Appendix II

Inertia and E = Mc²

The physical basis of inertia and so of the well-known formula E = Mc² resides in the Principle of Conservation of Energy and, contrary to what many physicists believe, the unwillingness of an electron to radiate and so shed the only attribute that accounts for its existence, its electric charge and the energy intrinsic to that charge.

Electron acceleration in company with other electrons accelerated by the same electric field will engender a collective action by which energy can be said to be dispersed by setting up electromagnetic wave propagation. If N electrons are involved then the rate of energy radiation is, by Larmor theory, said to be proportional to N². Physicists who see this as applying to radio transmission from antenna in which numerous electrons are caused to oscillate in synchronism with one another must, however, ask themselves whether the radiation might be proportional, not to N² but to (N² – N), whereby we exclude radiation of the very energy that keeps the electrons alive.

Why, I ask, should physicists just declare that an electron is accelerated without, as they do, factoring into their analysis the external electric field that causes that acceleration?

Now I could write many, many pages in support of my concern, but see no reason for replicating and developing what I have already published elsewhere. My book: ‘Physics Unified’ includes a discussion in chapter 4 where I refer to 17 authors who seem to be troubled by this problem. These authors include Dirac and Einstein.

If you think Einstein’s theory is rigorous, ask yourself how we measure relativistic mass increase of a fast-moving electron unless it is rapidly accelerated. Then note Einstein’s words in a famous paper of his entitled: ‘On the Electrodynamics of Moving Bodies’ : “As the electron is to be slowly accelerated, and consequently may not give off any energy in the form of radiation, the energy withdrawn from the electrostatic field must be put down as equal to the energy of motion of the electron.”

The reference is Annalen der Physik, 17, 891 (1905).

If you respect the work of Nobel Laureate Paul Dirac, just look up the paper in which, in discussing the classical theory of energy radiation by accelerated charge to accommodate relativistic principles, he stated: “It would appear that we have a contradiction with elementary ideas of causality”

Here the reference is Proc. Roy. Soc., A167, 148 (1938).

So my case is simple. Just go back to see how the formula for electron energy radiation was derived in the first place. I will not repeat the analysis here but will present the mathematical integral on which it is based:

This is derived at p. 81 in my book: ‘Physics Unified’. There are two important points to notice about this formulation. Firstly, it contains no symbol which represents the intensity of the electric field which must be present in order to set up the acceleration f. Secondly, that factor of 2 before the integral sign is put there because it is assumed that the electric field disturbance that is propagated must, by virtue of our understanding of Maxwell’s theory of wave propagation, be matched by the propagation of an equal magnetic field disturbance.

Now I can declare quite categorically that once the accelerating electric field of strength V is included that integral above becomes zero, provided we have: Ve/f = e²/2c²(ct) and since e²/2(ct) is a measure of the electric field energy outside the radius ct, t being time, that remains to be accelerated as the disturbance progresses at speed c, it is evident that here we have a formula that tells us that an electron will accelerate in just such a way as to avoid shedding its electric energy, the condition being that the inertial mass is the electric field energy involved divided by c². So we have E = Mc² derived by classical electron theory and a physical insight into the nature of inertia.

The reason I am delving into this subject here is my concern about the theory of the Hubble constant in relation to the classical formula for the Thomson scattering cross-section of the electron. The theory for this depends upon the above formula for the rate of energy radiation by the electron deemed to be accelerated by the passage of an electromagnetic wave intercepted by the electron. If there can be no radiation of electric field energy by the isolated electron accelerated by such a wave then, in that respect, the scattering cross-section of the electron must be zero. However, I can see the case for the magnetic disturbance, or rather the kinetic energy disturbance implied, to still ripple through as part of the resulting wave propagation. To that extent, and bearing in mind that there is a measure of qualitative evidence supporting the Thomson scattering attributed to electrons, I tend to the opinion that an electromagnetic wave encountering isolated electrons in space does confront an obstructing cross-section that is half that indicated by the classical Thomson formula.

In this case there is logic in looking to the transient creation of electrons by the aether’s failed attempts to create protons as a reason accounting for the attenuation of intensity and frequency of waves coming from distant stars. It is just that the theoretical determination (chapter 8) of the magnitude of the Hubble constant is affected, but surely we do have here a profound insight into some fascinating aspects of the physics that underpin our universe and I just hope that physicists will see enough reason to revise their opinions in the light of these comments.

As to revising one’s opinions, readers may find it of interest to read what I have recorded ahead in Appendix VI concerning Einstein’s notion of time dilation and also concerning the significance of what I have said in Chapter 9 about Fechner’s hypothesis and its bearing upon electrodynamic interaction. Our understanding of the physics of Creation is, to be sure, an evolving theme, but one can be equally sure that what is reported here in this work is closer to the truth than a physics based on Einstein’s theory.

© HAROLD ASPDEN, 2003

APPENDIX I

The Exclusion Zone of Interaction Energy

When two particles interact owing to the interaction of their mutual electrostatic or gravitational fields, the fact that field energy density is proportional to field intensity squared means that there is a cross-product component of energy density separate from the self-energy components of either particle. This cross-product component is what is meant here by ‘interaction energy’. Our task now is to prove the following theorem:

Given two particles separated by a distance r and subject to an inverse square of distance law of force, prove that their interaction energy sums to zero within a sphere of radius r centred on either charge.

Why is this important? It is important because in physics we face the problem of working out how fast that energy can transfer from its distribution over the whole of the field enveloping the particles to satisfy the Principle of Conservation of Energy by moving to or from the kinetic energy state which is seated with and shares the motion at the particle location.

For the electrostatic case, consider a charge Q in the figure above as developing a radial electric field VQ at radius x. Imagine then a charge q distant r from Q developing a radial electric field Vq at the radius x from Q. Let y denote the distance from q and a point P under consideration at radius x from Q. Then, with φ as the angle between VQ and Vq, we know that the interaction energy density component at P is:

Also, VQ is Q/x² and Vq is q/y². Now consider the volume of an elemental section of a spherical shell of thickness dx at the radius x, as subtended at P by a small solid angle from q. The elemental volume is y²/cosφ times this angle per unit thickness of the shell. If this is multiplied by the above expression after replacing VQ and Vq by the above terms in x and y, we find that the cosφ term is eliminated as is the y² term, to leave us with an expression for the energy attributable to that elemental section dx of shell is Qq(dx)/4πx² times the solid angle mentioned. Since this does not depend upon y, we can evaluate the total energy component dE for the full solid angle of 4π to obtain:

Provided x is greater than r, the fields VQ and Vq are in the same direction. With x less than r the two regions of the spherical shell intercepted by the same solid angle have opposite and cancelling interaction energies owing to the change of direction of VQ relative to Vq. Thus within the radius r the interaction field energy sums to zero and so the proposition is proved.

Now we could have replaced Q and q by masses M and m and, by introducing G, the constant of gravitation, as a coefficient, reached exactly the same conclusion, namely that there is no net interaction energy within a sphere centred on M or, by the logic of symmetry, centred on m.

Accordingly, whether we have in mind the electrostatic interaction or the gravitational interaction, the energy transfer for change of separation distance involves energy having to traverse the distance r in going to or from the kinetic energy state in the process of transfer with field energy.

A separate mathematical analysis based on the use of MacClaurin’s Theorem can show that the interaction field energy distribution at radii beyond r is inversely proportional to the square of x. From that one concludes that the precise distance energy travels in its transfer between field and kinetic energy is the distance r. The latter analysis is of record in chapter 1 of my book: ‘Physics Unified’ and also in a paper of mine published by the U.K. Institute of Physics [‘The Inverse Square Law of Force’, J. Phys. A: Math Gen., 13, 3649–3655 (1980)].

The importance of the theorem here presented is evident from the analysis in chapter (9) where we derived the equation (9.6) which led us to a physical foundation for the Neumann potential in terms of the Coulomb interaction. An equally important result, however, is that afforded by the clear analogy with the gravitational interaction, because equation (9.6) has a gravitational counterpart that is the basis of the point made in chapter 5 by reference to the expression (5.1). Physicists whose minds are entrenched in relativistic doctrine would do well to take stock of what has just been stated here, because one can see, from the argument that follows expression (5.1) in chapter 5, that the perihelion motion of planet Mercury can be explained by the simple classical logic of classical field theory, once the theorem presented in this Appendix I is given due attention.

© HAROLD ASPDEN, 2003