Einstein's Law of Motion

Physical Law

The force and acceleration on a body of constant rest mass are related by the eqn:

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

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.

Proof

Into Newton's Second Law of Motion:

$\displaystyle \mathbf F = \frac{\mathrm{d}}{\mathrm{d}{t}} \left({m \mathbf v}\right)$

we substitute Einstein's Mass-Velocity Equation:

$\displaystyle m = \frac {m_0}{\sqrt{1 - \dfrac {v^2}{c^2}}}$

to obtain:

$\displaystyle \mathbf F = \frac{\mathrm{d}}{\mathrm{d}{t}} \left({\frac {m_0 \mathbf v}{\sqrt{1 - \dfrac {v^2}{c^2}}}}\right)$

Then we perform the differentiation WRT time:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \frac{\mathrm d}{\mathrm d t} \left({\frac {\mathbf v}{\sqrt{1 - \dfrac {v^2}{c^2} } } }\right)\)	\(=\)	\(\displaystyle \frac{\mathrm d}{\mathrm d v} \left({\frac {\mathbf v}{\sqrt{1 - \dfrac {v^2}{c^2} } } }\right) \frac{\mathrm d v}{\mathrm d t}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Chain Rule
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \mathbf a \left({\frac {\sqrt{1 - \dfrac {v^2}{c^2} } - \dfrac v 2 \dfrac 1 {\sqrt{1 - \dfrac {v^2}{c^2} } } \dfrac{-2 v}{c^2} } {1 - \dfrac {v^2}{c^2} } }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Chain Rule, Quotient Rule, etc.
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \mathbf a \left({\frac {c^2 \left({1 - \dfrac {v^2}{c^2} }\right) + v^2} {c^2 \left({1 - \dfrac {v^2}{c^2} }\right)^{3/2} } }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \mathbf a \left({\frac 1 {\left({1 - \dfrac {v^2}{c^2} }\right)^{3/2} } }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Thus we arrive at the form:

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

$\blacksquare$

Comment

Thus we see that at low velocities (i.e. much less than that of light), the well-known eqn $\mathbf F = m \mathbf a$ holds to a high degree of accuracy.

Source of Name

This entry was named for Albert Einstein.

Sources

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