Work needed to lift a mass in Schwarzschild spacetime

Abstract We determine the work needed to lift a mass from $r_2$ to $r_1$ in Schwarzschild spacetime in three ways. First we use a winch and a light rope. Then we use a photon collision. Finally we let the mass decay into photons and reconstruct it at the higher radius. All three methods give the same local work at $r_1$, $W_1=mc^2[1-N(r_2)/N(r_1)]$, where $N(r)=\sqrt{1-r_s/r}$. Thus, independently of $r_1$, the work tends to $mc^2$ as $r_2$ approaches the horizon. The rope example also gives a useful general-relativistic version of a simple machine: a rope in equilibrium need not have equal local forces at its two ends. The redshift factor plays the role of the lever arm. The examples are intended as teaching material for undergraduate relativity courses.