Uniqueness of electric-magnetic spacetimes with massive particle

Uniqueness of the four-dimensional static, asymptotically flat, Einstein-Maxwell spacetime with both electric and magnetic charges, containing non-extremal massive particle sphere, being an inner boundary in it, has been proved. It is isometric to Reissner-Nordstr\"om spacetime with electric/magnetic charges. In contrast to the previous results concerning the classification of photon spheres, it describes the existence of the entire set of spacetme foliations, a set of massive particle sphere addressed to the various energies of the particles. The conformal positive energy, positive mass theorem and adequate conformal transformations constitute the mail tools in the proof.