Photon region boundary for stationary axisymmetric black holes

Abstract

The black hole shadow is fundamentally connected to the structure of light rings and the photon region in the background geometry. We investigate the photon region boundary in a generic asymptotically flat, stationary, axisymmetric black hole spacetime that admits spherical photon orbits (SPOs). Explicit expressions possessing real solutions are provided for the photon region's boundary purely in terms of the background metric functions, independent of the photon's parameters like energy or angular momentum, which are applicable to both separable and non-separable spacetimes. We further analyze its common features, including overlap with the ergoregion and rotation sense of SPOs. Additionally, light rings are identified at the extrema of the photon region boundary curves in the $(r,θ)$ plane. Our approach is validated against a few exact black hole solutions. Implications are discussed.

Figures (9)

• Figure 1: 3D illustrations showing the photon region in orange, ergoregion in cyan, and the event horizon in black for Kerr black holes.
• Figure 2: The red and blue dashed lines show the curves $r_i(\theta)$ and $r_o(\theta)$ of the inner and outer boundaries. The yellow line represents the curve $r_c(\theta)$. The black dashed lines show the stationary limit surface $r_g(\theta)$.
• Figure 3: Intersection points $\theta_{in}$ for black hole mass $m=1,1.5$ and $2$ and the spin parameter $a<m$
• Figure 4: 3D illustrations showing the photon region in orange, ergoregion in cyan, and the event horizon in black for Kerr-Newman black holes.
• Figure 5: The red and blue dashed lines show the curves $r_i(\theta)$ and $r_o(\theta)$ of the inner and outer boundaries. The yellow line represents the curve $r_c(\theta)$. The black dashed lines show the stationary limit surface $r_g(\theta)$.