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Centrifugal force is an inertial effect which is induced by motion through the Minkowski 4D space-time continuum. While it can act in opposition to gravity, there is evidence from Einstein's General Theory of Relativity, that gravity, if strong enough, can affect the physical structure of the 4D space-time continuum in such a way as to destroy the centrifugal force and convert it into an electrostatic force of attraction that augments the gravity. The physical nature of centrifugal force and the manner in which it can be altered by gravity will now be investigated.
Centrifugal Force in the Schwarzschild Field
Frederick David Tombe,
Belfast, Northern Ireland,
United Kingdom,
15th September 2021
Abstract. Centrifugal force is an inertial effect which is induced by
motion through the Minkowski 4D space-time continuum. While it
can act in opposition to gravity, there is evidence from Einstein’s
General Theory of Relativity, that gravity, if strong enough, can
affect the physical structure of the 4D space-time continuum in such
a way as to destroy the centrifugal force and convert it into an
electrostatic force of attraction that augments the gravity. The
physical nature of centrifugal force and the manner in which it can
be altered by gravity will now be investigated.
Leibniz’s Equation Corrected for General Relativity
I. When corrected for Einstein’s General Theory of Relativity,
Leibniz’s radial planetary orbital equation, [1], takes on the form,
̈ = −GMm/r2 + L2/mr3 3GML2/mc2r4 (1)
where L is angular momentum, GMm/r2 is the familiar Newtonian
gravity term, L2/mr3 is the centrifugal force, and 3GML2/mc2r4 is the so-
called relativistic term. Note that the relativistic term, which is negligible
in most cases, is an attractive term like gravity and hence it detracts from
the repulsive effect of the centrifugal force. This leads to a deviation from
the classical Keplerian solution to planetary orbits, and this deviation
starts to become noticeable in strong gravitational fields.
On initial inspection, the relativistic term appears somewhat
mysterious since it is a hybrid effect involving characteristics of both
gravity and centrifugal force, each of which has a completely different
physical cause. Gravity is an inverse square law force indicating that it
arises in connection with a monopole field, whereas centrifugal force is
an inverse cube law force indicating that it arises in connection with a
dipole field. We can however get a better insight into the physical
interaction between gravity and centrifugal force which underlies this
hybrid force if we rearrange equation (1) so as to reduce the right-hand-
side into two distinct forces, one being an attractive monopole force and
the other being a dipole force which can be either repulsive or attractive.
Equation (1) then takes on the form,
̈ = −GMm/r2 + L2/mr3[1 3/2(v2/c2)] (2)
where c is the speed of light, and v is the escape velocity as per the
v2 = 2GM/r (3)
The dipole inverse cube law force is primarily a repulsive centrifugal
force, but with gravity acting as a reducing factor which can make it
become attractive in a strong gravitational field, hence augmenting the
gravity. It can be seen from equation (2) that the greater the escape
velocity, and hence the greater the gravitational field strength, the lesser
will be the magnitude of the inverse cube law force in the positive
direction. Indeed, when the escape velocity is such that the ratio of v2/c2
is 2/3, this force will disappear altogether, and as v increases further still,
approaching the speed of light, the inverse cube law force will have
reversed its direction and will now be working to bolster the gravitational
force of attraction rather than opposing it.
Maxwell’s Sea of Molecular Vortices
II. In order to better understand how gravity acts to reduce centrifugal
force, we will identify the Minkowski 4D space-time continuum with
Maxwell’s sea of molecular vortices [2], as amended by “The Double
Helix Theory of the Magnetic Field, [3], [4]. This means that Maxwell’s
vortices become dipolar vortices, each containing an aether sink
(electron) and an aether source (positron). The Minkowski 4D space-time
continuum then becomes a dipole field, and hence it only relates to the
inverse cube law inertial force in equation (2), and not to gravity. The
role of the dipolar vortex sea in causing the inertial forces has already
been covered in earlier papers, [5], [6], and the objective now will be to try
and explain how gravity acting upon this dipolar vortex sea, when strong
enough, converts the inverse cube law force from a centrifugal repulsion
into an attractive force.
The general principle behind the following proposed explanation is
that the orientation of the rotation axes of the individual dipolar vortices
within the vortex sea is determined by either an externally applied
gravitational field or by something in the vicinity moving through the sea.
This orientation will in turn determine whether these rotating dipoles face
their immediate neighbours in their equatorial planes or in their axial
planes. This can make the difference between mutual repulsion or mutual
attraction between neighbouring dipoles across the interface between two
gravitational fields.
If gravitating bodies, just like electrons, are also aether sinks, but on
the large scale, then when a gravitational field is superimposed upon the
sea of tiny dipolar vortices, these vortices will experience a torque which
will cause them to precess about an axis that is parallel to the
gravitational field lines. In a weak gravitational field, this won’t cause too
much impact on the already existing background double helix magnetic
alignment of this electron-positron dipole sea, but when the gravitational
field strength increases significantly, this creates gravitational tubes of
force comprised of precessing dipoles which repel their immediate
neighbours laterally due to fine-grained centrifugal force emanating from
the equatorial plane of the tiny dipoles as they strive to dilate. Increasing
the gravitational field strength further will increase this centrifugal force,
and as argued in section II of “Induction of Electrostatic Repulsion by
Strong Gravity” [7], the centrifugal force will increase at a greater rate
than the gravity does, and therefore a reversal threshold will be reached in
which two gravitating bodies will actually repel each other. This is
perhaps what we observe when two like-electric charges repel each other,
suggesting that an electrostatic field, in the case where the charge is sink-
based, is simply a strong gravitational field.
The explanation for convectively induced centrifugal force is
extrapolated from Maxwell’s explanation for Ampère’s Circuital Law,
whereby the rotation axes of the tiny molecular vortices form concentric
solenoidal rings of force around the path of motion of the electric current,
[5], [8]. This constitutes a magnetic field, equivalent to a centrifugal force
field. When this principle is extended to all moving objects, then when
two bodies pass each other transversely, the tiny vortices at the interface
between the two centrifugal force fields will repel each other [5]. The
same principle will apply at the interface between two gravitational fields
undergoing mutual transverse motion, and this will give rise to orbital
centrifugal force on the large scale. But if the gravitational field strength
is large enough, the precessional angle of the tiny electron-positron
vortices in the gravitational tubes of force will be such that the orientation
will cause the centrifugal force alignment to collapse, and the centrifugal
force will fail. Instead, the electrons and positrons in each dipole will
attract the positrons and electrons across the interface between the two
gravitational fields.
At the interface between the gravitational fields of two orbiting
bodies, where the gravitational field of one of them is very strong, there
will be no centrifugal force if the gravitational torque has reorientated the
tiny dipoles such as to cause an electrostatic force of attraction instead.
This interface attractive force will now be augmenting the gravitational
force. We will then have a gravitating body, so strong, that below a
certain radius there can never be a centrifugal force to protect another
body from falling into it. This radius is known as the Schwarzschild
Aether Terminologies
III. Einstein used the term luminiferous aether for what James Clerk
Maxwell termed “the luminiferous medium”. But what Einstein referred
to as the luminiferous aether, [9], should not be confused with the pure
aether itself. The former is the latter, only when the latter is rendered into
a state of tiny dipolar whirlpools [10], [11], [12]. The former is Minkowski’s
4D space-time continuum while the latter is the inflowing medium
associated with the gravitational field, and from which the tiny dipolar
vortices in the former are comprised. The interaction between these two
inter-related physical media interferes with the inertial forces. We will
equate the speed of inflow of the pure aether (electric fluid) in a
gravitational field with the escape velocity mentioned in section I.
It is considered that gravity, of which the ultimate cause remains
unknown, is mediated by the flow of this fundamental aether (electric
fluid) as it flows through the sea of tiny aether whirlpools, and into
gravitating bodies. The sea of tiny aether whirlpools, which constitutes
Minkowski’s 4D space-time continuum, does not in itself flow into
gravitating bodies. In the sea of tiny whirlpools, the aether flows into
sinks (electrons) and out of sources (positrons). These electrons and
positrons orbit each other, and when subjected to a gravitational field,
there will be a prevailing net flow of the same pure aether, towards and
into a gravitating body of ponderable matter. Hence, contrary to common
belief, gravity is not explained as being a warp in the 4D space-time
continuum as is taught in conjunction with General Relativity. We don’t
actually have an explanation for the vital force that draws the aether into
the sinks, and which is the ultimate cause of gravity. Meanwhile, gravity
is something external to the 4D space-time continuum, but which
superimposes upon and distorts its internal structure, hence affecting
optical and inertial phenomena. It’s possible that in turn, the 4D space-
time continuum, being Maxwell’s sea of tiny aethereal whirlpools, also
has an effect on gravity by acting like a rotationally elastic sponge which
absorbs some of the large scale vorticity in the velocity field of the aether
that flows into gravitating bodies.
IV. Einstein’s General Theory of Relativity, published in its final form
in 1916, is about fine-grained gyroscopics in the Minkowski 4D space-
time continuum. In the abstract of his earlier 1905 paper on “Special
Relativity”, [9], Einstein discarded the luminiferous aether and he was
quite wrong to do so. He was wrong because this only resulted in
paradoxical absurdities, [13], in what otherwise would have been a very
ingenious theory. For a start, we need the luminiferous aether as a
physical standard of absolute rest. Einstein did however back-pedal
somewhat during his address at the University of Leiden in 1920 [14]. He
realized that to deny the aether is to assume that space has no physical
qualities whatsoever, but again he erred by failing to identify
Minkowski’s 4D space-time continuum specifically with Maxwell’s
luminiferous medium. 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.
For example, while gravity opposes centrifugal force in a planetary
orbit, it also damages it by virtue of the gyroscopic torque effect that it
has on the tiny aethereal vortices that are the cause of the centrifugal
force. Normally this damage is negligible, but if the gravity is strong
enough it will not just oppose the centrifugal force but will destroy it
completely and convert it into a fine-grained electrostatic force of
attraction at the interface between the two gravitational fields. The
gravity will then be augmented, and there will be no escape from the
[1] Tombe, F.D., “Leibniz’s Radial Planetary Orbital Equation” (2017)'s_Radial_Planetary_Or
[2] Maxwell, J.C., “On Physical Lines of Force”, Philosophical Magazine, Volume
XXI, Fourth Series, London, (1861)
[3] Tombe, F.D., “The Double Helix Theory of the Magnetic Field” (2006)
Galilean Electrodynamics, Volume 24, Number 2, p.34, (March/April 2013)
[4] Tombe, F.D., “The Double Helix and the Electron-Positron Aether” (2017)
[5] Tombe, F.D., “Aether Friction in the Planetary Orbits” (2021)
[6] Tombe, F.D., “Magnetic Repulsion and the Gyroscopic Force”, (2015)
[7] Tombe, F.D., “Induction of Electrostatic Repulsion by Strong Gravity” (2017)
[8] Tombe, F.D., “Straight Line Motion” (2018)
[9] Einstein, Albert, “Zur Elektrodynamik bewegter Körper”, Annalen der Physik
322 (10) pp. 891-921, Bern, (1905)
[10] O’Neill, John J., “PRODIGAL GENIUS, Biography of Nikola Tesla”, Long
Island, New York, 15th July 1944, Fourth Part, paragraph 23, quoting Tesla from his
1907 paper “Man’s Greatest Achievement” which was published in 1930 in the
Milwaukee Sentinel,
“Long ago he (mankind) recognized that all perceptible matter comes from a primary
substance, of a tenuity beyond conception and filling all space - the Akasha or
luminiferous ether - which is acted upon by the life-giving Prana or creative force,
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
primary substance”.
[11] Whittaker, E.T., “A History of the Theories of Aether and Electricity”, Chapter
4, pages 100-102, (1910)
“All space, according to the younger Bernoulli, is permeated by a fluid aether,
containing an immense number of excessively small whirlpools. The elasticity which
the aether appears to possess, and in virtue of which it is able to transmit vibrations,
is really due to the presence of these whirlpools; for, owing to centrifugal force, each
whirlpool is continually striving to dilate, and so presses against the neighbouring
[12] Lodge, Sir Oliver, “Ether (in physics)”, Encyclopaedia Britannica,
Fourteenth Edition, Volume 8, Pages 751-755, (1937)
In relation to the speed of light, The most probable surmise or guess at present is
that the ether is a perfectly incompressible continuous fluid, in a state of fine-
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
waves of the same general nature as light waves i.e., periodic disturbances across
the line of propagationand would transmit them at a rate of the same order of
magnitude as the vortex or circulation speed”
[13] Dingle, H., “The Case Against Special Relativity”, Nature, Volume 216, pages
119-122, (1967)
[14] Einstein, Albert, “Ether and the Theory of Relativity”, University of Leiden,
But on the other hand there is a weighty argument to be adduced in favour of the
ether hypothesis. To deny the ether is ultimately to assume that empty space has no
physical qualities whatever. The fundamental facts of mechanics do not harmonize
with this view.
[15] Tombe, F.D., “Einstein’s Big Mistake” (2020)'s_Big_Mistake
[16] Tombe, F.D., The Positronium Orbit in the Electron-Positron Sea (2020)
[17] Tombe, F.D., Wireless Radiation Beyond the Near Magnetic Field (2019)
[18] Tombe, F.D., The Fine Structure of Four-Dimensional Space-Time” (2020)
Appendix I
(A Note on Entrainment of the Luminiferous Medium by Planetary Bodies)
In the case of small objects where their gravitational fields are too weak to entrain the
luminiferous medium, then when in motion, the luminiferous medium will pass
through the interstitial spaces between their constituent atoms and molecules, just like
water flowing through a basket. As explained in section II above, centrifugal force is
then caused by the solenoidal vortex rings that form around these bodies when in
motion through this electron-positron vortex sea.
In the case of large planetary bodies however, as also explained in section II
above, when gravity acts on the electron-positron sea such as to cause the tiny
vortices to precess, hence forming fine-grained gyroscopic tubes of force consisting of
the tiny precessing vortices, then the entire vortex sea within the gravitational field
will be bonded to the planet. Centrifugal force is generated in the equatorial planes of
these tiny precessing vortices, and the gravitational field strength will determine the
precessional angle and hence the orientation of the centrifugal force on the large scale.
While precessing, the centrifugal force being generated by the tiny vortices, will have
components acting both parallel to the gravitational tubes of force and perpendicular
to them. It’s the perpendicular component that gives rise to orbital centrifugal force on
the large scale.
When the gravitational field strength is very strong, the centrifugal force can
become orientated almost entirely parallel to the tubes of force, and this will prevent
the tiny vortices themselves from being drawn down into the planet by the gravity,
since the vortices will now be repelling each other along the line of the gravitational
tubes of force. Meanwhile, the orbital centrifugal force on the large scale will have
been destroyed.
... VII. In the classical limit, ignoring Einstein's "General Theory of Relativity", [15], the elasticity in space that is involved in electromagnetism and electromagnetic waves must conform to Newton's laws of motion and to Kepler's laws of planetary motion, otherwise the planets would spiral into the Sun. The inertial forces are closely related to the convective aspect of electromagnetic induction where, = μ × , while Newton's second law of motion is closely related to time-varying electromagnetic induction where, = − / , with only the latter being involved in electromagnetic waves. ...
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