Kerr-like black holes shadow surrounded by dark matter halos: Comparison between various dark matter profiles

Abstract: The study of shadows of static and rotating black holes immersed in various dark matter halo profiles has gained significant attention in recent years. In this paper, we consider the static black hole solution immersed in three dark matter halo profiles (King, Hernquist, and Moore) and, by using the Newman-Janis algorithm, obtain the corresponding rotating black hole solutions. The main goal of the paper is to study the influence of the characteristic density, characteristic radius, and the spin parameter on the inner and outer horizons, the ergoregion, and the shadow radius, and to compare the results with Kerr black holes and other dark matter profiles. Our findings indicate that the shadow radius of rotating black holes immersed in dark matter increases compared to that of the Kerr black hole in the absence of dark matter; however, the impact on the size and shape of the shadow is negligible for King, Hernquist, and Moore profiles, and for all other dark matter halo profiles considered, and is independent of whether the halo is cuspy or cored. Therefore, at least for these Kerr-like black holes, the black hole shadow is not a suitable tool for distinguishing the nature of the dark matter distribution in galactic centers.

• The paper develops rotating Kerr-like black hole metrics surrounded by dark matter halos using the Newman–Janis algorithm to extend static solutions to spinning cases.
• Detailed analysis of photon orbits shows that increasing dark matter parameters slightly enlarges the shadow radius while the black hole spin predominantly governs shadow distortion.
• Comparative results indicate that shadow differences across King, Hernquist, and Moore profiles are negligible, limiting the use of shadow imaging for dark matter discrimination.

Kerr-like Black Hole Shadows Surrounded by Dark Matter Halos: Comparative Analysis of Multiple Halo Profiles

Introduction

This study systematically investigates rotating Kerr-like black holes (BHs) immersed within various dark matter (DM) halo environments, focusing on the King, Hernquist, and Moore DM profiles. The analysis extends previous static BH solutions to the rotating case via the Newman–Janis algorithm and examines the resulting influence of DM parameters and the BH spin on key geometrical and observable quantities, with an emphasis on the BH shadow. By comprehensively comparing shadow properties across both cored and cuspy halo profiles, the paper critically evaluates the viability of the shadow as a discriminator of DM distribution in galactic centers.

Kerr-like Black Holes in Dark Matter Halos

The DM densities are parametrized by the generalized Dekel-Zhao profile, with characteristic density ($\rho_s$), scale radius ($r_s$), and power-law indices $(\alpha, \beta, \gamma)$. Specific index choices recover common cored (e.g., King, Burkert) and cuspy (e.g., NFW, Moore) models. For each profile, the static metrics are constructed, and then rotated using the Newman–Janis method, producing axisymmetric metrics parameterized additionally by BH spin $a$. The mass function $m(r)$ incorporates both the central BH and the DM distribution.

The King, Hernquist, and Moore models—previously lacking comprehensive rotating solutions—are constructed explicitly. For all, in the $\rho_s \rightarrow 0$ limit, the metrics reduce to the standard Kerr solution.

In-depth analysis demonstrates that, for fixed $a$, increasing $\rho_s$ or $r_s$ shifts the location of the event horizon to larger radii. Conversely, higher $a$ reduces the outer horizon radius, in line with expectations from the pure Kerr solution. Ergosphere morphologies are analyzed as a function of DM and spin parameters, revealing consistent shrinking with increasing $r_s$0.

Photon Geodesics and Circular Photon Orbits

The authors derive the null geodesic equations for photons in the ambient metric using the Hamilton–Jacobi formalism. Constants of motion (energy $r_s$1, angular momentum $r_s$2, and the Carter constant $r_s$3) allow separation of variables and reduction to radial and polar effective potentials. The paper details conditions for unstable circular photon orbits (“photon spheres”), identifying their radii via extremization of the effective potential.

Analytical expressions for the critical impact parameters $r_s$4 (normalized angular momentum) and $r_s$5 (normalized Carter constant) for photon spheres are obtained for both generic DM environments and the special Kerr case, which serve as input for shadow computation.

Shadow Computation and Parametric Dependencies

The shadow is treated as the set of end-points $r_s$6 in celestial coordinates for photons from infinity, with $r_s$7 and $r_s$8. Varying DM parameters modifies the underlying space-time, thus altering critical geodesics and the resulting shadow boundary.

(Figure 1)

Figure 1: Celestial coordinates define the observer's image plane for shadow projections (2605.18205).

Parametric studies show that for fixed BH spin, the shadow radius increases monotonically with either $r_s$9 or $(\alpha, \beta, \gamma)$0. The shadow distortion—quantified by departure from circularity—grows with higher spin and for observers at lower inclination angles (closer to the equatorial plane). Across all studied cases, however, the variation in both shadow size and shape is extremely minor relative to the pure Kerr scenario. Notably, the effect magnitude is not substantially different between cored and cuspy profiles, nor between the King, Hernquist, and Moore models.

Numerical Results and Comparative Analysis

The enlarged shadow in DM-immersed spacetimes is demonstrable but numerically small. For realistic DM profile parameters, the shift in shadow radius remains well below current or near-future observational sensitivity. Furthermore, when directly comparing the shadow boundaries for several DM profiles (isothermal, NFW, Burkert, King, Hernquist, Moore, Dehnen, etc.), the distinctions become virtually indistinguishable, even under significant variations of profile parameters.

Key results include:

• For both cored (King, Burkert) and cuspy (Moore, NFW) profiles, the DM-induced shadow modifications are minuscule and statistically insignificant for plausible parameter ranges.
• Increasing the DM characteristic density or scale slightly increases the shadow radius, with more pronounced changes in $(\alpha, \beta, \gamma)$1 than in $(\alpha, \beta, \gamma)$2 at fixed spin parameter.
• Shadow shape distortion remains dictated primarily by BH spin and observer inclination, not by the DM halo properties.

Theoretical and Observational Implications

The analysis undermines the practical utility of BH shadow images as probes of galactic-scale DM profile distinctions. The DM-induced changes are subdominant to both intrinsic BH spin effects and current astrophysical measurement uncertainties. Consequently, no meaningful discrimination between cored and cuspy DM models can be achieved by shadow analysis for Kerr-like BHs embedded in DM halos.

These findings have direct implications for the interpretation of EHT and other VLBI imaging results: the apparent BH shadow is insensitive to the presence or detailed form of galactic DM halos. Consequently, attempts to infer galactic-scale DM properties from shadow observations in the Kerr regime are fundamentally limited by the underlying space-time structure.

Future Directions

While this work thoroughly addresses rotating BHs in equilibrium DM environments, several directions remain open:

• Consideration of non-equilibrium DM distributions, DM accretion or dynamical self-interactions, which might amplify observable signatures.
• Extensions to non-Kerr metrics arising from modified gravity, which could in principle introduce larger deviations.
• Analysis of alternative observables, such as ringdown gravitational waves, high-energy photon rings, or lensing time delays, to discern DM effects beyond the shadow.

Conclusion

This paper provides a comprehensive formal derivation of the metrics, photon dynamics, and shadow boundaries for rotating Kerr-like black holes in both cored and cuspy DM halos. The strong numerical result is that for all realistic DM profile choices, the BH shadow is not a viable discriminator for the nature of the DM distribution in galactic centers. The impact of the DM halo on shadow size and distortion is negligible relative to both theoretical expectations and observational capabilities. These outcomes reinforce the notion that alternative astrophysical or gravitational wave probes are required to advance the characterization of DM in the vicinity of supermassive black holes.