A fully relativistic twisted disc around a slowly rotating Kerr black hole: derivation of dynamical equations and the shape of stationary configurations

Abstract

(abbreviated) In this paper we derive equations describing dynamics and stationary configurations of a twisted fully relativistic thin accretion disc around a slowly rotating black hole. We find that the disc dynamics and stationary shapes are determined by a pair of equations for two complex variables describing orientation of the disc rings and velocity perturbations in the disc. We analyse shapes of stationary twisted configurations. It is shown that the stationary configurations depend on two parameters - the $\alpha $parameter and $\tilde \delta = \delta_{*}/\sqrt a$, where $\delta_{*}\sim h/r$ is the disc opening angle (h is the disc halfthickness) and $a$ is the black hole rotational parameter. When $a > 0$ and $\tilde \delta \ll 1$ the shapes depend drastically on value of $\alpha$. When $\alpha $ is small the disc inclination angle oscillates with radius with amplitude and radial frequency of the oscillations dramatically increasing towards the last stable orbit. For moderate values of $\alpha $ the oscillations do not take place but the disc does not align with the equatorial plane at small radii. Its inclination angle is either increasing towards $R_{ms}$ or exhibits a non-monotonic dependence on the radial coordinate. Finally, when $\alpha $ is sufficiently large the disc aligns with the equatorial plane at small radii. When $a < 0$ the disc aligns with the equatorial plane for all values of $\alpha $. The results reported here may have implications for determining structure and variability of accretion discs close to $R_{ms}$ as well as for modelling of emission spectra coming from different sources, which are supposed to contain black holes.