Dynamics of accretion discs - the force analysis and related conformal geometries

Fluid circling in gravitational and electromagnetic fields around a central compact object can form an equilibrium toroidal structure - a scenario representing a basic model for studying accretion discs orbiting around a black holes or compact stars. For mapping of possible typical shapes and physical properties of such structures, we commonly use a general relativistic magneto-hydrodynamic model based on the energy-momentum conservation written in a standard representation, which works for neutral as well as for electrically charged fluids. Moreover, we introduce this model in terms of two covariant force representations, both based on a proper hypersurface projection of the energy-momentum conservation. Then, space-like forces following from a decomposition of the fluid four-acceleration can be defined in the related hypersurface. These representations provide us with an insight into a fluid flow. Moreover, they are also well reflected in the related conformal hypersurface geometries; especially, behavior of the centrifugal forces is directly related to geodesics of the conformal hypersurfaces and their embedding diagrams. In this respect, we present a correspondence between the charged fluid flow world-lines from an ordinary spacetime and the world-lines determined by the charged test particles equation of motion in a conformal spacetime. The introduced force formalism is very general, i.e. it is not restricted only to the circling fluids. Since it is based on the 3+1 splitting of the fundamental equations, it is very convenient for a general fluid flow investigation where an application of numerical procedures is necessary. We illustrate the most important results by considering the circling fluid taking shape of a torus settled in the equatorial plane of the Schwarzschild spacetime accompanied by an asymptotically uniform magnetic field.