Trajectories of light beams in a Kerr metric: the influence of the rotation of an observer on the shadow of a black hole

TL;DR

This work analyzes how light beams propagate in the Kerr spacetime and how a rotating observer modifies the black hole shadow. It develops a Hamiltonian framework for null geodesics, exploits Carter’s constant to achieve separability into radial and angular potentials $R(r)$ and $\Theta(\theta)$ with parameters $\lambda=L/E$ and $\eta=Q/E$, and constructs a detailed bifurcation diagram to classify trajectory types. It then derives boundary relations for the shadow seen by stationary observers with angular velocity $\Omega$, providing explicit formulas for boundary angles and a stereographic projection visualization, including simplifications for Carter observers. The results clarify how shadow shape and image topology depend on observer position and rotation, with implications for interpreting high-resolution black-hole images like M87* and Sgr A*.

Abstract

This paper investigates the trajectories of light beams in a Kerr metric, which describes the gravitational field in the neighborhood of a rotating black hole. After reduction by cyclic coordinates, this problem reduces to analysis of a Hamiltonian system with two degrees of freedom. A bifurcation diagram is constructed and a classification is made of the types of trajectories of the system according to the values of first integrals. Relations describing the boundary of the shadow of the black hole are obtained for a stationary observer who rotates with an arbitrary angular velocity about the axis of rotation of the black hole.

Figures (10)

• Figure 1: Bifurcation diagram for the fixed $a=0.97$. Gray denotes the values of the integrals for which inequalities \ref{['eq_OVD']} are satisfied.
• Figure 2: For fixed $\lambda=-0.6$ and $a=0.97$ (a) the function $\Theta(\theta)$, (b) the function $R(r)$, and (с)-(d) the phase portraits, versus $\eta$. The colored curves in these figures are plotted for the values of the first integrals lying on the bifurcation curves shown in the same colors in Fig. \ref{['fig2']}.
• Figure 3: Typical circular trajectories (red) and separatrices (light blue) for the fixed parameter $a=0.97$ and the initial conditions $r(0)=10$ and $\varphi(0)=0$, which correspond to point $N$.
• Figure 4: (a) Schematic diagram of the horizon $\mathcal{S}_h$ and (b) schematic diagram of the sphere $\mathcal{S}_N$ and the plane $\mathcal{P}$ relative to the spatial components of the vectors $\boldsymbol{e}_{(r)}$, $\boldsymbol{e}_{(\theta)}$ and $\boldsymbol{e}_{(\varphi)}$.
• Figure 5: The position of the surface $\Theta_*(r_c)=0$ relative to the surfaces $r_c=r_1(a)$ and $r_c=r_2(a)$.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Proposition 1
• proof
• Remark 4