 Solution

The apple's proper time $\tau$ is the time read by a clock carried with it, with $d{\tau }^2=-d{s}^2/c^2$ 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 givesMultiplying by the inverse coordinate Jacobian matrix yields the exact geodesic equationHere $x^{\mu }$ 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 $u^0=cdt/d\tau$. In the weak-field approximation and at small spatial speed compared with $c$, the leading spatial geodesic equation isIt 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 $2{\Gamma }_{0j}^{\mu }{u}^0{u}^j$ and ${\Gamma }_{ij}^{\mu }{u}^i{u}^j$ 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 $d^{2}r/d{\tau }^2=v^2{\gamma }^2{b}^2/r^3\ne 0$, although ${\Gamma }_{00}^r=0$. The missing angular term ${\Gamma }_{\varphi \varphi }^r(d\varphi /d\tau {)}^2$ supplies that radial coordinate acceleration.