= Solution
Put $F(r)=1-r_s/r$ and $r_s=2GM/c^2$. For radial motion $u^\theta=u^\phi=0$. The given <Christoffel symbols> imply
$$
\boxed{a^0=\frac{du^0}{d\tau}
+\frac{2GM}{c^2r^2F}u^0u^r,}
$$
and
$$
\boxed{a^r=\frac{du^r}{d\tau}
+\frac{GM}{c^2r^2}\left[F(u^0)^2-F^{-1}(u^r)^2\right].}
$$
The <four-velocity> normalization of part <1/c> is
$$
-F(u^0)^2+F^{-1}(u^r)^2=-c^2.
$$
Its bracketed combination is therefore exactly $c^2$, independently of the instantaneous <velocity> or the thrust history. Consequently
$$
\boxed{a^r=\frac{du^r}{d\tau}+\frac{GM}{r^2}.}
$$
The derivative here is with respect to <proper time>. This expression does not say that the ship measures a gravitational acceleration $GM/r^2$ during <free fall>; setting $a^r=0$ gives its radial proper-time equation of motion in <Schwarzschild spacetime>.
Back to article page
