"If my theory of relativity is proven successful, Germany will claim me as a German and France will declare that I am a citizen of the world. Should my theory prove untrue, France will say that I am a German and Germany will declare that I am a Jew."  Albert Einstein ( 1879  1955 )
§ Listen to Albert Einstein ^{[1]} giving a short lecture about e=mc2 :
"It followed from the special theory of relativity that mass and energy are both but different manifestations of the same thing  a somewhat unfamiliar conception for the average mind. Furthermore, the equation E is equal to m csquared, in which energy is put equal to mass, multiplied by the square of the velocity of light, showed that very small amounts of mass may be converted into a very large amount of energy and vice versa. The mass and energy were in fact equivalent, according to the formula mentioned above. This was demonstrated by Cockcroft and Walton in 1932, experimentally."  Prof. Albert Einstein ( excerpts from 1947 film, "Atomic Physics" )
source: American Institute of Physics ( https://www.aip.org/history/exhibits/einstein/voice1.htm )
Special Relativity physics was first published in 1905 by Albert Einstein at age 26 working quietly in the Swiss Patent Office, Bern, Switzerland, under the title "On The Electrodynamics Of Moving Bodies", translated from "Zur Elektrodynamik bewegter Körper", Annalen der Physik, volume 17: 891, Bern June 1905, a downloadable copy of which is available here in pdf. German version: "Zur Elektrodynamik bewegter Körper", von A. Einstein.
And, "Does the Inertia of a Body Depend upon its Energy  Content?", by A. Einstein, Annalen der Physik volume 18: 639, Bern September 1905. German version: "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?", von A. Einstein.
Also read "On The Relativity Principle And The Conclusions Drawn From It", by A. Einstein, translated from Jahrbuch der Radioaktivität und Elektronik volume 4 (1907): 411  462. German version: "Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen", von A. Einstein, 4 Dezember 1907
§ Listen to a reading of Einstein's "Relativity, The Special and General Theory", December, 1916, as translated by Robert W. Lawson:
"Relativity  The Special and the General Theory", by Albert Einstein, December, original version 1916, translated by Robert W. Lawson, The Physics Laboratory, The University of Sheffield, June 12, 1920, Copyright © 2004, Barnes & Noble Publishing, Inc.
§ Define:
§ Some Derivations:
1).
.
2).
§ The Problem:
However the entire classical Newtonian physics derived above is predicated upon the concept of mass as an invariant constant. But we now know differently, namely that mass is a speed  dependent variable quantity owing to the Addition of Relativistic Velocities, where
is the relationship between rest ( or proper ) mass undergoing velocity and its equivalent dilated mass .
§ More Definitions:
§ The Solution:
But, whoa! Look,
§ More Simple Algebraic Derivation:
note: see another quick and dirty matheamtical derivation
§ ^{[1]} Einstein's Interpretation:
The interpretation that Einstein therefore applied is as follows:
Nevertheless it still should always be remembered that
On the other hand, applying a relativistic kinetic energy concept, we can arrive at the following:
§ The Law of Inertia of Energy:
Or,
is the equation for matter in the form of relativistic ( dilated ) mass which can be derived from a given amount of energy whose capability for performing work is given by
.
§ Derivation of classical Newtonian kinetic energy:
§ Derivation of relativistic kinetic energy:
However for , relativistic mass dilation as a function of velocity,
where is rest mass ( proper mass ) within a given inertial frame of reference, we still have
However using a dummy variable trick for integrating,
And therefore,
§ 2nd Derivation of relativistic kinetic energy:
§ Derivation of classical kinetic energy:
§ Here we now have these important energy definitions:
Therefore,
§ Derive the law of conservation of total energy, relativistic and non  relativistic:
Finally, using the binomial series to derive the law of conservation of total energy, relativistic and non  relativistic:
§ Deriving mass dilation using Richard Fehnman's suggested equations from his "Lectures on Physics  Vol. I " ( although this derivation is somewhat recursive ):
Epilogue: Some Final Questions
§ Was Einstein the original discoverer of e=mc2?:
Friedrich ( Fritz ) Hasenöhrl ( 1874  1915 )
Friedrich ( Fritz ) Hasenöhrl proposed for the heat ( energy ) radiation of a moving black box ( or "cavity" ) the following equation:
However, Hasenöhrl's derivation was an entirely classical derivation in the manner of Newton and James Clerk Maxwell without the overall context of relativistic physics which Einstein brought into full fruition by his 'principle of relativity' mathematics.
The other couple of lines of intellectual attack against Einstein for his relativistic understanding of the inertial mass of all electromagnetic energy propagation arose first by Nazi Nobelist Philipp Lenard in his anti  semitic Deutsche Physik movement by which Lenard attempted to ascribe special relativity credit away from Jewish Albert Einstein and onto Johann Georg von Soldner and Paul Gerber but was immediately and totally rebutted by Max von Laue, recipient of the Nobel Physics Prize in 1914 for the diffraction of x  rays by crystals; and whereas the second line of attack arose by those who supported Henri Poincaré's mathematics which, however, ultimately lacked the fullness of the intimate nexus of spacewithtime that Einstein's special relativity physics provided and wholly elaborated upon in the light  sphere geometry of "Space and Time", by Hermann Minkowski, Cologne 1908. Hence, Poincaré's mathematics was "pre  relativistic", he having discovered the few remaining Lorentz velocity transformations still outstanding including those for Maxwell's electromagnetic equations. Read: "On The Dynamic of the Electron", by Henri Poincaré, 1905, English translation.
"On the Theory of Radiation in Moving Bodies", by Friedrich ( Fritz ) Hasenöhrl, Annalen der Physik 15, 344  370, 1904. German version: "Zur Theorie der Strahlung in bewegten Körpern"
"Dismissing renewed attempts to deny Einstein the discovery of special relativity", by Roger Cerf, Université Louis Pasteur, Strasbourg, France, June 2006
§ What meaning did Einstein ascribe to e=mc2?:
Einstein abjured as late as his 1934 Gibbs Lecture at the Carnegie Institute of Technology ( now Carnegie  Mellon University ) the well established, modern consensus for the concept of "speed  dependent" mass otherwise known as "relativistic mass" when he continued to insist on writing 'mass' as an invariant scalar ( tensor rank zero ) as follows:
as opposed to the modern relativistic physics of
That Einstein maintained such an interpretation is amply shown in the following couple of theses:
"Einstein's 1934 two  blackboard derivation of energymass equivalence", by David Topper, Dwight Vincent, American Journal of Physics volume 75, issue 11, pp. 978  983,
July 2007 © 2007 American Association of Physics Teachers
"Einstein on mass and energy", by Eugene Hecht, Department of Physics, Adelphi University, Garden City, New York, June 2009 © 2009 American Association of Physics Teachers
§ ^{[1]} Caveat: The name 'Einstein' in the above passages of this web page essay is to be understood more generally as encompassing the mathematical works of Max Planck, especially as regards "relativistic mass" which
Einstein himself avoided in favor of inertial or, equivalently, rest ( proper ) mass, as revealed above in the Epilogue: Some Final Questions, particularly regarding the question "...as to what meaning did Einstein
attach to e=mc2?"
§ References:
"Imagination is more important than knowledge"  Albert Einstein ( 1879  1955 )
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