Centrifugal Force in the Schwarzschild Field

Frederick David Tombe,

Belfast, Northern Ireland,
United Kingdom,
sirius184@hotmail.com
15th September 2021

Abstract. Centrifugal force is an inertial effect which is induced by

motion through the luminiferous medium. 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 luminiferous medium 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,

mr̈ = −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

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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,

mr̈ = −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
equation,

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 luminiferous medium 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 luminiferous medium 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

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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 exceed ninety degrees, and

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so their direction of rotation will reverse. At the interface with another
gravitational field, unless that other gravitational field is equally
excessive, the neighbouring vortices will instead bond, electron to
positron, across the interface. The centrifugal repulsion will therefore be
replaced by an electrostatic force of attraction. We can therefore have a
gravitational field that is so strong, that below a certain radius there can
never be a centrifugal force to protect another body from falling into it,
unless that other body has an excessively strong gravitational field also.
This radius is known as the Schwarzschild Radius.

Aether Terminologies

III. Einstein used the term luminiferous aether for what James Clerk
Maxwell termed “the luminiferous medium”, [9]. But what Einstein
referred to as the luminiferous aether 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
Maxwell’s sea of molecular vortices 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
the luminiferous medium, 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 4-D 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 luminiferous medium, but which superimposes upon and
distorts its internal structure, hence affecting optical and inertial
phenomena. It’s possible that in turn, the luminiferous medium, 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

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large scale vorticity in the velocity field of the aether that flows into
gravitating bodies.

Conclusion

IV. Einstein’s “General Theory of Relativity”, published in its final form

in 1916, is about fine-grained gyroscopics in the medium for the
propagation of light, which Einstein referred to as the luminiferous
aether. 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 the luminiferous aether
specifically with Maxwell’s sea of molecular vortices. This is despite the
fact that special relativity can be made correct, and electron-positron pair
annihilation explained too, simply by re-introducing this fine-grained
vortex sea, [15], [16], [17]. What Einstein saw as the Minkowski’s 4-D
space-time continuum is in fact an 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
dipolar electron-positron vortex sea, and the effect that this has on optical
and inertial phenomena.
Gravity exerts a torque on the tiny electron-positron vortices, hence
causing them to precess about the radial gravitational field lines. This in
turn generates centrifugal pressure perpendicular to the field lines. If,
however, the gravity happens to be very strong, the precession cone can
exceed ninety degrees, hence reversing the direction of spin so as to
undermine the centrifugal pressure that is generated at the interface with
the gravitational field of another body. An electrostatic force of attraction
then acts, between electron and positron, across the interface. There will
therefore be a certain radius below which Kepler’s laws break down to
the extent that there can be no escape. However, if both gravitational
fields are excessively strong, then the rotation reversal will have occurred
in the tiny vortices on both sides of the interface, and so a centrifugal
force of repulsion will still exist. It would seem that between two
comparable masses, there is always a self-standing centrifugal force of
repulsion, independent of any angular momentum on the large scale, but
that it is always overridden by gravitational attraction.

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 References
[1] Tombe, F.D., “Leibniz’s Radial Planetary Orbital Equation”, (2017)
https://www.researchgate.net/publication/320694417_Leibniz's_Radial_Planetary_Or
bital_Equation

[2] Maxwell, J.C., “On Physical Lines of Force”, Philosophical Magazine, Volume
XXI, Fourth Series, London, (1861)
http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf

[3] Tombe, F.D., “The Double Helix Theory of the Magnetic Field”, (2006)
Galilean Electrodynamics, Volume 24, Number 2, p.34, (March/April 2013)
https://www.researchgate.net/publication/295010637_The_Double_Helix_Theory_of
_the_Magnetic_Field

[4] Tombe, F.D., “The Double Helix and the Electron-Positron Aether”, (2017)
https://www.researchgate.net/publication/319914395_The_Double_Helix_and_the_El
ectron-Positron_Aether

[5] Tombe, F.D., “Aether Friction in the Planetary Orbits”, (2021)

https://www.researchgate.net/publication/350873624_Aether_Friction_in_the_Planeta
ry_Orbits

[6] Tombe, F.D., “Magnetic Repulsion and the Gyroscopic Force”, (2015)
https://www.researchgate.net/publication/283225757_Magnetic_Repulsion_and_the_
Gyroscopic_Force

[7] Tombe, F.D., “Induction of Electrostatic Repulsion by Strong Gravity”, (2017)

https://www.researchgate.net/publication/322211308_Induction_of_Electrostatic_Rep
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[8] Tombe, F.D., “Straight Line Motion”, (2018)

https://www.researchgate.net/publication/325472420_Straight_Line_Motion

[9] Einstein, Albert, “Zur Elektrodynamik bewegter Körper”, Annalen der Physik
322 (10) pp. 891-921, Bern, (1905)
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[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

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matter; the force subsiding, the motion ceases and matter disappears, reverting to the
primary substance”.
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http://www.ascension-research.org/tesla.html

[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
whirlpools.”

[12] Lodge, Sir Oliver, “Ether (in physics)”, Encyclopaedia Britannica,

Fourteenth Edition, Volume 8, Pages 751-755, (1937)
http://gsjournal.net/Science-
Journals/Historical%20PapersMechanics%20/%20Electrodynamics/Download/4105
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 propagation—and 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)
https://www.nature.com/articles/216119a0

[14] Einstein, Albert, “Ether and the Theory of Relativity”, University of Leiden,
(1920)
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.
https://mathshistory.st-andrews.ac.uk/Extras/Einstein_ether/

[15] Tombe, F.D., “Einstein’s Big Mistake”, (2020)

https://www.researchgate.net/publication/344461744_Einstein's_Big_Mistake

[16] Tombe, F.D., “The Positronium Orbit in the Electron-Positron Sea”, (2020)
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_Electron-Positron_Sea

[17] Tombe, F.D., “Wireless Radiation Beyond the Near Magnetic Field”, (2019)
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e_Near_Magnetic_Field

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[18] Tombe, F.D., “The Lorentz Aether Theory”, (2020)
https://www.researchgate.net/publication/339696770_The_Lorentz_Aether_Theory

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. While 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, the orbital centrifugal force on the large scale will now
have been destroyed, unless a similar situation has occurred in the other body.