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Connecting E = mc2 to the Lorentz Transformation
Frederick David Tombe,
Belfast, Northern Ireland,
United Kingdom
sirius184@hotmail.com
21st November 2023
Abstract. The Lorentz transformation and the equation E = mc2 are two separate issues.
This article will examine the individual physical contexts in which each of them apply
separately, and when they apply simultaneously.
The Lorentz Transformation
I. The Lorentz transformation is observed in optics as the return-path
longitudinal Doppler effect [1], and the maths also applies to the optical
transverse Doppler effect. It further applies to the fields surrounding charged
objects and how these fields alter when the source charge is in motion relative
to the wave-carrying medium. An electrostatic field morphs into a magnetic
field as the source charge accelerates towards the speed of light, [2].
The commonality between the optical Doppler effect and the transformation
of fields in motion must therefore be rooted in the fabric of the physical medium
within which electromagnetic waves propagate, and within which a magnetic
field is constructed.
Radiation Pressure
II. The equation E = mc2 has got no immediate connection to the Lorentz
transformation. It relates to the energy in wireless electromagnetic radiation,
and it can be derived by considering the momentum of the radiation in
conjunction with Maxwell’s equation for radiation pressure, [3].
Conclusion
III. Except in the case of tiny rotating electron-positron dipoles whose
circumferential speed is equal to the speed of light, the Lorentz transformation
cannot be applied to the kinematics of particle motion, at least not without
introducing some kind of scaling factor to take account of the complexities of
the atoms and molecules of ponderable matter. Both E = mc2 and the Lorentz
transformation only relate either to wireless electromagnetic radiation or to the
tiny dipolar vortices that comprise the wave-carrying medium, known as
luminiferous medium, [4], [5], [6]. These tiny vortices are in fact the tiny rotating
electron-positron dipoles mentioned above, [3], [7]. The equation E = mc2 applies
to the centrifugal pressure in the equatorial plane of the vortices [3], [8]. and as
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well as deriving it from Maxwell’s radiation pressure equation, it can also be
derived from electron-positron pair annihilation and production, because this
equation is ultimately just Newton’s equation for the speed of a wave in an
elastic solid, the elastic solid in question being comprised of electrons and
positrons, [9], [10].
As for the Lorentz transformation, it is not a coordinate frame
transformation at all. It relates to the precession of the tiny dipolar vortices in
the fields surrounding charged objects that are accelerating through the
luminiferous medium, [11]. There is no time dilation. The time terms refer
exclusively to the angular frequency of the tiny vortices that fill all of space,
and without the presence of this sea of vortices, the Lorentz transformation is
meaningless.
In the case of EM radiation being absorbed by targets that are in motion
through the luminiferous medium, E = mc2 and the Lorentz transformation
combine together into the energy-momentum four-vector to yield the relevant
energy changes. This is important when it comes to atomic clocks in orbit in the
Global Positioning System, [12].
The so-called relativistic mass, which is inferred from the energy-
momentum four-vector, is merely a velocity dependent product involving actual
mass. It should not be read that the actual mass itself increases with speed.
References
[1] Gupta, P.D., “Exact derivation of the Doppler shift formula for a radar echo without
using transformation equations”, Am. J. Phys., vol. 45, No. 7, pp. 674-675, (July 1977)
[2] Tombe, F.D., “The Lorentz Aether Theory”, (2020)
https://www.researchgate.net/publication/339696770_The_Lorentz_Aether_Theory
[3] Tombe, F.D., “The Double Helix Theory of the Magnetic Field”, (2006)
Galilean Electrodynamics, vol. 24, number 2, p.34, (March/April 2013)
See Section IV for centrifugal pressure and Section VI for radiation pressure.
https://www.researchgate.net/publication/295010637_The_Double_Helix_Theory_of_the_M
agnetic_Field
[4] 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.”
[5] Maxwell, J.C., “On Physical Lines of Force”, Philosophical Magazine, Volume XXI,
Fourth Series, London, (1861)
http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf
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[6] Lodge, Sir Oliver, “Ether (in physics)”, Encyclopaedia Britannica, Fourteenth Edition,
vol. 8, pp. 751-755, (1937)
See pp. 6-7 in the pdf file in the link below, beginning at the paragraph that starts with,
Possible Structure. −, and note that while the quote suggests that the ether is incompressible,
this is almost certainly not the case. The quote in question, in relation to the speed of light,
reads,
“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.”
http://gsjournal.net/Science-
Journals/Historical%20PapersMechanics%20/%20Electrodynamics/Download/4105
[7] Tombe, F.D., “The Double Helix and the Electron-Positron Aether”, (2017)
https://www.researchgate.net/publication/319914395_The_Double_Helix_and_the_Electron-
Positron_Aether
[8] Tombe, F.D., “Radiation Pressure and E = mc² ”, (2018)
https://www.researchgate.net/publication/325859308_Radiation_Pressure_and_E_mc
[9] Simhony, M., “The Electron-Positron Lattice Space, Cause of Relativity and Quantum
Effects”, Physics Section 5, The Hebrew University, Jerusalem, (1990)
http://web.archive.org/web/20040606235138/www.word1.co.il/physics/mass.htm
[10] Tombe, F.D., “The Positronium Orbit in the Electron-Positron Sea”, (2020)
https://www.researchgate.net/publication/338816847_The_Positronium_Orbit_in_the_Electr
on-Positron_Sea
[11] Tombe, F.D., “The Deception Inherent in Coordinate Frame Transformations”,
(2023)
https://www.researchgate.net/publication/372533786_The_Deception_Inherent_in_Coordinat
e_Frame_Transformations
[12] Tombe, F.D., “Atomic Clocks and Relativity”, (2023)
https://www.researchgate.net/publication/372165749_Atomic_Clocks_and_Relativity