These notes teach the mathematical basis of Aether Science theory


© Harold Aspden, 1997

Faraday, Ampere, Neumann, Gauss, Weber and Fechner are names one finds amongst the pioneers of early and mid-19th century electrical science. That was the period during which the empirical foundations of the electrodynamic interaction of currents in separate circuits were thoroughly established. Key to this work was a potential function which came to be known as the Neumann potential. It can be expressed in various ways, for example as the energy potential expressed as the integral of two current circuit elements i.ds and i'.ds', presented in the form:

where the (ds.ds') term is the scalar product of the two interacting circuit elements. If ds and ds' are segments of circuits separated by the distance r and mutually inclined by the angle A, then (ds.ds') is (ds)x(ds')x(cosA).

Another way of expressing the potential is in the form that applies to two separate electric charges in motion:

where e, e' are the two electric charges and v, v' are their respective velocities, r being their separation distance.

Now whereas the integral of that first Neumann potential expression, as a circuit integral, represents the mutual potential energy of an electrodynamic interaction involving a closed current circuit and is something that can be established empirically, the other elemental version cannot be proved experimentally because the electrodynamic interaction of two isolated charges in motion has not, as yet, been possible. Accordingly, the validity of the charged particle version has only been inferred from the closed circuit tests of the current circuit version.

The issue is extremely important scientifically, because gravitation, if it is to yield to unification with electrodynamic force law, concerns action between discrete elements and not circuital systems. The task we confront, therefore, is that of determining from very first principles the true force law and the true electrodynamic potential that applies to the action between two discrete electric charges in motion. Even that is only part of the full solution to the gravitational problem, but it is an essential part.

Now I do wish in these tutorials to adhere to proving the grounds on which I rely by building from first principles, but it helps in the presentation here to advance by using a hypothesis that is well documented in science history. To prove it I would need to show that when an electron is deemed to move from A to B it really is part of a team active in the aether owing to the induction at B of electron-positron pairs from 'nowhere'. The electron moves only half way from A to B as the positron moves from B to that half-way point. They meet and are annihilated to disappear into that 'nowhere' world of the aether. This leaves the electron at B ready to move on in such quantum steps. This is my interpretation of the actual physical process which is implicit in Fechner's hypothesis, a hypothesis advanced on empirical evidence even before the electron was discovered and long before the positron was discovered. Well, this does not amount to a 'proof', but you will see that it serves as an adequate foundation for our onward analysis. As to that materialization of electron-positron pairs from the vacuum, albeit thanks to there being some energy present, that is something physicists accept in spite of their disbelief in the existence of the aether. It is a feature of quantum electrodynamics.

Fechner's hypothesis requires acceptance that the flow of electric current in a circuit element is really a counterflow of electric charge of opposite polarity. Thus a charge e moving at velocity v/2 and a charge -e moving at velocity -v/2 is equivalent to a single charge e moving at velocity v, so far as its electrodynamic effect or effect as an electric current are concerned. The notion of electrons and positrons was unheard of in the 19th century, but those who pioneered electrical science in the latter part of that century could, guided by the Fechner hypothesis, derive the Neumann potential from that formula we deduced in Tutorial No. 3.

Consider two electric charges, e, -e having velocities v/2, -v/2, respectively at P distant r from Q and interacting with e', -e' having velocities v'/, -v'/2, respectively, at Q. Use that force formula involving the relative velocity V:

F = ee'(V/c)2/2r2

and you should be able to show that the addition of the four force components set up by the particle interactions have velocity squared terms which cancel, but have cross product terms which sum to give a net force:
F = -(ee')(v.v')/rc2

This is a force and, if it were expressed as an energy potential by multiplying by r, it would be the Neumann potential duly derived by using the Fechner hypothesis. F is a force acting directly between the charges and directly proportional to their separation distance r, regardless of the relative disposition of those charges in relation to the separation vector r. Furthermore the force is always an attractive force whose strength is independent of the relative disposition of those charges in relation to the separation vector, if those charges move in mutually parallel directions. This, it is stressed, applies regardless of the angle between the separation vector r and those velocity vectors and this is extremely important in our quest to resolve the riddle of unification of gravity and electromagnetism. Our quest to develop that link with the force of gravity has got easier, since we now know that the requirement is a mutually parallel charge system at the very heart of the action accounting for the phenomenon of gravitation.

Now, physicists had reached this position, but not appreciated it, well before the end of the 19th century but there was something they missed in their efforts to understand actions in real electric circuits. They were determined to adhere to that law of Newton which requires action and action to be equal and opposite. They had seen how the Neumann potential affected charge interaction and that seemed to work well provided they restricted attention to interactions involving a closed circuit, but they did not make that scientific leap across the gap that was then directly before them.

They had, in fact, forgotten the reality of their problem, which is that two charges never, ever, exist in isolation from other charge unless they restrict their interactions to oscillations in modes that ensure their respective motions are mutually parallel. In the real world of electrical engineering and laboratory science tests on electricity the Neumann potential and its equivalent force formulation are strictly component forces of a partial system. Newton's Third Law of Motion need not apply to each and every pair of charges in such a case, meaning that other forces can be acting on the individual charges, as set up by the influence of other electric charge in motion in the environment. The reason is that energy is pooled as between the separate charge interactions. The sole governing requirement is that the energy of the Neumann potential is conserved overall in its deployment into and from the kinetic energy of the motion of the charges involved in setting up that potential.

Now, I am going to try, in presenting these tutorials, to avoid reference to textbook back-up but I do mention at this stage that much of what I will be presenting is of convenient record in my book 'Physics Unified' which is available and can be ordered from booksellers or as indicated in the book and report section of these Web pages. At this point in developing the onward argument I shall be following fairly closely the text to be found on pp. 3-17 of that book, though some of that detail that adds considerable weight to what I say will be omitted here. Indeed, the points I am making are so simple that it really does not need such treatment. It is just that the task of getting scientists to wake up to the realities of where Einstein went wrong has proved to be such a struggle that proof and over-proof seemed warranted when I wrote 'Physics Unified' and its earlier version 'Physics without Einstein'.

The Electrodynamic Force

Merely by taking full account of the conservation of energy there are certain general aspects of the force which acts on a charged particle in motion that we can investigate.

Referring to the figure below, imagine two electric particles q, Q of mass m, M moving at velocity v, V, respectively and subject to a mutual force F acting directly between them along their line of separation. Note that F is not the only force acting on the particles, because we will be taking into account inertial reaction forces and extraneous interaction effects owing to the presence of other charge in the near environment.

Consider next the energy deployment as charge Q moves under the action of the force F in the direction -r. This is depicted in the next figure:

Note that force is merely a manifestation of an effect which occurs as energy seeks to redeploy as a function of time and distance, taking into account the energy package wrapped up as 'mass' in the intrinsic state of the particle on which it acts.

Now, key to the argument I am following here with regard to the above figure is the assumption that V is a velocity which, for some reason, is sustained at a constant value. Therefore, if the action is deemed to be purely electrodynamic in origin, we simply cannot have the charge Q moving at a constant velocity V solely under the action of the force F. The force F expends work at a rate expressed by the scalar product (F.V) and the energy has to go somewhere. We might expect V to change, but we are considering what happens if V does not change, namely the circumstance prevailing if there are energy transfer processes at work within the electrodynamic system itself. This implies the 'field', but I prefer to avoid use of that term in this analysis.

The consequence of this, if we are to assure energy conservation, is that Nature must assert another force component on Q. We denote this as the force Z as shown in the next figure and write the energy conservation equation:

(F.V) + (Z.V) = 0

We now take note that, whereas F acts through the centre of mass of the two-body system formed by q and Q, the force Z must assert a turning moment on Q about that centre of mass. Z cannot act through that mass centre because, if it did, then, to satisfy the above equation, it would merely cancel F completely and there would be no electrodynamic action to consider.

Now, at this point I am going to declare that no material body in its completeness can develop a spin of its own accord, meaning by the agency of its own internally produced forces. It can develop a spin if, somehow, it can push in a rotational sense against something non-material, meaning the aether. I believe that is possible for reasons explained elsewhere in these Web pages (notably in my Lecture No. 5 where I discuss the creation of stars and planets), but that action is basically seated in the electrostatic displacement state in the vacuum medium and is not a function of what can be termed electrodynamic action. So far as the electrodynamic action is concerned there is no way in which that two-body charged particle system can develop spin, which means it cannot acquire angular momentum by virtue of its self-interaction and the induction of forces such as that we term Z above. Remember also that I shall, as we proceed in these tutorials, be proving that gravitation is of electrodynamic origin.

It is well established by experiments on the measurement of gyromagnetic properties in magnetized pivotally-mounted rods that when the direction of intrinsic ferromagnetism is reversed so the electrons in the atoms within that rod impart a rotational kick on the rod, the reason being that angular momentum is conserved. If the rod is seen to spin clockwise, the electrons spin unseen in the anticlockwise sense and this is detected by measurements which relate to the individual charge to mass ratios of those electrons. Now you know what I mean by the word 'completeness' as used above. That rod and those electrons within it must be considered together as a whole system. The rod may spin and lead you to think that a law of physics has been disproved, but angular momentum is still conserved because you need to take account of the change of angular motion of those electrons.

Reverting to our problem of the two charges q, Q, to balance the turning action of Z, there has to be a third force component acting on Q in the above figure. This third force P is an extraneous force arising as the inertial reaction. As is usual with reaction phenomena, this reaction force is that associated with a maximization of the amount of energy transferred, corresponding to a minimization of the potential energy associated with the primary action. Thus, for optimum reaction involving maximum energy transfer as Q is displaced, the force P has to be in line with the velocity vector V.

The figure shows both charges with forces Z' and P' designated as the counterparts of Z and P that act on charge q. Now, to avoid any turning effect owing to the self-interaction of q and Q, the forces shown must combine to accelerate the two particles in the same direction and at the same linear rate. When formulated, this condition just deduced leads to:

Z = (M/m)P'
Z' = (m/M)P
with Z parallel to P' and Z' parallel to P.

In this analysis I have avoided discussing the change of kinetic energy associated with the forces P and P' acting in those directions v and V, respectively. Analysis on those lines is found in my book 'Physics without Einstein' [1969b] or in my Journal of the franklin Institute paper [1969a]. The result is the same as we find by proceeding from the equations already formulated.

I did, on pp. 7-10 of my book 'Physics Unified' include an argument based essentially on symmetry considerations by which I derived the form of the Neumann Potential and so the force F. However, I later discovered how to prove the true origin of that force and it was published in Hadronic Journal. See reference [1988a] and note that a full copy of that paper is reproduced in my 1996 book 'Aether Science Papers'. For our purposes here it suffices to proceed by writing the force F attributable to the Neumann Potential as:

F = -K(v.V)r
K = qQ/r3
and then, from the energy balance equation involving Z and F above derive:
-K(v.V)(V.r) = (Z.V) = 0
From this:
Z = K(V.r)v


Z' = -K(v.r)V
Replacing Z' by (m/M)P then gives:
P = -K(M/m)(v.r)V
and, as a result, the total force acting on Q, which is F+Z+P, is:
FQ = (qQ/r3)[(V.r)v - (M/m)(v.r)V -(v.V)r]

This is the complete and general law of electrodynamics to which we have been led by straightforward analysis. It will form the basis of the unified theory by which we shall explain gravitation as an electrodynamic force.

I may add here that some detailed background which refers to Clerk Maxwell's study of this same problem can be found in my Lecture No. 5 in these Web pages. That M/m term is interesting from the viewpoint of plasma experiments where there are anomalous interaction forces asserted between electrons and heavy ions. It is also of interest in connection with the prospect of extracting energy from the aether, which is the subject of the research findings of Dr. Correa and Alexandra Correa, as described in my Energy Science Report No. 8. However, from the viewpoint of gravity, the first two terms in that general law of electrodynamics cancel to leave only the last term. The first term, incidentally, when combined with the last term, gives the Lorentz force law.

Remember here that you are taught to think that electromagnetic action on moving charge can only be at right angles to the charge motion. You ought to ask your teacher how the charge can lose or gain energy by transfer to the magnetic field if its reaction with that field prevents it from slowing down or speeding up. If Faraday's name is then mentioned or that of Lenz, then ask how that affects the form of the law of electrodynamics, which should stand on its own to explain the phenomenon of electrodynamic action.

On the gravity theme, we shall soon see in these tutorials that the aether includes electric charges that share an organized synchronous motion on a universal scale and it also contains energy in the form of electric charges that migrate around at random. The organized system is in two parts which are dynamically balanced. Any matter present shares the motion of one part and, in spite of that motion, is effectively at rest in the electromagnetic frame of reference, because that 'part' of the aether constitutes that frame of reference. The other 'part' comprises charges that I term 'gravitons' because they are the seat of the gravitational action. They move relative to the electromagnetic reference frame and always share motion that is mutually parallel as between all the gravitons. They are held in place by powerful electrostatic forces which keep them in step with limited freedom of movement. They are not 'free' in the sense that their masses can affect electrodynamic interaction as opposed to dynamic balance in the permitted degree of freedom. In short, the first two terms in the general law of electrodynamics are ineffective and this leaves the force:

F = -(qQ/r3)(v.V)r
which establishes the form of law we seek for correspondence with Newton's law of gravitation.

Note that the general form of the law of electrodynamics or the Lorentz version of that law, meaning the versions adding terms to the equation just presented, play no role in the theory of interactions between charges within atoms. Nor can they affect interaction between moving atoms (as opposed to ions). Those charges are not free to move solely under electrodynamic constraints. They are not akin to the effects of steady state current flow through electrical circuits, where electrostatic interactions are fully neutralized. The dominant forces in atomic systems are electrostatic in origin and the same applies to the aether, except for that one type of interaction as between the gravitons in that half of the vacuum medium which provides dynamic balance for matter and the aether's related electromagnetic reference frame. If those gravitons can attract one another, that attraction is communicated to the matter they are balancing and we see that as gravitation. Gravitation is not an electrodynamic force acting directly on matter. Its effect is indirect and is communicated by the dynamic linkage with the graviton system.

I must now conclude this tutorial. I set out to explain the Neumann Potential and the role it plays in determining the form of the law of electrodynamics. This is part of my plan to work towards that account of gravitation by which we shall evaluate G, the constant of gravity in terms of the electron's charge/mass ratio. However, I have gone just a little too far in opening the door to show how the aether performs on the gravity stage. We must halt that discussion now to examine in Tutorial No. 5 a traditional feature of the aether which concerns the speed of light.

To progress to the next Tutorial press:

Tutorial No. 5