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Abstract

The Lorentz transformations are best known for the relativistic Lorentz factor, γ = 1/√(1 – v^2/c^2), 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 = μεv×E, and also the magnetic force in the form E = 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, E = 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.
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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 Poincarés “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,
x1 = γ(x1 + ivxx4/c) (14)
x2 = x2 (15)
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x3 = x3 (16)
x4 = 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 B along the z-axis will take the form,
Bꞌz = ∂Aꞌy/∂x − ∂Aꞌx/∂y (18)
In four-vector notation this becomes,
Bꞌz = ∂Aꞌ2/∂x1 − ∂Aꞌ1/∂x2 (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/∂t − ∂ψꞌ/∂z (27)
Hence, we can write,
Eꞌz = ic[ ∂Aꞌ4 /∂x3 ∂Aꞌ3/∂x4] (28)
From equation (17), while still deliberately omitting the gamma
factor, γ, we can apply the Lorentz transformations,
Aꞌ4 = (A4 ivxA1/c) (29)
and
∂/∂x4 = (∂/∂x4 ivx/c.∂/∂x1) (30)
Then trivially, since motion is only along the x-axis, it follows from
equation (16) that Aꞌ3 = A3 and ∂/∂x3 = ∂/∂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
14
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,
(Av) = 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,
(Av) = v×B + (v)A (36)
Hence,
(v)A = −v×B + (Av) (37)
Since by the theorem of total derivatives,
15
dA/dt = ∂A/∂t + (v)A (38)
it then follows that,
dA/dt = ∂A/∂t − v×B + (Av) (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
(Av) 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 =
xЄ½, where x = (x vt). He also wrote dt1 = dtЄ½, where t = 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 Є½x,
y, z, and Є½t (v/c2) Є½x.
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,
x = γ(x vt) (44)
y = y (45)
z = z (46)
t = 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/∂t = ∂/∂y[γ(Bz v/c.Ey)] − ∂/∂z[γ(By + v/c.Ez)]
1/c.∂/∂t[γ(Ey v/c.Bz)] = ∂Bx/∂z ∂/∂x[γ(Bz v/c.Ey)]
1/c.∂/∂tꞌ[γ(Ez + v/c.By)] = ∂/∂x[γ(By + v/c.Ez)] − ∂Bx/∂y
1/c.∂Bx/∂t = ∂/∂z[γ(Ey v/c.Bz)] − ∂/∂y[γ(Ez + v/c.By)]
1/c.∂/∂t[γ(By + v/c.Ez)] = ∂/∂x[γ(Ez + v/c.By)] − ∂Ex/∂z
1/c.∂/∂t[γ(Bz v/c.Ey)] = ∂Ex/∂y ∂/∂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.
References
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XXI, Fourth Series, London, (1861)
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20
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21
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calling into existence, in never ending cycles, all things and phenomena. The primary
substance, thrown into infinitesimal whirls of prodigious velocity, becomes gross
matter; the force subsiding, the motion ceases and matter disappears, reverting to the
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grained vortex motion, circulating with that same enormous speed. For it has been
partly, though as yet incompletely, shown that such a vortex fluid would transmit
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27, pp. 915 (1845)
22
[25] Tombe, F.D., “Radiation Pressure and E = mc2 (2018)
https://www.researchgate.net/publication/325859308_Radiation_Pressure_and_E_mc
... This is despite the fact that special relativity can be made correct, and electron-positron pair annihilation explained too, simply by re-introducing Maxwell's sea of molecular vortices [15], [16], [17]. Minkowski's 4D space-time continuum is the electron-positron sea [18], while General Relativity is simply a case of applying special relativity to the escape velocity in a gravitational field. General relativity is ultimately about the manner in which gravity distorts the physical structure of the 4D Minkowski space-time continuum and the effect that this has on optical and inertial phenomena. ...
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