Uniqueness of photon spheres in electro-vacuum spacetimes

In a recent paper (Cederbaum C and Galloway G J 2015 Commun. Analysis Geom. at press), the authors established the uniqueness of photon spheres in static vacuum asymptotically flat spacetimes by adapting Bunting and Masood-ul-Alam's proof of static vacuum black hole uniqueness. Here, we establish uniqueness of suitably defined sub-extremal photon spheres in static electro-vacuum asymptotically flat spacetimes by adapting the argument of Masood-ul-Alam (1992 Class. Quantum Grav. 9 L53-5). As a consequence of our result, we can rule out the existence of electrostatic configurations involving multiple 'very compact' electrically charged bodies and sub-extremal black holes.