Off-Equatorial Orbits around Magnetically Charged Black Holes

TL;DR

The paper investigates stable off-equatorial circular orbits of charged test particles around magnetically charged black holes (MBHs), addressing both static and rotating spacetimes. For the static MBH, it derives an exact analytic expression for the orbital latitude $\theta(r)$, links latitudinal extrema to the photon sphere and the neutral ISCO via $\frac{d}{dr}\Bigl[r\sqrt{\frac{\frac{M}{r}-\frac{P^2}{r^2}}{1-\frac{3M}{r}+\frac{2P^2}{r^2}}}\Bigr]=0$, and demonstrates stability against synchrotron radiation, allowing sizable latitudinal deviations even for small magnetic charge $P$. In rotating MBHs, prograde and retrograde branches are computed numerically, with frame-dragging modifying their structure and maintaining the photon ring as the inner boundary; such off-equatorial orbits are absent in the electrically charged Kerr–Newman spacetime. The results hint at distinctive observational signatures in black hole imaging and polarimetry and open avenues for further work on stability, radiation reaction, and spin effects around MBHs.

Abstract

We present a complete characterization of stable, off-equatorial circular orbits around magnetically charged black holes (MBHs). For a static, spherically symmetric MBH, we derive an exact analytic expression for the orbital latitude theta as a function of radius r. We establish a direct connection between these orbits and the spacetime fundamental structure, and demonstrate their stability against synchrotron radiation. We show that charged particles such as electrons and protons can exhibit O(1) latitude deviations at the ISCO radius and remain stable under synchrotron emission even for extremely small values of the black hole magnetic charge. We then extend the analysis to rotating MBHs, numerically computing the prograde and retrograde orbital branches and demonstrating how frame-dragging modifies their structure and stability regions. We show that these off-equatorial orbits are a unique feature of the magnetic charge, being forbidden in the analogous electrically charged Kerr-Newman spacetime. Our results suggest that environments surrounding magnetically charged black holes can exhibit distinctive phenomenological signatures, with potential implications for black hole imaging and polarimetric observations.

Figures (8)

• Figure 1: Circular orbit of a charged particle around an MBH. The magnetic force shifts the particle to a nontrivial polar angle.
• Figure 2: $\theta_+(r)$ of a $q/m=-1$ particle, for multiple $P/M$ MBHs. $\theta_+<\pi/2$ since the magnetic force on a prograde, negatively charged particle points above the equator. The y-axis is inverted so that north pole direction ($\theta = 0$) appears at the top of the figure, matching the intuitive orientation.
• Figure 3: Cross-sectional view of circular orbits around a magnetically charged black hole with $P/M=0.6$. The black disk marks the event horizon, while the grey and blue dashed circles denote the photon sphere and the ISCO of neutral particles, respectively. The orange and purple curves indicate a series of orbits of prograde particles with $q/m=-1$ and $q/m=-10$.
• Figure 4: The parameter space of circular orbits for a charged particle with charge-to-mass ratio $|q/m|$ orbiting a massive black hole (MBH) with charge-to-mass ratio $|P/M|$. The dashed lines indicates where the circular orbit exhibits a significant latitudinal deviation ($|\tan\theta| < 1$). The red region marks the values of the parameters for which the orbit becomes unstable under the emission of synchrotron radiation. The extent of this unstable region depends on the ratio $m/M$, as shown in Eq. \ref{['eq:scalingratio']}. In this figure, we take $m$ to be the electron mass and $M = 10^{6}\, M_\odot$.
• Figure 5: $\theta(r)$ of a $q/m=-1$ particle prograding a $P/M=0.6$ spinning MBH, for multiple $P/M$ values. The peak and smallest allowed radius decrease monotonously with $a/M$.