AdS Black Hole Solution with a Dark Matter Halo Surrounded by a Cloud of Strings

Abstract: We derive and analyze a Schwarzschild-like Anti-de Sitter (AdS) black hole (BH) obtained as a static, spherically symmetric solution of Einstein's equations sourced by a cloud of strings (CoS) and a dark matter (DM) halo modeled by a Dehnen-type density profile. We first study the geodesic motion of massless and massive test particles, emphasizing how the CoS parameter $\alpha$ and the DM halo parameters $(\rho_s, r_s)$ influence photon spheres, circular orbits, the BH shadow, and the innermost stable circular orbit (ISCO). We then examine scalar perturbations via the effective potential and the associated quasinormal-mode (QNM) spectra, showing how $\alpha$ and $(\rho_s, r_s)$ deform oscillation frequencies and damping rates, thereby affecting stability diagnostics. Furthermore, we investigate the thermodynamics in the extended phase space, deriving the Hawking temperature, equation of state, Gibbs free energy, and specific heat capacity, and establishing a consistent first law and Smarr relation with natural work terms for $\alpha$ and $(\rho_s, r_s)$. We find that the interplay between the CoS and the DM halo produces quantitative and sometimes qualitative changes in both dynamical and thermodynamical properties, including shifts of the Hawking--Page transition and heat-capacity divergences, thus reshaping the phase structure of Schwarzschild--AdS BHs.

• The paper presents an exact solution of a Schwarzschild-AdS black hole modified by a Dehnen dark matter halo and a cloud of strings, highlighting their combined effects on observable features.
• It employs a range of analyses—from geodesic motion and photon spheres to scalar perturbations and quasinormal modes—to quantify impacts on shadow size and phase structure.
• Results suggest that astrophysical observations, such as those from the Event Horizon Telescope and gravitational wave detections, could constrain the parameters of the string cloud and dark matter halo.

Schwarzschild-AdS Black Holes with a Dehnen Dark Matter Halo and Cloud of Strings: Geodesics, Perturbations, and Thermodynamics

Introduction and Motivation

This work presents a comprehensive analysis of a static, spherically symmetric Schwarzschild-AdS black hole (BH) solution modified by two physically motivated external structures: a Dehnen-type dark matter (DM) halo and a cloud of strings (CoS). The construction is motivated by astrophysical evidence for supermassive black holes embedded in DM halos and the theoretical relevance of topological defects such as string clouds. The resulting metric function is derived as an exact solution to Einstein’s equations with a composite source, and the study systematically explores the impact of the CoS parameter $\alpha$ and DM halo parameters $(\rho_s, r_s)$ on geodesic structure, photon spheres, BH shadows, scalar perturbations, quasinormal modes (QNMs), and extended thermodynamics.

Spacetime Structure and Metric Properties

The metric function is given by

$$f(r) = 1 - \alpha - \frac{2M}{r} - \rho_s r_s^2 \ln\left(1 + \frac{r_s}{r}\right) + \frac{r^2}{\ell_p^2}$$

where $M$ is the BH mass, $\ell_p$ is the AdS curvature radius, $\alpha$ encodes the string cloud, and $(\rho_s, r_s)$ characterize the Dehnen DM halo. The solution is shown to satisfy Einstein’s equations for the composite energy-momentum tensor, with the DM halo density decaying as $r^{-4}$ at large radii and yielding a finite asymptotic mass offset.

Figure 1: The metric function $f(r)$ as a function of $r$ for different values of the CoS parameter $(\rho_s, r_s)$0 ($(\rho_s, r_s)$1, $(\rho_s, r_s)$2).

The metric interpolates between Schwarzschild-AdS, Letelier-AdS, and the DM-halo solution depending on the limits of $(\rho_s, r_s)$3 and $(\rho_s, r_s)$4. The interplay between the CoS and DM halo introduces nontrivial modifications to the causal structure and horizon properties.

Geodesic Structure: Null and Timelike Orbits

Photon Dynamics and Shadows

The effective potential for null geodesics is

$(\rho_s, r_s)$5

where $(\rho_s, r_s)$6 is the angular momentum. The presence of $(\rho_s, r_s)$7 and $(\rho_s, r_s)$8 lowers the potential barrier, enlarges the photon-sphere radius $(\rho_s, r_s)$9, and increases the shadow size $$f(r) = 1 - \alpha - \frac{2M}{r} - \rho_s r_s^2 \ln\left(1 + \frac{r_s}{r}\right) + \frac{r^2}{\ell_p^2}$$0 and capture cross-section.

Figure 2: Effective potential for null geodesics as a function of $$f(r) = 1 - \alpha - \frac{2M}{r} - \rho_s r_s^2 \ln\left(1 + \frac{r_s}{r}\right) + \frac{r^2}{\ell_p^2}$$1 for varying $$f(r) = 1 - \alpha - \frac{2M}{r} - \rho_s r_s^2 \ln\left(1 + \frac{r_s}{r}\right) + \frac{r^2}{\ell_p^2}$$2 and $$f(r) = 1 - \alpha - \frac{2M}{r} - \rho_s r_s^2 \ln\left(1 + \frac{r_s}{r}\right) + \frac{r^2}{\ell_p^2}$$3 ($$f(r) = 1 - \alpha - \frac{2M}{r} - \rho_s r_s^2 \ln\left(1 + \frac{r_s}{r}\right) + \frac{r^2}{\ell_p^2}$$4, $$f(r) = 1 - \alpha - \frac{2M}{r} - \rho_s r_s^2 \ln\left(1 + \frac{r_s}{r}\right) + \frac{r^2}{\ell_p^2}$$5).

Figure 3: Photon-sphere and shadow diagnostics: (a) $$f(r) = 1 - \alpha - \frac{2M}{r} - \rho_s r_s^2 \ln\left(1 + \frac{r_s}{r}\right) + \frac{r^2}{\ell_p^2}$$6, (b) $$f(r) = 1 - \alpha - \frac{2M}{r} - \rho_s r_s^2 \ln\left(1 + \frac{r_s}{r}\right) + \frac{r^2}{\ell_p^2}$$7, (c) heatmap of $$f(r) = 1 - \alpha - \frac{2M}{r} - \rho_s r_s^2 \ln\left(1 + \frac{r_s}{r}\right) + \frac{r^2}{\ell_p^2}$$8.

Photon trajectories, deflection angles, and the transition between capture and scattering regimes are explicitly computed, demonstrating that both $$f(r) = 1 - \alpha - \frac{2M}{r} - \rho_s r_s^2 \ln\left(1 + \frac{r_s}{r}\right) + \frac{r^2}{\ell_p^2}$$9 and $M$0 systematically increase the shadow radius and the strong-lensing region.

Figure 4: Null geodesics with a common impact parameter for different $M$1 values, showing the effect on photon trajectories.

Figure 5: Null geodesics for fixed $M$2 and varying $M$3, illustrating capture, near-critical, and scattering cases.

Figure 6: Deflection angle $M$4 as a function of impact parameter for different $M$5.

Figure 7: Capture cross-section $M$6 as a function of $M$7 for several $M$8 values.

Figure 8: Effective radial force experienced by photons as a function of $M$9 for different $\ell_p$0 and $\ell_p$1.

Timelike Geodesics and ISCO

For massive particles, the effective potential and the conditions for circular orbits and the innermost stable circular orbit (ISCO) are derived. The ISCO radius increases with $\ell_p$2 and decreases with $\ell_p$3, indicating a competition between the angular deficit and the DM-induced logarithmic term.

Figure 9: Effective potential for timelike geodesics as a function of $\ell_p$4 for different $\ell_p$5 and $\ell_p$6.

Figure 10: Specific angular momentum $\ell_p$7 as a function of $\ell_p$8 for varying $\ell_p$9 and $\alpha$0.

Figure 11: 3D plot of the ISCO radius as a function of $\alpha$1 and $\alpha$2 ($\alpha$3, $\alpha$4, $\alpha$5).

Topological Properties of Light Rings

The topological analysis employs the potential function $\alpha$6 and the associated unit vector field to classify light rings. The critical points of $\alpha$7 correspond to photon spheres, and the structure of the vector field is shown to be continuously deformed by $\alpha$8 and $\alpha$9.

Figure 12: Potential function $(\rho_s, r_s)$0 as a function of $(\rho_s, r_s)$1 for different $(\rho_s, r_s)$2 and $(\rho_s, r_s)$3.

Figure 13: Potential function $(\rho_s, r_s)$4 as a function of $(\rho_s, r_s)$5 for different $(\rho_s, r_s)$6 and $(\rho_s, r_s)$7.

Figure 14: Unit vector field $(\rho_s, r_s)$8 on the $(\rho_s, r_s)$9-$r^{-4}$0 plane for varying $r^{-4}$1.

Figure 15: Unit vector field $r^{-4}$2 on the $r^{-4}$3-$r^{-4}$4 plane for varying $r^{-4}$5.

Scalar Perturbations and Quasinormal Modes

The scalar field perturbation equation is reduced to a Schrödinger-like form with an effective potential $r^{-4}$6 that depends on $r^{-4}$7 and $r^{-4}$8. The potential barrier decreases as either parameter increases, leading to lower QNM frequencies and longer damping times.

Figure 16: Scalar perturbative potential $r^{-4}$9 as a function of $f(r)$0 for dominant mode $f(r)$1 and varying $f(r)$2, $f(r)$3.

Figure 17: 3D plot of $f(r)$4 for $f(r)$5 as a function of $f(r)$6 and $f(r)$7 for different $f(r)$8.

Figure 18: Contour plot of the scalar perturbative potential for $f(r)$9 and varying $r$0.

The QNM spectra, computed via the WKB approximation, show that both the real and imaginary parts of the frequencies decrease with increasing $r$1 and $r$2, indicating slower oscillations and weaker damping.

Extended Thermodynamics and Phase Structure

The extended thermodynamic analysis treats the cosmological constant as pressure and includes natural work terms for $r$3 and $r$4. The Hawking temperature, entropy, enthalpy, and thermodynamic volume are derived in closed form. The first law and Smarr relation are generalized to include the CoS and DM halo contributions.

Figure 19: Hawking temperature $r$5 as a function of $r$6 and entropy $r$7 for varying $r$8 and $r$9.

Figure 20: BH mass $(\rho_s, r_s)$00 as a function of $(\rho_s, r_s)$01 and $(\rho_s, r_s)$02 for varying $(\rho_s, r_s)$03.

Figure 21: Specific heat capacity $(\rho_s, r_s)$04 as a function of $(\rho_s, r_s)$05 and $(\rho_s, r_s)$06 for varying $(\rho_s, r_s)$07.

Figure 22: Gibbs free energy $(\rho_s, r_s)$08 as a function of $(\rho_s, r_s)$09 and $(\rho_s, r_s)$10 for varying $(\rho_s, r_s)$11.

The presence of $(\rho_s, r_s)$12 and $(\rho_s, r_s)$13 lowers the Hawking temperature, shifts the divergence of the specific heat, and deforms the Gibbs free energy and Hawking-Page transition. The phase structure is thus quantitatively and qualitatively altered compared to standard Schwarzschild-AdS, with the unstable small-BH branch and the onset of global stability displaced by the matter sector.

Implications and Future Directions

The analysis demonstrates that the combined effect of a string cloud and a Dehnen DM halo produces measurable modifications in geodesic structure, shadow size, QNM spectra, and thermodynamic phase transitions. The growth of the shadow radius and capture cross-section with $(\rho_s, r_s)$14 and $(\rho_s, r_s)$15 suggests that EHT observations can constrain these parameters. The opposite trends of the ISCO radius with respect to $(\rho_s, r_s)$16 and $(\rho_s, r_s)$17 have implications for accretion disk efficiency and X-ray spectra. The QNM results indicate that gravitational wave ringdown signals are sensitive to the near-horizon matter environment.

Potential extensions include the study of rotating (Kerr-AdS) analogs, alternative halo profiles, higher-order perturbations, and a full mapping of the $(\rho_s, r_s)$18 phase diagram including criticality and Joule-Thomson expansion. The framework provides a controlled setting for probing the interplay between astrophysical environments and strong-field gravity, with direct relevance for multi-messenger observations.

Conclusion

This work establishes a detailed theoretical framework for Schwarzschild-AdS black holes surrounded by a Dehnen-type dark matter halo and a cloud of strings. The analysis covers geodesic motion, topological properties of light rings, scalar perturbations, QNMs, and extended thermodynamics, elucidating the quantitative and qualitative impact of the matter sector on observable and thermodynamic properties. The results provide a basis for interpreting astrophysical data in the context of nontrivial BH environments and motivate further studies of more general configurations and observational signatures.