= Solution
{c}
The apple's <proper time> $\tau$ is the time read by a clock carried with it, with $d\tau^2=-ds^2/c^2$ for <metric signature> $-+++$. At any event choose a <local inertial frame> with coordinates $\xi^\alpha$, in which free motion satisfies $d^2\xi^\alpha/d\tau^2=0$ at that event. The <chain rule> gives
$$
0=\frac{\partial\xi^\alpha}{\partial x^\mu}\frac{d^2x^\mu}{d\tau^2}
+\frac{\partial^2\xi^\alpha}{\partial x^\nu\partial x^\lambda}
\frac{dx^\nu}{d\tau}\frac{dx^\lambda}{d\tau}.
$$
Multiplying by the inverse coordinate <Jacobian matrix> yields the exact <geodesic equation>
$$
\boxed{\frac{d^2x^\mu}{d\tau^2}+\Gamma^\mu_{\nu\lambda}u^\nu u^\lambda=0,\qquad
u^\mu=\frac{dx^\mu}{d\tau},\qquad
\Gamma^\mu_{\nu\lambda}=\frac{\partial x^\mu}{\partial\xi^\alpha}
\frac{\partial^2\xi^\alpha}{\partial x^\nu\partial x^\lambda}.}
$$
Here $x^\mu$ are the observer's coordinates, $u^\mu$ is the apple's <four-velocity>, and the <Christoffel symbols> describe the <Levi-Civita connection>. The last expression applies when the $\xi$ coordinates are locally inertial at the event considered; one such chart need not remain inertial throughout a finite region.
Choose $x^0=ct$, so $u^0=c\,dt/d\tau$. In the <weak-field approximation> and at small spatial <speed> compared with $c$, the leading spatial <geodesic equation> is
$$
\boxed{\frac{d^2x^i}{d\tau^2}+c^2\Gamma^i_{00}\left(\frac{dt}{d\tau}\right)^2\simeq0.}
$$
It is exact at the release event when all spatial $u^i$ vanish, and is the intended approximation to the displayed expression in the source. Terms $2\Gamma^\mu_{0j}u^0u^j$ and $\Gamma^\mu_{ij}u^iu^j$ cannot generally be dropped; the time component must also be retained to find changing $dt/d\tau$. For a concrete counterexample to the unrestricted expression, a straight free trajectory $x=v\gamma\tau$, $y=b$ in flat <spacetime> has polar radius $r=\sqrt{b^2+v^2\gamma^2\tau^2}$ and $d^2r/d\tau^2=v^2\gamma^2b^2/r^3\ne0$, although $\Gamma^r_{00}=0$. The missing angular term $\Gamma^r_{\phi\phi}(d\phi/d\tau)^2$ supplies that radial coordinate <acceleration>.
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