Black hole shadows and invariant phase space structures

Utilizing concepts from dynamical systems theory, we demonstrate how the existence of light rings, or fixed points, in a spacetime will give rise to families of periodic orbits and invariant manifolds in phase space. It is shown that these structures can define the shape of the black hole shadow as well as a number of salient features of the spacetime lensing. We illustrate this through the analysis of lensing by a hairy black hole.

Figure 1

Stable (green) and unstable (red) manifolds of a periodic orbit (black).

Figure 2

Lyapunov families $\mathfrak{L}\mathfrak{O}$ (solid) of light rings $\mathfrak{L}$ (dots) and their envelopes (dashed).

Figure 3

Projection of Lyapunov orbits with $\eta =-2.6$ from ${\mathfrak{L}\mathfrak{O}}_{2.3}$ (bright green) and ${\mathfrak{L}\mathfrak{O}}_1$ (bright red) and their unstable manifolds (light green, light red).

Figure 4

Poincaré section of unstable manifolds of Lyapunov orbits with different $\eta$ in the $r=r_{\text{observer}}$ plane. The interior of each intersections is highlighted in the same color.

Figure 5

Unstable manifold of Lyapunov orbit in ${\mathfrak{L}\mathfrak{O}}_{2,3}$ for $\eta =-2.6$ in $(X,\theta ,p_{\theta })$ space.

Figure 6

Intersections of the unstable manifolds of ${\mathfrak{L}}_1$, ${\mathfrak{L}}_2$, and ${\mathfrak{L}}_3$ (white squares) as well as of their Lyapunov families ${\mathfrak{L}\mathfrak{O}}_1$ and ${\mathfrak{L}\mathfrak{O}}_{2,3}$ with the image plane (white dots). Lines of constant $\eta$ corresponding to ${\eta }_1$, ${\eta }_2$, and ${\eta }_3$ are also shown.

Figure 7

Time delay (logarithmic scale) and unstable manifold intersections with the image plane. Manifolds of ${\mathfrak{L}}_1$ (${\mathfrak{L}}_2$, ${\mathfrak{L}}_3$) are marked by a red square (green diamonds). Manifolds of members of the Lyapunov families ${\mathfrak{L}\mathfrak{O}}_1$ (${\mathfrak{L}\mathfrak{O}}_{2,3}$) are marked by red circles (green triangles).