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A twisted disc forms around a rotating black hole each time when the disc outskirts are not aligned with the black hole’s equatorial plane. We derive equations describing the evolution of the shape of twisted discs and perturbations of density and velocity necessarily arising in such a disc. This is done under the following simplifying assumptions: a small aspect ratio of the disc, a slow rotation of the black hole, and a small tilt angle of the disc rings with respect to the black hole equatorial plane. Nevertheless, the GR effects are considered accurately. Additionally, an analysis of particular regimes of non-stationary twist dynamics (the wave and diffusion regimes) is presented both in the framework of the Newtonian dynamics and taking into account Einstein’s relativistic precession. At the end of the chapter, a calculation of the shape of a stationary relativistic twisted accretion disc for different values of free parameters of the model is done.

In this chapter we present a model of a standard accretion disc around a rotating black hole taking general relativity effects into full account. This model was first described in the paper by Novikov and Thorne (Black holes (Les Astres Occlus). Gordon and Breach, New York, 1973) and has since then been used in many studies to obtain convincing evidence of the existence of black holes, in both stellar binary systems and active galactic nuclei. It remains topical since a full account of the general relativistic properties of the motion of matter in the disc, and the generation of disc emission, allows the position of the inner disc radius and hence the black hole spin to be inferred from observations. In addition, the standard accretion disc is the basis for more complicated theories of warped (twisted) accretion discs, which are formed when the accreting matter moves outside the equatorial plane of a rotating black hole.