Relativity Science Calculator - E=mc2


"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 )

§ Read Albert Einstein's words [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 c-squared, 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 ( )

Special Relativity 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

§ Define:


§ Some Derivations:


derivation 1.png.


derivation 2.png

§ 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 

variable mass.png

is the relationship between rest ( or proper ) mass  undergoing velocity and its equivalent dilated mass .

§ More Definitions:

§ The Solution:

derivation 3.pngbinomial expansion for e=mc2

But, whoa! Look,


§ More Simple Algebraic Derivation:

note: see another quick and dirty matheamtical derivation

binomial expansion series for e=mc2

§ Einstein's Interpretation [ see: 'Caveat' down below ]:

The interpretation that Einstein therefore applied is as follows: 

reinterpretation of energy

Nevertheless it still should always be remembered that 


On the other hand, applying a relativistic kinetic energy concept, we can arrive at the following: 

relativistic k.e.binomial expansion for e=mc2

§ The Law of Inertia of Energy:

Law of Energy Inertia


inertial mass

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:

classical 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,

integral calculus

And therefore,

relativistic kinetic energy

§ 2nd Derivation of relativistic kinetic energy:

§ Derivation of classical kinetic energy:

mathematical references

§ Here we now have these important energy definitions:


§ 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:

binomial expansion for e=mc2


§ 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:

Friedrich ( Fritz ) HosenöhrlFriedrich ( Fritz ) Hosenöhrl

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 space-with-time 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.

"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 energy-mass 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

§ Caveat [ see above: 'Einstein's interpretation' ]:

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 " to what meaning did Einstein attach

to e=mc2?"

§ References:

  1. "Einstein's comprehensive 1907 essay on relativity", parts I, II, and III, by H.M. Schwartz, Department of Physics, University of Arkansas. These essays also provide some of Einstein's embryonic ideas on gravity. © 1977 American Association of Physics Teachers

  2. "Gravitational Mass, Its Mechanics - What it is; How it Operates", by Roger Ellman ( 1932 - ), Scientist retired, MS Engineering Stanford University, Graduate West Point as well as having taught atomic physics and electrical engineering there, founder - director of The - Origin Foundation, Inc., Santa Rosa, Cal.

  3. "Inertial Mass, Its Mechanics - What it is; How it Operates", by Roger Ellman ( 1932 - )