Light-ring pairs from A -discriminantal varieties

Abstract

When geodesic equations are formulated in terms of an effective potential $\mathcal{U}$, circular orbits are characterized by $\mathcal{U}={\ensuremath{\partial}}_{a}\mathcal{U}=0$. In this paper, we consider the case where $\mathcal{U}$ is an algebraic function. Then, the condition for circular orbits defines an $A$-discriminantal variety. A theorem by Rojas and Rusek, suitably interpreted in the context of effective potentials, gives a precise criteria for certain types of spacetimes to contain at most two branches of light rings (null circular orbits), where one is stable and the other one unstable. We identify a few classes of static, spherically symmetric spacetimes for which these two branches occur.