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Critical Behavior of Photon Rings in Kerr-Bertotti-Robinson Spacetime

Xi Wan, Zhenyu Zhang, Fang-Stars Wei, Yehui Hou, Bin Chen

Abstract

In this work, we investigate the critical behavior of photon rings in the Kerr-Bertotti-Robinson spacetime, describing a rotating black hole immersed in a background magnetic field. We analyze the radial and angular motions of photons under the small magnetic field approximation. Focusing on unstable spherical orbits, we determine three key parameters, $γ$, $δ$, and $τ$, which characterize radial compression, azimuthal advancement, and time delay. We then examine how these parameters depend on the black hole spin, magnetic field strength, and observer inclination for both on-axis and off-axis observers, and we further analyze the properties of higher-order images through near-critical lens equations. The results show that the magnetic field modifies the geodesic structure, and leads to observable changes in the fine structure of photon rings, providing a useful framework for probing magnetized black hole environments.

Critical Behavior of Photon Rings in Kerr-Bertotti-Robinson Spacetime

Abstract

In this work, we investigate the critical behavior of photon rings in the Kerr-Bertotti-Robinson spacetime, describing a rotating black hole immersed in a background magnetic field. We analyze the radial and angular motions of photons under the small magnetic field approximation. Focusing on unstable spherical orbits, we determine three key parameters, , , and , which characterize radial compression, azimuthal advancement, and time delay. We then examine how these parameters depend on the black hole spin, magnetic field strength, and observer inclination for both on-axis and off-axis observers, and we further analyze the properties of higher-order images through near-critical lens equations. The results show that the magnetic field modifies the geodesic structure, and leads to observable changes in the fine structure of photon rings, providing a useful framework for probing magnetized black hole environments.

Paper Structure

This paper contains 16 sections, 129 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The KBR spacetime
  3. Detailed analysis of the geodesic integrals
  4. Angular integrals
  5. Radial roots
  6. Critical rays and key parameters
  7. Key parameters results
  8. Off-axis observer
  9. On-axis observer
  10. Near-critical equations
  11. Polar-axis Observer
  12. Off-axis Observer
  13. Summary
  14. Expansion for critical parameters
  15. Radial integrals for near-critical rays
  16. ...and 1 more sections

Figures (5)

  • Figure 1: Critical parameters $\gamma$, $\delta$ and $\tau$ evaluated along the critical curve of $\theta_o=80^{\circ}$, parameterized by the screen polar angle $\varphi$. Top row: $a=0.5$. Bottom row: $a=0.9$.
  • Figure 2: Critical parameters evaluated along the critical curve of $a=0.9$ and $\theta_o=20^{\circ}$, parameterized by the screen polar angle $\varphi$.
  • Figure 3: For an on-axis observer, the three parameters vary with the magnetic field strength for fixed black hole spin. From left to right, the panels correspond to $a=0$, $a=0.5$, and $a=0.9$, respectively. The green, blue, and red curves correspond to $\delta_0$, $\tau_0/3\sqrt{3}$, and $\gamma_0$, respectively.
  • Figure 4: The deviations of the three parameters relative to the Kerr case. The blue and red curves correspond to the non-rotating case $a=0$ and the high-spin case $a=0.99$, respectively.
  • Figure 5: For an on-axis observer, the three parameters vary with the black hole spin for fixed magnetic field strength. From left to right, the panels correspond to $B=0$, $B=0.08$, and $B=0.16$, respectively. The green, blue, and red curves correspond to $\delta_0$, $\tau_0/3\sqrt{3}$, and $\gamma_0$, respectively.