Abstract
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 Poincaré 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 Mi 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 ∼ {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 Poincaré 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 are of order unity in Planck units. At large l, there is a power law spectrum of fluctuations that extends up to l ∼ {M}_i^2/M , beyond which the fluctuations fall off exponentially, with corresponding total rms shear tensor fluctuations ∼ MiM−3/2.
Highlights
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 Poincaré 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
The results are, in Planck units: (i) The black hole position has a uncertainty of ∼ Mi2 at late times, where Mi is the initial mass. (ii) The black hole mass M has an uncertainty of order the mass M itself at the epoch when M ∼ Mi2/3, well before the Planck scale is reached
Summary
We define a stochastic process that gives a crude model of the evaporation of a black hole, including fluctuations in its Poincaré conserved charges. It incorporates only one spatial dimension, and treats only the Poincaré conserved charges of the black hole, neglecting the additional charges associated with the BMS algebra and its extensions The key feature of the model is that independent, uncorrelated fluctuations are introduced into the black holes conserved charges at each timestep or each emission event. Those fluctuations originate in incoming modes of quantum fields at past null infinity, which are orthonormal and so uncorrelated for the incoming vacuum state
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