Fundamental photon orbits: Black hole shadows and spacetime instabilities

The standard black holes (BHs) in general relativity, as well as other ultracompact 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, axisymmetric spacetimes, nonplanar 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 Poincaré maps, determine a criterion for their stability. For the Kerr BH, all FPOs are unstable (similar 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 generalization of LRs beyond spherical symmetry and should generalize the LRs key role in different spacetime properties.

Figure 1

Illustration of some FPOs in the $(r,\theta )$ plane and their classification. The grey areas represent forbidden regions with $V>0$. The left and right panels show a typical unstable LR and a stable planar orbit, respectively.

Figure 2

Kerr(-like) FPOs and shadow, illustrated for a Kerr BH with dimensionless spin $j\simeq 0.820$, ${\eta }_{-}=-6.70$, ${\eta }_{+}=3.17$. Left panel: $r_{\text{Peri}}$ and $\mathrm{\Delta }\theta$ for FPOs vs $\eta$. Lines with $\eta =\text{const}$ take the values of the LRs or three selected FPOs, $\eta =-5.10$, 0, 2.90. Middle panel: Spatial trajectories of these three FPOs and two LRs, in Cartesian coordinates defined from BL coordinates. Right panel: BH shadow, in the same observation conditions as Fig 3. Almost vertical (solid) lines have $\eta =\text{const}$ and horizontal (dotted) lines have a fixed Carter’s constant $Q$, both with the values of the three selected FPOs. Observe how the FPOs ($\eta$, $Q$) values correspond to points at the edge of the shadow. The same colors are used in all panels for the same FPOs.

Figure 3

Left panel: Lensing of the hairy BH with a cuspy shadow, obtained with the same setup as in Ref. [41]. Right panel: The cuspy shadow in the same observation conditions as the ones for the Kerr BH [which has the same $(M,J)$] in Fig. 2. Almost vertical lines have constant $\eta$ and in this case there is no analogue of Carter’s constant. The small (pink) eye lashes correspond to a particular lensing pattern connecting to the cusp, which can be observed in the inset.

Figure 4

Non-Kerr(-like) FPOs, illustrated for the hairy BH described in the text. Left panel: $r_{\text{Peri}}$ and $\mathrm{\Delta }\theta$ for FPOs vs $\eta$. We selected ten FPOs (A1–A4,B1–B3,C1–C3), including the two LRs. The line $\eta \simeq -1.71$ takes the value at which the cusp in the shadow occurs (see Fig. 3). Middle panel: Spatial trajectories of these ten FPOs, in Cartesian coordinates defined from the spheroidal coordinates in Ref. [44]. The A4 (blue) and B3 orbits (yellow), at the intersection between stable and unstable branches are repeated to convey a sense of scale. Right panel: One unstable [stable] FPO of the group A (top) [B (middle)] and a neighboring perturbed orbit which diverges from [oscillates around] the FPO, together with the Poincaré map (on $\theta =\pi /2$) of B2, showing rotation about the fixed point $(r,\dot{r})=(\tilde{r},0)$.