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± = Φ0 0 ± lvert Ψ0 rvert, where Φ0 0 is a Ricci scalar and Ψ0 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 Φ0 0 > lvert Ψ0 rvert there. Assuming the Einstein field equations with generic stress--energy tensor Ta b, we recast this as 4 πTa b pa pb > lvert Ψ0 rvert, where pa 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.
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BibTeX
@misc{shipley2026stability,
title = {{Stability conditions for light rings in circular stationary axisymmetric spacetimes}},
author = {Jake O. Shipley},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14747},
eprint = {2610.14747},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14747},
}
RIS
TY - UNPB
TI - Stability conditions for light rings in circular stationary axisymmetric spacetimes
AU - Jake O. Shipley
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14747
UR - https://arxiv.org/abs/2610.14747
AB - 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_± = Φ_0 0 ± 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.
ER -