the constant χ. If both δ0 /|ω| and η are of the order of 10−15 meters, the spinning frequency ν = c|ω|/2π is around 1022 Hertz. This prediction sounds correct. On the other hand, there is also the
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possibility that single stable isolated neutrinos do not exist in nature, but only as parts of the more complicated structures (see figure 6.3) or as basis for charged particles (see below). According to the rules of relativity, neutrinos with zero mass should move at the speed of light. If this was not true, one would find a reference frame where the spin is zero. This argument does not apply with our notion of spin. Therefore, as far as we are concerned, neutrinos are allowed to travel at any speed, removing in this way an unpleasant and unrealistic constraint. Figure 5.7 and the definition of E and B given in (5.3.24) refer to the case k = 2 (i.e. the number of oscillations along the rotation angle), but we said that other values of k > 2 should work as well. According to this, we could construct other similar versions of neutrinos. In fact, it is known that there are at least three types of neutrinos, although we do not have any evidence that they are characterized by displaying different values of the integer k. The possibility of building annular solutions by connecting the circular extremities of a cylinder, after applying a finite number of twists in the longitudinal direction, has also been investigated. To this end, we examined solutions similar to those of section 5.3, where the rotation angle is shifted by a phase depending on z. The research was not successful, but this does not exclude the existence of more complicated 3-D solutions. Conjectures about the existence of gravitational entities having toroidal structure and generated by revolving electromagnetic waves were made some time ago. They were called geons (see Wheeler (1955)). A qualitative asymptotic study was carried out by simplifying the shape of these objects and by using spheres. The results were disappointing and the idea was not developed further. The main drawback is that the discussion relied heavily on the stationary Kerr-Newman topology, that we believe not to be well-suited to these purposes. Now that we have at our disposal new theoretical material, a more serious analysis can be carried out. We believe that the theory of geons can be reviewed, acquiring greater solidity. Indeed, the arguments provided in Wheeler (1955) can lead to a better understanding of the size and the properties of our neutrinos. In addition, they can help the study of interactions between neutrinos at a gravitational level. The neutrino is the primary building block for constructing other particles and anti-particles. In fact, we know that neutrinos are by-products of high-energy interactions at the nuclear level, where the governing rules are well understood experimentally. We can mention, for instance, the pair an-
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Fig. 5.11 Displacement of the fields of particles and anti-particles in the toroid Σ. Above, there is an electron and a proton (both left-handed). Below, there are the corresponding mirror images, where we recognize an anti-proton and a positron (both right-handed).
nihilation of an electron and a positron, giving rise to pure electromagnetic emission, represented by a couple of gamma rays. In our theory, in order to obtain a charged particle it will be enough to “add” to the neutrino a stationary component carrying a suitable density ρ. We did the same in section 5.3 (see (5.3.32), (5.3.33), (5.3.34) and figure 5.8). At the end of section 5.3, we also argued that the stationary part cannot survive alone, but needs the support of a suitable dynamical neutrino component, in order to generate the gravitational environment that glues everything together. The price we have to pay, if we want to discard the time-dependent component, is the existence of points of singularity at infinity. Annular-shaped particles have been considered by various authors. We mention for instance Kanarev (2000), where the displacement of the fields
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(not justified by rigorous theory) is different from the one here examined. A qualitative ring model for particles and molecules, which is claimed to agree very well with experiment, has been considered in Lucas and Lucas (2002) (see also Bergman (1991) and subsequent papers by the same author). Earlier (and prophetic?) ring models of atoms and electrons, having some affinity with ours, were proposed in Thomson (1867) and Parson (1915). Let Γ be the ring of radius η, lying on the plane z = 0. Inside Σ, the electric field is parallel to the line pointing toward Γ and the magnetic field is parallel to Γ. The situation is schematically illustrated in figure 5.11. Now, a field V can be naturally defined inside Σ, oriented as the cross-product ± c(E × B)/|E × B| of the stationary fields. As specified in the caption of figure 5.11, four cases can occur. They correspond to the electron, the proton and their respective anti-particles. Recalling the remarks at the beginning of section 5.2, if we want to deal with matter or anti-matter in the model equations we have to correctly adjust the sign of s (introduced in (3.3.13)). By examining (3.3.14) we conclude that the quantity |E + sV × B| is equal for the electron (s = −1) and the positron (s = 1 and E → −E). The same can be said for the proton (s = −1) and the anti-proton (s = 1). From (5.3.36), we realize that the mass of the first two particles is going to be different from the mass of the other two particles. The displacement of the field B recalls the one generated inside the loops of a perfectly conductive wire wrapped around Σ, when a current is flowing along the direction determined by V (Biot-Savart law). Although only fields are involved, this is another example reproducing a simple laboratory experiment (see also the caption of figure 2.1). As remarked in section 2.3, the validity of our model is once again confirmed, since our equations are able to predict basic laws of physics even if no matter is effectively present. In the 2-D case it was discovered that, inside Ω, the divergence ρ must be constant. We do not know if this is true in 3-D. We set the basis for the analysis. We start by taking a stationary potential Φ, not depending on φ, and satisfying: −∆Φ = − Φ + Φr + Φ = ρ in Σ rr zz r (5.4.11) Φ = 0 on ∂Σ
where ρ is given.
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Then, we define: E = −∇Φ = −(Φr , Φz , 0)
B = (0, 0, B3 )
V = (V1 , V2 , 0)
where B3 , V1 and V2 do not depend on φ. Note that E is of radial type. One has: divE = ρ, divB = 0, curlE = 0. In addition (5.2.1) implies c curlB = ρV, from which we deduce: c2 ∂B3 B3 c2 ∂B3 (5.4.12) V2 = − + V1 = ρ ∂z ρ ∂r r 2
According to (5.1.4), in the stationary case, we need to impose that V is orthogonal to E. Up to a multiplicative constant, this may happen only if: B3 = (Φ + γ)/r, where γ is another arbitrary constant. We must then show that a pressure p actually exists. In other terms, by writing: DV (5.4.13) Dt we need to find conditions in order to ensure that the right-hand side is a gradient. By known properties (see also (3.4.1)), recalling that there is no dependence on time, we get: ∇p = − µρ (E + V × B) − ρ
∇p = µρ ∇Φ − µρ V × B − ρ(V · ∇)V = µρ ∇Φ − = ρ∇ µΦ −
1 2ρ
∇|V|2 + ρ V × (curlV − µB)
1 2 2 |V|
− V × (c2 ∆B + µρB) (5.4.14) where we used that V × ρcurlV = V × curl(ρV) − ∇ρ × V = c2 V ×
curl(curlB) = −c2 V × ∆B. Finally, the last expression in (5.4.14) can be written in terms of Φ and ρ, by using (5.4.12).
Unfortunately, we are unable to proceed, since better knowledge of the properties of function Φ is necessary. These properties are related to those of the set Σ and its section Ω. We guess that there are chances to adjust the function ρ in such a way that p turns out to be a gradient. If η is relatively large compared to the size of Ω, we expect ρ to be nearly constant. The problem can be approached numerically. Further conditions can only be recovered from the resolution of the corresponding Einstein equation, that also needs to be solved numerically, with greater additional effort. We have no idea of what is going to happen to ρE in the new metric and what the role of the mass tensor in the dynamical assets of the particle is. In particular, ρE is not necessarily positive (in a left-hand side universe), which leads to some confusion, since we are trying to replace a density of mass with a
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density of charge. We are confident that, once a complete solution becomes available, we shall be able to enjoy a clearer understanding. For the moment, we shall not study what happens outside Σ, leaving the discussion until section 6.1. Our “bare” particles will finish at the boundary of Σ. This kernel constitutes a charged and massive part. Externally, there is the radiating part, whose intensity decays as the square of the distance from Σ (see also section 5.1). As in (5.3.32) and (5.3.33) the fields may be joined on ∂Σ in a continuous way. This will not be true for ρ. However, according to what we said at the end of section 5.3, possible discontinuities should not be noted in the space-time surroundings modified by the dynamical evolution of the electromagnetic fields. Indeed, we learnt that understanding the secrets of matter is something that cannot be done from the privileged framework of a global flat geometry. Our dissertation can acquire more credibility if we provide quantitative results. As we said, we are unable to fully solve the complete set of equations. Incidentally, if we simply restrict our analysis to the stationary case, some help comes from the Kerr-Newman solution introduced in section 5.2. For example, we can study the situation outside Σ, getting rid of the singularities of the electric charge on the ring Γ. Recalling (5.2.17), we introduce the constants: r χ µe MG q = m = 2 c 2 4πǫ0 c2
(5.4.15)
where M is a mass, G is the gravitational constant, e is a charge and ǫ0 is the permittivity of free space. In this way, both m and q have the dimension of a distance (concerning m see for instance Fock (1959), p. 194), so that Ξ in (5.2.17) is adimensional. This agrees with the requirement that the metric tensor must be adimensional. For the Kerr-Newman metric tensor, at a certain distance, we have: 2 µe χ 1 2M G (5.4.16) − g00 ≈ 1 − 2 2 c r 2 4πǫ0 c r2 Based on Misner, Thorne and Wheeler (1973), p. 891, this setting corresponds to a black-hole having mass M and charge e. Note that the gradient of A0 in (5.2.15) agrees, to a first approximation, with the classical radial field of a point-wise charge: −
∂A0 e ≈ ∂ rˆ 4πǫ0 r2
for r ≈ rˆ large
(5.4.17)
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If M is the mass of a particle, the corresponding m is going to be quite a small number. On the other hand, from (5.4.15), one has: √ χ µ × (1.60 10−19 ) meters q ≈ √ 2 × 4π × (8.85 10−12 ) × (3.00 108 )2 ≈
√ χ µ × (1.13 10−26 ) meters
(5.4.18)
It is known that the radius η of a particle is of the order of 10−15 meters. By taking into account (5.2.18), we obtain the following estimate: √ χ µ ≥ 1011 Coulomb/kilograms (5.4.19) The above inequality is reasonable if we think that, in the micro-world, charges expressed in Coulombs are much larger than weights expressed in kilograms, of 10 to 12 orders of magnitude. In section 4.3, at least in the case of free photons, we proved that an electromagnetic wave of frequency ν produces a gravitational wave, also with frequency ν. According to equation (4.3.13), an adimensional relation between the amplitude of these two waves is given by: √ χµ u (5.4.20) σ = ν By recalling (5.4.19), a remarkable coincidence is obtained when the size of u is of the order of the electron charge and ν ≈ 1022 . Incidentally, by this choice, the size of σ is of the order of the electron mass. Models of the electron based on the Kerr geometry have been examined in Carter (1968), Burinskii (2000), Arcos and Pereira (2004), Burinskii (2005), also in conjunction with quantum effects, through the Dirac equation. The results of those papers were limited by the severe restriction m2 ≥ q 2 + η 2 , that we were able to remove here (see (5.2.18)). In the theory of black-holes the size of the ring Γ can also be put in relation with the magnetic moment. In our particle, however, the displacement of the fields does not agree with the one of a simple stationary black-hole. This makes it difficult to find other estimates. Note that, in our particle, the line forces of the magnetic field form closed bounded loops inside Σ. Hence, we are not in the presence of little magnetic dipoles, as is usually assumed by virtue of their rotation. Nevertheless, the particle should react in a classical way to solicitations from external fields, by means of a mechanism that will be outlined in section 6.1. In order to know more, we need to examine the interior of Σ. For simplicity we assume that Ω is a circle. This is going to be true if η (radius
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of the circumference Γ) is relatively large with respect to the diameter of Ω. Let us write η = η0 /|ω| and assume that η0 /δ0 is of the order of unity (see for example (5.3.23), but remember that this was the value obtained for the 2-D case). In the light of (5.3.32), we propose the following expression for the stationary radial component of the electric field in Ω: 2 qω |ω| s δ03
Eradial =
q s2
for 0 ≤ s ≤ δ0 /|ω| (5.4.21) for s ≥ δ0 /|ω|
p where s = y 2 + z 2 and q = e/4πǫ0 . In this way, if e is the electron charge, the behavior of the field at a distance from Σ is similar to that of a concentrated electric charge. Note that Eradial is a continuous function and, outside Σ, there is no dependence on the parameter ω. Moreover, by integrating the divergence of the stationary field E over the volume Σ, the resulting expression does not contain ω (see also (5.3.35)). We are concerned with studying what is happening inside Σ. To this end, we compute the energy of the electric field: Z Z Z δ0 /|ω| ǫ0 E = |E|2 = πǫ0 η |Eradial |2 sds |E|2 = 2π 2 ǫ0 η 2 Σ 0 Ω 2π 2 ǫ0 ηq 2 ω 6 = δ06
Z
δ0 /|ω|
0
s3 ds =
π 2 ǫ0 ηq 2 ω 2 ηω 2 e2 η0 |ω|e2 = = 2δ02 32 δ02 ǫ0 32 δ02 ǫ0
For the magnetic part, the corresponding energy does not differ too much from the one above (compare with (5.3.31)), thus, for simplicity, we shall pass over it. By virtue of the equation E = mc2 , where m is the electron mass, we can recover ω: |ω| =
32 mc2 δ02 ǫ0 ≈ δ0 × 1014 η0 e2
(5.4.22)
Hence, the value η0 /|ω| is not too far from the particle radius. Finally, by evaluating ν = c|ω|/2π ≈ 1022 Hertz, we actually get something of the order of frequency of the gamma rays. The fact that the charge outside Σ is not influenced by the value of ω tells us that the value of e is stable independently of the perturbations applied to the particle (see section 6.1).
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Substituting (5.4.22) in the expression of E, one gets: πη0 e2 kilograms × (meters)2 E ≈ 10−35 = < h 2 ν 16 cδ0 ǫ0 seconds
(5.4.23)
Also in the last formula the constant ω disappears. The Planck constant h could then be defined according to the stationary geometrical properties of the electron. If m is the mass of the proton, more than 1800 times that of the electron, we guess that η0 is also going to be bigger. For example, if η0 was magnified 60 times with respect to δ0 , we would get a new radius η0 /|ω|, which is about double that of the electron (note that now |ω| is amplified 30 times). Therefore, we end up with a larger toroid displaying a smaller section. Roughly, the volume and the mass of the toroid particle are proportional to ηδ 2 and η/δ 2 , respectively, where η and δ are the radii of the large and the small circumferences. Note, however, that the equilibrium configuration of a proton is quantitatively different from the one of the electron. In addition, we only took in consideration stationary fields. Hence, these conclusions should be interpreted with some care. We believe that more accurate results can be obtained with a better knowledge of the shape of Σ and of the intensity of the electromagnetic fields there defined.
5.5
Connections with a Dirac type equation
This last section is devoted to a brief inspection of the possible links between our approach and some of the equations governing quantum mechanical phenomena. In particular, we study how to introduce quantization by showing that specific solutions of our set of model equations have close relations with those of Dirac’s equation (see for instance Bjorken and Drell (1964) or Greiner (1997)). The analysis is only intended to be a preliminary step, since a deeper investigation is not within the scopes of the present manuscript. Additional qualitative considerations about the connections with quantum theories will be made in chapter 6. In section 2.5, we were able to derive the following differential relation (see (2.5.8)), involving the electromagnetic potentials A and Φ: ⊔ ⊓A = J ⊔ ⊓Φ
where
⊔ ⊓ =
∂2 − c2 ∆ ∂t2
(5.5.1)
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By taking the divergence of (5.5.1) and recalling the Lorenz condition (2.5.7), one easily arrives at the continuity equation: ∂ (⊓ ⊔Φ) = − c div J(⊓ ⊔Φ) (5.5.2) ∂t which is exactly equation (2.1.4) valid for ρ = divE = c−2 (⊓ ⊔Φ) (use the definition of E in terms of the potentials). In section 2.5, we also proved that, if relation A = ΦJ holds (see (2.4.12)), then the Lorenz condition implies the continuity equation also for Φ (see (2.5.9)). Thus, both quantities Φ and ⊔ ⊓Φ are conserved during transport. Putting these facts together, it is reasonable to consider the eigenvalue problem: ⊔ ⊓Φ = ± λ2 Φ (5.5.3) where the real number λ has the dimension of frequency. In particular, λ can be expressed as energy divided by the Planck constant. As far as free-waves are concerned, one can compute Φ from (5.5.3), and, by knowing the direction J of evolution, determine the other potential A by integrating (5.5.1).
Some considerations have to be made regarding the sign on the righthand side. Suppose that we are looking for solutions shifting in the direction of the z-axis. A possible candidate is the plane√wave solution (see, e.g., Bjorken and Drell (1964), section 3.1): Φ = sin(t 1 ∓ λ2 − z/c). Nevertheless, we are concerned with the propagation of light rays and this wave moves at speed c if and only if λ = 0 (technically corresponding to the case of a massless particle). We have some possibility, however, to find solitonlike solutions with compact support, amongst the ones studied in the previous chapters, by looking for functions of the form: Φ = F (x, y)g(t − z/c), for an arbitrary function g. These will evolve at the speed of light. Substituting in (5.5.3), it is straightforward to arrive at the equation: 2 ∂2F ∂ F 2 = ± λ2 F (5.5.4) + −c ∂x2 ∂y 2 Since the Laplacian is a negative definite operator, we are free to impose suitable conditions on the boundary of a given domain of the plane (x, y) only by assuming the sign + on the right-hand side of (5.5.4) and (5.5.3). We can double check the above computations in the case of the photon described by (2.4.14), whose magnetic field satisfies divB = 0. For example, we choose the solution: Φ = cF (r)g(t − z/c). Imposing (5.5.3), in cylindrical coordinates, we recover: 2 ∂2Φ ∂ Φ 1 ∂Φ ∂2Φ 2 ⊔ ⊓Φ = − c + + ∂t2 ∂r2 r ∂r ∂z 2
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F′ g = ± λ2 cF g = ± λ2 Φ = − c3 F ′′ + r
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(5.5.5)
yielding the equation: λ2 F′ ± 2 F = 0 (5.5.6) r c which is solvable with the help of Bessel functions, provided one chooses the sign + (also corresponding to + in (5.5.3)). Namely, up to multiplicative constant, we get: F (r) = J0 (r|λ|/c). Since the electric field is E = (cF ′ g, 0, 0), by looking for the zeros of J0′ , we find circles outside of which E1 can be continuously prolonged to zero. This procedure introduces a constraint upon the shape of the photon. Depending on λ, we can adjust its magnitude and influence its energy. Of course, this is equivalent to introducing a kind of quantization. We will better clarify our position in section 6.1. No bounded solutions with compact support are obtainable by selecting the minus sign in equation (5.5.6). Finally, we compute A by taking A2 = J2 Φ (with J2 = 1) or, equivalently, by solving (5.5.1). The result agrees with the expression in (2.4.14). F ′′ +
We infer that Φ entirely characterizes the evolution of a single photon. When instead the wave is constrained, we cannot use relation A = ΦJ (see section 3.4). Therefore, in order to guarantee (2.5.9), equation (5.5.3) needs some adjustments. It is natural to associate equation (5.5.3) with the Klein-Gordon equation: 2 ∂2ψ 2πmc2 −⊓ ⊔ψ = c2 ∆ψ − ψ (5.5.7) = ∂t2 h where m is a mass, h is the Planck constant and ψ is a probability density. Therefore, up to multiplicative dimensional constant, the two scalar fields ψ and Φ are assimilated. This is an interesting fact, because a deterministic role is now assigned to ψ. In fact, from ψ ≈ Φ, we get the potential A and the corresponding electromagnetic fields (using (2.4.1)). Unfortunately, (5.5.7) has a given sign, corresponding to a minus on the right-hand side of (5.5.3). Hence, self-contained photons, as those examined up to this moment, cannot be generated from (5.5.7), unless we modify, giving a physical explanation, the sign of its right-hand side. We know that the Klein-Gordon equation is a relativistic invariant and well suited to describing a certain number of quantum phenomena. However, a better model is provided by the Dirac equation (see Dirac (1928)),
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that we now introduce. In this more general context, we will examine its relationship with our model. Dirac’s equation involves four unknowns, Ψ = (ψ1 , ψ2 , ψ3 , ψ4 ), which are complex functions of real space-time variables. Roughly speaking, it √ ⊔ ⊓. We examine corresponds to an eigenvalue problem for the operator the situation for Cartesian coordinates in Minkowski space. First of all, we define the so-called γ-matrices: 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 γ (0) = γ (x) = 0 0 −1 0 1 0 0 0 0
γ (y)
0 0 = 0 i
0
0
−1
0 0 −i 0
0 i 0 0
−i 0 0 0
1
γ (z)
0 0 = 1 0
0
0
0
0 0 0 −1
1 0 0 0
0 −1 (5.5.8) 0 0
where i denotes the imaginary unit. Moreover, let I denote the identity matrix. We have: [γ (∗) ]2 = I
for ∗ = 0, x, y, z
(5.5.9)
For γ-matrices, additional relations can be proven. They are important for expressing the Dirac equation in relativistic form. We leave the discussion of these properties, however, to specialized texts. Thus, for a real eigenvalue λ, representing a frequency, the Dirac equation reads as follows: ∂Ψ ∂Ψ ∂Ψ ∂Ψ (5.5.10) + icγ (x) + icγ (y) + icγ (z) = λγ (0) Ψ iI ∂t ∂x ∂y ∂z In explicit form we have: i ψ1,t + cψ4,x − ciψ4,y + cψ3,z = λψ1 i ψ2,t + cψ3,x + ciψ3,y − cψ4,z = λψ2 i ψ3,t + cψ2,x − ciψ2,y + cψ1,z = − λψ3 i ψ4,t + cψ1,x + ciψ1,y − cψ2,z = − λψ4
where the subscript letter denotes the partial derivative.
(5.5.11)
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An important property is the possibility of obtaining a sort of continuity equation. We first take the complex conjugate of equation (5.5.10): ¯ ¯ ¯ ¯ ∂Ψ ∂Ψ ∂Ψ ∂Ψ ¯ − icγ (x) + icγ (y) − icγ (z) = λγ (0) Ψ (5.5.12) −iI ∂t ∂x ∂y ∂z where we notice that γ¯ (y) = −γ (y) . Then, let us multiply (scalar vec¯ and subtract equation (5.5.12) tor multiplication) equation (5.5.10) by Ψ multiplied by Ψ. This yields: ¯ ¯ ∂Ψ ¯ ∂Ψ ∂Ψ ¯ ∂Ψ i ·Ψ + · Ψ + ic γ (x) · Ψ + γ (x) ·Ψ ∂t ∂t ∂x ∂x ¯ ¯ (y) ∂ Ψ (z) ∂Ψ ¯ (z) ∂ Ψ (y) ∂Ψ ¯ ·Ψ − γ · Ψ + ic γ ·Ψ + γ ·Ψ + ic γ ∂y ∂y ∂z ∂z = i
∂|Ψ|2 ∂ (x) ¯ + ic ∂ i(γ (y) Ψ) · Ψ ¯ + ic ∂ (γ (z) Ψ) · Ψ ¯ + ic (γ Ψ) · Ψ ∂t ∂x ∂y ∂z h i ¯ − (γ (0) Ψ) ¯ ·Ψ = 0 = λ (γ (0) Ψ) · Ψ (5.5.13)
At this point, we define the positive probability density:
Φ = |Ψ|2 = |ψ1 |2 + |ψ2 |2 + |ψ3 |2 + |ψ4 |2
(5.5.14)
and a probability current vector, whose components are: ¯ = 2R(ψ1 ψ¯4 ) + 2R(ψ¯2 ψ3 ) ΦJ1 = (γ (x) Ψ) · Ψ ¯ = − 2I(ψ1 ψ¯4 ) − 2I(ψ¯2 ψ3 ) ΦJ2 = i(γ (y) Ψ) · Ψ ¯ = 2R(ψ1 ψ¯3 ) − 2R(ψ¯2 ψ4 ) ΦJ3 = (γ (z) Ψ) · Ψ
(5.5.15)
where R(∗) and I(∗) are the real and imaginary parts of the complex number ∗. All the above quantities are real. In the end, from the last but one line in (5.5.13) we get (2.5.9), i.e., the continuity equation for Φ. We can build explicit solutions in the case of free-waves. As before, we consider a photon travelling in the direction of the z-axis. Let us define the two functions: Hm = Fm (x, y)g(t − z/c), where Fm , m = 1, 2, are complex ∂ ∂ and g is real. Of course, we have: ∂t Hm + c ∂z Hm = 0, for m = 1, 2. Then, we set: ψ 1 = H1
ψ2 = iH2
ψ 3 = H1
ψ4 = − iH2
(5.5.16)
By substitution into (5.5.11), the system reduces to two equations: c
∂F2 ∂F2 − ic = λ F1 ∂x ∂y
(5.5.17)
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∂F1 ∂F1 + ic = λ F2 (5.5.18) ∂x ∂y We can recover F1 from the first equation and substitute into the second one. This yields: 2 ∂ 2 F2 ∂ F2 = − λ 2 F2 (5.5.19) + −c2 ∂x2 ∂y 2 c
A similar equation is valid for F1 . Both of them are equivalent to equation (5.5.4), displaying the negative sign on the right-hand side. Again, we are imposing some constraints upon our photon, although we would have preferred the positive sign, in order to be able to come out with solitons having compact support. We can modify a little bit the model equations, with the aim of switching the sign of the right-hand side of (5.5.19). We can simply redefine γ (0) in (5.5.8): 0 0 i 0 0 0 0 −i (5.5.20) γ (0) = −i 0 0 0 0 i 0 0
In this way, we still have (5.5.9). The new version of the equations reads as follows: ψ1,t + cψ4,x − ciψ4,y + cψ3,z = λψ3 ψ2,t + cψ3,x + ciψ3,y − cψ4,z = − λψ4 ψ3,t + cψ2,x − ciψ2,y + cψ1,z = − λψ1 ψ4,t + cψ1,x + ciψ1,y − cψ2,z = λψ2
(5.5.21)
¯ − (¯ ¯ · Ψ] = 0, we can review the comBy noting that [(γ (0) Ψ) · Ψ γ (0) Ψ) putations in (5.5.13) and conclude that (2.5.9) is still true (Φ and ΦJ are defined as in (5.5.14) and (5.5.15)). Therefore, the changes we have made do not compromise the energy conservation properties of Dirac’s equation. Taking into account the same set of solutions proposed in (5.5.16), this time we get: ∂F2 ∂F2 − ic = iλ F1 (5.5.22) c ∂x ∂y c
∂F1 ∂F1 + ic = iλ F2 ∂x ∂y
(5.5.23)
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In practice, in (5.5.17) and (5.5.18), λ2 hasbeen replaced by iλ. The result∂ 2 ∂2 ing equations are − c ∂x2 Fm + ∂y = λ2 Fm , for m = 1, 2, which is 2 Fm exactly what we were looking for. Now we are allowed to impose suitable constraints at the boundary of a bounded set of the plane (x, y). Since g is arbitrary, we are able to construct examples of travelling waves occupying a bounded region of space. Successively, by (5.5.14) and (5.5.15), one obtains: Φ = 2|H1 |2 + 2|H2 |2 ψ1 ψ¯4 + ψ¯2 ψ3 = 0
⇒
ΦJ1 = 0 and ΦJ2 = 0
ΦJ3 = 2R(ψ1 ψ¯3 − ψ¯2 ψ4 ) = 2|H1 |2 + 2|H2 |2
⇒ J3 = 1
This confirms that the photon is shifting along the direction (0, 0, 1). Since we are dealing with a free-wave, it is enough to impose Ak = ΦJk , k = 1, 2, 3, to re-obtain the same electromagnetic fields we found in the case of equation (5.5.3). The analogy between the probability density |Ψ|2 and the scalar electromagnetic potential Φ is evident and opens the door to a deterministic interpretation of quantum mechanics, as it will be illustrated in section 6.1 with a series of heuristic arguments. In addition to this, Dirac’s equation allows for the treatment of more general situations than the Klein-Gordon equation. In fact, in the Lorenz gauge, we automatically have (5.5.2) and (2.5.9), without necessarily assuming the condition A = ΦJ. More exactly, we may take into consideration the case of the 2-D rotating photons studied in section 5.3. We are now dealing with a constrained wave, therefore the relation A = ΦJ is not valid, as one may directly check by examining (5.3.1). Although we do not have a direct verification, we guess that it is possible to build solutions of (5.5.10) similar to the fields introduced in section 5.3. In this context, the quantization properties become of primary importance. As a matter of fact, we have already shown that the radial shape of a rotating photon cannot be arbitrary, but determined by an eigenvalue problem (see for example (5.3.9) or (5.3.21)). Of course, the situation is even more complicated in 3-D. In the geometry of figure 5.10, the arrows show the displacement of the field J. The scalar potential Φ is transported along the stream-lines and, if we satisfy (5.5.10), this is done in a conservative way. The practical difficulty is to determine the corresponding functions (ψ1 , ψ2 , ψ3 , ψ4 ). By knowing Φ, the vector potential A is also determined, since we can get it from (5.5.1). We
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said that the size and the shape of our elementary particles is unique and it is related to a situation of equilibrium balancing electromagnetic and gravitational forces. We approximate the section Ω of the toroid Σ by a circle of radius rM ≈ 10−15 meters. Due to (5.3.18), rM turns out to be inversely proportional to ω (see also (5.4.22)). The eigenvalue problem (5.5.10), with γ (0) modified as in (5.5.20), can be reduced to a boundary value problem on Ω. The eigenvalue λ depends on the size of Ω and attains approximately the value c/rM ≈ 1022 Hertz, corresponding to the energy of the rotating photons divided by the Planck constant. More detailed results are not available. Without doubt, an abundant dose of numerical computations can delineate a more distinct picture. Spin and parity are prerogatives of quantum theories. These concepts have already been mentioned several times in the book. Indeed, a study of these aspects, from the point of view of spinor analysis, is welcome, but beyond the current scope. We do not exclude promising developments in the future. We only point out a strange coincidence. According to (5.3.10) for γˆ0 = 0, we take the stationary part of the potentials in (5.3.1), i.e.: A = (0, 0, γ0 r3 /8) and Φ = γ0 r2 /2ω. Considering that J = V/c = (0, 0, ωr/2), in the case of the rotating photons, we get: A = SΦJ
(5.5.24)
with S = 12 . Since for a free photon one has S = 1, we may ask ourselves if there is any formal implication relating the spin to the value of S in (5.5.24). The Dirac type equation we introduced in this section may replace, in specific situations of energy resonance, the far more complicated Einstein equation. Thanks to considerations that will be made in section 6.1, these eigenstates not only influence the magnitude and the stability of an elementary particle, but should affect the entire structure of an atom and regulate the interactions between photons and particles. The above observations raise a couple of questions. Can we see the gravitational field produced by a stable particle as an eigenstate of Einstein’s equation? If this is true, is there any direct relation between the unknown Ψ of (5.5.10) and the entries of the metric tensor associated with our particle model? Right now, we do not have any serious mathematical analysis in support of these conjectures.
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Final speculative considerations
6.1
Towards deterministic quantum mechanics
In this chapter, we add more considerations, which, although not rigorously validated by the theory, find nevertheless correspondences with observation of the real world. We have shown that a stable elementary particle (electron or proton) consists of a central annular core Σ, made of fast-rotating photons, whose diameter matches standard predictions. According to the definition given in Batchelor (1967), section 2.6, the surface of Σ is a vortex-tube (see also section 7.2 and the plates 20 and 21 of the same book; some articles on vortex rings are: Maxworthy (1972), Norbury (1973), Pullin (1979), Shariff and Leonard (1992), Tung and Ting (1967), Wakelin and Riley (1997)). The object has finite size, since the space-time metric is capable of guaranteeing the rigid rotation of the sections Ω of Σ up to a prescribed limit. Now we can try to explain what happens beyond such a limit. The solution described refers to a situation of equilibrium which is expected to have a congruous basin of stability. Realistically, Σ should resist suitably small perturbations, altering the shape of the stream-lines and resulting in accelerations of the Coriolis type. It is known that Coriolis forces give elastic properties to rotating fluids. The restoring effects of such forces are discussed, for instance, in Batchelor (1967), section 7.6. Thus, our particles resemble small vibrating jelly-like doughnuts. A plausible picture of a perturbed section of Ω is shown in Batchelor (1967), p. 533. In sections 5.3 and 5.4, we claimed that there are reasons to believe that the charge is not modified by these disturbances. The fluctuations are within the range of the Planck constant h (see (5.4.23)) and should be related to the theory of 169
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Vortices, made of photons, developing around a spinning electromagnetic vortex. The angular velocity decreases departing from the center. Fig. 6.1
so-called zero-point radiation (see for instance de la Pe˜ na and Cetto (1996), chapter 4), where the space is supposed to be filled by random fields (of unknown origin) having zero average. The zero-point energies are in the limit of the atomic scale and continue to persist even at zero temperature. Due to the Heisenberg principle, most of these phenomena escape direct observation, but their role is considered to be decisive in explaining many aspects of the quantum world. The study of the zero-point radiation fields is usually approached from a stochastic point of view, but what we are going to say in this chapter may provide the foundations for a deterministic analysis. In practice, a particle is not a totally isolated system. It is suspended in an ocean of wiggles forming an electromagnetic background evolving and interacting at the speed of light. These photons in casual motion are
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part of a huge scenario, an unstructured immensity of “waste”, carrying energy and deforming geometries, that does not have the necessary reputation to be called “matter”, but can be put in relation to what we call “temperature”. In proximity of Σ, this debris is captured and organized, forming new external shells. In a very similar way, the eye of a tornado is surrounded by concentric layers evolving at different angular velocities. The toroid structure of our vortex makes things more complicated to study (one can have a vague idea by looking at figure 2.6.2 in Batchelor (1967)). However, arguing in 2-D, we can refer to figure 6.1. The inner disk Ω is rotating at a prescribed angular velocity. From a mathematical viewpoint there is no diffusion, so that the disk has no possibility to communicate with the exterior. On the other hand, due to unavoidable perturbations, we expect small radial oscillations, so that electromagnetic radiations randomly fluctuating around the disk are dragged in a rotatory motion, but with a lower angular velocity. Locally, one can see the tip of each single vector of the electric field rotating like a small (elliptic) clock. The first external shell has a finite diameter too. This depends on the maximum peripheral velocity achievable, resulting from the compensation properties of space-time geometry (induced by the same photons of the shell via the Einstein equation). Other layers are then generated on the same principle. Their diameters are inversely proportional to the frequency of oscillation of the fields (compare with (5.3.11), where ζ grows as c|ω| and rM decays as δ/|ω|). The passage from one shell to the next appears to be discontinuous in the classical flat geometry, however, if locally examined in the geometry induced by the electromagnetic fields the question is not pertinent; the central core and the shells are coexisting, independent and non-communicating microscopic worlds. Note that, as in (5.3.32) and (5.3.33), we can set the stationary component of the electric and magnetic fields to be continuous at the interfaces. This will not be true for the divergence of E. There is no loss of energy, because there are no diffusive terms in the model as for instance in viscous fluid dynamics. Nevertheless, there is loss of information when travelling from a shell at higher frequency to one containing lower harmonics. As mentioned at the end of section 3.1, this could be put in relation to the second law of thermodynamics. If we compare our particles to sets of fluid vortices, depending on various circumstances (for example the encounter between two or more particles), a complicated dynamics with intricate symmetry-breaking bifurcations is expected to be generated. In the framework of fluid dynamics, most of these questions have been investigated in depth and the literature is vast.
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We mention here only a few papers: Braun, Feudel and Guzdar (1998), Marques, Gelfgat and Lopez (2003), Hirsa, Lopez and Miraghaie (2002), Moisy, Pasutto and Rebaud (2003). Of course, due to the complexity of these problems, especially in 3-D, we are unable at the moment to figure out a clear picture. The situation is further complicated by the lack of viscous terms in the model. Certainly, at a quantitative level, the Planck constant h and the fine structure constant α should be involved. In this regard, some papers (see for instance Binder (2002a), Binder (2002b)), by analyzing interference, precession, and other geometric properties of waves, present interesting links with the constant α. In Kanarev (2000) it is noted that h is dimensionally equivalent to a moment of inertia divided by a unit of time. Therefore, requiring h to be constant leads us to the conservation of angular momentum, independently of the shell under observation. The predominant role of fluid mechanics, also at the level of the micro world, should not be a surprise. In his celebrated experiment, R. A. Millikan measured the elementary charge using equations recovered from the physics of fluid motion. The transition from the fluid properties of electromagnetic waves to the oil drops of the Millikan test (orders of magnitude greater) follows a continuous path. The more important aspect, however, is not the global behavior, but what happens locally to the fields of the various shells. Upon careful examination, we discover that they oscillate with a frequency decaying with distance. This corresponds to the innate idea of relating larger objects to lower frequencies. In our opinion, a body is potentially furnished with an extended range of coexisting frequencies (like strings), starting from those of gamma rays. Depending on the stimulation applied, the body responds by appearing in an appropriate fashion (recall the De Broglie theory, where the wavelength is related to the particle momentum). The heart of an elephant and the wings of a mosquito are made precisely of the same kind of particles, yet they beat within different regimes. From our viewpoint, this does not descend directly from the difference of their masses (as experience would suggest). Our explanation is that each of them belongs to an electromagnetic environment, resonating at a frequency naturally associated to the physical extension of the whole system. Only by introducing the concept of mass does everything become intelligible. For instance, by examining the relation (5.4.20), it turns out that the amplitude of the gravitational wave oscillations is inversely proportional to the corresponding frequency. Thus, in presence of low frequencies, we expect the gravitational effects to dominate the electromagnetic ones. Later, in this section, we propose a
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way of defining the inertial mass of a particle, based on the frequencies of the surrounding shells. In reality, the theory introduced in this manuscript was motivated by the above observations. In the beginning, the aim was to develop a differential mathematical model, able to produce, from a given oscillatory source, other induced oscillations, exhibiting a frequency decay (small diameter, high frequency, and vice versa). It is not difficult to check that the passage from high to low frequencies cannot be done continuously, since it would give rise to solutions having space derivatives growing with time (take for instance the function sin(t/r), whose frequency is inversely proportional to r). This problem turned out to be so crucial (and so difficult) that we had to subvert the entire superstructure of physics. In the case of an electron (or a proton), we may assume that the timevarying electric fields (similar to those of a neutrino) of the various shells are superposed to a radial stationary component (compare with figure 5.8), with a corresponding density ρ (not necessarily constant). Moreover, let us suppose that, globally, the function |E| decays as the inverse of the square of the distance when moving away from the central kernel. Thus, we have found the mechanism by means of which a particle carries its charge: it develops shells by capturing the circulating waste, transmitting the polarity to them. The shells alone are not stable entities, and, deprived of the central core, they have a very short life, after which they disintegrate. In this process they produce photons (we will say more about this mechanism later). During their brief existence, they may behave as unstable particles, carrying their own mass and charge (could they be related to mesons?). We propose a conjecture, maybe a bit speculative, about the significance of the fine structure constant (α ≈ 1/137). Its definition is not purely geometric, but involves the dynamics of a stable particle, around the point of equilibrium. We expect the particle to have a proper frequency of resonance, so that the entire structure vibrates in the neighborhood of a fixed displacement. We believe that this is done with a rate that is α times the frequency of the rotating photons forming the particle itself. Hence, the oscillations communicated to the next outer shell are reduced by a factor α. In addition, we can make the following hypothesis about the frequency decay rate: αc (6.1.1) ν ≈ 2πr where r is the distance from the source. The relation above is just an estimate, since we said that ν is not continuous but tends to zero in a
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quantized way. There are a couple of remarkable reference points. When r is of the order of 10−15 meters (the diameter of a stable particle), we get that ν is around 1021 Hertz (the frequency range of gamma rays). When r = h2 ǫ0 /πme2 is the first Bohr orbit of hydrogen (h is the Planck constant, m is the electron mass and e its charge), recalling that α = e2 /2ǫ0 hc, we obtain ν = e4 m/4ǫ20 h3 , which is exactly the ground state frequency of the atom. If r is of the order of the distance earth-moon, ν corresponds to periods of about 15 minutes, but at this regime we do not think the formula is still reliable. Note also that, assuming a decay proportional to 1/r of the angular velocity of the vortices of figure 6.1, the intensity of the vector G = DV/Dt decreases as the inverse of the square of the distance. Alternatively, a discontinuous version, asymptotically equivalent to (6.1.1), can be introduced as follows. We define a sequence of positive numbers rk , for k ≥ 0. The frequency assigned to the first interval 0 < r ≤ r0 is: ν = c/2πr0 . The other values are determined by: cα (6.1.2) νk = 2πrk for rk−1 < r ≤ rk and k ≥ 1. For example, in the 2-D case, the values rk can be adjusted in such a way that the energy associated with the k-th shell is proportional to νk (through a constant not depending on k). If the intensity of the electric field decays as 1/r, then one finds out that the difference rk − rk−1 is asymptotically constant. The 3-D case is certainly more difficult and predictions are hazardous. We attempt a computation in the case of a spherical environment. If the radial electric field decays as e/4πǫ0 r2 , the energy of the k-th shell is obtained from: 2 Z rk ǫ0 e2 1 e 1 2 Ek = r dr = (6.1.3) 4π − 2 4πǫ0 r2 8πǫ0 rk−1 rk rk−1
We now impose the condition that this is equal to hνk : 1 hαc 1 e2 1 1 Ek = = = ⇒ − ⇒ rk = 3rk−1 = 3k r0 2πrk 4πǫ0 2 rk−1 rk rk where we use α = e2 /2ǫ0 hc. Thus, the size of the shells grows geometriP∞ cally. Note that this implies that the sum k=1 Ek is finite. Such an important property would not be true if we assume that the frequency decays in a continuous way, as suggested in (6.1.1). Our estimates may be totally unreliable when approaching the particle, also considering that, at short distances, the value of the fine structure constant looks different (see
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Burcham and Jobes (1997), section 7.5). We think that this is due to the behavior of the inner-most shells, corresponding to what is otherwise called vacuum polarization. We would like to draw the reader’s attention to the wonderful pictures in Lim and Nickels (1995), dealing with water vortex rings. The agreement with our particle model is really amazing. Note, however, that the layers developing in the experiments are not symptoms of quantization, but depend on the way the dye has been injected in the liquid. As far as numerical experiments are concerned, we refer for instance to Krasny and Nitsche (2002), where layers of various periodic orbits and resonant bands are detected in the evolution of a vortex ring. Let us note that our model only depends on three parameters (c, µ, χ). Hence, if we believe that the equations are sufficiently accurate for a theoretical description of the constitutive properties of matter, at least at atomic level, all the other related physics constants should descend from (c, µ, χ) and some geometrical characteristic values deducible from certain eigenfunctions. It is important to remark again that the quantized behavior is peculiar to some special solutions and does not explicitly belong to the set of model equations. For example, apparently in contrast to what has been anticipated in section 5.5, such a quantization is not present in the case of free-waves, so unperturbed photons are not directly associated with any quantum effect. At first sight, this may sound strange and unconventional. On the other hand, a solitary wave is a very poor object, only characterized by having a prescribed (but in principle arbitrary) shape and travelling undisturbed at the speed of light. The quantum properties that are attributed to photons actually come from their interaction with matter, that, being made of elementary particles, displays a quantum-like behavior. Thus, what we see (for instance in the diffraction through a small aperture) is not due to the properties of pure light, but to the strong influence that matter (constituting the rim of the aperture and the screen where we observe the experiment) has on incoming photons. This happens when light approaches the site of impact, which is covered by an invisible cloud of layers made of gravito-electromagnetic fields (a white noise at first sight; a well-structured stencil, if observed without interfering, at distances of the order of the Planck constant). More or less, our point of view agrees with the one expressed in Little (1996) (and successive papers), where, by introducing suitable elementary
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waves, the author reviews, from an original viewpoint, a series of problems and paradoxes in quantum mechanics. The elementary waves are “reverse waves” emitted by matter, by means of an only vaguely specified procedure, allowing for the transfer of quantum properties (for instance from matter to photons). They fill the space, move in all directions and fit all possible particle states. They are always present and emitted with no uncertainty. The results in Little (1996) are a bit speculative, since they do not give formal description of the nature of elementary waves and the way they are generated. Nevertheless, if combined with our approach, the picture starts to become clear: elementary waves are an intrinsic by-product of electromagnetic nature, ruled by our set of equations. Consequently, quantum effects turn out to be of a deterministic type. Although we do not comply with all the statements in Little (1996), the paper is stimulating and certainly provides a starting point for finding the link between our theory and more serious quantum mechanical problems. Continuing with our heuristic analysis, we may now ask ourselves why and how a charged particle is stimulated to move and accelerate, when immersed in an electric field. Such a property is taken for granted, but the question is not trivial at all. We shall try, nevertheless, to come up with some explanation. At rest, the orbits of the light rays constituting the particle are closed and, following figure 5.10, lie on the vertical sections of the torus. Under the action of an external field, the triplets (E, B, V), via the model equations, modify their orientation a little. As a consequence, the trajectories of the rays start precessing (like in a shifting tornado) and the entire body is seen to move at speeds lower than that of light. The motion of groups of vortices, from the point of view of the mechanics of fluids is examined for instance in Batchelor (1967), p. 531. Let us provide further justification for this phenomenon. Suppose that the applied electric field is E0 = −∇Φ0 for some scalar stationary potential Φ0 . Fields perfectly constant in time do not exist in our theory, but we can assume the time-dependent component to be negligible. Then, the contribution of E0 is added to E, which is the time-varying field of the particle, at the right-hand side of (5.2.3). We also assume that ρ = divE is nearly constant, while divE0 = 0. Note that E0 can be included in the other two equations, (5.2.1) and (5.2.2), yielding no effect. Instead, in (5.2.3), the new added term can be compensated by a variation of pressure p0 , defined by: ∇p0 = µρ∇Φ0 . But, due to the equation of state (5.2.8), a change in the scalar curvature must occur. The entire space-time topology is then modified and the light rays have to follow the new geodesics. Let us
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observe that the sign of the gradient ∇p0 depends on that of the product ρ∇Φ, i.e.: the variation tends to be positive or negative depending on the charge of the particle and the applied electric field. A positively charged particle produces, far away from the central core, slow-varying fields that can “attract” a negative particle. At closer distance, the system of the two resonant particles has to be studied as a whole and the quantum effects prevail. Indeed, these arguments are far from being a rigorous proof. A validation can come only after a more serious analysis. By measuring the acceleration rate, according to the energy introduced in the system by the external field, one might be able to estimate the inertial mass of a particle. Of course, the acceleration rate is proportional to the constant µ. In the unrealistic case in which the trajectories are totally stretched, becoming straight-lines, the particle translates at speed c. When imparting acceleration to a particle, other consequences may be inferred. For reasons difficult to formalize, we can expect that the photons circulating in the outer shells are subjected to lower accelerations. Therefore, the central core tends to escape faster than the next outer shell, and the mechanism is reproduced in cascade on the other shells. If the action of the external field ceases, the shells are recreated with the original structure by the bare particle. For an observer, the particle now moves at constant speed, displaying a distribution of shells which is undistinguishable from the one of the particle at rest (Galilean principle). Theoretically, by taking a picture of a complex system of particles one could figure out, from the way the various shells are placed, the instantaneous accelerations and the stresses involved. However, if during acceleration most inner shells have time to reach higher velocities, they may leave the farthest outer shells behind forever (see the qualitative 2-D pictures of figure 6.2). These last are unstable structures and decompose to what they are made of, i.e. pure photons. Recall that, in section 4.3, we noticed that the elementary photon also has a toroid shape. This explains how an accelerated particle emits photons (as for example in the bremsstrahlung process). In some respects, these are quantized photons, but only because they are associated with the packets of energies pertaining to those shells. Hence, a photon has a “color”, inherited from the contingent situation, which is not an intrinsic property of being a photon. Of course, in photon emission, the global energy is preserved. However, for an accurate analysis, one has to consider that the destruction of a shell also disrupts the gravitational setting that, in view of (3.4.12), was subtracting energy from the electromagnetic component.
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Phases of the acceleration of a particle: particle at rest, the particle starts increasing its speed, the particle emits a photon. Fig. 6.2
At this point, we can boldly introduce an innovative definition of inertial mass for a particle: hν (6.1.4) m = √ v c2 − v 2 where h is the Planck constant. We have to specify what ν = ν(v) is. Suppose that the particle accelerates from velocity zero to velocity v, so that it loses some of the outer shells, which are expelled as photons. Then, ν is the frequency of the last shell still attached to the particle. Note that m will depend on the velocity v, but this is not surprising in special relativity. Note also that ν grows with v, since imparting higher accelerations, in the same unit of time, leaves the particle with fewer shells (carrying higher frequencies). The mass at rest is going to be defined as: m0 = h limv→0 (ν(v)/v). It is important to observe that, by evaluating the De Broglie wavelength
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λ = c/ν = h/m0 v, we end up with the well-known relation: m = p
m0 1 − (v 2 /c2 )
(6.1.5)
We also deduce that the ratio ν(v)/v is effectively bounded with respect to v. As a consequence we have discovered how and why a wave is associated with a particle in motion. What we have done seems to be consistent, but also very imaginative. The validity of this approach calls for experimental verifications. In the hydrogen grounded atom, the electron and the proton stay aligned along their spin axes, vibrating gently at a distance calibrated in such a way that the frequencies associated with two of their outer shells resonate. This can only happen for a very specific frequency value, that characterizes the atom at its ground state. The space between the two cores turns out to be filled by vibrating electromagnetic patterns that, in more complex molecules, are usually called phonons (see for instance Bleaney and Bleaney (1965), p. 522). Note that the spin vectors of the two particles could have either convergent or opposing directions (hence originating orthohydrogen or parahydrogen molecules). As ocean waves, phonons transfer energy with no net movement. The particles are kept at a distance through a kind of Venturi effect (the flow is choked since its velocity grows with the inverse of the distance and, due to Bernoulli’s principle, the attractive force, proportional to pressure, diminishes). Due to the nonlinearity of the system of differential equations, the configuration is certainly very complicated, so that, to validate this conjecture, numerical experiments are needed (see for instance Martin (2004), section 2.5). Schr¨odinger equations would work as well, but the reader should not forget that we are trying to give a deterministic meaning to quantum mechanics (see for instance the remarks in Eisberg and Resnick (1985), section 3-6), at least in the framework of local type theories. When a suitable amount of new energy is added to the atom, the structure “inflates”. The two particles jump to a new resonant configuration. After a while, perhaps due to other external fluctuations, the atom returns to the ground state. The electron, having less mass compared to the proton, accelerates more vigorously and gives back the energy in the form of photons, through the mechanism explained above, which is experimentally recognized by the scientific community. Regarding more complex molecular structures, large-scale numerical simulations could better clarify how energy is carried by phonons and explain, in a deterministic way, the conductive properties of metals. Our model should be able to reproduce
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Fig. 6.3 Section of a neutron n and a deuteron d. In the first case, we can see three counter-rotating rings (an electron surrounded by a neutrino and a proton). The total spin is 12 . The electron-neutrino couple forms a negative pion π − (spin =0). In the second picture, we have four rings and the total spin is 1. We also recall the reactions π + + d → p + p and π − + d → n + n, that are perfectly compatible with our pictures.
all these phenomena up to the finest level of detail. However, when dealing with a large number of particles, statistic considerations prevail. Accordingly, the source tensors on the right-hand side of Einstein’s equation must be modified, bringing us back, depending on the size of the object under study, to more familiar and consolidated theories. Finally, some speculations about the assembling of a nucleus. We recall that, according to our theory, the naked core of a particle has no charge in the way it is commonly thought. Such a kernel may transmit the properties of its internal fields by means of the electromagnetic background energy surrounding it (if available). Anyway, bare particles are mathematically acceptable if no other energy is floating around (perfect zero temperature).
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Hence, it is not scandalous to assume that more bare particles could be packed into a small space, finding a unique and solid state of equilibrium. We do not wish to go too far on such slippery ground, so we refer the reader to figure 6.3, where a hypothetical versions of a neutron and a deuteron are shown. As results from scattering experiments (see for instance Eisberg and Resnick (1985), p. 629), the deuteron actually displays an asymmetric shape with a negatively charged core. Outside a complex nucleus, due to its specific typology, the fields are driven in a unique characteristic dynamical configuration. Thus, the nucleus is not only a charged lifeless agglomerate, but takes an active part in constituting the corresponding atom by generating well-marked patterns, where the electrons find exact stable locations, according to a pre-ordered design (implying for instance the Pauli exclusion principle).
6.2
Conclusions
We have presented a model of the universe where electromagnetism is the principal (indeed the sole) ingredient. The 3-D space is filled by triplets (E, B, V) wandering around at the speed of light. A sort of duality allows us to distinguish between the wave-fronts (where the electromagnetic information lies) and the rays (or stream-lines), which are basically governed by the laws of fluid dynamics. If the reader has followed the details, it should be clear that what we have achieved and justified is essentially the coupling of mechanical and electromagnetic forces, and not just a mere combination of tensors. We stress once more that this unification has been made possible by a full review of the theory of electromagnetism, since working with the usual set of Maxwell equations does not lead anywhere. We wish to emphasize that the most important aim of this book was to provide a practical mathematical description of photons, the primary building blocks for successive developments. We believe that the main goal of clarifying the old and troubled mystery of light emission has been achieved, giving a new perspective to modern physics. The triplets (E, B, V) not only define the displacement of the electromagnetic fields, but locally describe the orientation of privileged reference frames, tightly related to the local metric. Therefore, there is not a global stationary and easy description of the geometric properties of the whole universe, but a complex dynamical structure (rich in substructures) mani-
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festing itself with different appearances at various scales of magnitude (see the caption of figure 6.1). We guess that most of this agitated motion of photons is practically invisible to us and may be detected only indirectly. It would be incorrect to define ether as such a fluctuating background, because it is made of pure light and, of course, it is not the means of propagation of light-waves. For us, ether is the hypothetical fluid whose stream-lines are tangent to V. Under specific circumstances, photons give rise to real matter. They show themselves better in this form. Incidentally, stable particles are not immersed in pure void. They continue to be surrounded by an ocean of “dark” gravito-electromagnetic energy vibrating with an extended range of frequencies. Inside atoms, due to the strong nonlinearity of the ruling equations, the fields follow an extremely complicated evolution. Although complex, the phenomenon is deterministic nonetheless. For small molecules, the use of large numerical computations could help us to understand what is really happening in the interspace between nuclei and electrons. For larger structures, statistical considerations prevail, thus leading to the basic assumption of quantum mechanics (in the framework of local theories), where the solutions are in general expressed as probability distributions. For even larger structures, the space surrounding “real objects” is filled with a dynamical electromagnetic component which is zero on average, while the gravitational aspect dominates. The passage through the various stages of complexity is totally unknown at the moment. For this reason, the concept of mass for a large molecule is still to be clarified. It is also possible that, going from the micro to the macro world, further adjustments to the model equations are necessary. As far as we are concerned, general relativity exists for the purpose of allowing electromagnetic photons to interact with each other, preserving the rules of the dynamics of bodies. We are not aware of possible extra corrections to be applied when dealing with macro structures, such as those examined in astronomy. At the various levels of complexity, suitable theories, with an indisputable degree of reliability, already exist and are able to achieve satisfactory results. Our goal however was to investigate the secrets of the intimate structure of matter and set up a plausible (perhaps definitive?) model for the micro universe. Our view has philosophical implications that will certainly be a source of debate. We are comforted by the fact that the theory is heavily supported by irrefutable mathematical passages, and not just by generic hypotheses. In addition, all the instruments used to carry out the analysis are classical and widely accepted by physicists. Indeed, no eccentric or sophisticated
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Final speculative considerations
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new assumptions and mathematical tools have been introduced. Everything was there already, waiting to be used. Those who were expecting a complete explanation of the basic components of our universe to require fancier theoretical arguments will have been disappointed. The last issue in our discussion is the convenience of setting up experiments validating our theory. The problem is left to the experts. However, we think that many convincing arguments, also based on a multitude of practical observations, have already been collected, showing that our model is adequate. The real breakthrough would be in predicting the realization of an electromagnetic device capable of producing a tangible gravitational field.
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Bibliography
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Index
action function, 37, 40, 72, 77, 118 Amp`ere law, 24, 121 anti-particle, 33, 151, 153, 156
92, 95, 107, 127 electromagnetic tensor, 37, 38, 92, 98, 145, 152 electron, 68, 122, 144, 156 elementary wave, 175 energy tensor, 25, 45, 84, 87, 108, 129 equation of state, 71, 89, 113, 120, 176
bare particle, 177, 180 Bernoulli’s principle, 112, 117 Bessel function, 16, 130, 134, 147 bifurcation, 59, 171 Casimir effect, 114 characteristic curve, 28, 29, 35 chirality, 122 Christoffel symbol, 80, 91 circularly-polarized wave, 57, 87, 96 constrained wave, 55, 65 continuity equation, 25, 44, 70, 84, 89, 112 contravariant tensor, 37, 84, 88 covariant form, 42, 101, 120
free-wave, 4, 41, 53, 62, 73, 119, 125 gauge, 37, 94, 114 geodesics, 62, 83, 95, 102, 124, 146, 176 geometrical optics, 10, 44, 61, 62 gravitational field, 35, 36, 82, 104, 183 gravitational wave, 92, 95, 100, 172 horizon, 128 Huygens principle, 8, 9, 19, 31, 38, 45, 56, 103 hyperbolic equation, 28, 59, 92, 146, 148
deterministic, 59, 170, 176, 179, 182 deuteron, 181 diffraction, 59, 61, 65, 117 dipole, 11, 21, 41, 117 Dirac equation, 163
inertial mass, 128, 152, 173, 178 inviscid fluid, 62, 70 irrotational fluid, 71 isentropic flow, 74
eikonal, 10, 44, 55, 71, 82, 112, 121 Einstein’s equation, 62, 91, 92, 95, 101, 112, 116, 127, 152, 171 electromagnetic potential, 36, 41, 44, 77, 86, 114, 127, 130, 139, 147 electromagnetic stress tensor, 84, 87,
Kerr-Newman solution, 126, 129, 154, 158 Klein-Gordon equation, 163 189
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Lagrange multiplier, 38 Lagrangian, 7, 38, 43, 72, 77, 87, 118 left-handed, 68, 119, 122 Lorentz invariance, 46 Lorentz law, 41, 67, 68, 114, 121, 130, 148 Lorentz transformation, 46 Lorenz condition, 43 magnetic monopole, 64, 109, 124 mass density, 70, 71, 88 mass tensor, 88, 89, 92, 108 Maxwellian, 7, 10, 14 metric tensor, 80, 81, 87, 91–93, 99, 113, 127 Minkowski space, 80, 103, 111, 114, 119 monochromatic wave, 2, 13, 53, 60, 101 neutrino, 22, 117, 153, 155 neutron, 181 no slip condition, 117 particle, 2, 114, 121, 122, 124, 126, 152–154, 169 Pauli exclusion principle, 181 periodic solution, 8, 146 phonon, 179 photon, 35, 44, 55, 80, 92, 101, 122, 136, 137, 151, 169, 175, 177, 181 photon emission, 177 Planck constant, 102, 151, 161, 169, 178 plane wave, 2, 8, 22, 53, 57, 71, 88, 93, 99 Poincar´e disk, 102 polarization, 3, 5, 27, 56, 64, 71, 87, 107 positron, 156 Poynting vector, 3, 8, 9, 11, 19, 25, 27, 62 pressure tensor, 111 proton, 123, 144, 156 quantized eigenvalues, 26
Quantum Electro-Dynamics, 36, 44 quantum phenomena, 26, 44, 121, 161, 170, 175, 176, 179 red-shift, 133 reflection, 53, 54, 58, 65 Ricci tensor, 91, 93, 96 saddle point, 43 scalar curvature, 91, 93, 113, 120 Schr¨ odinger equation, 137, 179 signal-packet, 6, 30 singular solution, 2, 4, 13, 24, 29, 129, 134, 155 soliton, 7, 62, 65, 71, 83 space signature, 80, 91, 92, 126, 127 space-time, 29, 58, 77, 80, 81, 83, 93, 100, 113, 120, 124, 146, 169 special relativity, 46 spherical wave, 11, 21, 27, 39, 64, 71, 97, 152 spin, 127, 139, 151, 154, 179 stationary solution, 29, 32, 72, 116, 140, 146 stream-line, 58, 62, 66, 69, 77, 103, 121, 136, 151, 169 Thomas precession, 134 transport equation, 31 Venturi effect, 179 vortex ring, 151, 169 wave equation, 15, 16, 22, 26, 44, 82, 131, 147 wavelet, 9, 10, 103 zero-point radiation, 170
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