# 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 $M$ as given by the equation:

- $E = M c^2$

where $c$ is the speed of light.

## Proof

From Einstein's Law of Motion, we have:

- $\mathbf F = \dfrac {m_0 \mathbf a} {\paren {1 - \dfrac {v^2} {c^2} }^{\tfrac 3 2} }$

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.

Without loss of generality, assume 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:

- $a = \dfrac {\d v} {\d t} = \dfrac {\d v} {\d x} \dfrac {\d x} {\d t} = v \dfrac {\d v} {\d x}$

Then from the definition of energy:

- $\displaystyle E = \int_0^x F \rd x$

which leads us to:

\(\displaystyle E\) | \(=\) | \(\displaystyle m_0 \int_0^x \frac a {\paren {1 - v^2 / c^2}^{\tfrac 3 2} } \rd x\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle m_0 \int_0^v \frac v {\paren {1 - v^2 / c^2}^{\tfrac 3 2} } \rd v\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle m_0 \paren {- \frac {c^2} 2} \int_0^v \paren {1 - \frac {v^2} {c^2} }^{-\tfrac 3 2} \paren {- \frac {2 v \rd v} {c^2} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \intlimits {m_0 c^2 \paren {1 - \frac {v^2} {c^2} }^{- \tfrac 1 2} } 0 v\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle m_0 c^2 \paren {\frac 1 {\sqrt {1 - \frac {v^2} {c^2} } } - 1}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle c^2 \paren {\frac {m_0} {\sqrt {1 - \frac {v^2} {c^2} } } - m_0}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle c^2 \paren {m - m_0}\) | Einstein's Mass-Velocity Equation | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle M c^2\) |

$\blacksquare$

## Also known as

This is usually known as **Einstein's equation**, but there are a number of such equations that Einstein deduced.

However, this is the most famous one, and has caught the imagination of the general public.

Hence, if you refer to **Einstein's equation** at a party, for example, everyone will know which one you mean, and it's this one.

## Source of Name

This entry was named for Albert Einstein.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): Miscellaneous Problems for Chapter $2$: Problem $32 \ \text{(b)}$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.7$: A Simple Approach to $E = M c^2$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Einstein's equation**