A unified topological classification of circular orbits for charged particles in black hole spacetimes

TL;DR

This work develops a unified topological framework for circular orbits of charged test particles around static, spherically symmetric black holes with flat, AdS, and dS asymptotics using Duan's φ-mapping theory. By constructing a two-component vector field from the effective potential and defining a topological current with winding number W, the authors classify the existence and stability of both null and timelike circular orbits for fixed angular momentum and charge. The main findings show that charge qualitatively alters the topological classification, with a universal W=-1 in inter-horizon regions of multi-horizon spacetimes and asymptotic-boundary-dependent behavior outside the outer horizon; explicit RN, RN-AdS, and RN-dS examples validate these results. The work suggests potential relevance for astrophysical plasmas with effective charge dynamics and lays a foundation for extending the approach to rotating black holes, potentially linking topology to observable signatures such as QPOs and black-hole shadows.

Abstract

The study of circular orbits offers profound insights into the structure of spacetime around black holes. While the topological properties of these orbits are well-established for neutral particles, the influence of electric charge-particularly for massless particles-remains a subject of exploration. In this work, we employ a topological current $φ$-mapping approach to systematically investigate the circular orbits of charged test particles in static, spherically symmetric black hole spacetimes with flat, anti-de Sitter (AdS), and de Sitter (dS) asymptotics. We demonstrate that the particle's charge significantly alters the topological classification of both timelike and null circular orbits. A key finding is that for multi-horizon black holes, if a circular orbit with fixed angular momentum and charge exists between two neighboring horizons, there will always be at least one unstable null and one unstable timelike circular orbit. Outside the outermost horizon, the asymptotic behavior of spacetime and the specific charge ratio crucially determine the topological charge $W$, dictating the existence and stability of orbits. Our results, validated through Reissner-Nordström (RN), RN-AdS, and RN-dS examples, extend the topological orbit classification framework and provide a foundation for potential applications in environments where effective charge dynamics may be relevant, such as magnetized plasmas around black holes.

Figures (16)

• Figure 1: Representation of the contour $C=\sum_i\cup l_i$ (which encloses $\Sigma$) on the $(r,\theta)$ plane. The curve $C$ has a positive orientation, marked with the red arrows. $r_{\mathrm{in}}$ and $r_{\mathrm{out}}$ have different meanings in different cases. The angles $\Omega_i$ represent the directional change of the vector $\phi^a$ at the joints between adjacent segments $l_i$, used to compute the winding number via Eq. (\ref{['W']}).
• Figure 2: The behavior of $f(r)$ in the region between two neighboring horizons.
• Figure 3: Representation of the contour $C=\sum_il_i$ (which encloses $\Sigma$) on the $(r,\theta)$ plane. The curve $C$ has a positive orientation, marked with the red arrows. The black, bule and yellow arrows indicate the approximate directions of the vector $\phi$ at the boundaries.
• Figure 4: The behavior of $f(r)$ in an asymptotically flat black hole. At $r_h$, it is observed that $f'(r)>0$.
• Figure 5: The behavior of $f(r)$ in RN black hole. At $r_-$, one has $f'(r)<0$; at $r_+$, one has $f'(r)>0$.