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

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 $10.8μ$as (Kerr) to $12.02μ$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 $14.4μ$as to $16μ$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 $B$.

💡 Research Summary

The paper investigates strong gravitational lensing by magnetized Kerr black holes (MKBHs), which are exact Kerr‑Bertotti‑Robinson solutions describing a rotating black hole immersed in a uniform magnetic field of strength B. Unlike the traditional Kerr‑Melvin spacetimes, the MKBH geometry is Petrov type D, free of conical singularities, and asymptotically approaches a uniform magnetic background rather than flat space. This makes it possible for photons to travel to infinity, rendering the model astrophysically realistic.

The authors begin by presenting the MKBH metric in Boyer‑Lindquist coordinates, introducing the functions Σ, P, Q, and Δ that depend on the mass m, spin a, and magnetic field B. The event‑horizon locations are obtained from Δ = 0, and the extremal condition shows that a non‑zero B reduces the maximal allowed spin for a given mass.

To study lensing, the paper adopts the strong deflection limit (SDL) formalism originally developed by Bozza. Because null geodesics are conformally invariant, the overall conformal factor Ω⁻² can be ignored, and the analysis proceeds with the effective metric \tilde g_{μν}. The motion is restricted to the equatorial plane (θ = π/2) where the metric simplifies considerably: Σ = x², P = 1, Q = (1 + B²x²)Δ, and Δ becomes a quadratic function of the dimensionless radius x = r/m.

Conserved quantities (energy E and angular momentum L) lead to an impact parameter u = L/E. Using the null condition, the radial equation is cast into an effective potential form. The photon‑sphere radius r_ps is found by solving dV_eff/dr = 0; numerical results show that r_ps grows with B for prograde orbits and exhibits a milder increase (or even a slight decrease) for retrograde orbits, reflecting the interplay between spin‑induced frame dragging and magnetic‑field‑induced modifications of the g_{tφ} term.

The critical impact parameter u_c is then evaluated at r_ps, yielding larger values as B increases. This directly translates into a larger angular position of the relativistic images, θ_∞ = u_c/D_{OL}, where D_{OL} is the observer‑lens distance. The SDL coefficients \bar a and \bar b, which control the logarithmic divergence of the deflection angle α(θ) = −\bar a ln