Letelier-AdS Black Hole Surrounded by a Perfect Fluid Dark Matter in the presence of Quintessence

TL;DR

This work analyzes a static Schwarzschild–AdS black hole dressed by a cloud of strings, surrounded by perfect-fluid dark matter and a quintessence-like field. The authors derive the Letelier–AdS metric with $f(r)=1-\alpha-\frac{2M}{r}+\frac{\lambda}{r}\ln\frac{r}{|\lambda|}-\frac{N}{r^{3w+1}}+\frac{r^2}{\ell_p^2}$ and study null and timelike geodesics to determine the photon sphere via $2f(r)-r f'(r)=0$ and the shadow through $R_{sh}=r_{ph}\sqrt{\frac{f(r_O)}{f(r_{ph})}}$. They extend the analysis to black-hole thermodynamics in extended phase space, computing $M(r_h)$, $T_H=f'(r_h)/(4\pi)$, entropy $S=\pi r_h^2$, Gibbs energy $G$, and specific heat $C_p$, and they develop a thermodynamic-topology framework using a generalized Helmholtz free energy, finding a single physical branch with topological charge $W=-1$ and a unit unstable photon-sphere charge $Q=+1$. The external fields nontrivially shift the photon sphere, horizon radii, ISCO, and shadow size, while the radiative efficiency of thin disks remains near the Schwarzschild value across wide parameter ranges. The results establish a practical baseline for connecting external dark-sector fields to observable black-hole optics and thermodynamics, with implications for future imaging, timing, and spectral analyses.

Abstract

This study investigates a Schwarzschild-anti de Sitter black hole coupled to a cloud of strings featuring only the electric-like component of the string bivector, embedded in a perfect fluid dark matter and a quintessence field. We examine the dynamics of photons and massive particles, focusing on trajectories, photon spheres, BH shadows, their topological characteristics, and innermost stable circular orbits (ISCOs), and emphasizing the influence of string cloud, perfect-fluid dark matter, and quintessence-like field parameters. Additionally, we explore the black hole's thermodynamics, deriving the Hawking temperature, Gibbs free energy, and specific heat, and discuss the modified first law of thermodynamics and thermodynamic topology under external matter fields. We demonstrate that the presence of a string cloud, perfect-fluid dark matter, and a quintessence-like field together modifies the geodesic structure and thermodynamic properties, thereby shifting the results relative to the standard Schwarzschild BH solution.

Figures (16)

• Figure 1: Behavior of the metric function for different values of string parameter $\alpha$, perfect fluid dark matter parameter $\lambda$, and the normalization constant $N$ of quintessence-like field. Here $M=1,\,w=-2/3,\,\ell_p=10$.
• Figure 2: Behavior of the effective potential governs the dynamics of photon particles as a function of the radial coordinate $r$ for different values of $\alpha$, $\lambda$ and $N$. Here $M=1,\,w=-2/3,\,\ell_p=10,\,\mathrm{L}=1$.
• Figure 3: Behavior of the effective radial force for different values of $\alpha$, $\lambda$ and $N$. Here $M=1,\,w=-2/3,\,\ell_p=10,\,\mathrm{L}=1$.
• Figure 4: Null geodesics in the equatorial plane for the AdS black hole with a cloud of strings and perfect fluid dark matter in the presence of a quintessence-like field. We fix $M=1$, $w_q=-2/3$, $\alpha=0.10$, $\lambda=0.30$ and choose $(N,\ell_p)$ such that a photon sphere exists outside the event horizon (in the example shown, the auto-tuned configuration gives $N=0$ and $\ell_p=4$, which yields an event-horizon radius $r_h\simeq 1.471$, a photon-sphere radius $r_{\rm ph}\simeq 2.450$, and the critical impact parameter $b_c\simeq 2.895$). The solid black disk depicts the horizon, the dotted gray circle shows the photon sphere, and the colored curves are null orbits with impact parameters $b=f\,b_c$ for representative factors $f\in\{0.85,0.97,1.01,1.20,1.60,2.20\}$. Orbits with $b<b_c$ are captured (red/orange), the near-critical one ($b\approx b_c$) asymptotically approaches the photon sphere (blue), while for $b>b_c$ the trajectories undergo scattering (green) according to the AdS "return/escape" criterion described in the text.
• Figure 5: Three-dimensional plot of the photon sphere radius $r_{\rm ph}$ as a function of combination of $\alpha$, $\lambda$ and $N$. Here $M=1,\,w=-2/3$.