In response to a Quora question, I wrote the following:

In his September, 1905 paper entitled *Does the inertia of a body depend upon its energy-content?* (*Ist die Trägheit eines Körpers von seinem Energiegehalt abhängig?*, Annalen der Physik. 18:639, 1905) Einstein investigates the energy $l$ (in the notation used in the 1923 English publication of the paper) of a system of plane electromagnetic waves, as measured by one observer. Another observer, moving relative to the first with velocity $v$ directed at an angle $\phi$ relative to the direction of the waves, sees the energy

$$l^*=l\frac{1-\frac{v}{c}\cos\phi}{\sqrt{1-v^2/c^2}}.$$

He then investigates a body that sends out two light signals with energy $\frac{1}{2}L$ in opposite directions (such that its momentum does not change). If the body's energy is $E_0$ before and $E_1$ after the emission, we have (due to energy conservation)

$$E_0=E_1+\frac{1}{2}L+\frac{1}{2}L.$$

In the other reference frame, let the body's energy be, before and after, $H_0$ and $H_1$. Then

$$H_0=H_1+\frac{1}{2}L\frac{1-\frac{v}{c}\cos\phi}{\sqrt{1-v^2/c^2}}+\frac{1}{2}L\frac{1+\frac{v}{c}\cos\phi}{\sqrt{1-v^2/c^2}},\tag*{}$$

or, after simple algebra,

$$H_0=H_1+\frac{L}{\sqrt{1-v^2/c^2}}.$$

Subtracting the two yields

$$(H_0-E_0)-(H_1-E_1)=L\left\{\frac{1}{\sqrt{1-v^2/c^2}}-1\right\}.$$

Now Einstein notes that $H_0$ and $E_0$ refer to the energy of the same body in the same state, in two different inertial systems; same goes for $H_1$ and $E_1$. On the other hand $H-E$ is the difference in kinetic energy, as seen in two systems that move relative to each other, up to some additive constant $C$ that is really just a matter of how the kinetic energy is defined in the two systems:

$$\begin{align*}H_0-E_0=K_0+C,\\

H_1-E_1=K_1+C.\end{align*}$$

So then,

$$K_0-K_1=L\left\{\frac{1}{\sqrt{1-v^2/c^2}}-1\right\}.$$

When the velocity is small, the square root in the denominator can be series expanded, and terms that contain higher powers of $v$ can be dropped, which leaves

$$K_0-K_1=\frac{1}{2}\frac{L}{c^2}v^2.$$

Note that the body's velocity does not change, yet its kinetic energy (which is $\frac{1}{2}mv^2$ when the velocity is small) changed by this amount after the emission. From this Einstein concludes that if a body gives off the energy $L$ in the form of radiation, its mass diminishes by $L/c^2$. He also notes that the fact that the energy withdrawn is in the form of radiation makes no difference, which leads to the conclusion that the mass of a body is a measure of its energy-content, with $c^2$ being the conversion factor between the two. (Note that the actual formula, $E=mc^2$, doesn’t actually appear in Einstein’s paper! But it is implied trivially by his result and is described in words.)

This was Einstein's fourth paper during his *annus mirabilis* (1905), and the second on relativity theory. Another example of an amazing early Einstein paper (see https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf): short (less than three pages), no references, and so much to the point, it is almost impossible to summarize.