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 ΩHBH-ring\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^{\text {BH-ring}}_{\text {H}}$$\end{document} of the black-hole horizon in the composed black-hole–ring system is no longer related to the black-hole angular momentum JH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{\text {H}}$$\end{document} by the simple Kerr-like (vacuum) relation ΩHKerr(JH)=JH/2M2RH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^{\text {Kerr}}_{\text {H}}(J_{\text {H}})=J_{\text {H}}/2M^2R_{\text {H}}$$\end{document} (here M and RH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{\text {H}}$$\end{document} 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 ΩHBH-ring(JH=0,JR,R)=2JR/R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^{\text {BH-ring}}_{\text {H}}(J_{\text {H}}=0,J_{\text {R}},R)=2J_{\text {R}}/R^3$$\end{document} for the angular velocity of a central black hole with zero angular momentum, where JR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{\text {R}}$$\end{document} 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 ΩHBH-ring(JH=0,JR,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^{\text {BH-ring}}_{\text {H}}(J_{\text {H}}=0,J_{\text {R}},R)$$\end{document} implies a smooth transition of the central black-hole angular velocity from its asymptotic near-horizon value ΩHBH-ring(JH=0,JR,R→RH+)→2JR/RH3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^{\text {BH-ring}}_{\text {H}}(J_{\text {H}}=0,J_{\text {R}},R\rightarrow R^{+}_{\text {H}})\rightarrow 2J_{\text {R}}/R^3_{\text {H}}$$\end{document} (that is, just before the assimilation of the ring by the central black hole), to its final Kerr (vacuum) value ΩHKerr(JHnew)=JHnew/2Mnew2RHnew\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^{\text {Kerr}}_{\text {H}}(J^{\text {new}}_{\text {H}})= J^{\text {new}}_{\text {H}}/2{M^{\text {new}}}^2R^{\text {new}}_{\text {H}}$$\end{document} [that is, after the adiabatic assimilation of the ring by the central black hole. Here JHnew=JR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{\text {new}}_{\text {H}}=J_{\text {R}}$$\end{document}, Mnew\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^{\text {new}}$$\end{document}, and RHnew\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^{\text {new}}_{\text {H}}$$\end{document} 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 ΩHBH-ring=ΩHBH-ring(JH,JR,R→RH+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^{\text {BH-ring}}_{\text {H}}=\Omega ^{\text {BH-ring}}_{\text {H}}(J_{\text {H}},J_{\text {R}},R\rightarrow R^{+}_{\text {H}})$$\end{document} of generic (that is, with JH≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{\text {H}}\ne 0$$\end{document}) black-hole–ring configurations. Remarkably, we find the simple universal relation ΔΩH≡ΩHBH-ring(JH,JR,R→RH+)-ΩHKerr(JH)=JR/4M3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \Omega _{\text {H}}\equiv \Omega ^{\text {BH-ring}}_{\text {H}}(J_{\text {H}},J_{\text {R}},R\rightarrow R^{+}_{\text {H}})-\Omega ^{\text {Kerr}}_{\text {H}}(J_{\text {H}})={{J_{\text {R}}}/{4M^3}}$$\end{document} 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.

(J H = 0, J R , R) = 2J R /R 3 for the angular velocity of a central black hole with zero angular momentum, where J 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 BH-ring H (J H = 0, J R , R) implies a smooth transition of the central black-hole angular velocity from its asymptotic nearhorizon value BH Remarkably, we find the simple universal relation H ≡

Introduction
The gravitational two-body problem has attracted much attention over the years from both physicists and mathematicians. In particular, it is highly important to explore the physics of a central black hole surrounded by an orbiting ring: it is expected that this composed two-body system may be formed as an intermediate stage in the gravitational collapse of a compact spinning star to form a black hole [1][2][3]. Likewise, the coalescence of two compact objects may produce a composed black-hole-ring system [1][2][3]. In addition to these astrophysical motivations, it is highly interesting to explore the composed black-hole-ring system in order to understand how an exterior matter configuration affects the physical properties of central black holes [1][2][3][4][5].
The general-relativistic problem of a slowly spinning black hole surrounded by a thin orbiting ring was studied perturbatively by Will [4,5] (see also [6][7][8]). It was shown in [4] that, due to the well-known phenomenon of dragging of inertial frames by the orbiting ring, the angular velocity BH-ring H 1 of the central black hole in the composed black- 1 We use the symbol BH-ring H to denote the angular velocity of the black-hole horizon in the composed black-hole-ring system. hole-ring system is no longer related to the black-hole angular momentum J H by the simple Kerr-like (vacuum) relation [Here M and R H = M + (M 2 − J 2 H /M 2 ) 1/2 are the mass and horizon-radius of the black hole, respectively]. In particular, Will [4] has demonstrated explicitly that, in the composed black-hole-ring system, one can have a central black hole with zero angular momentum but with a non-zero angular velocity 2 where J R and R are, respectively, the angular momentum of the orbiting ring and its proper circumferential radius. To the best of our knowledge, no exact (analytical) calculations of the frame-dragging effect have been performed for generic black-hole-ring configurations (that is, for the case of central black holes with non-negligible angular momenta).

The continuous (smooth) behavior of the black-hole angular velocity
The main goal of the present paper is to generalize the result (2) of [4] to the regime of composed black-hole-ring configurations in which the central black holes possess non-zero angular momenta. In particular, we shall use a simple continuity argument in order to provide a concrete analytical prediction for the angular-velocity/angular-momentum asymptotic functional relation H ) of generic (that is, with J H = 0) central black holes in the composed black-hole-ring system.
Our approach here is based on a continuity argument for the behavior of the black-hole angular velocity in an adiabatic process in which the orbiting ring is assimilated (adiabatically lowered) into the central black hole. In order to demonstrate the idea, we shall first analyze the analytical relation (2) of [4] for the angular velocity of a zero angular momentum (J H = 0) central black hole.
Let us first consider a sequence of black-hole-ring configurations with adiabatically varying (decreasing) circumferential radii. Inspection of Eq. (2) reveals that, for a given value of the ring angular momentum J R , the central blackhole angular velocity increases as the ring approaches the black-hole horizon (that is, as R decreases). In particular, taking the near-horizon limit R → R + H in (2), one finds 3

BH-ring H
for the angular velocity of the central black hole just before it assimilates the ring. Let us now calculate the new angular velocity Kerr H (J new H ) of the resulting Kerr (vacuum) black hole after it absorbed the ring. The adiabatic assimilation of the rotating ring by the central black hole produces the following changes in the black-hole physical parameters: where the energy E R of the rotating ring at the absorption point R = R H is given by [9] 4,5,6 Substituting (4) and (5) into (1), one finds (see footnote 3) Kerr for the angular velocity of the final (vacuum) Kerr black hole. 7 Comparing the near-horizon asymptotic (R → R + H ) expression (3) for the angular velocity of the central black hole in the composed black-hole-ring system just before the assimilation of the ring, with Eq. (6) for the angular velocity of the resulting Kerr (vacuum) black hole after it assimilated the ring, one concludes that the black hole is characterized by a smooth (continuous) evolution of its angular velocity during the adiabatic assimilation process. That is, 3 Here we have used the leading-order relation R H = 2M for the horizon-radius of the zero angular-momentum black hole. 4 We consider here an adiabatic assimilation process of the ring into the central black hole. In this idealized scenario the assimilated ring has a zero radial momentum at the point of capture. 5 It is worth emphasizing that the mass-energy of the ring as measured by asymptotic observers, √ g 00 μ, is completely red-shifted ( √ g 00 = 0) at the absorption point (at the black-hole horizon). 6 This results is valid in the perturbative regime J R /R 2 1. There are also small (subleading) correction terms of order O(J 2 R /R 3 , μ 2 /R) on the r.h.s. of this relation. 7 Note that the black-hole surface area 8π M R H plays the role of an adiabatic invariant (see, Bekenstein [10]). It therefore acquires small (subleading) O(J 2 R /M 2 , μ 2 ) corrections, which are only second-order in the ring parameters.

The angular-velocity/angular-momentum relation for generic black-hole-ring configurations
In the present section we shall analyze the angular-velocity/ angular-momentum asymptotic functional relation for the evolution of the black-hole angular velocity during an adiabatic assimilation process of the orbiting ring by the central black hole. We consider a composed black-hole-ring system which is characterized by the physical parameters J H , J R , and R. The adiabatic absorption of the ring by the central black hole produces a final Kerr (vacuum) black hole with the following parameters: where H (R → R + H ) ≡