Einstein's Mass-Energy Equation

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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