= Solution
Use <metric signature> $(-,+,+,+)$ and $x^0=ct$. At an event in the ship's instantaneous rest <local inertial frame>, $u^\alpha=(c,0,0,0)$. Since its squared norm is a scalar,
$$
\boxed{u^\alpha u_\alpha=-c^2}
$$
in every coordinate system. Equivalently it follows directly from $ds^2=-c^2d\tau^2$ and $u^\alpha=dx^\alpha/d\tau$.
<Metric compatibility> lets us differentiate this identity covariantly:
$$
0=\frac D{d\tau}(g_{\alpha\beta}u^\alpha u^\beta)
=2u_\alpha a^\alpha,
\qquad\boxed{u_\alpha a^\alpha=0.}
$$
In the instantaneous rest <local inertial frame>, the last equation gives $a^0=0$. The spatial components are the acceleration recorded by an onboard accelerometer, $\mathbf a_{\rm rest}$, whose magnitude is $g$. Therefore
$$
\boxed{a^\alpha a_\alpha=|\mathbf a_{\rm rest}|^2=g^2.}
$$
This <proper acceleration> is zero in ideal <free fall>, even when the coordinate position has a nonzero second derivative. A coordinate acceleration and an accelerometer reading are different quantities.
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