Lower bound on the time-to-mass ratio of circular trajectories in spherically symmetric curved spacetimes

Abstract

It is proved that the Einstein-matter field equations supplemented by the weak (positive) energy condition yield the dimensionless lower bound $T_{\infty }/M\left(r\right)\ge 6\sqrt{3}\pi$ on the orbital-time-to-mass ratio of closed circular motions in curved spacetimes, where $\{r,T_{\infty },M(r\left)\right\}$ are respectively the radius of the circular trajectory, the orbital period as measured by asymptotic observers, and the gravitational mass contained within the sphere of radius r. We explicitly prove that the bound is valid for circular motions of test particles in black-hole spacetimes as well as in spatially regular horizonless spacetimes.