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The Lorentz Aether Theory

Frederick David Tombe,

Belfast, Northern Ireland,

United Kingdom,

sirius184@hotmail.com

27th February 2020

Abstract. The Lorentz transformations are best known for the

relativistic Lorentz factor, γ = 1/√(1 – v2/c2), which appears in the

equations of special relativity. It is also known that the Lorentz

transformations can be used to derive the Biot-Savart law in the form

B = μ0ε0v×E, and also the magnetic force in the form EC = v×B.

What is not so well-known, however, is that the emergence of

these two cross-product equations from a Lorentz transformation has

got no bearing on the Lorentz factor itself. It is often argued that the

magnetic force, EC = v×B, is a relativistic effect, yet aside from the

very obvious fact that magnetism is observable at laboratory speeds,

it will be demonstrated in this article that magnetism is a

consequence of the physical structure of 4-D space-time, and that it is

definitely not a relativistic effect.

The Lorentz Force

I. The Lorentz force first appeared as an electromotive force term at

equation (77) in Part II of Scottish physicist James Clerk Maxwell’s 1861

seminal paper entitled “On Physical Lines of Force”, [1]. It appeared in

the basic form,

E = μ0v×H + ∂A/∂t − ∇ψ (1)

and in 1864 he listed it as one of the original eight “Maxwell’s

Equations” in his paper entitled “A Dynamical Theory of the

Electromagetic Field”, [2]. The term, E, on the left-hand side is the force

acting on unity of the electric particles within his proposed dielectric sea

of molecular vortices. Today, E is referred to as “force per unit charge”

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while the term electromotive force is generally used for voltage, which of

course plays the exact same role in electric circuit theory. In modern

textbooks, the charge term, q, is taken over to the right-hand side, hence

making equation (1) appear as,

F = q[v×B + ∂A/∂t − ∇ψ] (2)

where,

B = μ0H (3)

The quantity, B, is the magnetic flux density which will be referred

to as the magnetic field, while μ0 is the magnetic permeability which is

closely related to the density of the sea of aethereal vortices, and H is the

magnetic intensity. The second and third terms on the right-hand side of

equation (1) are the electric field terms. The second term, which arises

from Faraday’s Law, is the partial time derivative of the electromagnetic

momentum, A, nowadays referred to as the magnetic vector potential. The

second term arises due to a changing magnetic field, while the third term

is the electrostatic field as per Coulomb’s law. Collectively they are

represented as,

E = ∂A/∂t − ∇ψ (4)

It’s common to have a negative sign in front of the ∂A/∂t term in

order to take account of Lenz’s Law, but it will be left out here so as to

maintain consistency with Maxwell’s original convention. In modern

textbooks, equation (1) is generally written in the familiar Lorentz force

format,

F = q[E + v×B] (5)

As regards the convective term, v×B, which is often referred to as the

Lorentz force on its own, it was derived by Maxwell in conjunction with

his sea of molecular vortices long before Lorentz got involved in the

topic, and so in this article it will be referred to as the Maxwell-Lorentz

Force. The v×B force is like a kind of compound centrifugal force arising

from the differential pressure acting on either side of an element of

electric current as it flows through the sea of tiny aethereal vortices. This

differential pressure causes a deflection in the path of motion.

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Prior to introducing electric particles into his sea of aethereal vortices

in Part II of his 1861 paper, Maxwell had relied exclusively on three-

dimensional aether hydrodynamics, and he operated in terms of force per

unit volume. Let us consider the two force terms which appear as parts 3

and 4 on the right-hand side of equation (5) in part I of his 1861 paper,

E =

0v×(∇×v) (6)

where E refers to force per unit volume. Since the velocity term, v,

is the fine-grained circumferential velocity of the tiny vortices, then the

vorticity must be expressed as,

∇×v = 2ω (7)

where ω is the angular velocity at the edge of the vortices, and since

the magnetic permeability, μ0, is related to the density of the sea of

vortices, this brings us to,

F = 2mv×ω (Coriolis Force) (8)

which is the familiar Coriolis force. Maxwell identified the

circumferential velocity, v, with the magnetic intensity, H. Substituting

equations (3) and (7) into equation (6) therefore leads to,

E = B×(∇×B/

0) (9)

Comparing with Ampère’s Circuital Law, and substituting J = ρv,

where ρ is the charge density, we obtain,

E = B×ρv (10)

which is the familiar Maxwell-Lorentz force. A subtle difference

between the Coriolis force when observed in atmospheric cyclones on the

one hand, and the Maxwell-Lorentz force when observed in a magnetic

field on the other hand, is that in the atmospheric case, we are observing

inertial motion in a vortex, whereas in the electromagnetic case, we are

observing motion between vortices. See “The Coriolis Force in

Maxwell’s Equations”, [3].

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The Three-Dimensional Aether Hydrodynamical Analysis

II. In Maxwell’s hydrodynamical analysis in Section I above, he bases

the magnetic intensity, H, and the magnetic flux density, B, on the fine-

grained circumferential aether circulation in his sea of tiny vortices.

However, all papers relating to “The Double Helix Theory of the

Magnetic field”, [6], [7], instead apply H directly to the vorticity of the

vortices, and hence Maxwell’s electromagnetic momentum, A, becomes

his displacement current, [4], [5]. It’s upon this basis that we will operate in

this article. The scalar potential, ψ, represents hydrostatic aether pressure,

and together with the dynamic momentum/displacement current, A, they

make up the two important ingredients of Bernoulli’s Principle within the

sea of fine-grained aethereal vortices. The Maxwell-Lorentz term, EC =

v×B, follows naturally from three-dimensional aether hydrodynamics as

shown in Section I above, and also in Appendix A at the end.

Relativists claim that Einstein’s Special theory of Relativity covers

all these hydrodynamical relationships, hence rendering Maxwell’s

pioneering works redundant. This is despite the fact that relativity

completely removes the aether from the entire picture, whereas it’s very

hard to comprehend how a force such as the Maxwell-Lorentz force, EC =

v×B, which follows from differential pressure within a sea of tiny

aethereal vortices could still exist after we have removed the aether itself.

Could an aeroplane fly if we removed the atmosphere?

It’s true though that the Lorentz transformations can indeed produce

the Maxwell-Lorentz force, EC = v×B, from the combined electric field

term, ∂A/∂t − ∇ψ, on the right-hand side of equation (4). But can they do

it without the aether? And while Maxwell’s approach introduces the

speed of light into the equations of electromagnetism through the 1855

Weber-Kohlrausch experiment, [8], the Lorentz transformations introduce

it through optics and relative speeds. The commonality between the two

methods will now be investigated.

The Lorentz Transformations

III. In a letter entitled “The Ether and the Earth’s Atmosphere” written

by the Anglo-Irish physicist George Francis Fitzgerald, dated 2nd May

1889 and published in the “Science” illustrated weekly journal, [9],

Professor Fitzgerald expressed great interest in the recent experiment

carried out in 1887 by Messrs. Michelson and Morley at Cleveland, Ohio,

in the USA. Michelson and Morley, using an interferometer in an attempt

to measure the speed of the Earth relative to the luminiferous medium,

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had obtained no significant fringe shifts. Professor Fitzgerald stated that

the only hypothesis that could be reconciled with this negative

observation is that the length of moving bodies changes due to electrical

interaction with the aether.

In 1892 the Dutch physicist Hendrik Lorentz began working on a

project aimed at reconciling Maxwell’s equations with certain

experiments involving optics and relative motion, [10]. Initially attention

was focused on stellar aberration and Fizeau’s 1851 experiment involving

the speed of light in a moving column of water, but Lorentz extended the

investigation to include inter-molecular forces and length contraction in

order to try and explain the null result of the 1887 Michelson-Morley

experiment. In 1897 the Ulster physicist and mathematician Sir Joseph

Larmor extended Lorentz’s work further, paying particular attention to

the electrical interaction between matter in motion and the luminiferous

aether, [11]. Lorentz on the other hand was largely working on an ad hoc

mathematical basis. It was Larmor who first suggested that length

contraction would have to also involve some kind of retardation of local

time, a concept which he never clearly explained but which is understood

by some to relate to the frequency of natural processes, and it was Larmor

who is said to have been the first to have arrived at the Lorentz

transformations as we know them today, [12]. See Appendix B.

In 1905, French physicist Henri Poincaré formulated equivalent

transformation equations and named them the Lorentz transformations as

it seems that he was unaware of Larmor’s contributions, [13]. Shortly

afterwards, Albert Einstein published an alternative derivation of the

Lorentz transformation equations which gave them a brand-new

interpretation not involving the aether at all, [14]. It was this act of

mindless vandalism on Einstein’s part which is central to a major

controversy which rages to this day.

The Luminiferous Aether

IV. Maxwell never overtly mentioned electric charge in his 1861 paper.

Instead, he talked about the density of free electricity, a concept which he

never elaborated upon. This is highly significant since the term free

electricity is likely to refer to the electric fluid, which is the primordial

aether from which everything is made. This would equate electric charge

to the state of compression of space itself, space being something that can

be compressed or stretched, and which can also flow. Maxwell

considered the luminiferous medium to be a sea of tiny aethereal vortices

exhibiting 3-D cylindrical symmetry and obeying the classical laws of

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hydrodynamics, and from the arguments in Sections I and II above, there

would seem to be no compelling reason to see the matter any differently.

The idea however, that the luminiferous medium is actually a 4-D

space-time continuum, originates in optics, and it can be argued from the

equation,

s2 = x2 + y2 + z2 – c2t2 (11)

where the quantity, s, is Lorentz invariant, meaning that it doesn’t

change under a Lorentz transformation. This equation is the cornerstone

of Einstein’s special theory of relativity, and the assertion that the speed

of light, c, is a universal constant is what gives rise to the controversial

concept of time dilation. Meanwhile, we all know that when the Earth has

performed a complete orbit of the Sun, relative to the background stars,

that one year will have passed for everybody in the universe, and that

therefore time dilation is a nonsense concept.

Equation (11) is closely related to the return-path longitudinal

Doppler shift in the frequency of a ray of light, [15], but it would appear to

have no obvious connection to the hydrodynamics of a sea of tiny

vortices. If, however, we could somehow connect this equation to the

electric and magnetic fields that are involved in a ray of light, we might

get to see the picture at a deeper level, hence enabling us to discard

absurd ideas like time dilation. Throughout the 1890s, Sir Joseph Larmor

was working directly on this problem in connection with a rotationally

elastic aether and the manner in which the frequency of rotating dipoles is

altered as they move through this aether, [11], [12]. Larmor’s papers would

be an ideal place to begin in order to investigate this matter further.

Larmor talked about positive and negative electrons being singularities in

the aether and he connected this idea with electromagnetic radiation. See

page 211 in his 1897 paper, [11], and Section 114, pages 179-180, in his

1900 paper, [12]. This line of research would be relevant to the effect of

motion through the aether on atomic clocks, such as in the case of atomic

clocks in GPS satellites, where it can be shown how the ensuing

frequency dilations relate to conservation of energy, and where the

associated equations can be shown to approximate to the Lorentz

transformation equations, [16],

We need to investigate the deeper origins of the electric and magnetic

fields that are involved in the electromagnetic wave propagation

mechanism, and to this end it is proposed that the wave carrying medium

is in fact Maxwell’s sea of molecular vortices, as modified by “The

Double Helix Theory of the Magnetic Field”, [6], [7], which replaces his

molecular vortices with rotating electron-positron dipoles. These tiny

vortices press against each other with centrifugal force while striving to

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dilate, [17], [18], [19]. The sea of tiny aether vortices may itself be

incompressible, but the pure aether of which these vortices are comprised

will certainly be compressible and stretchable.

The Four-Dimensional Space-Time Continuum

V. In Einstein’s 1905 paper, [14], he derived the kinematical Lorentz

transformations in his own way and then went on to apply them to

Ampère’s Circuital Law and Faraday’s Law. On page 907, Einstein wrote

out these two curl equations side by side in a perfectly dual format

involving three rows and two columns. He split each equation into three,

one for each of the three Cartesian components, and he used Gaussian

units so that the speed of light was overtly displayed. Einstein then

applied the Lorentz transformations to these two dual curl equations and

wrote out the solutions. The components of these solutions in the plane

perpendicular to the motion, are E = γ(1/c)v×B and B = γ(1/c)v×E, where

γ = 1/(√1 – v2/c2). In the same year, French mathematician and theoretical

physicist Henri Poincaré devised an analytical tool known as four-vectors

which showed how Einstein could have arrived at these solutions. This

was written up in Poincaré’s “Palermo paper”, [20], and the same idea

was later developed further in 1908 by German mathematician Hermann

Minkowski whose name is now associated with the 4-D space-time

continuum. It’s not clear how Einstein could have arrived at his solutions

without using four-vectors. See Appendix C.

The Lorentz transformation equations can be converted into four-

vector format by treating time as a fourth dimension. We can then re-

write equation (11) as,

s2 = x12 + x22 + x32 + x42 (12)

The term x4 involves the imaginary number, i2 = −1, such that,

x4 = ict (13)

The Lorentz transformations for motion exclusively along the x-axis

then take on the form,

x1ꞌ = γ(x1 + ivxx4/c) (14)

x2ꞌ = x2 (15)

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x3ꞌ = x3 (16)

x4ꞌ = γ(x4 – ivxx1/c) (17)

To whatever extent these equations are rooted in the optical return-

path longitudinal Doppler effect, we will treat x1 as referring to

wavelength and x4 as referring to frequency. This will be in line with

Larmor’s and Lorentz’s concept of local time, as understood in terms of

the frequency of the system under investigation. Time dilation in the

broader sense, as per Einstein’s special theory of relativity, which follows

from his disregarding of the luminiferous aether, will be ruled out on the

basis that it is not a realistic option.

We will now apply these transformation equations directly to the tiny

dipolar vortices (rotating electron-positron dipoles) that fill all of space

and which form the basis of the electromagnetic wave propagation

mechanism. While the return-path longitudinal Doppler effect is

something that normally applies to waves, we will see if it can be

extended into the context of a rotating dipole in a state of translational

motion through a larger sea of such rotating dipoles.

The Biot-Savart Law

VI. The circumferential momentum density of a rotating electron-

positron dipole is A, where ∇×A = B. Consider such a dipole, of random

orientation, undergoing translational motion along the x-axis through a

larger sea of such rotating dipoles (vortices). The physical interaction will

distort B, and the component of Bꞌ along the z-axis will take the form,

Bꞌz = ∂Aꞌy/∂xꞌ − ∂Aꞌx/∂yꞌ (18)

In four-vector notation this becomes,

Bꞌz = ∂Aꞌ2/∂xꞌ1 − ∂Aꞌ1/∂xꞌ2 (19)

And now we will introduce the Lorentz condition, more accurately

known as the Lorenz gauge after Danish physicist Ludvig Lorenz. It takes

the form,

∇∙A + 1/c2∂ψ/∂t = 0 (20)

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This is in effect the equation of continuity of the aether with the

scalar potential, ψ, representing aether pressure. The speed of light has

been introduced into the proceedings, with the benefit of hindsight. With

the Lorentz condition satisfied, the four-vector for A and ψ takes the form

(A1, A2, A3, iψ/c). The z-component of the momentum, Az, becomes A3,

while the scalar potential (pressure), ψ, becomes iψ/c. The four-vector for

∇ and ∂/∂t becomes, (∂/∂x, ∂/∂y, ∂/∂z, i/c.∂/∂t).

Using the Lorentz transformation equations (14) to (17), but

deliberately omitting the gamma factor, 1/(√1 – v2/c2), equation (19) then

expands to,

Bꞌz = (∂/∂x1 + ivx/c.∂/∂x4)A2) − ∂/∂x2(A1 + ivxA4/c) (21)

Hence,

Bꞌz = (∂A2/∂x1 − ∂A1/∂x2) + (ivx/c)(∂A2/∂x4 − ∂A4/∂x2) (22)

The first bracketed term on the right-hand side of equation (22)

should already be recognizable as the z-component of ∇×A, which is Bz.

As regards the second bracketed component on the right-hand side of

equation (22), we must remember that ∂/∂x4 is (i/c)∂/∂t while A4 is iψ/c.

Hence,

Bꞌz = Bz – (vx/c2)(∂Ay/∂t − ∂ψ/∂y) (23)

Hence,

Bꞌz = Bz – vxEy /c2 (24)

A reciprocal result for Bꞌy leads us to,

Bꞌ = B – μ0ε0v×E (25)

where μ0ε0v×E is restricted to the yz-plane. This suggests that a

rotating dipole, when in translational motion along the x-axis, precesses

about that axis. The gamma factor, 1/(√1 – v2/c2), was deliberately

omitted from the analysis in order to explicitly demonstrate that it has no

involvement in these classical electromagnetic relationships.

From the 1856 Weber-Kohlrausch experiment, [8], we can write,

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c2 = 1/μ0ε0 (26)

where μ0 is the magnetic permeability of space, ε0 is the electric

permittivity of space.

In order to establish the meaning of E in equation (25), we will now

perform another Lorentz transformation, this time on, E = ∂A/∂t − ∇ψ.

The z-component of Eꞌ is,

Eꞌz = ∂Aꞌz/∂tꞌ − ∂ψꞌ/∂zꞌ (27)

Hence, we can write,

Eꞌz = ic[ ∂Aꞌ4 /∂xꞌ3 − ∂Aꞌ3/∂xꞌ4] (28)

From equation (17), while still deliberately omitting the gamma

factor, γ, we can apply the Lorentz transformations,

Aꞌ4 = (A4 – ivxA1/c) (29)

and

∂/∂xꞌ4 = (∂/∂x4 – ivx/c.∂/∂x1) (30)

Then trivially, since motion is only along the x-axis, it follows from

equation (16) that Aꞌ3 = A3 and ∂/∂xꞌ3 = ∂/∂x3. Applying these

transformations to equation (28) leads to,

Eꞌz = ic[∂/∂x3(A4 − ivxA1/c) – (∂/∂x4 – ivx/c.∂/∂x1)A3] (31)

therefore,

Eꞌz = ic(∂A4/∂x3 − ∂A3/∂x4) + vx(∂A1/∂x3 − ∂A3/∂x1) (32)

By comparing the first bracketed term on the right-hand side of

equation (32) with the starting equation (28), it simply becomes Ez. The

second bracketed term on the right-hand side applies purely within 3-D

space and it is readily identifiable as the y-component of the curl of A.

It’s of interest to note that curl is a purely spatial operation which exists

only in three and seven dimensions. There can be no curl in four

dimensions, [21], but curl can still operate in tandem with time in 4-D

space-time. Hence,

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Eꞌz = Ez + vxBy (33)

Repeating this exercise across all the Cartesian components, we end

up with,

Eꞌ = E + v×B (34)

where v×B is exclusively in the yz-plane, just like the additional

component of the magnetic field in equation (25). The equation B =

−μ0ε0v×E is the Biot-Savart Law in its most fundamental form, and we

now know that the E field is due to the centrifugal force, EC = v×B, that

emanates from the equatorial plane of a rotating electron-positron dipole,

[1], [8], [17]. See Appendices D, E, and F. We have established a clear

bridge between Maxwell’s sea of aethereal vortices and the Lorentz

aether theory. Two specific points of interest are, (1) that this derivation

could not have been done without invoking Hermann Minkowski’s

concept of 4-D space-time, and (2) that there was no need to invoke the

gamma factor for this particular purpose. The magnetic field, B =

−μ0ε0v×EC, and the magnetic force, EC = v×B, contrary to popular

opinion, are not relativistic effects.

The Physical Interpretation

VII. A Lorentz transformation should not be considered in the manner of

a Galilean transformation whereby we are viewing the same event from a

different frame of reference. A Lorentz transformation is intricately tied

up with the elasticity of the luminiferous medium, which is the carrier of

electric and magnetic fields, as well as electromagnetic waves, and so we

are studying the physical effects of absolute motion, and not simply

relative motion.

In the previous section, we saw how a Lorentz transformation

appears to have the effect of applying a gyroscopic force to a rotating

dipolar vortex, such as to cause it to precess. While it was assumed that

the translational motion in question was relative to the larger sea of tiny

vortices that fill all of space, it is now proposed that the same effect

ensues when the motion is relative to the pure vortex fluid itself (the

aether), when it is in a state of acceleration. Consider a charged sphere on

the large scale and its surrounding radial electrostatic field. According to

whether the charge is negative or positive, this large-scale electrostatic

field will involve an inflow or an outflow of pure aether which will flow

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through the tiny dipolar vortices in the vicinity. It is proposed that this is

equivalent to a Lorentz transformation based on the aether velocity field,

and so we should expect the tiny vortices to precess about the lines of

force of the electrostatic field on the large scale, and when the sphere is at

rest in the sea of tiny vortices, we will have a state of spherical symmetry.

If, however, the charged sphere is then caused to move translationally, the

tiny vortices will begin to re-align so that their rotation axes trace out

concentric vortex rings around the path of motion, much in the likeness of

smoke rings. In the vicinity of the tiny vortices (rotating electron-positron

dipoles) themselves, an electrostatic field on the tiny scale exists in the

axial direction while a fine-grained centrifugal repulsion field exists in

their equatorial planes. Hence the re-orientation of these tiny vortices

interferes with the electrostatic field on the large scale in a manner such

as to undermine it in the direction parallel to the path of motion and to

convert it into a magnetic field perpendicular to the path of motion. The

Biot-Savart law, B = −μ0ε0v×EC, describes the magnetic field lines that

form concentrically around the moving sphere, and we now know that

this solenoidal magnetic field involves a radial centrifugal force field, EC

= v×B, pressing inwards on the moving source. Hence, due to the agency

of the all-pervading sea of dipolar vortices, the electrostatic field that

surrounds a charged sphere on the large-scale, is converted into a

magnetic field as the body accelerates linearly. As the sphere approaches

the speed of light, its radial electrostatic field will have been largely re-

aligned into a disc-shaped magnetic field perpendicular to the path of

motion. When we introduce the gamma factor, 1/√(1 – v2/c2), this will

account for the tendency of the electrostatic field on the large-scale to

diminish in the direction of motion while increasing the magnitude of the

magnetic field perpendicular to the direction of motion, particularly as the

charged body approaches the speed of light. This is the relativistic effect

which is additional to the classical electromagnetic relationships.

The Maxwell-Lorentz force, EC = v×B, is actually more familiar in

the context in which it deflects a moving charged particle in an already

existing background magnetic field, such as to cause it to undergo helical

motion. In this context, it behaves more like a Coriolis force. This

Coriolis-like aspect of the Maxwell-Lorentz centrifugal force will be due

to the fact that when a charged particle moves through the background

sea of tiny vortices, since these vortices are all spinning in nearly the

same direction as their immediate neighbours, the moving charged

particle will experience a differential centrifugal pressure at right angles

to its direction of motion, hence causing it to deflect.

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Lorentz-Fitzgerald Contraction in Electric Currents

VIII. American physicist Edward Mills Purcell wrote a book in 1963

entitled “Electricity and Magnetism”, [22]. There is a belief that Purcell

demonstrated that a magnetic field in one frame of reference is an

electrostatic field in another frame of reference. Purcell involved the

concept of Lorentz-Fitzgerald contraction in an electric circuit in order to

provide a source charge for the electrostatic field, where only a magnetic

field existed from the perspective of a stationary observer. The

application of the Lorentz-Fitzgerald contraction to electric current, as a

source of charge density, seems to have progressed into the myth that the

Maxwell-Lorentz force, EC = v×B, as viewed in a stationary frame of

reference is equivalent to an electrostatic force, ES = −∇ψ, as viewed in a

moving frame of reference. Purcell’s theory is based on the relativistic

gamma factor, truncated to first order binomial approximation, and the

essence of the equality in Purcell’s analysis is based on the beta squared

(β2) factor, v2/c2, within the gamma factor. The equality which Purcell

then relies on has a superficial resemblance to the equality which Weber

used when arguing that the speed of light is a reducing factor in his 1846

force law, [8]. Purcell has amazingly managed to transport a classical

electromagnetic relationship into a relativistic context. The most

important thing though is, that the Lorentz contraction is being applied

selectively as between the positive particles and the negative particles in

the conducting wire, hence creating the equivalent of the clock paradox.

This context is hence unrealistic, and so Purcell’s theory must be ruled

out.

Conclusion

IX. The luminiferous medium of the Lorentz aether theory is specifically

Maxwell’s sea of tiny aether vortices and it is one and the same thing as

Minkowski’s 4-D space-time continuum. The four-dimensional aspect

has been shown to be of crucial importance. The classical

electromagnetic relationships which unfold from a Lorentz

transformation, arise through aether hydrodynamics and are not due to the

Lorentz factor, γ = 1/(√1 – v2/c2), itself. Magnetism, contrary to popular

opinion, is not a relativistic effect.

Where the Lorentz factor does become relevant is in matters relating

to wireless radiation and the speed of light. As well as being a factor in

the return-path longitudinal Doppler effect in electromagnetic waves, the

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Lorentz factor is an asymptotic effect which implies the existence of an

upper speed limit for matter in motion in the luminiferous medium.

The claim that Maxwell’s equations have been subsumed by

Einstein’s theories of relativity is patently false. The connection between

Maxwell and Lorentz is through the aether, and when we remove the

aether, as Einstein did, we remove the linkage between Maxwell and

Einstein, leaving Einstein with no physical basis whatsoever to justify his

theories. With Einstein’s interpretation, we have no rest frame upon

which to base the Lorentz transformations and we end up in an absurd

universe where waves propagate in empty space, and where two clocks

can both tick slower than each other, [23]. Meanwhile, all experimental

results which are claimed for Einstein are, at least to a reasonable

approximation, a vindication of the Lorentz aether theory in connection

with Maxwell’s sea of molecular vortices.

As regards Lorentz himself, he need have had no worries about

vortices forming high up at the interface of Stokes’s entrained aether, [24],

since vortices are actually the essence of the electromagnetic wave

propagation mechanism in the first place, and they already exist

everywhere.

Appendix A

(Three-Dimensional Aether Hydrodynamics)

The gradient of the scalar product of two vectors can be expanded by the

standard vector identity,

(A∙v) = A×(×v) + v×(×A) + (A∙)v + (v∙)A (35)

Let us consider only the vector A to be a vector field. If v represents

arbitrary particle motion, the first and the third terms on the right-hand

side of equation (35) will vanish, and from the relationship ×A = B, we

will obtain,

(A∙v) = v×B + (v∙)A (36)

Hence,

(v∙)A = −v×B + (A∙v) (37)

Since by the theorem of total derivatives,

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dA/dt = ∂A/∂t + (v∙)A (38)

it then follows that,

dA/dt = ∂A/∂t − v×B + (A∙v) (39)

Using the vector identity for the curl of a cross product in

conjunction with the same reasoning as per the derivation of equation

(36) above, we can safely conclude that,

×(v×B) = –(v∙)B (40)

Hence taking the curl of equation (39) leads to,

dB/dt = ∂B/∂t + (v∙)B (41)

since the curl of a gradient is always zero, hence eliminating the

(A∙v) term. Then with reference to equation (40), if we take the curl of

Maxwell’s equation (1) at the beginning of the article, which is broadly

the same as equation (39), we obtain,

×E = ∂B/∂t + (v∙)B (42)

This time it was the electrostatic term that was eliminated by the fact

that the curl of a gradient is always zero. From equation (41) this is

equivalent to,

×E = dB/dt (43)

which when the negative sign is added to take account of Lenz’s

Law, is a complete total time derivative version of Faraday’s Law

covering for both convective and time-varying electromagnetic induction.

Faraday’s Law is therefore equivalent to Maxwell’s electromotive force

equation, known today as the Lorentz Force.

Appendix B

(The Lorentz Transformations)

In 1897, Ulster physicist Sir Joseph Larmor presented equations in a

paper which was published in Philosophical Transactions of the Royal

16

Society [11]. On page 229, Larmor wrote x1 = xЄ½, where the more familiar

gamma factor, γ, appears in the form Є½. He probably meant to write, x1 =

xꞌЄ½, where xꞌ = (x – vt). He also wrote dt1 = dtꞌЄ−½, where tꞌ = t – vx/c2.

These equations approximate to what we know today as the Lorentz

transformations. Then in the year 1900, on page 174 in his article entitled

“Aether and Matter” [12], Larmor transformed x1, y1, z1, and t1 into Є½xꞌ,

yꞌ, zꞌ, and Є−½tꞌ − (v/c2) Є½xꞌ.

Whatever the finer details are, because they are not always very

clear, Lorentz and Larmor were the two pioneers who first worked on the

problem throughout the 1890s. They achieved what they believed to be

justification for length contraction, but as regards their twin aim of

finding a transformation that would make Maxwell’s equations invariant,

this wasn’t possible until Henri Poincaré invented four-vectors in 1905.

In that same year, Einstein re-derived the Lorentz transformations in the

form below, which is unequivocally that which is used in modern

textbooks,

xꞌ = γ(x – vt) (44)

yꞌ = y (45)

zꞌ = z (46)

tꞌ = γ(t – vx/c2) (47)

Appendix C

(The Advent of Four-Vectors)

On page 907 of his 1905 Bern paper, [14], Einstein purported to subject

Ampère’s Circuital Law and Faraday’s Law to Lorentz transformations.

He wrote these two curl equations out in a perfectly dual format, using

Gaussian units, which expose the speed of light, and he expanded them

into their three Cartesian components, hence resulting in six equations in

total. The primed versions were then displayed on pages 907-908 as seen

below, with the solutions shown within the curved brackets. However,

even though these solutions are correct, it should not have been possible

for Einstein to have arrived at them by using the kinematical Lorentz

transformations, which he had derived on page 902. With these

transformations alone, he would not have been able to introduce the beta

factor, v/c, so symmetrically. It’s only by invoking the concept of 4-D

space-time that the beta factor multiplies out correctly. Meanwhile, the

17

appropriate mathematical tool, known as four-vectors, invented by

Poincaré, [20], wasn’t published until after Einstein had already published

the solutions below in the form,

1/c.∂Ex/∂tꞌ = ∂/∂yꞌ[γ(Bz – v/c.Ey)] − ∂/∂zꞌ[γ(By + v/c.Ez)]

1/c.∂/∂tꞌ[γ(Ey – v/c.Bz)] = ∂Bx/∂zꞌ − ∂/∂xꞌ[γ(Bz – v/c.Ey)]

1/c.∂/∂tꞌ[γ(Ez + v/c.By)] = ∂/∂xꞌ[γ(By + v/c.Ez)] − ∂Bx/∂yꞌ

1/c.∂Bx/∂tꞌ = ∂/∂zꞌ[γ(Ey – v/c.Bz)] − ∂/∂yꞌ[γ(Ez + v/c.By)]

1/c.∂/∂tꞌ[γ(By + v/c.Ez)] = ∂/∂xꞌ[γ(Ez + v/c.By)] − ∂Ex/∂zꞌ

1/c.∂/∂tꞌ[γ(Bz – v/c.Ey)] = ∂Ex/∂yꞌ − ∂/∂xꞌ[γ(Ey – v/c.Bz)]

Appendix D

(The Biot-Savart Law in the Coulomb Gauge)

“The Double Helix Theory of the Magnetic Field” [6], is essentially

Maxwell’s sea of aethereal vortices but with the vortices replaced by

rotating electron-positron dipoles. Within the context of a single rotating

electron-positron dipole, the angular momentum can be written as H =

D×v, where D is the displacement from the centre of the dipole and v is

the circumferential velocity. When elastically bonded to other dipoles

within the wider electron-positron sea, the displacement D will be related

to the transverse elasticity through Maxwell’s fifth equation, D = εE. A

full analysis can be seen in the articles “Radiation Pressure and E =

mc2” [25], and “The 1855 Weber-Kohlrausch Experiment” [8]. If we

substitute D = εE into the equation H = D×v, this leads to,

H = −εv×EC (48)

See Appendix E regarding why the magnitude of v should

necessarily be equal to the speed of light. Equation (48) would appear to

be equivalent to the Biot-Savart Law if EC were to correspond to the

Coulomb electrostatic force. However, in the context, EC will be the

centrifugal force, EC = µv×H, and not the Coulomb force. If we take the

curl of equation (48) we get,

∇×H = −ε[v(∇∙EC) – EC(∇∙v) + (EC∙∇)v – (v∙∇)EC] (49)

Since v is an arbitrary particle velocity and not a vector field, this

reduces to,

18

∇×H = −ε[v(∇∙EC) – (v∙∇)EC] (50)

Since v and EC are perpendicular, the second term on the right-hand

side of equation (50) vanishes. In a rotating dipole, the aethereal flow

from positron to electron will be cut due to the vorticity, the separate

flows surrounding the electron and the positron will be passing each other

in opposite directions, and so the Coulomb force of attraction will be

disengaged. Hence, the two particles will press against each other with

centrifugal force while striving to dilate, since the aether can’t pass

laterally through itself, and meanwhile the two vortex flows will be

diverted up and down into the axial direction of the double helix, [7].

Despite the absence of the Coulomb force in the equatorial plane, EC is

still nevertheless radial, and like the Coulomb force, as explained in

Appendix F, it still satisfies Gauss’s Law, this time with a negative sign

in the form,

∇∙EC = −ρ/ε (51)

Substituting into equation (50) leaves us with,

∇×H = ρv = J = A (52)

and hence since B = µH then,

∇×B = µJ = µA (53)

which is Ampère’s Circuital Law in the Coulomb gauge.

Appendix E

(The Speed of Light)

Starting with the Biot-Savart law in the Coulomb gauge, H = −εv×EC,

where EC = µv×H, means that we can then write H = −εµv×(v×H). It

follows therefore that the modulus│H│is equal to εµv2H since v, EC, and

H are mutually perpendicular within a rotating electron-positron dipole.

Hence, from the ratio εµ = 1/c2, it follows that the circumferential speed v

must be equal to c within such a rotating dipole. In other words, the ratio

εµ = 1/c2 hinges on the fact that the circumferential speed in Maxwell’s

molecular vortices is equal to the speed of light.

19

Appendix F

(Gauss’s Law for Centrifugal Force)

Taking the divergence of the centrifugal force, EC = µv×H, we expand as

follows,

∇·(µv×H) = μ[H·(∇×v) − v·(∇×H)] (54)

Since v refers to a point particle that is in arbitrary motion, and not to

a vector field, then ∇×v = 0, and since ∇×H = J = ρv, it follows that,

∇·(µv×H) = −μρv·v (55)

then substituting v = c as per Appendix E,

∇·(µv×H) = −μρc2 (56)

and substituting c2 = 1/με, this leaves us with,

∇·(µv×H) = −ρ/ε (57)

which is a negative version of Gauss’s law for centrifugal force.

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