A Constraint of Black Hole Mass and the Inner Edge Radius of Relativistic Accretion Disc

A constraint to black hole (BH) accretion has previously been derived for the inner edge fixed at the innermost stable circular orbit (ISCO) and the innermost bound circular orbit (IBCO). This constraint is referred to as the mass-radius (MR) relation in this study, and the validity of the MR relation is discussed for different cases. It is shown that the product of the BH mass and the inner edge radius decreases monotonically in the accretion process for the inner edge located between IBCO and ISCO. In addition, we discuss the validity of the MR relation by considering the magnetic coupling (MC) effects of a Kerr BH with its surrounding disc. Although theoretically the product of the BH mass and the radius of ISCO increases (decreases) with time for a BH spin greater (less) than some critical value in the MC process, this relation is approximately valid for an Eddington accretion rate persisting for a rather long time, such as more than 10⁶ years. Finally, we discuss the possible application of the MR relation to astrophysics.