Einstein's Mass-Energy Equation

Theorem
The energy imparted to a body to cause that body to move causes the body to increase in mass by a value $$M$$ is given by the equation:
 * $$E = M c^2$$

where $$c$$ is the speed of light.

Proof
From Einstein's Law of Motion, we have:


 * $$\displaystyle \mathbf F = \frac{m_0 \mathbf a}{\left({1 - \dfrac{v^2}{c^2}}\right)^{\tfrac 3 2}}$$

where:
 * $$\mathbf F$$ is the force on the body;
 * $$\mathbf a$$ is the acceleration induced on the body;
 * $$v$$ is the magnitude of the velocity of the body;
 * $$c$$ is the speed of light;
 * $$m_0$$ is the rest mass of the body.

Assume WLOG that the body is starting from rest at the origin of a cartesian coordinate plane.

Assume the force $$\mathbf F$$ on the body is in the positive direction along the x-axis.

To simplify the work, we consider the acceleration as a scalar quantity and write it $$a$$.

Thus, from the Chain Rule:
 * $$\displaystyle a = \frac{\mathrm{d}{v}}{\mathrm{d}{t}} = \frac{\mathrm{d}{v}}{\mathrm{d}{x}} \frac{\mathrm{d}{x}}{\mathrm{d}{t}} = v \frac{\mathrm{d}{v}}{\mathrm{d}{x}}$$

Then from the definition of energy:
 * $$\displaystyle E = \int_{0}^{x} F \mathrm{d}{x}$$

which leads us to: