Stability conditions for light rings in circular stationary axisymmetric spacetimes

Abstract

We consider the stability of light rings in four-dimensional circular stationary axisymmetric spacetimes. Using a Hamiltonian formalism for null geodesics and the Killing symmetries of the spacetime, we show that the transverse Hessian of the geodesic potential, which controls the light ring's stability, reduces there to a pure curvature expression. Its eigenvalues are $μ_{\pm} = Φ_{0 0} \pm \lvert Ψ_{0} \rvert$, where $Φ_{0 0}$ is a Ricci scalar and $Ψ_{0}$ a Weyl scalar of a complex null tetrad adapted to the light ring. Through the Sachs optical equations, these quantities are responsible for focussing and shear of a null geodesic congruence. Stability requires both eigenvalues to be positive, so a light ring is stable if and only if $Φ_{0 0} > \lvert Ψ_{0} \rvert$ there. Assuming the Einstein field equations with generic stress--energy tensor $T_{a b}$, we recast this as $4 πT_{a b} p^{a} p^{b} > \lvert Ψ_{0} \rvert$, where $p^{a}$ is the light ring's tangent vector: for stability, the focussing by matter must dominate the local tidal shear. The trace of the Hessian alone provides an inexpensive test of stability, requiring no knowledge of the Weyl curvature, since a light ring at which the strict null energy condition is violated cannot be stable. Applying the test to specific matter models, we recover known results for vacuum and electrovacuum, with and without a cosmological constant. We also demonstrate that stable light rings are forbidden for a minimally coupled real scalar field with any self-interaction potential, since such a field must be invariant under both isometries. We show that a complex scalar field escapes this obstruction whenever it carries harmonic dependence on the Killing coordinates.