Schwarzschild Black Hole Coupled with a Cloud of Strings Immersed in King Dark Matter Halo

TL;DR

This paper analyzes a Schwarzschild black hole threaded by a cloud of strings (Letelier) and immersed in a King-type dark matter halo, systematically exploring geodesic, shadow, and thermodynamic properties. By combining effective-potential methods for photon and massive-particle trajectories with Duan’s $\phi$-mapping topological current theory, it reveals how the string-cloud parameter $\alpha$ and halo parameters reshape the photon sphere, shadow, ISCO, and the thermodynamic landscape, including $T_H$, $G$, and $C_P$. A consistent topological classification is established for both photon rings (topological charge $Q=-1$) and thermodynamic critical points via $\Phi=T_H/\sin\theta$ and the generalized free energy $\mathcal{F}$, linking dynamical and thermodynamic structures. The findings indicate that increasing $\alpha$ enlarges the photon sphere and shadow while reducing the horizon scale in the thermodynamic topology, with observable implications for horizon-scale imaging and accretion spectra, and point to extensions to rotation, QNMs, and data-driven constraints.

Abstract

In this paper, we examine the geodesic and thermodynamic properties of a Schwarzschild black hole with a cloud of strings (known as the Letelier black hole) immersed in a King dark matter (KDM) halo under an isotropic configuration. The dynamics of both photons and massive particles are analyzed in detail using the effective potential formalism, including particle trajectories, the photon sphere, black hole shadow, and the innermost stable circular orbits (ISCOs). Particular emphasis is placed on how the presence of the KDM halo modifies these geometric and dynamical features. Furthermore, we explore the topological characteristics of photon rings by constructing a normalized vector field, following Duan\textquotesingle{}s topological current $φ$-mapping theory, and demonstrate how this field is influenced by both the string cloud and the KDM halo. In the thermodynamic context, we analyze the impact of the KDM halo and the string cloud on the Hawking temperature, Gibbs free energy, thermal stability, and phase transitions of the black hole. Finally, we examine the thermodynamic topology of the system using a theoretical framework that incorporates a generalized Helmholtz free energy and topological current theory. In both the photon sphere and thermodynamic analyses, we show that the string cloud parameter shifts the location of the zero point of the vector field in the equatorial plane. Specifically, in the photon sphere case, the radius of the photon sphere increases with increasing values of the string cloud parameter, while in the thermodynamic topology, the horizon radius decreases as the string cloud parameter increases.

Figures (18)

• Figure 1: Behavior of the metric function $f(r)$ as radial coordinate $r$ for different values of string parameter $\alpha$ and the scale parameter $R/M$. Here $\rho_0\,M^2=0.5$.
• Figure 2: Behavior of the effective potential $V_{\rm eff}$ that governs the photon dynamics for different values of the CoS parameter $\alpha$ and the scale parameter $R/M$. Here $\rho_0\,M^2=0.5$ and $\mathrm{L}/M=1$.
• Figure 3: 3D plot of the photon sphere radius $r_{\rm ph}/M$ as a function of $(\alpha, R/M)$ and $(\alpha,\rho_0\,M^2)$.
• Figure 4: Behavior of the effective radial force $MF_{\rm eff}$. Here $\alpha=0.1$ and $\mathrm{L}/M=1$.
• Figure 5: Behavior of the effective potential $V_{\rm eff}$ that governs the dynamics of test particles for different values of the CoS parameter $\alpha$ and the scale parameter $R/M$. Here $\rho_0\,M^2=0.5$ and $\mathrm{L}/M=1$.