Proper acceleration from coordinate velocity and acceleration (${\alpha }^2={\gamma }^4{\mathbf{a}}^2+{\gamma }^6(\mathbf{v}\cdot \mathbf{a}{)}^2/c^2$)

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 givesIn 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 ${\alpha }^2={\gamma }^6{a}_{\parallel }^2+{\gamma }^4{a}_{\perp }^2$. In particular, constant coordinate acceleration and constant proper acceleration are different prescriptions.