Relativistic Dynamics and Bondi-Hoyle-Lyttleton Accretion onto Rotating Embedded Black Hole Models

In this paper, we examine the motion of test particles and relativistic accretion mechanisms within the spacetime of a rotating and embedded BH. In this case, the geometric properties of the metric and their dynamical consequences for particle trajectories are systematically studied, with a specific focus on circular orbits together with their existence criteria and stability constraints. Also, the effective potential and the corresponding effective force are constructed to quantify the influence of rotation and embedding parameters on the attractive and repulsive sectors of the gravitational interaction. Closed-form expressions for orbital frequencies as measured by a distant observer are derived, enabling a quantitative analysis of relativistic precession phenomena, including periastron advance and Lense-Thirring precession. Furthermore, we conduct general-relativistic hydrodynamic simulations of BHL accretion onto rotating embedded BHs. In addition, within the framework of the BHL accretion mechanism, the numerical solution of the GRH equations shows that the embedding parameter αsystematically modifies the morphology of the shock cone formed around embedded BHs compared to the Kerr model. In particular, a wider opening angle of the cone is produced, the compression of matter in the post-shock region is weakened, and the dynamical variability of the flow is enhanced. The time-dependent mass accretion rate exhibits increasing oscillation amplitudes and long-term variations with increasing α, while these amplitudes are found to be suppressed by the frame-dragging effect associated with the BH spin parameter. At the same time, increasing values of $α$ lead to a strengthening of the QPO frequencies formed around embedded BHs in the LFQPO regime, enhancing their observability and increasing the likelihood of detecting commensurate frequency ratios such as 3:2.

💡 Research Summary

The paper investigates both the particle dynamics and the relativistic accretion processes in the spacetime of a rotating black hole that is “embedded’’ in an external geometric background, characterized by two key parameters: the spin a and an embedding parameter α. The metric (Eqs. 1‑4) reduces to Kerr when α = 0 and to Schwarzschild when both a = 0 and α = 0, confirming consistency with known solutions. The embedding modifies the effective mass function to f(r)=M−4αr, thereby introducing deviations from the vacuum Kerr geometry.

Using a Hamiltonian formalism, the authors derive the conserved specific energy E and angular momentum L for neutral test particles confined to the equatorial plane. By imposing the circular‑orbit conditions (dV_eff/dr = 0, d²V_eff/dr² > 0) they obtain closed‑form expressions for E(r,a,α) and L(r,a,α). The effective potential V_eff is shown to deepen as α increases, indicating stronger gravitational binding for a given radius. Consequently, the innermost stable circular orbit (ISCO) radius shifts inward with larger a (as in Kerr) and further inward with larger α, demonstrating that the embedding enhances the relativistic confinement of matter.

The authors also compute the three fundamental orbital frequencies measured at infinity: the azimuthal frequency Ω_φ, the radial epicyclic frequency Ω_r, and the vertical epicyclic frequency Ω_θ. From these they derive the periastron‑advance Δϖ and the Lense‑Thirring precession Ω_LT = Ω_φ − Ω_θ. The analysis reveals that increasing α reduces both Ω_r and Ω_θ while leaving Ω_φ relatively less affected, thereby altering the ratios Ω_φ/Ω_r and Ω_φ/Ω_θ. These altered ratios make the 3:2 resonance—commonly observed in high‑frequency QPOs—more readily attainable, suggesting that the embedding parameter could leave an observable imprint on QPO spectra.

The second major component of the work is a suite of general‑relativistic hydrodynamic (GRH) simulations of Bondi‑Hoyle‑Lyttleton (BHL) accretion onto the rotating embedded black hole. The simulations employ a 2‑D axisymmetric grid with a Mach‑number‑controlled inflow and solve the full set of GRH equations. The results show that the shock cone formed downstream of the black hole widens as α grows; the post‑shock compression is weakened, and the overall morphology becomes less collimated. The mass‑accretion rate Ṁ(t) exhibits quasi‑periodic oscillations whose amplitude increases with α but is damped by larger spin a, reflecting the competing influences of embedding‑induced potential softening and frame‑dragging. Long‑term variations in Ṁ also become more pronounced for higher α.

Spectral analysis of the simulated light curves uncovers low‑frequency quasi‑periodic oscillations (LFQPOs) in the 10–30 Hz range. The LFQPO amplitude rises with α, and the frequency ratios tend toward simple integer ratios such as 3:2, especially when the spin is moderate. This suggests that the embedding modifies the effective epicyclic frequencies in a way that favours resonant amplification of LFQPOs.

In summary, the paper demonstrates that the embedding parameter α, alongside the spin a, has a substantial impact on both test‑particle orbital structure (energy, angular momentum, ISCO location, precession rates) and on the dynamics of accretion flows (shock‑cone geometry, accretion‑rate variability, QPO characteristics). These theoretical predictions provide concrete, potentially observable signatures—such as altered QPO frequencies, shock‑cone opening angles, and variability patterns—that could be used to test the existence of embedded (or regular) black holes with future high‑precision X‑ray timing missions. The work also opens avenues for further studies, including three‑dimensional magnetohydrodynamic simulations, exploration of negative α values, and coupling to quantum‑gravity motivated corrections.