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

Shahar Hod

doi:10.1140/epjc/s10052-015-3787-9

2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2015) 75: 541
https://doi.org/10.1140/epjc/s10052-015-3787-9

Regular Article - Theoretical Physics

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

Shahar Hod¹^,2^*

¹ The Ruppin Academic Center, 40250, Emeq Hefer, Israel
² The Hadassah Institute, 91010, Jerusalem, Israel

^* e-mail: shaharhod@gmail.com

Received: 25 June 2015
Accepted: 6 November 2015
Published online: 19 November 2015

Abstract

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.