Pressure anisotropy and magnetic field geometry in magnetorotational instability of Kerr black hole accretion disks

We formulate the dispersion relation for magnetorotational instability (MRI) in the accretion disk of a Kerr black hole, incorporating general relativistic effects and pressure anisotropy. By linearizing the general relativistic magnetohydrodynamic (GRMHD) equations in Boyer-Lindquist coordinates, we derive the MRI dispersion relation while explicitly accounting for frame-dragging effects, the gravitational potential, and the interaction between the magnetic field and the rotating plasma. Our analysis considers both toroidal and poloidal magnetic field components, allowing us to explore how different field geometries influence the MRI growth rate across three regimes corresponding to varying frame-dragging effects. The results show that the MRI growth rate is strongly influenced by plasma beta, pressure anisotropy, and the black hole spin parameter. Specifically, we find that pressure anisotropy alters the MRI dispersion relation by introducing additional instability criteria, which can either enhance or suppress MRI growth, depending on the alignment of the magnetic field components. These findings have important implications for electron acceleration in black hole accretion disks, as MRI-driven turbulence plays a key role in energy dissipation and particle energization. Our results provide a theoretical foundation for understanding plasma instabilities in relativistic accretion flows and their impact on high-energy astrophysical phenomena.