Aether Science Papers: Part I: The Creative Vacuum
Pages 3-9

Copyright © 1996 Harold Aspden


THE AETHEREAL ROLE OF FERMAT'S LAST THEOREM

As we approach the end of the 20th century we should pause to examine our achievements in science and technology in the past 100 years. They are indeed remarkable and there is reason to wonder whether there is much left to conquer as we enter the 21st century. After all, the discovery of new territory in a geographic sense came to an end once exploration had completed the survey of the Earth's surface, so one day soon science, at least physical science, should reach its zenith.

We will then still have to ponder on our incomplete knowledge of space that we cannot easily explore and still need to confront the few never-to-be solved mysteries that science has bequeathed to us even from centuries past. To be sure there is much for us yet to discover in the medical and biochemical field, but physics should by now have yielded the answers to all the secrets that Nature is willing to reveal.

We will never understand what lies beyond our comprehension, such as why the universe exists and what there is beyond its bounds in the context of time and space. Indeed, whatever we might foresee in the long range future of the universe, is not really relevant to mankind, because it seems probable that, on a more limited time scale, our planet is destined to encounter catastrophy sufficient to terminate human and animal life on Earth.

So let us take stock and reflect a little on what the transition to a new millenium can mean for physical science.

Yes, indeed, there are very few challenges now left to tax the mind of a true physicist. However, in saying this, I, as author, am speaking from my own knowledge as a physicist and each of us has our own different and limited perception. I am all too conscious of the fact that, if the scientific community on Earth were to be eradicated and all the books on science were to be destroyed, then, even though the human race might survive, so science as we know it would have to evolve again from nothing. It would then take several more centuries, perhaps another millenium, to bring us back to where we are today.

However, would it really matter and would science develop in the same way as our history indicates? Would we have another Newton and another Einstein? Surely, there would be another Pythagoras! The Pythagoras theorem has a unique quality. It is a survivor, a fact of science on which one can build and, though taught as mathematics, one can even wonder whether it is a statement in physics. Once a surviving remnant of mankind can reason sufficiently to rediscover and find interest in the theorem of Pythagoras then science, including physical science, has been reborn.

Thinking along such lines might seem to be pure fantasy, but let me make my point a little differently. Suppose that I were to say that I know how to formulate the Unified Field Theory and how to explain the true nature of gravitation and certain other still unsolved fundametal issues in physics. Suppose I were to die, as is inevitable, and my writings on these subjects were to be ignored, as seems not unlikely. Then how would that impact the world at large?

It would not even be noticed. Nor, I submit, would the loss of much of the knowledge that takes up space on our university library bookshelves. The simple truth is that mankind in general is not concerned with the understanding of the kind of physics or mathematics that fills the minds of many of our university professors.

However, technology has become important to our daily lives and there are certain basic teachings that physics in its applied form does contribute to that spectrum of activity, so I must not decry what physics at its applied level does offer to our well being. The major problem ahead of us in the 21st century is the need to discover a new and abundant non-polluting source of energy. I am convinced that this is a problem that could easily have been avoided if some aspects of the 20th century could be erased from our memories.

We have ventured into the realm of nuclear power whereas we should have been 'burning the midnight oil' in studious endeavour and probing the energy secrets of the aether. We erred because Einstein outlawed the aether, closing off access to the power source which created the universe. We erred by adopting Einstein's belief in a mathematically abstruse philosophy of so-called four-space, a four dimensional distortion of reality, a virtual world that has become a drug to which theoretical physicists have now become addicted.

Einstein took us into a mental world which had no Pythagoras. The two space dimensions of a flat surface on which one can draw a triangle with two sides and a hypotenuse were replaced, not by the three-dimensional space of the curved surface of the Earth we inhabit, but by an illusory scheme we cannot picture in our mind's eye. We are even being told today that, thanks to Einstein, we can look forward to 'time travel' as we exit through 'worm holes' in a 'time warp' to leap into the past and perhaps into the future. That surely tells us that Einstein's theory is a drug we can best do without!

Yet, in their hearts, all of those Einstein-drug-addicted theoreticians must know that they have draped the universe in a web so fine that it cannot be seen or felt or serve any useful purpose. Does it really need a child to cry out: "The Emperor wears no clothes?" Does it not suffice, after 80 years since Einstein enunciated his General Theory of Relativity, for us to ask what it does for mankind?

Why would God create four-dimensional space and give us a perception of it in three dimensions? Why, even, in applying General Relativity, do we always need to transform its results back into three dimensions to give them meaning?

So, as I say above, if we could erase all this from science as we know it, the world would be unaffected and a new generation of physicists could begin anew in developing a theory which says that the universe was created from energy shed by the aether. After all, if something is created there has to be something serving as a source for what is created.

In saying this I am reminded that Sir Edmund Whittaker, author of 'A History of the Theories of Aether and Electricity', quoted Spinoza to introduce his work as 'The intellectual love of God'. This was a way of saying that to understand the aether is to understand the Creator.

The purpose of this work is to show that the 20th century did, in fact, provide most of the answers to the primary unsolved problems of fundamental physics, including discovering that Holy Grail we call the 'Unified Field Theory'. Sadly, however, that drug-addicted community of relativists which regards such theory as their private province has refused to listen to those not sharing their addiction and so I am seeking to interest those outside that community who have retained their senses and their sanity.

It is the author's intention to show elsewhere, under the title of 'Energy Science Reports', that the 20th century has also delivered a solution to the impending energy crisis by the discovery of ways of extracting energy from the aether. This touches upon the beliefs of a more practical scientific community, but one responsive to what can be demonstrated, whereas this work is strictly concerned with reason and theory, something far more difficult to project into the minds of others than is the reality of the new energy scene. This work describes that aether and its creative role.

It may be that if this account is ignored by the scientific community then it may take several centuries before some future scientist rediscovers what is here presented. Take note that even knowing that someone once did prove something in scientific history does not make the task of rediscovery any easier.

Witness the centuries of effort in trying to solve the problem of Fermat's Last Theorem. This was Pythagoras converted to a power higher than 2, with integer sides to a notional 'triangle', the impossible dream! Fermat assured us that he could prove it was impossible but his secret was somehow lost.

Modern opinion, today, is that Fermat may have been deceived in thinking he had proved his theorem. Very probably that is valid opinion, because if there were a simple proof it would, undoubtedly, have been discovered by now. As will be seen below I do have reason for connecting an aspect of the aether problem with Fermat's Last Theorem, but first note that in 1995 it was announced that, after centuries of effort, a Professor of Mathematics, Andrew Wiles, at Princeton had at long last discovered a proof of Fermat's Last Theorem.

It was this reference to Princeton, the university where Einstein had spent many years as a professor, that aroused my interest. I knew how to connect Fermat's Last Theorem with the nature of electricity and thereby introduce the aether in a way that could be a challenge to Einstein's theory. With my Cambridge background and my anti-Einstein disposition, I then thought of introducing this theme in this work.

It was also the memory I had from 1981 when my wife and I passed through Princeton on our way south for a weekend in colonial Williamsburg. This was before going back north to attend a conference on fundamental physical constants at the Bureau of Standards at Gaithersburg near Washington D.C.

My wife popped into the university bookstore at Princeton and persuaded them to stock my book 'Physics Unified', published just a few months earlier. How long, I wondered, would the book be reordered, once the relativistic community on the teaching staff woke up to what their students might see in my book. Indeed, it took a while before the inevitable happened and orders stopped, but a similar venture at the university bookstore at Stanford in California has led to a sustained inflow of orders for stock, even to this day.

I had, incidentally, already seen a brief mention of Andrew Wiles for his achievement in solving Fermat's Last Theorem in the pages of the Michaelmas Term 1995 issue of CAM, the University of Cambridge Alumni Magazine. There it was explained how, according to John Coates, Sadleirian Professor of Mathematics, "Cambridge has always produced some of the most original and gifted mathematical minds in the world." The report declared that 'notable amongst them is number theorist Andrew Wiles who sparked worldwide press interest when he cracked one of the great conundrums in all mathematics: Fermat's Last Theorem'.

It went on to quote Fermat as noting on a Greek mathematical text found after his death in 1665: "I have a truly marvellous demonstration of this proposition which the margin is too narrow to contain." Then the report further declared "Today's scholars doubt that he had. But, says Coates, over the centuries pure mathematicians have developed deep mathematical ideas trying to resolve the problem", followed by "I did not expect to see it happen in my lifetime".


It was later reported by Marcus du Sautoy in the British newspaper THE TIMES on Monday April 8th 1996 that Andrew Wiles, 'for his solution of Fermat's Last Theorem was rewarded in the Knesset (Israel's parliament) with one of mathematics' highest accolades, the Wolf prize worth $100,000, which he shares with his colleague at Princeton, Robert Langlands.' The headline caption of that report read: 'The solving of a famous condundrum will lead to new challenges. Is this solution the end of maths?'

Well, Marcus du Sautoy, it may not be the end of mathematics but it might well become the beginning of a new age in physics as we see its scope for uprooting Einstein's theories. The event described is a reminder that Albert Einstein was offered the Presidency of the State of Israel, whilst scientists at large still seek that Holy Grail, their Unified Field Theory which eluded Einstein.

Curiously, there was something in Marcus du Sautoy's report that reveals an extrasensory perception because I had already written the text which appears ahead on pages 12 and 49. He suggested that the next challenge would concern the 'Riemann Hypothesis' concerning prime numbers. "Those numbers are in some sense the harmonics of the 'Riemann zeta function'. It is these harmonics which tell you all about prime numbers. Riemann conjectured what these harmonics look like. If true, it could imply that the music of the primes is far from being just noise."

Well, true or false, the harmonics of the primes do feature in this author's theory as outlined ahead, but I did not know I was treading the holy ground of the mathematician when I confronted the electrodynamic resonances in my study of the subject. I still think that the discipline of mathematics is a tool designed to help us to understand Nature, rather than to fashion it by shaping it to fit what we want to believe. The challenge ahead is not one to be classified as mathematics.

Curiously, one senses history beginning to repeat itself, because it was the Riemann tensor which was applied to underpin the mathematics of Einstein's Theory of General Relativity. My attack using 'the music of the primes' will be aimed at proving the aether exists and that the concert hall in which Nature plays that music is one having three space dimensions.

To migrate from the numerology of Fermat's Last Theorem to the physics of electrical phenomena we need now to consider physical dimensions and how we incorporate electrical phenomena in this system of dimensions.

The standard physical dimensions used when expressing measured values are mass M, length L and time T plus something that has an electrical connection, the dielectric constant k. To bridge the gap between inertia and electricity it is not mass that has primary significance, but energy E, inasmuch as the inertia of any electric charge is the property by which it conserves its energy to avoid continuous (non-quantum) loss by radiation when accelerated. See my paper in International Journal of Theoretical Physics, v. 15, p. 631, 1976 or see section 7 of the last of the fourteen appended papers.

This introduces us to the problem of understanding the true nature of electricity and in particular why it comes in positive and negative forms. The answer is similar to there being odd and even numbers. They represent alternate states in a sequence. In the binary number system we see the last digit as either 1 or 0, this being the odd or even condition. In electricity we have (+) or (-) as the polarity of electric charge which I envisage as having spherical form. There is no zero charge state at the truly fundamental level because that only arises where electric particles combine into a neutral aggregation. It is, however, possible for two charges, a particle and its antiparticle, to annihilate one another and shed energy, a quantum event leaving no electrical form or normal electromagnetic wave that we can trace, which is why physicists invented the 'neutrino', but the root question we face is 'what attribute determines whether a charge is positive or negative?'

Mass M has dimensions EL-2T2 and it is appropriate to seek to explain all phenomena in terms of E, L and T as the primary physical dimensions and, as we are probing fundamental physics rather than applied physics, to use the esu system in which the dielectric constant k of the vacuum medium is unity. Thinking in terms of energy E, length L and time T, the way forward is to regard an electric charge as a package of energy E which occupies a volume of space L3 but oscillates at a frequency 1/T by exchanging some of that volume with a similar package of energy, albeit also with with energy transfer to and fro between them. This means that there will be two types of charge, or rather states, which differ in character only according to the instant at which we observe them. One will be expanding and the other will be contracting. One, the positive charge, will be in 'phase' with whatever charge form we take as our positive reference and the other at the same moment will be in anti-phase and so be a negative charge.

Do note here that Einstein's declaration that space and time were intermeshed precluded him from ever accepting the concept of instantaneous action at a distance, thereby excluding the synchrony and phase-locked oscillations which we shall use as the key to understanding electric charge polarity. Einstein lost his way with the first step he took on his path of relativity.

We, following a different path, can now, if we wish to go to really fundamental levels, explore how electric particles develop into different families, the conservation of energy and the space they occupy being key features of the transmutation process. That will lead us automatically to the point where we see how to solve the problem of linking gravitation and electrical action. The task in sight is no less than that of meeting the challenge posed by Unified Field Theory, but from there we can move even further ahead and come to terms with the very nature of electricity.

Fermat's Last Theorem can play a role in this pursuit.

When an electric charge is compressed into a sphere of radius a the charge occupies a volume of space 4(pi)a3/3 and it has, if under uniform pressure within the sphere, an energy E inversely proportional to a. Now, given the hypothesis that charge polarity depends upon the phase of an oscillation under conditions where volume of space occupied by charge is conserved, we see that a group of particles in close proximity can only change form subject to the combined volume (pi)a3 being constant. In energy terms this means that the summation of (1/E)3 is constant, so if two fundamental particles could merge to become one single particle, which adopts one or other charge polarity then, using x,y z as their energy parameters:

x-3 + y-3 = z-3

If this had an integer solution then, by multiplying throughout by x3y3z3, we could use the numbers yz,xz and xy of that solution as an integer solution to a Fermat equation for which n=3. This is impossible and so, if we were to assume that the energy quantities really do comprise integer multiples of a basic energy quantum and the space taken up by the particles is conserved, the merger of two such electrical forms vibrating in anti-phase can never result in the creation of a single particle of unitary charge. We know this without appeal to empirical fact concerning how charges of different polarities are seen to interact. In effect, we have given meaning to the polarity of electrical charge by logical argument based on the physical dimensions E, L and T and the use of Fermat's Last Theorem.

So far as this author is aware this is the only application of Fermat's Last Theorem to a truly physical problem.

It does, however, open the question of whether, if one searches to find integer solutions to equations such as:

xn + yn = zn + wn
or:
xn + yn + zn = wn
for n equal to -3, such solutions will be found which bear upon the issue of the hypothetical fundamental quantum of energy.

To satisfy simple equations of the above form, such an energy quantum would, of course, be extremely small in relation to the mass energy of the electron and we would then need to see the neutrino as comprising large quantities of such quanta. However, since the neutrino is surely a figment of imagination, just something invented as a `bookkeeping' exercise to keep the energy and momentum balance as between matter and aether, the aether itself becomes the storehouse for energy which, in its ultimate form, may well be quantized in units of the notional energy quantum.

Unless we pursue this possibility we cannot but wonder whether an avenue of science remains unexplored, and it may well be that there is no integer solution to these equations which has any special significance.

It is to be noted that two of the appended papers use the following equation:

x-3 + y-3 = Nz-3
with N=5 expressing the merger of 5 muons to form a kaon, but that did not extend to a search for the fundamental energy quantum. [Hadronic Journal, v.9, p.137, 1986] and [Hadronic Journal, v.12, p.101, 1989]. These are the second and eighth of the appended papers. Note that the equation as used in those papers has a special meaning in that the kaon was portrayed as a particle oscillating constantly between two states, spending half of the time in each state, one being that of a particle of unitary composition (x) and the other being that of a three-particle form, (x) plus two (y), so that on average it comprised simply a single (x) and a single (y) form.

With N=6 the latter equation does have integer solutions, as one sees reported by Mike Mudge in Personal Computer World, p. 614, April 1995. Values x=357, y=777 and z=629, satisfy the equality, but these do not relate in any way helpful in our search for the fundamental energy quantum. Solutions, if any, for N=2 are of primary interest.

The idea that conservation of three-dimensional space is the determining factor governing the properties of a fundamental electric charge, whereas the phase of the pulsating state of this space volume determines the charge polarity, may seem quite revolutionary. Physicists have, it seems, spent less time pondering the question of why electric charge comes in negative and positive forms than they have in hypothesizing about the imaginary notions of negative mass, negative energy and negative time. Concern about the nature of electric charge proper, rather than worrying about the speed of light, is important because it provides a more appropriate line of demarcation between the features of aether theory and relativity. Historically, the investigations of C. A. Bjerknes (c. 1877) on spheres pulsating in antiphase in an enveloping medium to set up mutually attractive or repulsive forces give us a lead. See references on p. 284 of Sir Edmund Whittaker's 'A History of the Theories of Aether and Electricity: The Classical Theories' (Nelson, 1951).

This, therefore, is this author's justification for arguing that Fermat's Last Theorem has real relevance to physics. It concerns the physics of three space dimensions and three physical dimensions, such as energy, length and time. The three dimensional world is the real world which the true scientist should be exploring, not the imaginary mathematical jungle which followers of Einstein have adopted.











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