Light-ring pairs from $A$-discriminantal varieties

TL;DR

This paper explores the mathematical conditions under which certain spacetimes admit one or two light rings, using algebraic geometry tools to classify static, spherically symmetric solutions.

Contribution

It applies $A$-discriminantal varieties and a theorem by Rojas and Rusek to characterize light ring configurations in static, spherically symmetric spacetimes.

Findings

Spacetimes with non-degenerate horizons lack stable light rings.

Identified classes of spacetimes with exactly two light ring branches.

Provided algebraic criteria for the number and stability of light rings.

Abstract

When geodesic equations are formulated in terms of an effective potential $U$, circular orbits are characterised by $U=\partial_a U=0$. In this paper we consider the case where $U$ 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 and show that the spacetimes with non-degenerate horizons do not have stable light rings.