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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle E\)	\(=\)	\(\displaystyle m_0 \int_{0}^{x} \frac a {\left({1 - v^2 / c^2}\right)^{\tfrac 3 2} } \mathrm{d}{x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle m_0 \int_{0}^{v} \frac v {\left({1 - v^2 / c^2}\right)^{\tfrac 3 2} } \mathrm{d}{v}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle m_0 \left({- \frac {c^2} 2}\right) \int_{0}^{v} \left({1 - \frac {v^2} {c^2} }\right)^{-\tfrac 3 2} \left({- \frac {2 v \mathrm{d}{v} } {c^2} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[{m_0 c^2 \left({1 - \frac {v^2} {c^2} }\right)^{- \tfrac 1 2} }\right]_0^v\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle m_0 c^2 \left({\frac 1 {\sqrt {1 - \frac {v^2} {c^2} } } - 1}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle c^2 \left({\frac {m_0} {\sqrt {1 - \frac {v^2} {c^2} } } - m_0}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle c^2 \left({m - m_0}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	from Einstein's Mass-Velocity Equation
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle M c^2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Source of Name

This entry was named for Albert Einstein.

Sources

• George F. Simmons: Calculus Gems (1992), Chapter $\text{B}.7$