In this paper, we study the trajectories of massive and massless particles in four-dimensional static and spherically symmetric black holes in the de Rham–Gabadadze–Tolley massive gravity theory via phase-plane analysis and point out several novel features. In particular, we show the existence of a static sphere, a finite radial distance outside the black holes in these theories, where a massive particle can be at rest, as seen by an asymptotic zero-angular-momentum observer. Topological arguments show that stable and unstable static spheres, which come in pairs, have opposite charges. In the presence of angular momentum, we first study the behavior of massless particles and find the presence of stable and unstable photon spheres in both neutral and charged black holes. Subsequently, we study the motion of massive test particles around these black holes, and we find one pair of stable and unstable timelike circular orbits (TCOs), such that the stable and unstable TCOs are disconnected in certain regions. Computing the angular velocity ${\mathrm{\Omega }}_{\mathrm{CO}}$ of the TCOs, measured by a static observer at rest, shows the unusual nature of its monotonic increase with the radius of the TCO near the location of a stable photon sphere. This confirms the existence of the Aschenbach effect for spherically symmetric black holes in massive gravity, which was only found to exist in rapidly spinning black holes, with the only other exception being the rare example of gravity coupled to quasitopological electromagnetism.

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• Classical black holes
• General relativity
• General relativity equations & solutions