Static spheres around spherically symmetric black hole spacetime

Unique features of particle orbits produce novel signatures of gravitational observable phenomena and are quite useful in testing compact astrophysical objects in general relativity or modified theories of gravity. Here we observe a representative example that a static, spherically symmetric black hole solution with nonlinear electrodynamics admits static points at finite radial distance. Each static point thus produces a static sphere, on which a massive test particle can remain at rest at arbitrary latitudes with respect to an asymptotic static observer. As a result, the well-known static Dyson spheres can be implemented by such orbits. More interestingly, employing a topological argument, we disclose that stable and unstable static spheres (if they exist) always come in pairs in an asymptotically flat spacetime. In contrast to this, the counterpart naked singularity has one more stable static sphere than the unstable one. Our results have potential applications in testing black holes in standard Maxwell and nonlinear electrodynamics, as well as in uncovering the underlying astronomical observation effects in other gravitational theories beyond general relativity.