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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 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,

mr

̈ = −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,

mr

̈ = −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 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

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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

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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

Radius.

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

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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.

Conclusion

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

field.

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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

ulsion_by_Strong_Gravity_The_Link_between_Gravity_and_Electromagnetism

[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)

http://users.physik.fu-berlin.de/~kleinert/files/1905_17_891-921.pdf

https://en.wikisource.org/wiki/Translation:On_the_Electrodynamics_of_Moving_Bod

ies

[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”.

http://www.rastko.rs/istorija/tesla/oniell-tesla.html

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)

https://www.researchgate.net/publication/338816847_The_Positronium_Orbit_in_the

_Electron-Positron_Sea

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

https://www.researchgate.net/publication/335169091_Wireless_Radiation_Beyond_th

e_Near_Magnetic_Field

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[18] Tombe, F.D., “The Fine Structure of Four-Dimensional Space-Time” (2020)

https://www.researchgate.net/publication/339696770_The_Fine_Structure_of_Four-

Dimensional_Space-Time

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.