Stimulated-emission effects in particle creation near black holes

It has recently been shown that if a black hole is formed by gravitational collapse, spontaneous particle creation will occur and a thermal spectrum of all species of particles will be emitted to infinity if the quantum matter was initially in the vacuum state. In this paper we investigate the stimulated-emission effects which occur if particles are present initially. We show in general that for a Hermitian scalar field in an external potential or in curved, asymptotically flat spacetime, stimulated-emission effects can occur precisely in those modes for which there is spontaneous particle creation from the vacuum. For the case of a Schwarzschild black hole, this result appears paradoxical, since spontaneous emission occurs at late times but there is no classical analog of stimulated emission at late times. The resolution of this paradox is that in order to induce emission of particles which emerge at late times one must send in particles at early times, so that they reach the black hole very near the instant of its formation. However, enormous energy is required of these incoming particles in order to stimulate emission of particles which emerge at late times. Thus, for a Schwarzschild black hole, even if particles are initially present (with limited energy) they will induce emission only at early times; at late times one will see only the spontaneously emitted blackbody thermal radiation. For the case of a Kerr black hole stimulated emission can be induced by particles sent in at late times with the appropriate frequencies and angular dependence. If the number of incoming particles is large, this quantum stimulated emission just gives the classical superradiant scattering.