Dragging of inertial frames in the composed black-hole–ring system

A well-established phenomenon in general relativity is the dragging of inertial frames by a spinning object. In particular, due to the dragging of inertial frames by a ring orbiting a central black hole, the angular velocity $$\Omega ^{\text {BH-ring}}_{\text {H}}$$ of the black-hole horizon in the composed black-hole–ring system is no longer related to the black-hole angular momentum $$J_{\text {H}}$$ by the simple Kerr-like (vacuum) relation $$\Omega ^{\text {Kerr}}_{\text {H}}(J_{\text {H}})=J_{\text {H}}/2M^2R_{\text {H}}$$ (here M and $$R_{\text {H}}$$ are the mass and horizon-radius of the black hole, respectively). Will has performed a perturbative treatment of the composed black-hole–ring system in the regime of slowly rotating black holes and found the explicit relation $$\Omega ^{\text {BH-ring}}_{\text {H}}(J_{\text {H}}=0,J_{\text {R}},R)=2J_{\text {R}}/R^3$$ for the angular velocity of a central black hole with zero angular momentum, where $$J_{\text {R}}$$ and R are respectively the angular momentum of the orbiting ring and its proper circumferential radius. Analyzing a sequence of black-hole–ring configurations with adiabatically varying (decreasing) circumferential radii, we show that the expression found by Will for $$\Omega ^{\text {BH-ring}}_{\text {H}}(J_{\text {H}}=0,J_{\text {R}},R)$$ implies a smooth transition of the central black-hole angular velocity from its asymptotic near-horizon value $$\Omega ^{\text {BH-ring}}_{\text {H}}(J_{\text {H}}=0,J_{\text {R}},R\rightarrow R^{+}_{\text {H}})\rightarrow 2J_{\text {R}}/R^3_{\text {H}}$$ (that is, just before the assimilation of the ring by the central black hole), to its final Kerr (vacuum) value $$\Omega ^{\text {Kerr}}_{\text {H}}(J^{\text {new}}_{\text {H}})= J^{\text {new}}_{\text {H}}/2{M^{\text {new}}}^2R^{\text {new}}_{\text {H}}$$ [that is, after the adiabatic assimilation of the ring by the central black hole. Here $$J^{\text {new}}_{\text {H}}=J_{\text {R}}$$, $$M^{\text {new}}$$, and $$R^{\text {new}}_{\text {H}}$$ are the new parameters of the resulting Kerr (vacuum) black hole after it assimilated the orbiting ring]. We use this important observation in order to generalize the result of Will to the regime of black-hole–ring configurations in which the central black holes possess non-zero angular momenta. In particular, it is shown that the continuity argument (namely, the characteristic smooth evolution of the black-hole angular velocity during an adiabatic assimilation process of the ring into the central black hole) yields a concrete prediction for the angular-velocity/angular-momentum asymptotic functional relation $$\Omega ^{\text {BH-ring}}_{\text {H}}=\Omega ^{\text {BH-ring}}_{\text {H}}(J_{\text {H}},J_{\text {R}},R\rightarrow R^{+}_{\text {H}})$$ of generic (that is, with $$J_{\text {H}}\ne 0$$) black-hole–ring configurations. Remarkably, we find the simple universal relation $$\Delta \Omega _{\text {H}}\equiv \Omega ^{\text {BH-ring}}_{\text {H}}(J_{\text {H}},J_{\text {R}},R\rightarrow R^{+}_{\text {H}})-\Omega ^{\text {Kerr}}_{\text {H}}(J_{\text {H}})={{J_{\text {R}}}/{4M^3}}$$ for the asymptotic deviation of the black-hole angular velocity in the composed black-hole–ring system from the corresponding angular velocity of the unperturbed (vacuum) Kerr black hole with the same angular momentum