Lensing, Shadow and Photon Rings in a Magnetized Black Hole Spacetime

TL;DR

This paper addresses how a uniform external magnetic field alters lensing, shadows, and photon rings in the Schwarzschild–Melvin (Ernst) spacetime. It leverages analytic access in two integrable sectors—the equatorial plane and meridional planes—together with the strong-deflection formalism to derive explicit expressions for shadow radii and photon rings, and introduces the gap parameter $\Delta_2$ to distinguish Ernst from Schwarzschild. The main results include a Schwarzschild-like vertical shadow radius $\sin^2\alpha=\frac{27 M^2 (r_O-2M)}{r_O^3}$, a subcritical-field horizontal radius $\beta$ in the equatorial plane with exact $\Lambda$-dependent formulas, and analytic photon-ring radii $b_n$ for a polar observer alongside the gap parameter $\Delta_2$ showing potential observational discrimination for $B> B_o$ (with $B_o\approx0.160/M$). These findings offer observational pathways to constrain near-horizon magnetic fields in black-hole environments using high-resolution imaging of shadows and photon rings.

Abstract

We analyze gravitational lensing, in particular the shadow and photon rings, in the Ernst spacetime, also known as the Schwarzschild-Melvin spacetime, which describes a Schwarzschild black hole immersed in a homogenous magnetic field. Although the geodesic equation in this spacetime is chaotic, there are some relevant features that can be determined analytically. Among other things, we give analytic formulas for the vertical diameter of the shadow for an observer at arbitrary inclination and for the horizontal diameter of the shadow for an observer in the equatorial plane. Moreover, we use the strong-deflection formalism for analytically calculating the so-called photon rings of order $\ge 2$ and we use the recently introduced gap parameter $Δ_2$ for distinguishing lensing of an Ernst black hole from that of a Schwarzschild black hole.

Figures (8)

• Figure 1: Equatorial effective potential $V_\mathrm{eff}$ from Eq. (\ref{['eq:Effective_potential']}) vs. $r$ for various $B$ values. Here we give $r$ in units of $M$ and $V_\mathrm{eff}$ in units of $1/M^2$.
• Figure 2: Effective potential $V_{E,L}$ from Eq. (\ref{['eq: Timelike_Effective_Potential']}) vs. $r$ for timelike geodesics for various $B$ values. Here we give $r$ in units of $M$ and $V_{E,L}$ in units of $M^2$. $E$ and $L$ were arbitrarily chosen as $1$ and $4M$.
• Figure 3: $L^2$ plotted as a function of $r$ for $B=0.15/M$. $L^2<0$ indicates no orbit. Here we give $r$ in units of $M$ and $L^2$ in units of $M^2$.
• Figure 4: $V_{E,L}"$ vs. $r$ for various $B$ values. $V_{E,L}"=0$ corresponds to the ISCO. Here we give $r$ in units of $M$ and $V_{E,L}"$ in units of $M^2$. For each $r$, the energy $E$ and angular momentum $L$ of the circular timelike orbit were used, as given by Eqs. (\ref{['eq:E(r)']}) and (\ref{['eq:L(r)']}), respectively.
• Figure 5: Equatorial effective potential $V_\mathrm{eff}$ from Eq. (\ref{['eq:Effective_potential']}) vs. $r$ for $B=0.12/M$. Here we give $r$ in units of $M$ and $V_\mathrm{eff}$ in units of $1/M^2$.