= Proper acceleration from coordinate velocity and acceleration
{title2=$\alpha^2=\gamma^4\mathbf a^2+\gamma^6(\mathbf v\cdot\mathbf a)^2/c^2$}
In a Cartesian <inertial frame>, let $\mathbf v$ and $\mathbf a=d\mathbf v/dt$ be the coordinate <velocity> and <acceleration> of a massive particle. With $\gamma=(1-v^2/c^2)^{-1/2}$, differentiating its <four-velocity> gives
$$
A^\mu=\left(\frac{\gamma^4\mathbf v\cdot\mathbf a}{c},\ \gamma^2\mathbf a+\frac{\gamma^4(\mathbf v\cdot\mathbf a)\mathbf v}{c^2}\right).
$$
In the $+---$ <metric signature>, its invariant <Minkowski norm> is $A^\mu A_\mu=-\alpha^2$. Expanding that contraction gives the displayed <proper acceleration>. Resolving the coordinate <acceleration> into components parallel and perpendicular to $\mathbf v$ yields the equivalent formula $\alpha^2=\gamma^6a_\parallel^2+\gamma^4a_\perp^2$. In particular, constant coordinate <acceleration> and constant <proper acceleration> are different prescriptions.
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