Distorted static photon surfaces in perturbed Reissner-Nordström spacetimes

Abstract

The photon surface is defined as a timelike surface $S$ such that any photon emitted in arbitrary tangential direction to $S$ from an arbitrary point on $S$ continues to propagate on $S$. In this paper, we examine whether a static photon surface can be present in distorted electrovacuum spacetimes with perturbative approach, by constructing analytic solutions to the equations for static perturbations of a Reissner-Nordström spacetime that are regular outside the background photon surface. For each of the $\ell\ge 2$ modes, there are two physical solutions to the perturbative equations that correspond to adding the multipole moments of mass and electric charge, respectively. By adjusting the ratio of the amplitudes of the two solutions appropriately, it is possible to realize a distorted photon surface. In the case of $\ell=1$, although there is only one physical solution to the perturbative equations, there is also a degree of freedom to shift the coordinate position of the photon surface, and this enables the formation of a photon surface. Such a photon surface has a spherically symmetric spatial section, but it is also distorted in the three-dimensional sense. Our results show that distorted static photon surfaces can be formed at least at the level of the first-order perturbations, and imply that the uniqueness of the photon surface may not hold in electrovacuum spacetimes.