Here we face the Ultimate Challenge: Explaining the value of G as now measured in the year 2000

Copyright, Harold Aspden, 2000

Over the past 40 years I have sought to interest the world of science in my theory of gravitation. It is a theory based on interpreting the quantum activity which underlies our immediate environment and pervades all space. There are energy transitions involving what are called 'leptons', constantly occurring, even in the vacuum, in an underworld scenario which some physicists refer to as the field of 'zero-point energy'.

My theory is a comprehensive 'unified field theory' in that it explains gravitation as an electrodynamic process and is successful in that it allows G, the Constant of Gravitation, to be determined quantitatively and qualitatively in terms of the electric charge to mass ratio of the electron.

The theory has developed in stages, but its phenomenological basis, the link with electrodynamics, stands on the firm foundations disclosed in the first edition of 'The Theory of Gravitation' published in 1960. I wrote that text in the latter part of 1959 to put my theory on record at a time when I had decided to make a career move by joining IBM in England as a Senior Manager. I had not opted for an academic university career following my Ph.D. research years at Cambridge. Instead I decided to develop specialist skills concerning technological innovation and its protection in an industrial environment.

The theory of gravitation was of interest to me, essentially because my research had shown how the electromagnetic energy we associate with magnetic induction is stored in what we term a 'field', whether within solid matter or in vacuo. I could see a basis for connecting that with the phenomenon of gravitation, but had come to see the need to believe in the existence of a real aether, something of a 'taboo' subject in the world of academic science. Physicists had come to accept the Einstein doctrine implicit in his theory of relativity and the aether has no place in that doctrine. 'Four-space', a concept that defies visualization in physical terms, had displaced the 'aether' and I well knew that, to project my theoretical notions I would have to contest territory that was the exclusive province of the physicists who were addicted to Relativity.

Over the years, I did update the my theory of gravitation as it advanced step-by-step, notably in 1966, 1969, 1975, 1980 and 1996. [See book titles in list of references]

However, the scientific community has remained unconcerned and has seemingly ignored my efforts. So, with two new major advances of this the millennium year 2000, the one reported in the preceding web page item and that I now present, I venture now upon what I presume will be the culminating step, namely the task of deriving G, the constant of gravitation, by theory which gives the precise value in accord with the 1.5 part in 100,000 precision of the year 2000 measurement, which has involved a quantum leap in the degree of experimental measurement precision of this very-difficult-to-measure quantity.

Contrary to conventional practice, and contrary to the natural and logical path I followed in reaching the result, I will present the formulation first and then show how it is derived. I want the numbers to point to the underlying physical picture, because it is all too easy, if I start by portraying a physical scheme, for the unbelieving reader to presume that that I rely on hypothesis and that the assumptions implicit in that scheme were deliberately chosen to give the correct numerical result.

Of course, even with this approach, critics will be critical. They will suggest that I am merely playing with numbers, pressing the buttons on a calculator until I find a mathematical relationship that gives the right answer. However, I think the mathematical odds against finding the formula I present below, as one fitting the known answer, are high enough to upset that argument, but you must judge. I can only say that I have been guided by what I see as the physics underlying the formulation and believe it portrays how Nature regulates the phenomenon of gravitation.

The formula defines the action of gravity as a quantum phenomenon, quantum in this case meaning 'as involving discrete units of action'. Two distinct quanta are needed, as I shall explain later, and these are odd integer numbers that signify a multiple of a unit based on the form of the electron. These numerical quanta are denoted A and B, respectively.

I shall formulate an expression for the square root of G, which I express as Gr, the reason being that I see the action as between systems each involving the same two basic quantum units, each such unit comprising one A quantum and two B quanta, for reasons that I shall explain in detail later. Then, without further ado, I present the formula:

Gr = (4π/3)(4.8032x10-10)/(108π)3(A)4(9.10939x10-28)X


X = [1 + 2(A/B)3]/[1 + 2(B/A)]

Now, if you use your calculator and substitute the numbers 5061 and 3485, for A and B, respectively, you will discover that G has the value 6.67435x10-8 and I ask you now to compare this with the latest, year 2000, value of the Constant of Gravity G reported in SCIENCE, v. 288, p. 944 (2000), which is 6.67423+/-0.00009 times 10-11

Note that my derivation, as formulated, involves the old-fashioned cgs system of units, as applied to a vacuum assigned unity permeability and permittivity. I prefer that because the vacuum is my reference base and the numbers are easier to work with if one bases the units on that as reference, given that the object is to understand physical phenomena occurring in vacuous space, rather than find a convenient way of dealing with practical problems by avoiding numerical expressions that put 4π into the calculation.

Those G values differ by very slightly more than one standard deviation of the year-2000 measurement. That is more than one order of magnitude more precise than any previous measurement of G.

So, I claim that my formula, as just presented, has something to say about the nature of the force of gravity, subject of course, to it having a true physical basis.

Well, first of all, if you are astute on matters relating to physics, you will already have noticed that the electron charge/mass ratio features in the equation for Gr. The charge of the electron in cgs esu is 4.8032x10-10 and the mass of the electron is 9.10939x10-28 gm.

So here is that 'Unified Field Theory' connection. The mutual force of electrodynamic attraction between two discrete electrical charges separated by unit distance is equal to the product of their charge values in esu, if they are moving mutually parallel at the limiting speed governing electromagnetic wave propagation in vacuo. That, in physical terms, raises certain questions, because we see particles generally as being close to a position of rest in the electromagnetic frame of reference. However, note that I am using the electron as a unit only. I do not say that gravitation stems from the interaction of electrons. Instead, I say that it arises from interactions between what I call gravitons, a kind virtual lepton that lives in a world that is ever moving at the speed of light, c, relative to the frame occupied by matter.

You see, the quantum energy activity underlying everything imparts a universal motion to matter (physicists call it the Zitterbewegung), a jitter associated with Heisenberg's Principle of Uncertainty. That system of matter in motion in an inertial frame defines the local electromagnetic frame of reference, and so the material universe would be dynamically out-of-balance inertially, were it not for a hidden counterpart system of gravitons moving in juxtaposition about the inertial frame. The relative motion, as between that graviton system and the matter it balances, is the speed of light c. However, we are dealing here with a system that is kept in step by synchronizing actions which override certain considerations that govern material particles moving linearly at high speed through the electromagnetic reference frame. The gravitons do not gain kinetic energy in the context of the relativistic mass increase that applies to material particles. I accept that this may seem a little complicated, but it is fully explained in my published works and here I am intent on justifying the numerical formulation just presented.

The formula contains four other quantities, namely 4π/3, 108π, 5061 and 3485.

As to 4π/3 you can guess that this is the numerical coefficient used to derive the volume of a sphere from the radius. So we are concerned with units of electric charge that sit within a spherical unit of space, or, alternatively, that, as part of a uniformly dense system of charge, are excluded from a spherical space.

Note then that 108π appears as a cubed expression. This also suggests a volume measure, where 108π is proportional to a side dimension of a cubic form. You will further notice that it has the effect of very substantially weakening the mutual attraction force implied by the interaction of two electrons. However, the force of gravity as based on the interaction of two electron masses is still far weaker than that reduction would imply. So we look, not to the unit charge of the electron, but rather to the unit charge excluded from a spherical form on the basis that the charge density is determined by e, the electron charge, as distributed uniformly over a cube of size involving that expression (108π)3.

Here I am going to digress for a moment to refer to a paper published in Physics Letters, 41A, 423-424 (1972), which I co-authored with Dr.D.M.Eagles. It was entitled 'Aether Theory and the Fine Structure Constant'. The fine-structure constant is a numerical quantity linking the three properties of the aether, namely c, the speed of light, e, the charge of the electron and h, the symbol of the energy quantum, Planck's action constant. Measurement indicates an approximate relationship:
hc/2πe2 = 137.0359
and, back in 1972, based on the development of my theory pertaining to gravitation at that time, I discovered the physical basis of this numerical quantity. My formulation was:
hc/2πe2 = 108π(8/1843)1/6

That number 1843 was determined by an energy minimization argument, subject to energy not becoming negative, but with the physical quantity represented by that 1843 number being the number of electrons and positrons that could fill the volume of space occupied by one unit charge particle in the aether. The number had to be an odd number. Its value was derived by rigorous analysis from first principles, as that paper shows.

The 108π term has the following meaning. The unit charge particle sat in a cubic volume of space with its charge balancing the opposite polarity of a uniform charge density filling that space. The aether is electrically neutral on a macroscopic scale but has electrical structure on a microscopic scale, as needed to account for Maxwell's displacement currents.

Analysis, based on explaining the energy quantum of the photon and deriving the action underlying Planck's radiation law, which includes h, told me that the side dimension of a unit cube of space is 72π times the classical radius of the electron, the latter being e2/mec2 in cgs units. Here you see that charge/mass ratio of the electron mentioned above. Now, the so-called classical radius of the electron is a notion only. In fact, the correct formulation of electron radius is that advocated by J.J.Thomson, which is two-thirds of the value of the classical radius. So one can say that the unit cube side of the space occupied by one aether charge e is actually 108π times the actual radius of the electron. You can, by cubing this quantity, and dividing by 4π/3 obtain a measure of the unit volume of a cubic cell of the aether in terms of electron volume.

It is, simply, 9,324,644, but one needs to divide this by that number 1843 to get a measure of the volume of that unit cube in terms of the volume of the unit aether particle that sits in that cube. The result is 5059.49. That explains how I first began to see something special about such a numerical quantity pertaining to the aether. You see, I could work out the electrical charge density that filled those cubic units of space, and I suspected that units of a spherical particle form in dynamic balance with matter would displace a tiny amount of charge owing to their finite volume, and I simply worked backwards on the assumption that this would account for gravity to estimate the physical size of that particle. I found that, using the J.J.Thomson formula, it would have a mass of approximately 5063 times that of the electron. Hence my interest when I saw that analysis of the photon by deriving the fine structure constant had led me to that 5059.49 quantity.

So I come to that 5061 term which appears in the gravity formula to a fourth power. The fourth power arises because, in electron unit terms, a non-composite particle of charge e that is N times the mass of the electron, has a volume that is 1/N3 that of the electron. Then, since volume determines the electrical charge involved, but gravitational action, in the Gr context, is scaled according to mass, we need to adjust that by the factor 1/N to obtain 1/N4 as the appropriate term to use.

Now, why have I used 5061, rather than 5059 or 5063? This is the real breakthrough in the development of my theory, guided now by this new year-2000 report of the latest G measurement.

The answer to this is that my theory has evolved in stages. Initially I was concerned with how gravity could arise from the influence of very tiny masses and yet satisfy the same constant of gravitation that applied to large masses. Even the energy content of electromagnetic waves gravitate, because we see them deflected in a gravitational field and I say this in the firm belief that Einstein is wrong in declaring that this arises because space is curved. My case, based on aether theory, is that the aether medium can offer the same G response to large systems of mass as it does for tiny units of mass, meaning the mass-equivalent of energy. [I note here that I adhere to an interpretation of the basis of the formula E=Mc2 as justified historically by energy conservation theory in classical electron theory, again rejecting the Einstein theoretical doctrine.]

The outcome of these considerations led me to picture a graviton system in which a component of unit charge of higher mass was partnered by a pair of unit charges of equal mass somewhat lower than that of their companion. The idea was that, consistent with quantum electrodynamic principles, the charge pair could be leptons which could mutually annihilate to vacate space and shed energy as needed should the populace of the particles of higher mass have to adjust slightly in volume to keep the energy balance and assure that the volume to mass ratio remains constant. The latter is essential for G to be a constant.

The simple equation governing G is:
G = (σV/M)2
where σ is the electric charge density filling those cubic cells of space and V/M is the volume to mass ratio of the graviton-lepton system that provides the dynamic balance for any matter present.

On this basis a mass M sitting at rest in the electromagnetic reference frame (the E-frame) has angular momentum owing to its motion about the inertial frame (I-frame) and its centrifugal action needs to be balanced by a corresponding mass M of gravitons and leptons of a gravitational frame (G-frame) moving with that frame in juxtaposed relationship.

So we can determine V/M if we know the mass values of the gravitons A and the leptons B in terms of units of electron mass. For a system with twice as many leptons B as gravitons A, one can calculate the volume V of charge they displace as:
V = (1/A)3 + 2((1/B)3
in electron charge volume units. Similarly, one can calculate the their combined mass M as:
M = A + 2B
in terms of the unit of electron mass.

The ratio V/M becomes:
V/M = [1 + 2(A/B)3]/[(A)4][1 + 2(B/A)]
which one can see has the form needed to justify the remaining components of our basic formulation for the value of the square root of G.

It follows that we can say that we have here a theory that can account for the precise value of G, the Constant of Gravitation, guided by the empirical data that A has the value 5061, meaning the graviton has a mass of 5061 times that of the electron, and B has the value 3485, which says that the lepton has a mass that is 3485 times the electron rest mass.

Now, again I digress, this time to explain that there are in physics essentially three basic forms of lepton that have mass. They are the (a) the electron, which in my theory has a prominent role in the E-frame, as part of matter, (b) the muon, which in its virtual form, has a major role to play in my theory by sitting in the I-frame and functioning as an omnipresent energy source and (c) the taon (the tau-particle), which, in its virtual form, and being the only remaining lepton, has to be the lepton form B that sits in the G-frame.

The theory developed from the initial calculation of the energy content of the aether particle that sits in every cubic cell of the aether. Using the J.J. Thomson formula:
E = 2e2/3a
for the electron of mass-energy E and charge e, a is electron charge radius, but applying this same formula to the aether particle, the energy scales by the factor (1843)-1/3 to a lower value. Here I am using the fact that the aether particle has a charge volume that is 1843 times that of the electron, as I knew from the photon theory mentioned above that gave the fine-structure constant. I regarded this energy of the aether particle as determining a pressure in the aether medium which signified a uniform energy density. Accordingly, I deduced the energy content of the unit cubic cell of aether, because I knew, as shown above, that the volume of that cell is some 5059.49 times that of the aether particle. Note then that:
(5059.49)(1843)-1/3 = 412.6656

In energy terms, as units of electron rest-mass energy, this is the energy needed to create two virtual muons, the leptons of the I-frame. Here was the component building block from which Nature creates protons and I did decipher that process by 1975, three years after the publication of that Physics Letters paper on the aether derivation of the fine-structure constant. It was duly published by the Italian Institute of Physics in Il Nuovo Cimento, v. 30A, pp. 235-238 (1975). It is entitled: 'Calculation of Proton Mass in a Lattice Model for the Aether'. The proton/electron mass ratio was found to be:
a value that was one part in two million too high according to measurements of that period, but which fitted even closer to the eventual measurements using a new direct method rather than merging two separately measured quantities.

The next advance relevant to this account was the discovery by which I deciphered the process of creation of the tau particle. It evolved from a sequence of reactions in which mesons are created, the subject of my paper in Hadronic Journal, v. 9, pp. 153-157 (1986). The paper is entitled:'An Empirical Approach to Meson Energy Correelation' Here I had shown that the common building block, the virtual muon, from which protons and tau-particles develop, gave simple formulated relationships from which one can deduce that the tau particle has a mass energy that is greater than that of the proton by a factor of 37/12.

This was a curious result, but, given that theoretical proton quantity 1836.1523, it told me that the tau particle should have a value close to 3485.21 in electron rest-mass energy units. Here, then you can see how I arrived at the odd integer value 3485 as used in the above gravity formula.

But still, what about that 5061 value? Here I struggled with a dilemma for quite a while, given that I had derived that value of 5059.49 from basic aether theory and the empirical evidence, also from certain particle decay reactions, was telling me that a particle form of that mass in electron terms had to exist in Nature, whereas I knew that the 5063 value had a link with G.

The latter connection arose because I presumed that such a particle existed in the G-frame and worked out that, if it expanded slightly to increase a certain amount in volume, that would increase its gravitational effect. I argued that it would thereby shed a commensurate amount of energy, which, since energy is conserved, would assume material form in the E-frame. The corresponding mass added to the E-frame would then have the gravitational property set up by the effect of that volume change in displacing charge in that G-frame. Now, you can easily work this out yourself. Given a particle satisfying the J.J.Thomson formula and having a certain V/M ratio as applies to its whole spherical form, how does a small increment of V relate to a change of M? The answer is that it is minus three times the V/M ratio.

Now, what this means is that, whereas the graviton A in the G-frame can engage in energy fluctuations as between itself and matter in the E-frame, and keep things in order gravitationally, there has to be a process by which, given a steady state matter presence in the E-frame, creation of the latter is accompanied by creation of the lepton-graviton form in the G-frame. The latter must comply with the V/M ratio. The volume V comes into being as M is created.

This is what we see with pair creation and annihilation in quantum-electrodynamics and I imagine a scenario where a pair of tau leptons group with a graviton, so that the tau leptons can feature in matter creation and annihilation reactions, whereas the gravitons serve in regulating minor energy fluctuations.

In any event it then becomes possible, owing to that minus three factor, to calculate the ratio of the mass g of the graviton to the mass τ of the tau particle. The ratio is simply that obtained by solving the equation:
(g/τ)3 - 3τ/g -1 = 0
which gives;
g = 1.452627τ

Given adoption of 3485 as the odd integer value of B based on the tau mass-energy calculation, one finds, using this formula, that g has to be close to 5062.4, but since A must be an odd integer and particle creation involves energy requirements that can exceed but not fall short of what is required, we need to opt for the next lower odd integer value, namely 5061 as the value of A.

Accordingly, I have, in physical terms, justified the values of A and B as used in the formulation of G, the Constant of Gravitation.

Before ending this summary discourse updating my theory of gravity, I need to clarify one important point that has crept into my writings over the years.

The question at issue has been the effect of an electric charge sitting in a surrounding sea of charge having a uniform charge density σ, given that the charge has a significant volume. I knew that the aether has a charge density that can be expressed as e/d3, where e is the charge of the proton and d is the side dimension of a cubic cell of aether. This assumes that the aether particle that sits in that cell and neutralizes that charge continuum is a negative charge of magnitude e that we associate with the electron. Above we have seen that knowledge of d allows one to calculate that charge density and we found that the volume of that cubic cell was 9,324,644 times the volume of the electron charge or 5059.49 times the volume of that aether particle. It follows, therefore, that the effect of a charge e sitting in what is, in effect, a hole in the continuum of charge density σ, will exhibit a net effect that is modified by the presence of that hole, meaning the absence of charge that otherwise would fill the space defined by that hole.

So I wondered if the calculation of the value of σ would need adjusting by one part in 5059 or so to keep the charge balance. I saw this as implying that I might need to increase the value I calculated for G, as a squared effect in terms of σ, by a factor of 2/5059, or 0.000395, which was enough to change a G value of 6.6700x10-8, say, to a value of 6.6726x10-8, which is an appreciable difference given that the standard deviation or degree of uncertainty recorded for G in tables of physical data was 0.00085x10-8 until this recent and far more precise measurement of G was announced.

It is necessary now to settle this issue. Note, however, that it really has little bearing upon the actions of charged particles of higher mass than the electron or the discrete aether charge sitting in the cubic cell of aether. The σ charge background would affect the electron charge by one part in ten million and so affect the fine-structure constant evaluation by 2 parts in ten million. The G calculation is where we see the primary effect of this problem.

Now, we take e as the unitary charge value by assumption and without thinking of any background charge continuum effects. Given no charge continuum in the background of space and ignoring the reality of Maxwell's displacement currents and their need to have a charge source, one can readily assume that a proton charge is equal in magnitude to the charge of an electron, but opposite in polarity. However, if we say there is that σ continuum present, then we should be declaring instead that the net effect in electric and magnetic field action stemming from the fundamental unit charges of opposite polarity is the quantity to be equated in magnitude. In other words, a positive charge of nominal value e and volume V sitting in a positive σ background really has a charge value equal to e plus σV because, when offset by the effect of the hole it occupies in the continuum charge, it is seen at a distance as having a net effect of a charge e. In contrast a negative charge of nominal value -e and volume V must really have a charge of minus e plus σV, but will be sensed at a distance as having the net charge -e.

Then, when the two charges are involved in a mutual annihilation process, given the slightly different values of their actual charge, they create a charge residue of 2σV which fills the 2V volume of the vacated space.

As to the neutral balance of continuum charge and the discrete aether charge sitting in one of those cubic cells of the aether, this then gives the relationship:
σ(d3-V) + (-e+σV) = 0
from which we see that:
e = σd3

Accordingly, it was rigorously correct to use this latter relationship in the formulation of G, without making any correction for the finite form of the discrete aether charges sitting in those cubic cells of the aether.


[1] The Theory of Gravitation, 2nd. Ed., (1966)
[2] Physics without Einstein (1969)
[3] Gravitation (1975)
[4] Physics Unified (1980)
[5} Aether Science Papers (1996)

H. Aspden August 29, 2000

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