The Influence of Uniform Magnetic Fields on Strong Field Gravitational Lensing by Kerr Black Holes

TL;DR

This work investigates strong gravitational lensing by magnetized Kerr black holes (MKBH), Kerr BHs embedded in a uniform magnetic field, using Bozza's strong deflection limit (SDL) framework. By deriving the photon sphere and critical impact parameter within the MKBH spacetime, the authors show that the magnetic field $B$ shifts photon orbits and enhances the relativistic image area, producing measurable changes in observables such as $\theta_\infty$, $s$, $r_{mag}$, and the time delays relative to Kerr. The study provides quantitative predictions for M87* and Sgr A*, illustrating that $B$ induces prograde/retrograde–dependent signatures that could help constrain both the spin $a$ and the magnetic field strength $B$ in near-horizon regions, albeit within a theoretical model. The results highlight strong lensing as a promising probe of magnetic environments around supermassive black holes and motivate extensions to off-equatorial paths, plasma effects, and full ray-tracing to connect with EHT/ngEHT observations. Acknowledging the model's limitations, the work lays groundwork for disentangling gravity from electromagnetism in the strong-field regime through lensing.

Abstract

We investigate strong gravitational lensing using magnetized Kerr black holes (MKBHs), which are accurate Kerr-Bertotti-Robinson solutions for Kerr black holes in a uniform magnetic field with additional magnetic field strength $B$ apart from mass $M$ and spin $a$. Unlike previous magnetized spacetimes, the MKBH geometry is Petrov type D, devoid of conical singularities, allowing photons to reach asymptotic infinity and making the concept astrophysically feasible. We use the strong deflection limit formalism to calculate the photon sphere radius, critical impact parameter, deflection angle, and lensing observables including the image position $θ_\infty$, angular separation $s$ and relative magnification $r_{\text{mag}}$, as well as their relationships with the parameters $a$ and $B$. Our results reveal that the relativistic image's photon sphere and angular size increase with $B$, whereas lensing observables deviate significantly from the Kerr scenario. For M87*, with $a=0.9$, the angular position of relativistic images increases from $10.8~μ$as (Kerr) to $12.02~μ$as, and the time delay between the first two images increases from $158.5$ h to $176$ h at $B=0.4$. Similarly, for Sgr A*, the image position increases from $14.4~μ$as to $16~μ$as, with time delays enhanced by approximately $0.7$ minutes. The relative magnification $r_{\text{mag}}$ grows with $B$ and deviates by $0.53$ from Kerr black holes at $B=0.4$. Our findings highlight strong gravitational lensing as a powerful tool to probe the presence of magnetic fields around astrophysical black holes, and in particular, we demonstrate that the MKBH spacetime enables constraints on the parameters $a$ and $B$.

Figures (7)

• Figure 1: Parameter space for the MKBH spacetime, plotted with the spin parameter $a$ against the magnetic field strength $B$). The solid black curve represents an extremal black hole, which separates the region containing a black hole with two horizons (light blue area below the line) from the region containing a naked singularity (area above the line).
• Figure 2: The radial effective potential $V_{\textrm{eff}}$ for photons is plotted against the radial distance $x$ for MKBHs (Upper Panel) and Kerr black holes (Lower Panel) with different values of the impact parameter $u$ and fixed magnetic field $B=0.1$. We keep spin $a=0.9$ (Left Panel) and $a=-0.9$ (Right Panel). The black-curve is for the critical impact parameter $u = u_\text{ps}$, or for an unstable photon orbit with the photon sphere radius $x = x_\text{ps}$. Photons with $u > u_\text{ps}$ (above the black curve) are deflected and scattered to infinity, creating lensed images. Photons with $u < u_\text{ps}$ (below the black curve) fall into the black hole horizon.
• Figure 3: Behaviour of the photon orbit parameters with respect to the spin parameter $a$ for different values of the magnetic field strength $B$. (Left) The radius of the photon sphere $x_\text{ps}$ decreases (increases) with increasing spin $\mid a \mid$ for prograde (retrograde) orbits, but increases monotonically with the magnetic field strength $B$. (Right) The critical impact parameter $u_\text{ps}$ shows a similar trend with spin $a$, but it decreases with $B$ for retrograde orbits and prograde orbits with smaller values of $a$, while it increases with $B$ for prograde orbits when the spin parameter is larger.
• Figure 4: The deflection angle $\alpha_D(u)$ of light in the SDL, plotted as a function of the impact parameter $u$ for different values of the magnetic field strength $B$. The left panel is for a spin parameter $a = 0.9$ (prograde) and the right panel for $a = -0.9$ (retrograde). The deflection angle diverges as $u \to u_\text{ps}$ (marked by dots on the horizontal axis). For a specified value of $u$, the deflection angle is smaller (larger) for MKBHs (non-zero $B$) than for a Kerr black hole ($B=0$) for retrograde (prograde) orbits. Furthermore, $u_\text{ps}$ increases with $B$, meaning the divergence happens at a larger impact parameter for prograde orbits compared to Kerr black holes, and $u_\text{ps}$ decreases with $B$ for retrograde orbits.
• Figure 5: A schematic diagram depicting the gravitational lensing. A black hole acts as the lens ($L$), bending the light rays from a distant source ($S$). An observer ($O$) sees multiple, highly magnified, and distorted relativistic images of the source. The angular positions of the source and images relative to the optical axis are given by $\beta$ and $\theta_n$, respectively. The distances between the observer, lens, and source are, respectively, denoted by $D_{OL}$ and $D_{LS}$, and $D_{OS} = D_{OL} + D_{LS}$.