Black holes and the adiabatic phase

An open system consisting of a scalar field bound to a Kerr black hole whose mass M and specific angular momentum a are slowly (adiabatically) perturbed is considered. The adiabatically induced phase and the conditions for the validity of the adiabatic approximation are obtained. The effect of closed cycles in parameter space (the a - M plane) on the energy levels of both stable and unstable scalar-field bound states, together with other quantities of interest, is illustrated. Lastly it is noted that the black-hole wavefunction will acquire an equal and opposite phase to that of matter, thus leading to a change of its effective action (entropy).