The geometry of massive particle surfaces

We propose a generalization of Claudel, Virbhadra, and Ellis photon surfaces to the case of massive charged particles, considering a timelike hypersurface such that any worldline of a particle with mass $m$, electric charge $q$ and fixed total energy $\mathcal{E}$, initially touching it, will remain in this hypersurface forever. This definition does not directly appeal to the equations of motion, but instead make use of partially umbilic nature of the surface geometry. Such an approach should be especially useful in the case of non-integrable equations of motion. It may be applied in the theory of non-thin accretion discs, and also may serve a new tool for some general problems, such as uniqueness theorems, Penrose inequalities and hidden symmetries. The condition for the stability of the worldlines is derived, which reduces to differentiation along the flow of surfaces of a certain energy. We consider a number of examples of electrovacuum and dilaton solutions, find conditions for marginally stable orbits, regions of stable or unstable spherical orbits, stable and unstable photon surfaces, and solutions satisfying the no-force condition.