Stable photon orbits in stationary axisymmetric spacetimes with an electromagnetic field and a cosmological constant

Stable light rings, which are associated with spacetime instabilities, are known to exist in four-dimensional stationary axisymmetric spacetimes that solve the Einstein-Maxwell equations (so-called electrovacuum solutions, with Faraday tensor $F_{\mu \nu }\mathrm{\ne }0$); however, they are not permitted in pure vacuum ($F_{\mu \nu }=0$). In this work, we extend this result to spacetimes with a non-zero cosmological constant $\mathrm{\Lambda }$. In particular, we demonstrate that stable light rings are permitted in $\mathrm{\Lambda }$-electrovacuum ($F_{\mu \nu }\mathrm{\ne }0$, $\mathrm{\Lambda }\mathrm{\ne }0$), but ruled out in $\mathrm{\Lambda }$-vacuum ($F_{\mu \nu }=0$, $\mathrm{\Lambda }\mathrm{\ne }0$).