Static spheres in black hole spacetimes: pairing, energy conditions, and an upper bound on the innermost radius

Static spheres – spherical surfaces, located outside the event horizon, on which massive test particles can remain at rest with zero angular momentum relative to asymptotic static observers – have recently attracted some attention. In this work, we investigate their existence, relation to the energy conditions, and radial bounds in general static, spherically symmetric, asymptotically flat black hole spacetimes within general relativity. By analyzing the behavior of a radial function constructed from the mass and radial pressure functions, we prove that, under the weak energy condition (WEC), a static sphere necessarily requires negative radial pressure (tension). Furthermore, for non-extremal black holes satisfying the WEC, we show that any exterior static spheres, if present and non-degenerate, occur in pairs, each consisting of an inner unstable sphere and an outer stable one. Assuming the WEC holds throughout the exterior, the inner static sphere is characterized by a violation of the strong energy condition (SEC), with ρ +p+2pT<0 ρ + p + 2 p T < 0 , where ρ ρ , p , and pT p T denote the energy density, radial pressure, and tangential pressure, respectively; the SEC inequality is restored ( ρ +p+2pT> 0 ρ + p + 2 p T > 0 ) at the outer sphere; a degenerate static sphere satisfies ρ +p+2pT=0 ρ + p + 2 p T = 0 . In addition, focusing on the innermost static sphere and assuming that the WEC holds while the SEC is uniformly violated between the event horizon and this sphere, we derive a rigorous upper bound on its radius. These results establish a direct, analytic link between the energy conditions and the existence of static spheres, and provide a quantitative constraint on the matter environment of black holes possessing such orbits.