A Simple Pendulum in Schwarzschild Spacetime

Abstract

We determine the small-oscillation period of a simple pendulum in Schwarzschild spacetime. After transients decay, the displaced rope coincides with a geodesic of the induced spatial metric. This determines the radial lift of the bob to quadratic order in the angular amplitude and leads to the period measured by an observer at the bob's equilibrium position, $r=r_2$, with the fulcrum at $r_1$, \[ T_{(2)}=4π\frac{r_2^2}{c r_s}\sqrt{N_2(N_1-N_2)}, \] where $N_i=N(r_i)$ and $N(r)=\sqrt{1-r_s/r}$ is the Schwarzschild lapse. The result is therefore expressed in terms of the same function that determines clock rates and gravitational redshifts. In the Newtonian limit, we reproduce the classical result $T=2π\sqrt{L_0/g}$. We interpret the period close to the horizon in terms of an effective pendulum length. We compare our treatment with the coordinate-length constraint adopted in an earlier study and argue that the two problems are inequivalent away from the classical, weak-field limit.

15 pages, 3 figures. Accepted for publication in General Relativity and Gravitation