The stress at the ISCO of black-hole accretion discs is not a free parameter

Radiation from the "plunging region" (the space between the innermost stable circular orbit (ISCO) and the black-hole (BH) surface) of an accretion flow onto a BH is supposed to add a small but significant contribution to the X-ray spectra of X-ray binary systems. The plunging region and its electromagnetic emission has been recently described by numerical and analytic calculations which lead to the conclusion that in the plunging region radiation is generated by the energy release through the action of a "leftover" stress in the vicinity of the ISCO and that the amount of this leftover can be chosen as a free parameter of the accretion-flow description. The present article aims to demonstrate that this is not true because the stress in the whole accretion flow onto a black hole is determined by its global transonic character enforced by the space-time structure of the accreting black hole. We use the slim-disc approximation (SDA) to illustrate this point. In our article we compare models obtained with the SDA with results of numerical simulations and of analytical models based on the assumption that accreting matter flows along geodesics. We show that the latter cannot describe adequately the physics of astrophysical accretion onto a BH because 1. particles on geodesics cannot emit electromagnetic radiation, 2. they ignore the global transonic character of the accretion flow imposed by the presence of a stationary horizon in the BH spacetime; a presence that fixes a unique value of the angular momentum at the BH surface for a solution to exist. Therefore the fact that geodesics-based models reproduce the trans-ISCO behaviour of GRRMHD simulations proves the physical reality of neither. We show also that the claimed detection of plunging-region emission in the spectrum of an X-ray binary is unsubstantiated.