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

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 BH-ring H of the black-hole horizon in the composed black-hole-ring system is no longer related to the black-hole angular momentum J H by the simple Kerr-like (vacuum) relation Kerr H (J H ) = J H /2M 2 R H (here M and R 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 BH-ring H(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 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 a e-mail: shaharhod@gmail.com ring into the central black hole) yields a concrete prediction for the angular-velocity/angular-momentum asymptotic functional relationof generic (that is, with J H = 0) black-hole-ring configurations. Remarkably, we find the simple universal rela-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.

Frame-Dragging: Meaning, Myths, and Misconceptions

“…In fact, these solutions are stationary. The impact of ring/disk's rotation is (via the dragging of the ZAMOs) causing a black hole with zero angular momentum to have a non-zero horizon angular velocity [18,45,72] and, conversely, a non-rotating black hole to have a non-zero angular momentum [45,72]-thereby even introducing a certain ambiguity in a black hole's spinning/non-spinning character. Let us see this in detail.…”

Perturbation of a Schwarzschild Black Hole Due to a Rotating Thin Disk

“…2 Recently the Will's black-hole-ring problem has been revisited by Sano & Tagoshi (2014), but using the perturbation approach of Chrzanowski, Cohen and Kegeles in which the metric is found on the basis of solution of the Teukolsky equation for the Weyl scalars. The Will's results have also been followed by Hod who analyzed the behaviour of the innermost stable circular orbit in the black-hole-ring field (Hod 2014) and the relation between the angular velocity of the horizon and the black-hole and ring angular momenta (Hod 2015).…”