## K6 Alternative Derivation of E = m • c^{2}

We derived this formula in **E4** in the usual way using kinetic energy and the amount of work invested in the acceleration of the mass. Similar to the Pythagorean Theorem, however, there are many proofs and derivations of this famous equation. I have already alluded to probably the most beautiful in the last paragraph of section **E5**. In this case Einstein used the conservation of momentum and the conservation of energy as well as the formulas for the energy and the momentum of photons. It is presented in [22-98ff].

Some other nice proofs (see [26-55ff], [15-131ff]) also use the momentum of light particles or the radiation pressure exerted by electromagnetic wave activity. This quantity was known in 1880 (i.e., ‘long’ before quantum theory) from the theory of Maxwell.

Max Born chose a different approach to represent Einstein's relativity theories in his book [42] for the general public first published in 1920. He derives the formula from the inelastic collision of two clumps moving at non-relativistic velocities, i.e., exactly as shown in **E3**. This method is, in principle, the same used by Sexl et al. in [11-24]: The formula for the increase in mass after its having been accelerated is expanded as a power series, whose fourth and higher order terms of v/c are dropped, thereby yielding E_{kin} = Δm • c^{2}.

Einstein used a similar approach in September 1905. In his essay entitled "Does the Inertia of a Body Depend on Its Energy Content?" he derives the famous equation from his transformation formulas for radiation energy, which we have not discussed. The essay is only four pages, but it can not be recommended for general reading. However, we can understand the last sections, in which Einstein shows that he is quite aware of the general implications of his formula. In the quote below, we have replaced the character L, which Einstein still used at that time for energy, with E and for the speed of light we have written c instead of V:

“If a body emits the energy E in the form of radiation, its mass decreases by E/c^{2}. Here it is obviously inessential that the energy taken from the body turns into radiant energy, so we are led to the more general conclusion:

The mass of a body is a measure of its energy content; if the energy changes by E, the mass changes in the same sense by E/(9 • 10^{20}) if the energy is measured in ergs and the mass in grams.

It is not excluded that it will prove possible to test this theory using bodies whose energy content is variable to a high degree (e.g., radium salts).

If the theory agrees with the facts, then radiation carries inertia between emitting and absorbing bodies.”

Bern, September 1905. [9-164]

Original publication in "Annalen der Physik", vol. 18 [1905], p.639-641. All of the works of Einstein in the journal "Annalen der Physik" can be downloaded here as pdf photocopies.