7. Fundamental unsolved problems

Analytic models assume the zero torque inner boundary condition: the viscous torque at the inner edge of a very thin accretion disc is vanishingly small. This assumption has been challenged by Krolik and others but defended by Paczynski and collaborators.

A very important numerical work that supports Paczynski's point of view was published in 2008 by Shafee and others. It describes the results of three-dimensional general relativistic magnetohydrodynamical simulations of a geometrically thin accretion disc around a non-spinning black hole. The disc has a relative thickness h ~0.005-0.1 over the radius range . In steady state, the specific angular momentum profile of the inflowing magnetized gas deviates by less than 2% from that of the standard thin disc model with the zero-inner-torque assumed. In addition, the magnetic torque at ISCO is only ~2% of the inward flux of the angular momentum at that radius. Both results indicate that magnetic coupling across the inner edge is relatively unimportant for geometrically thin discs around non-spinning black holes, which is in accordance with Paczynski's ideas. However, until a mathematically correct analytic model describing thin MHD accretion flows near the ISCO becomes available, the controversy is likely to continue.

The unresolved issue of the zero torque inner boundary condition is technically rather difficult, mostly because the coupled non-linear differential equations that describe the accretion flow have multiple critical points near . The issue is also fundamentally important because all detailed comparisons between theory and observations performed to date were based on the correctness of the zero torque inner boundary condition. A related fundamental problem, whether a non-zero torque at the event horizon may couple black holes with infalling matter, is also unresolved. While some authors claim that an electrodynamic torque is possible, others argue that "black holes have no hair" and thus, in particular, do not anchor magnetic fields. Therefore, any change in the three properties of a black hole (mass, angular momentum, charge) may occur only by a direct capture of matter by the black hole. Matter may gain some rotational energy of the black hole only if a particle or photon with negative energy would cross the event horizon.

Both thin and thick discs models are stationary and axially symmetric. They usually describe matter in the hydrodynamical approximation (with no magnetic field). In addition, however, it is still unknown whether it is physically legitimate to make all these assumptions and suppose that in some "averaged" sense accretion flows may be (approximately) described in terms of stationary and axially symmetric hydrodynamical equilibria. While observations seem to suggest that many real astrophysical sources experience periods in which this may be quite reasonable, several authors point out that results of recent numerical simulations indicate that the MRI and other instabilities could make the accretion flows genuinely non-steady and non-symmetric, and that the very concept of separate timescales may be questionable in the sense that locally it could be .

• S. Kato Basic Properties of Thin-Disk Oscillations Publ. Astr. Soc. Japan, 53, 1-24 (2001)