**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

^{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

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 of the black-hole horizon in the composed black-hole–ring system is *no* longer related to the black-hole angular momentum by the simple Kerr-like (vacuum) relation (here *M* and 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 for the angular velocity of a central black hole with zero angular momentum, where 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 implies a *smooth* transition of the central black-hole angular velocity from its asymptotic near-horizon value (that is, just *before* the assimilation of the ring by the central black hole), to its final Kerr (vacuum) value [that is, *after* the adiabatic assimilation of the ring by the central black hole. Here , , and 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 of *generic* (that is, with ) black-hole–ring configurations. Remarkably, we find the simple *universal* relation 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.

*© SIF and Springer-Verlag Berlin Heidelberg, 2015*