Schwarzschild-AdS Black Holes with Cloud of Strings and Quintessence: Geodesics, Thermodynamic Topology, and Quasinormal Modes

TL;DR

This work analyzes a Schwarzschild–AdS black hole threaded by a cloud of strings and surrounded by a quintessence-like fluid. It combines geodesic analysis (photon spheres, shadows, and ISCO), thermodynamic topology (free-energy framed classification with topological charges), and scalar perturbations (Klein–Gordon dynamics and quasinormal modes) to map how CoS and QF modify strong-field observables and the dual field theory dynamics. The authors show that CoS and QF shift photon-orbit structures and shadow sizes, induce distinct topological classifications of black-hole states, and alter the QNM spectrum, with potential implications for AdS/CFT thermalization and gravitational-wave phenomenology. The work highlights a rich interplay between spacetime geometry, topological charge conservation, and dynamical perturbations in a modified AdS black-hole background.

Abstract

In this study, we explore a Schwarzschild-anti de-Sitter black hole (BH) coupled with a cloud of strings (CoS) possessing both electric- and magnetic-like components of the string bivector, embedded in a Kiselev-type quintessence fluid (QF). We analyze the dynamics of photons and test particles, focusing on trajectories, photon spheres, BH shadows, and innermost stable circular orbits (ISCO), highlighting how CoS and QF parameters affect these features. We then examine the thermodynamic topology of the system by analyzing vector field zeros, showing that varying CoS leads to distinct topological configurations with total charges of either $0$ or $+1$, corresponding to known classes like RN and AdS-RN. Additionally, we study scalar field dynamics via the massless Klein-Gordon equation, reformulated into a Schr$\ddot{o}$dinger-like form to derive the effective potential. We compute the quasinormal modes (QNMs) of scalar perturbations, showing how CoS and QF influence oscillation frequencies and damping rates, with implications for gravitational confinement and thermalization in the AdS/CFT context.

Figures (9)

• Figure 1: Behavior of the metric function $f(r)$ by varying CoS parameter $\alpha$. Here, we select $M=1, b=0.2, \ell_p=10, \mathrm{N}=0.01$.
• Figure 2: Behavior of the effective potential for null and time-like geodesics by varying CoS parameter $\alpha$. Here, we set $M=1, b=0.2, \ell_p=10, w=-2/3, \mathrm{N}=0.01, \mathrm{L}=1$.
• Figure 3: The following set of figures presents the photon sphere (PSs) structures obtained for different combinations of key parameters: Figure (\ref{['200a']}) illustrates the photon sphere for $\alpha = 0.5$ and $b = 0.1$. Figure (\ref{['200b']}) shows the configuration when $\alpha = 0.3$ and $b = 0.15$. Figure (\ref{['200c']}) depicts the photon sphere corresponding to $\alpha = 0.3$ and $b = 0.25$. Figure (\ref{['200d']}) demonstrates the case with $\alpha = 0.5$ and $b = 0.25$. Figure (\ref{['200e']}) presents the structure for $\alpha = 0.1$ combined with $b = 0.05$. Figure (\ref{['200f']}) reveals the photon sphere arrangement at $\alpha = 0.1$ and $b = 0.1$. All these configurations are evaluated under the condition of a fixed charge parameter set as $\omega = -\frac{2}{3}$, along with a constant value of $N = 0.02$ and mass $M = 1$. Together, these images offer an in-depth depiction of how the photon sphere morphology responds to changes in the symmetry-breaking energy scale $\alpha$ and the coupling constant $b$, highlighting the sensitivity of photon trajectories and gravitational lensing effects to these parameters.
• Figure 4: Behavior of the effective radial force by varying CoS parameter $\alpha$. Here, we set $M=1, b=0.2, \ell_p=10, w=-2/3, \mathrm{N}=0.01, \mathrm{L}=1$.
• Figure 5: The behavior of the $\tau\text{-}r_h$ function for the Schwarzschild AdS BHs with a cloud of strings surrounded by quintessence-like fluid is analyzed within the framework of a normal vector field $n$, which is defined on the $(r_h, \Theta)$-plane. In this analysis, the zero points (ZPs) of the system are identified as the coordinates $(r_h, \Theta)$ at which the vector field vanishes. These zero points correspond to critical locations determined by the underlying structure of the spacetime and are influenced by the free parameters of the model. In particular, the study is conducted for a fixed value of the equation-of-state parameter, $\omega = -\frac{2}{3}$, which typically represents a quintessence-like field in cosmological models. Additionally, the pressure is chosen as $P = 0.001$, and the monopole charge (or topological charge) is set to $N = 0.02$. This specific set of parameters is used to explore the existence, position, and multiplicity of zero points in the $(r_h, \Theta)$-plane, allowing for insights into the geometric and thermodynamic properties of the BH solution under consideration.