Infrared Effects in the Late Stages of Black Hole Evaporation
As a black hole evaporates, each outgoing Hawking quantum carries away some of the black holes asymptotic charges associated with the extended Bondi-Metzner-Sachs group. These include the Poincar\'e charges of energy, linear momentum, intrinsic angular momentum, and orbital angular momentum or center-of-mass charge, as well as extensions of these quantities associated with supertranslations and super-Lorentz transformations, namely supermomentum, superspin and super center-of-mass charges (also known as soft hair). Since each emitted quantum has fluctuations that are of order unity, fluctuations in the black hole's charges grow over the course of the evaporation. We estimate the scale of these fluctuations using a simple model. The results are, in Planck units: (i) The black hole position has a uncertainty of $\sim M_i^2$ at late times, where $M_i$ is the initial mass (previously found by Page). (ii) The black hole mass $M$ has an uncertainty of order the mass $M$ itself at the epoch when $M \sim M_i^{2/3}$, well before the Planck scale is reached. Correspondingly, the time at which the evaporation ends has an uncertainty of order $\sim M_i^2$. (iii) The supermomentum and superspin charges are not independent but are determined from the Poincare charges and the super center-of-mass charges. (iv) The supertranslation that characterizes the super center-of-mass charges has fluctuations at multipole orders $l$ of order unity that that are of order unity in Planck units. At large $l$, there is a power law spectrum of fluctuations that extends up to $l \sim M_i^2/M$, beyond which the fluctuations fall off exponentially, with corresponding total rms shear tensor fluctuations $\sim M_i M^{-3/2}$.