Geometric properties of slowly rotating black holes embedded in matter environments

TL;DR

This work develops a self-consistent, slowly rotating black-hole spacetime embedded in an anisotropic dark-matter halo using a Hartle–Thorne–like expansion. By solving a single frame-dragging equation for ω(r) with a Hernquist-density halo, the authors quantify environmental corrections to equatorial geodesics, light rings, ISCOs, and epicyclic frequencies, highlighting a profile- and rotation-dependent frame-dragging effect. They show that halo rotation induces shifts in LR, ISCO, and ν_r, ν_θ, including a new feature—a local minimum in ν_θ/ν_r—altering resonance locations and potentially modifying EMRI and HFPO signals. The study provides a practical, extensible framework for incorporating realistic astrophysical environments into strong-field gravity tests and discusses the prospective to constrain dark-matter halos with future detectors like LISA via cumulative phase shifts in long-duration gravitational-wave signals.

Abstract

Astrophysical black holes are embedded in surrounding dark and baryonic matter that can measurably perturb the spacetime. We construct a self-consistent spacetime describing a slowly rotating black hole embedded in an external matter distribution, modeling the surrounding dark matter halo as an anisotropic fluid. Working within the slow-rotation approximation, we capture leading-order spin and frame-dragging effects while retaining analytic transparency. We show that the presence and rotation of the halo induce distinct deviations from the vacuum black hole geometry, modifying inertial frame dragging, equatorial circular geodesics, the light ring, the innermost stable circular orbit, and radial and vertical epicyclic frequencies. These effects produce systematic shifts in orbital constants of motion and the locations of epicyclic resonances. In particular, the epicyclic frequency ratios develop nonmonotonic behavior, such as local minima. We further demonstrate that these features depend on the angular velocity of the surrounding fluid, reflecting the interplay between environmental gravity and frame dragging. Our results demonstrate that environmental and rotational effects can leave observable imprints on precision strong-field probes, particularly extreme mass-ratio inspirals, where small corrections accumulate over many orbital cycles. This work provides a minimal and extensible framework for incorporating realistic astrophysical environments into strong-field tests of gravity with future space-based gravitational-wave detectors.

Figures (5)

• Figure 1: Radial profile of the frame-dragging angular velocity $\omega(r)$ for the slowly rotating spacetime in the presence of a surrounding dark matter halo. Different halo configuration is represented as $(i,j)\equiv (Loga_0,Log M_{\rm Halo})$. (a) $\omega(r)$ for zero-angular-momentum observers (ZAMOs) fluid. (b) $\omega(r)$ for a uniformly rotating matter configuration with constant angular velocity $\Omega=0$. The deviation from the vacuum behavior reflects the influence of the dark matter halo, described by a Hernquist-type density profile and modeled as an anisotropic matter distribution, on inertial frame dragging.
• Figure 2: Relative fractional deviation of the conserved energy per unit rest mass in percentage, $\Delta E/E_{\rm Vac}\%$, for equatorial circular timelike orbits in a slowly rotating spacetime surrounded by a dark matter halo, compared to the vacuum case. The upper panels correspond to prograde $(+)$ orbits and the lower panels to retrograde $(-)$ orbits. The left panels show ZAMO fluid, while the right panels correspond to a fluid with $\Omega=0$. Different curves represent distinct halo parameters. The splitting between prograde and retrograde branches and the departure from the vacuum behavior arise from halo-induced inertial frame dragging and the gravitational influence of the anisotropic dark matter distribution modeled by a Hernquist-type density profile.
• Figure 3: Relative fractional deviation of the conserved angular momentum per unit rest mass in percentage, $\Delta L/L_{\rm Vac}\%$, for equatorial circular timelike orbits in a slowly rotating spacetime surrounded by a dark matter halo, compared to the vacuum case. The upper panels correspond to prograde $(+)$ orbits and the lower panels to retrograde $(-)$ orbits. The left panels show ZAMO fluids, while the right panels correspond to a fluid with $\Omega=0$. Different curves represent distinct halo parameters. The observed radial dependence and the asymmetry between prograde and retrograde branches reflect the combined effects of halo gravity and inertial frame dragging induced by the anisotropic Hernquist-type dark matter distribution.
• Figure 4: Ratio of the vertical and radial epicyclic frequency is shown with respect to radius. The left panels correspond to prograde ($+$) orbits, while the right panels represent retrograde ($-$) orbits. Upper panels represent ZAMO fluid where $\Omega=\omega(r)$, while lower panels correspond to a fluid with $\Omega=0$. Purple, sky blue, green, and red correspond to $\chi = 0.01, 0.05, 0.1,$ and $0.3$, respectively. Solid, dashed, and dotted curves denote the density configurations $(2,1)$, $(3,2)$, and $(4,2)$. Horizontal black, brown, and blue lines indicate the $3\!:\!2$, $4\!:\!3$, and $5\!:\!4$ resonances, respectively. Colored disks on these lines mark the resonance positions for the corresponding spin in the absence of an environment. In the epilogue, we present the frequency ratios across the full radial range. The deviation from the vacuum behavior and the asymmetry between prograde and retrograde branches arise from inertial frame dragging modified by the dark matter halo.
• Figure 5: The conserved energy and angular momentum per unit rest mass, $E_\pm$ and $L_\pm$, for equatorial circular timelike orbits in a slowly rotating spacetime surrounded by a dark matter halo are shown, with the first and third rows corresponding to prograde $(+)$ orbits and the second and fourth rows to retrograde $(-)$ orbits. The left panels represent ZAMO fluid where $\Omega=\omega(r)$, while the right panels correspond to a static fluid with $\Omega=0$. The splitting between the prograde and retrograde branches, as well as the deviation from the vacuum case, originates from inertial frame dragging induced by the DM halo.