Zero-torque inner boundary condition

The zero-torque inner boundary condition sets the viscous torque to zero where nearly circular disk flow joins a plunging or stress-free region.

For constant kinematic viscosity $\nu$, use the phenomenological viscous torque in an accretion disk, specific angular momentum $h=rc\sqrt{m/(r-3m)}$, and $h_0=\sqrt{12}mc$ at the innermost stable circular orbit. Neglect radial kinetic motion in the clock relation and neglect differential light travel time. The Schwarzschild time to drift formally from $r_1\ge 6m$ to $6m$ is the displayed expression, withIndeed $\mathcal{T}(2)=0$ and ${\mathcal{T}}^{\prime }(X)=6X[X-2\sqrt{X-1}]/(X-1)\ge 0$ for $X\ge 2$. The zero-torque inner boundary condition makes the formal drift speed diverge as $r\downarrow 6m$, so the nearly circular approximation must give way to the plunging region of a black-hole accretion disk sufficiently close to that orbit. This formula describes the stipulated approximation, not an exact relativistic flow solution.

The plunging region begins near the innermost stable circular orbit, where nearly circular motion becomes unstable and radial inflow becomes rapid. If the inflow time is shorter than the stress-transport time, the gas approximately conserves its specific angular momentum there.

A thin disk with alpha disk viscosity $\nu \sim \alpha H^2\Omega$ has a viscous timescale much longer than its dynamical infall time in the plunging region of a black-hole accretion disk. At fixed nonzero accretion rate, rapid inflow reduces its surface density of a disk. The resulting viscous torque in an accretion disk is smaller by the displayed factor than in the slowly drifting circular disk. This estimate concerns radii outside the narrow matching layers.