Geodesics and Shadows in the Kerr-Bertotti-Robinson Black Hole Spacetime

TL;DR

This work analyzes geodesics and shadows in the Kerr–Bertotti–Robinson spacetime, a Kerr black hole embedded in a uniform magnetic field. By exploiting Hamilton–Jacobi separability for null geodesics, it derives radial and angular potentials and perturbative corrections to the photon region and ISCO in the weak-field limit, while timelike geodesics remain non-separable in general. The authors compute the black hole shadow using both analytic approximations and numerical ray tracing, introducing a deformation parameter to quantify deviations from Kerr as functions of the magnetic field strength $B$, observer distance $r_o$, and inclination $\theta_o$, and they explain these deviations via a near-zone Kerr-like and far-zone AdS-like asymptotic structure governed by the scale $R_B=1/B$. The results show that stronger fields, larger inclinations, and greater distances enhance shadow distortions, with a clear physical picture based on the asymptotic behavior of the spacetime. The study sets the stage for future explorations of accretion-disk imaging and polarization in KBR spacetimes and invites comparisons with Kerr–Melvin solutions.

Abstract

In this work, we investigate geodesics and black hole shadows in the Kerr-Bertotti-Robinson spacetime. We show that the equations of motion for null geodesics are separable and admit analytical treatment, whereas timelike geodesics are generally non-separable. Approximate analytical expressions for the photon sphere and the innermost stable circular orbit are derived via perturbative expansions in the magnetic field strength. We further explore the black hole shadow using both numerical and analytical methods, examining the effects of the magnetic field, the observer's inclination angle and radial position. Deviations from the standard Kerr shadow are quantified, and a physical interpretation is provided by introducing asymptotic regimes defined relative to the magnetic field strength.

Figures (3)

• Figure 1: Shadows of KBR black holes for $B=0$ (Left) and $B=0.01$ (Middle), with fixed parameters $a=0.94\,,r_o=300\,,\theta_o=90^\circ\,$. Right: Critical curves obtained via numerical ray tracing (blue curve) and the analytical method (red dashed curve). The parameters are fixed to $a=0.94\,,B=0.01\,,r_o=300\,,\theta_o=90^\circ\,$.
• Figure 2: Variations of the critical curve under different black hole parameters and observer locations.
• Figure 3: Variation of the dimensionless parameter $\sigma=\bar{\rho}/\bar{\rho}_0-1$ as a function of the magnetic field strength $B$ (left column), observer distance $r_o$ (middle column) and inclination angle $\theta_o$ (right column).