Viscous shear in the Kerr metric

Models of viscous flows on to black holes commonly assume a zero-torque boundary condition at the radius of the last stable Keplerian orbit. It is here shown that this condition is wrong. The viscous torque is generally non-zero at both the last stable orbit and the horizon itself. The existence of a non-zero viscous torque at the horizon does not require the transfer of energy or angular momentum across any spacelike distance, and so does not violate causality. Further, in comparison with the viscous torque in the distant, Newtonian regime, the viscous torque on the horizon is often reversed, so that angular momentum is viscously advected inwards rather than outwards. This phenomenon is first suggested by an analysis of the quasi-stationary case, and then demonstrated explicitly for a series of cold, dynamical flows which fall freely from the last stable orbit in the Schwarzschild and Kerr metrics. In the steady flows constructed here, the net torque on the hole is always directed in the usual sense; any reversal in the viscous torque is offset by an increase in the convected flux of angular momentum. Finally, a theorem relating the specific energy to the angular velocity in hot, viscously driven flows is derived and applied to a simple disc model.