Einstein $A$ and $B$ coefficients for a black hole

By quantum calculations in a classical background geometry, Hawking has shown that an isolated black hole emits thermal radiation spontaneously. Starting from Hawking's expectation value for the number of quanta emitted per mode, and using methods from statistical thermodynamics, one of us calculated earlier the probability distribution for the number of quanta per mode outgoing from a black hole placed in a thermal radiation bath. By the same methods we show here that this probability is not simply the combination of that for Hawking's spontaneous emission and that for pure scattering. From this we infer the existence of stimulated emission in all modes, even those which do not superradiate. We derive the probability that $m$ quanta go out in a given mode when precisely $n$ are incident. It satisfies a symmetry condition originally given by Hartle and Hawking for a special case. For all modes the average number of outgoing quanta contains a contribution from stimulated emission which shows up as a negative contribution to the effective absorptivity $\Gamma$. The situation is analogous to that for opacity in the theory of radiative transport. Superradiance occurs for modes in which the negative contribution dominates the pure absorptivity. We identify the Einstein $A$ and $B$ coefficients for a black hole. The $B$ coefficients satisfy the usual relation from atomic physics with the role of degeneracy factor played by the exponential of black-hole entropy. This agrees with the statistical interpretation of this quantity in terms of internal black-hole configurations. The relation between the $B$ coefficients suggests time reversibility of the radiative aspect of a black hole. This supports Hawking's view that a black hole and a white hole are essentially the same thing.