Past exam of the general relativity and cosmology course of the University of Oxford / 2018 / 1 / d / Solution

Put $F(r)=1-r_s/r$ and $r_s=2GM/c^2$. For radial motion $u^{\theta }=u^{\varphi }=0$. The given Christoffel symbols implyandThe four-velocity normalization of part c isIts bracketed combination is therefore exactly $c^2$, independently of the instantaneous velocity or the thrust history. ConsequentlyThe 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.