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

It is proved that the Einstein-matter field equations supplemented by the weak (positive) energy condition yield the dimensionless lower bound T ∞ / M ( r ) ≥ 6 3 π on the orbital-time-to-mass ratio of closed circular motions in curved spacetimes, where { r , T ∞ , M ( r ) } 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.