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

In this paper we derive equations describing the dynamics and stationary configurations of a twisted fully relativistic thin accretion disc around a slowly rotating black hole. We assume that the inclination angle of the disc is small and that the standard relativistic generalization of the alpha model of accretion discs is valid when the disc is flat. We find that similar to the case of non-relativistic twisted discs the disc dynamics and stationary shapes can be determined by a pair of equations formulated for two complex variables describing the orientation of the disc rings and velocity perturbations induced by the twist. We analyse analytically and numerically the shapes of stationary twisted configurations of accretion discs having non-zero inclinations with respect to the black hole equatorial plane at large distances r from the black hole. It is shown that the stationary configurations depend on two parameters - the viscosity parameter a and the parameter (delta) over tilde = delta(*)/root a, where delta(*) is the opening angle (delta(*) similar to h/r, where h is the disc half-thickness and r is large) of a flat disc and a is the black hole rotational parameter. When a > 0 and (delta) over tilde << 1 the shapes depend drastically on the 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, R(ms). When alpha has a moderately small value the oscillations do not take place but the disc does not align with the equatorial plane at small radii. The disc inclination angle either is 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 the 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.