Magnetorotational Instability around a Rotating Black Hole

Abstract

The magnetorotational instability (MRI) in the Kerr spacetime is studied on a 3 + 1 viewpoint. Maxwell's equations are expressed in a circularly orbiting observer's frame that corotates with matter in Keplerian orbits. The hydromagnetic equations are represented in a locally nonrotating frame (LNRF). There exist large proper growth rates in the MRI around a rapidly rotating black hole. The large ``centrifugal force'' and the rapid variations of magnetic fields are caused by the rotation of the spacetime geometry. As a result, in the extreme Kerr case the maximum proper growth rate at r = r(ms) becomes about 12 times as large as that in Schwarzschild case, where rms is the radius of a marginally stable orbit. The unstable range of the wavenumber expands according to the rotational speed of the spacetime geometry. The overstable mode of the instability becomes remarkable when the circular motion of the disk is quasi-relativistic and the strength of the magnetic field is so large that the Alfven velocity is nu(A) >= 0:1c. When the waves of perturbations propagate in the radial direction, these waves oscillate, and their amplitudes grow exponentially. This instability is caused by the differential rotation of the circularly orbiting frame. The universality of the local Oort A-value of the disk is discussed in curved spacetime. In the extreme Kerr geometry, the amplitude of the maximum growth rate in the dynamical shear instability (DSI) reaches to infinity at r = r(ms). The behaviors of maximum growth rates in MRI and DSI are remarkably different from each other.