# Fundamental photon orbits: black hole shadows and spacetime instabilities

## TL;DR

This paper classifies fundamental photon orbits in black hole spacetimes, analyzing their stability and impact on shadows, revealing new phenomena like stable orbits and cuspy shadows in non-Kerr black holes with Proca hair.

## Contribution

It introduces a classification of fundamental photon orbits in stationary, axi-symmetric spacetimes and explores their stability, extending the understanding of black hole shadows beyond Kerr solutions.

## Key findings

- All FPOs are unstable in Kerr black holes, determining their shadows.
- Stable FPOs can exist in non-Kerr spacetimes, leading to new instabilities.
- Non-Kerr black holes with Proca hair can have cuspy shadows due to stable and unstable FPO interplay.

## Abstract

The standard Black Holes (BHs) in General Relativity, as well as other ultra-compact objects (with or without an event horizon) admit planar circular photon orbits. These light rings (LRs) determine several spacetime properties. For instance, stable LRs trigger instabilities and, in spherical symmetry, (unstable) LRs completely determine BH shadows. In generic stationary, axi-symmetric spacetimes, non-planar bound photon orbits may also exist, regardless of the integrability properties of the photon motion. We suggest a classification of these fundamental photon orbits (FPOs) and, using Poincar\'e maps, determine a criterion for their stability. For the Kerr BH, all FPOs are unstable (similarly to its LRs) and completely determine the Kerr shadow. But in non-Kerr spacetimes, stable FPOs may also exist, even when all LRs are unstable, triggering new instabilities. We illustrate this for the case of Kerr BHs with Proca hair, wherein, moreover, qualitatively novel shadows with a cuspy edge exist, a feature that can be understood from the interplay between stable and unstable FPOs. FPOs are the natural generalisation of LRs beyond spherical symmetry and should generalise the LRs key role in different spacetime properties.