Dragging of inertial frames in the composed Kerr-Newman-orbiting-ring system

The dragging of inertial frames by an orbiting object implies that the horizon angular velocity ${\mathrm{\Omega }}_{\text{H}}^{\text{BH-ring}}$ of a central black hole in a composed black-hole-orbiting-ring system is no longer related to its angular-momentum $J_{\text{H}}$ by the familiar vacuum functional relation ${\mathrm{\Omega }}_{\text{H}}(J_{\text{H}})=J_{\text{H}}\mathrm{/}M\alpha$ (here $\{M,\alpha \}$ are respectively the mass and normalized area of the central spinning black hole). Using a continuity argument, it has recently been revealed that the composed Kerr-ring system is characterized by the universal (that is, spin-{\it independent}) relation $\mathrm{\Delta }{\mathrm{\Omega }}_{\text{H}}\equiv {\mathrm{\Omega }}_{\text{H}}^{\text{BH-ring}}(J_{\text{H}},J_{\text{R}},R\to R_{\text{H}}^{+})-{\mathrm{\Omega }}_{\text{H}}^{\text{Kerr}}(J_{\text{H}})=J_{\text{R}}\mathrm{/}4M^3$, where $\{R,J_{\text{R}}\}$ are respectively the radius of the ring and its orbital angular momentum and $R_{\text{H}}$ is the horizon radius of the central Kerr black hole. This intriguing observation naturally raises the following physically interesting question: Does the physical quantity $\mathrm{\Delta }{\mathrm{\Omega }}_{\text{H}}$ in a composed black-hole-orbiting-ring system is always characterized by the near-horizon functional relation $\mathrm{\Delta }{\mathrm{\Omega }}_{\text{H}}=J_{\text{R}}\mathrm{/}4M^3$ which is independent of the spin (angular momentum) $J_{\text{H}}$ of the central black hole? In the present compact paper we explore the physical phenomenon of dragging of inertial frames by an orbiting ring in the composed Kerr-Newman-black-hole-orbiting-ring system. In particular, using analytical techniques, we reveal the fact that in this composed two-body (black-hole-ring) system the quantity $\mathrm{\Delta }{\mathrm{\Omega }}_{\text{H}}$ has an explicit non-trivial functional dependence on the angular momentum $J_{\text{H}}$ of the central spinning black hole.