Understanding Special Relativity
and Maxwell’s Equations

With Implicattions for a Unified Field Theory

Preface

The order in which scientific laws are discovered is not pre-determined. It isn’t the case that one discovery inevitably leads to the next in a logical order.  Occasionally, a disruptive theory comes along – the special theory of relativity, quantum mechanics – that radically changes the way scientists think about the laws of physics.  New theories, based on these discoveries, are then followed down paths which may or may not lead to a unification of all physical theories.  Perhaps a wrong turn was taken some time ago, which is the cause of the inability, so far, to reconcile quantum mechanics with general relativity, for example, and create a unified field theory that explains everything.  And what ever happened to classical electromagnetic theory?  We know it is central to almost everything in our everyday lives – radio, TV, cell phones, and, in fact, everything electrical or magnetic.  Doesn’t it make sense that classical electromagnetic theory should play a central role in any unified field theory?  Instead, most of the effort has concentrated on combining general relativity, which is concerned with gravity – a much weaker force than electromagnetic forces – and quantum mechanics, of which many of its greatest contributors, including Albert Einstein and Erwin Schrodinger, were uncomfortable with its strange connection to physical reality.

As an example of the order of scientific discoveries, consider Maxwell’s discovery of the laws of electrodynamics in the 1860s.  He discovered his equations by studying the experiments of electrical phenomena carried out by Michael Faraday during the 1830s and 1840s.  Maxwell’s equations predict that electromagnetic waves travel in free space with a velocity equal to the speed of light.  Maxwell therefore believed that light was a form of electromagnetic waves, but most scientists at the time were skeptical.  No one knew how to generate or detect electromagnetic waves.  Maxwell died in 1879 at the age of 48, eight years before Heinrich Hertz generated and detected electromagnetic waves for the first time in his laboratory at the Institute of Technology in Karlsruhe, Germany.  In 1905, Einstein published his special theory of relativity.

Suppose that Einstein was born 100 years earlier.  Could he have developed his special theory of relativity in 1805 instead of 1905?  While it is true that Einstein’s 1905 paper was called On the Electrodynamics of Moving Bodies, and referred to Maxwell’s electrodynamics in the first sentence, the theory itself rests on only two assumptions:  The first, which he calls the Principle of Relativity, is that there is no such thing as absolute rest and that absolute uniform motion cannot be detected by any experiment.  The second assumption is that the speed of light is independent of the motion of the light source.  The speed of light had been estimated by James Bradley in 1728 using stellar aberration.  His estimated value was within about a half of one percent of the current value of the speed of light.  Inasmuch as Einstein’s 1905 paper contained no references, and was based to a large extent on mental experiments, it is conceivable that Einstein might have come up with it in 1805.  At that time, Einstein would have known of Charles Coulomb’s 1784 experiment showing the inverse square law for the force between two charged balls.

In this book, we will show how Maxwell’s equations of electrodynamics can be derived from only Coulomb’s law and Einstein’s special theory of relativity.  This means that if Einstein lived 100 years earlier, perhaps someone could have derived Maxwell’s equations without reference to any of Faraday’s experiments.  Who might that person be?  As we will see, the derivation requires some vector analysis, but today’s form of standard vector analysis was not introduced until the 1880s by Josiah Willard Gibbs and Oliver Heaviside.  However, their vector analysis was really a subset of the more general quaternions discovered by Sir William Rowan Hamilton in 1843.  Hamilton, who was born in Dublin, Ireland in 1805, was arguably one of the greatest mathematicians of his time.  Hamilton thought that quaternions were the best type of mathematics to describe all physical theories, and spent the last 22 years of his life studying quaternions.  So if Einstein lived 100 years earlier, perhaps Hamilton would have used his quaternions to derive Maxwell’s equations.   In fact, using quaternions, Maxwell’s normal four equations reduce to a single equation.  We will show how this is done in Section 4.7.

I first came across the derivation of Maxwell’s equations from Coulomb’s law and special relativity in 1966 in an article by R. S. Elliott in the IEEE Spectrum.  Elliott also included the derivation in his Electromagnetics book that same year.  I was just completing three years in the Air Force working in the Microwave Physics Laboratory at the Air Force Cambridge Research Laboratories in Bedford, MA.  A colleague of mine, Carl T. Case, and I were busy solving Maxwell’s equations for electromagnetic waves propagating through plasmas (ionized gases) in a magnetic field when we came across the derivation.  We redid the derivation using our own notation – which is the derivation given in this book.  In the latter part of 1966, I joined the newly formed engineering school at Oakland University in Rochester, Michigan.

I started teaching courses in electromagnetic theory at Oakland University and I would include this derivation of Maxwell’s equations in these classes.  Carl and I realized that the derivation assumed the assumption of special relativity that the moving reference frame was moving with a constant velocity.  Thus, we were pretty sure that Maxwell’s equations were valid for uniformly moving charges – that is, constant currents, which produce static magnetic fields.  But what about all of the accelerated charges that produce all of the electromagnetic waves, which we had been studying for so long?  We were convinced that Maxwell’s equations weren’t the final word, and that they must be an approximation for something more general.  We had some speculative ideas, which we wrote up and submitted as a paper to the American Journal of Physics in 1970.  The paper was rejected on the basis, not surprisingly, of being speculative.

I told the story of the paper rejection to a colleague of mine at Oakland University, Bob Edgerton, who had been a faculty member at Dartmouth College.  He told me, “I had a colleague at Dartmouth who had a similar experience.”  The person’s name was Miles V. Hayes and he was then an Associate Professor of Engineering at Dartmouth.  He had come up with a unified field theory, which was a single equation that he claimed could explain everything.  He had submitted three papers on the theory to the Physical Review, all of which were rejected on the grounds that the theory was “speculative.”  So in 1964, Miles Hayes decided to publish the theory himself as a small book – 70 pages long.  He published 400 copies and listed in the back of the book everyone who received one of the 400 copies.  He sent 33 copies to the leading physicists of the day, including de Broglie, Heisenberg, and Dirac.  He sent 166 copies to the Chairs of the Departments of Physics of major universities around the world.  He sent 89 copies to various libraries around the world, 15 copies to various publishing houses, 47 copies to a variety of other people, and he gave 39 copies to members of the faculty at Dartmouth, including one to Bob Edgerton.  Each book was numbered and Bob’s was number 309.  Miles Hayes died in 1995.

When Bob told me this story he said, “I have the book,” and he brought it to me.  On the front of the cover jacket, it read, “The universe consists of a complex quaternionic field which is a function of space-time such that its rate of change is proportional to the square of its magnitude.”  I opened the book and saw the field equation followed by a lot of quaternion algebra.  At the time, I didn’t know much about quaternions, (many years later, I would teach quaternions in the context of 3-dimensional rotations in computer graphics) so I closed the book, put it on my bookshelf, where it remained for 40 years.

Around 2010, I was going through my books on my bookshelf and pulled out the Miles Hayes book on A Unified Field Theory.  This time, I read it cover to cover and found it fascinating.  I searched on Amazon and found two used copies for sale through obscure small bookshops.  My colleague at Oakland University bought one, and my son bought the other.  I have never seen a copy available since.  There is almost no reference to this book anywhere on the web.  It is clear from all of the review articles that have been written about unified field theories, that almost no one knows this book exists.  I have never seen it referenced anywhere.  And yet, to me, it makes more sense than any of the unified field theories, or theories of everything, that are out there.

To make Hayes’ unified field theory more accessible, I will present the entire theory in Chapter 6.  The basics of quaternions are given in Appendix A and quaternions are used in Section 2.10 to give a quaternion representation of the Lorentz transformation and also in Section 4.7 to give a quaternion representation of Maxwell’s equations.  So by the time you get to Chapter 6, quaternions should not be such a mystery.

Returning now to Hamilton, who had just discovered quaternions in 1843 and perhaps, assuming Einstein lived 100 years earlier, may have just derived Maxwell’s equations from Coulomb’s law and special relativity.  Like Carl and I, he may have wondered how Maxwell’s equations could be correct for accelerating charges.  He would probably think that Maxwell’s equations were an approximation to some more general equation.  But he would have Maxwell’s equations as a single quaternion equation with a complex quaternion containing both the electric and magnetic fields on the left-hand side of the equation, and the charge and current densities on the right-hand side.  Perhaps, like Hayes 100 years later, he would be seeking a theory of the universe that was both unified and simple.  He would probably agree with Einstein that “there will be no place in the new physics for both fields and matter, that is, particles, because fields will be the only reality.”  Hamilton might also reason, as did both Einstein and Hayes, that particles should not be defined as the source of the field, but rather very small regions of space in which the field values are very high.  In fact, Hayes postulated that particles are standing light waves.  But he realized at once that Maxwell’s equations were linear and therefore could not account for a stable standing wave of finite size in the absence of material boundaries.  For this, and other reasons, Hayes recognized that a single unified field equation must be nonlinear.  Perhaps reasoning in a similar way, Hamilton might have come up with the same quaternion field equation as Hayes, in which the right-hand side is a quadratic function of a field made up of the electric and magnetic fields.  As we will see in Chapter 6, if the right-hand side is set to zero, the field equation reduces to Maxwell’s equations in free space.  If the right-hand side is set to a constant, the field equation reduces to Maxwell’s equations with charges and currents.  If the right-hand side is set to a linear function of the field, the field equation reduces to Dirac’s equation of quantum mechanics (properly interpreted).  This means the equations of classical electromagnetics and quantum mechanics are both approximations of the quadratic field equation of Hayes.  This quadratic field equation is also Lorentz invariant – a requirement of an equation which purports to be a theory of everything.

Let’s assume that Hamilton did, in fact, come up with the quadratic quaternion field equation 100 years before Hayes did.  This equation expands into eight coupled non-linear partial differential equations.  We still don’t know how to solve these equations analytically (only numerically), but perhaps Hamilton, as the leading mathematician of his day might have come up with solutions.  Suppose that he found that one solution is a Lorentz-invariant spectrum of fluctuating electromagnetic radiation (electromagnetic zero-point radiation) in the universe.  If so, someone could have derived the blackbody radiation spectrum without any quantum assumptions 100 years before Timothy Boyer did (see Phys. Rev., vol. 182, no. 5, 1969, pp. 1374-1383), and decades before Max Plank derived the blackbody radiation spectrum by introducing the quantum idea.  H. E. Puthoff has shown that the ground state of hydrogen can be modeled as a zero-point-fluctuation-determined state without resort to quantum mechanics (see Phys. Rev. D, vol. 35, no. 10, 1987, pp. 3266-3269).  Perhaps Hamilton would have found that exact solutions of the nonlinear field equation predict elementary particles as discrete, stable, oscillatory limit cycles in the nonlinear field as suggested by Hayes.  Perhaps if these derivations had occurred before Plank, Heisenberg, and Schrödinger, the development of quantum mechanics may have taken a different route.

What else might Hamilton have inferred from the Hayes field equation?  Hayes states:  “The theory predicts that light waves interact with light waves, or photons with photons.  The frequency, amplitude, phase, and other characteristics of light waves are modified by interaction with the other light waves through which they pass in coming from the stars.  The greater the distance travelled the greater the modification.”  Had Hamilton solved this problem and found that the field equation predicts that light from distant stars are red-shifted, then when such observations were actually made, perhaps everyone wouldn’t have rushed out and blamed it on the Doppler effect.  Perhaps the universe isn’t expanding after all.  Perhaps the big bang never occurred.  The consequences of possibly going down the wrong path for decades and decades boggle the mind.

What about gravity?  Hayes goes into some detail about how gravitational fields are fields, i.e. electromagnetic fields, between neutral particles.  If a solution of the field equation is the electromagnetic zero-point radiation described above, then H. E. Puthoff has shown that gravity can be modeled as a zero-point-fluctuation force (see Phys. Rev. A, vol. 39, no. 5, 1989, pp. 2333-3242).

To bring attention to the long-ignored and unknown Hayes book, I have written a novel called Peggy’s Discovery.  In this novel, Peggy, a high-school senior, with the encouragement of her engineering professor uncle, enters college on a quest to understand the nature of physical reality by challenging conventional wisdom.  In the process, she starts a company that revolutionizes higher education, leading her to uncover the secret to a theory of everything.  In this novel, Peggy watches her uncle derive Maxwell’s equations from Coulomb’s law and special relativity just as we do in Chapter 4 of this book.  After going through this derivation, Peggy realizes that the exact same derivation can be applied to Newton’s law of gravitation and special relativity, meaning that this would lead to equations identical with Maxwell’s equations except that the mass density would replace the charge density.  She then realizes that there is a second gravitational field, analogous to the magnetic field, which exerts a force only on moving bodies, that this force adds to the normal gravitational force acting on the outer stars of a galaxy and may therefore explain the higher velocities of these stars, meaning there would then be no need to invent dark matter.  We discuss Peggy’s new theory of gravitation in Chapter 5.  As Peggy points out in the novel, this would be the simpler and more obvious way to make Newton’s law of gravitation consistent with special relativity, rather than the much more complicated approach that Einstein took in his general theory of relativity, complete with curved four-dimensional space.  Perhaps general relativity could have been by-passed altogether as being unnecessary.  In fact, Peggy’s uncle tells her that Oliver Heaviside, in 1893, twelve years before Einstein published his special theory of relativity, suggested that gravity might be explained with two gravitational fields satisfying equations directly analogous to Maxwell’s equations of electromagnetics.  Now Peggy has derived these gravitational equations from first principles. 

As we will see in this book, Einstein’s special theory of relativity is central to understanding electromagnetics; and electromagnetics may be central to a simple unified field theory.  Might it be that Einstein’s general theory of relativity, with its emphasis on gravity, was a wrong turn down a long path, which has not been successful in leading to a simple unified field theory?   

It is too bad Einstein wasn’t born 100 years earlier!

                                                                                                            Richard E. Haskell

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