The Hawking cascade from a black hole is extremely sparse

The Hawking flux from a black hole, (at least as seen from asymptotic infinity), is extremely sparse and thin, with the average time between emission of the successive Hawking quanta being many times larger than the natural timescale set by the energies of the emitted quanta. While this result has been known for over 30 years, it has largely been forgotten. We shall focus on the early-stage low-temperature regime, and shall confront numerical estimates with semi-analytic approximations based on a naive Planck spectrum. First we shall identify several natural dimensionless figures of merit, and thereby compare the mean time between emission of successive Hawking quanta to several distinct but quite natural timescales that can be associated with the emitted quanta, demonstrating that very large ratios are typical for emission of massless quanta from a Schwarzschild black hole. Furthermore these ratios are independent of the mass of the black hole as it slowly evolves. We shall then show that the situation for the more general Reissner-Nordstrom and generic "dirty'' black holes is even worse, at least as long as the surrounding matter satisfies some suitable energy conditions. The situation for the Kerr and Kerr--Newman black holes (or even for charged particle emission from a Reissner--Nordstrom black hole) is considerably trickier, and depends on a careful accounting of the super-radiant modes. Overall, the Hawking quanta are seen to be dribbling out of the black hole one at a time, in an extremely slow cascade of 2-body decays. Among other things, this implies that the Hawking flux is subject to "shot noise''. Observationally, the Planck spectrum of the Hawking flux can only be determined by collecting and integrating data over a very long timescale. We conclude by connecting these points back to various kinematic aspects of the Hawking evaporation process.