Hawking radiation sets stringent constraints on primordial black holes (PBHs) as a dark matter candidate in the $M\sim {10}^{16}\mathrm{g}$ regime based on the evaporation products such as photons, electrons, and positrons. This motivates the need for rigorous modeling of the Hawking emission spectrum. Using semiclassical arguments, Page [Phys. Rev. D 16, 2402 (1977)] showed that the emission of electrons and positrons is altered due to the black hole acquiring an equal and opposite charge to the emitted particle. The Poisson fluctuations of emitted particles cause the charge $Z|e|$ to random walk, but since acquisition of charge increases the probability of the black hole emitting another charged particle of the same sign, the walk is biased toward $Z=0$, and $P(Z)$ approaches an equilibrium probability distribution with finite variance $⟨Z^2⟩$. This paper explores how this “stochastic charge” phenomenon arises from quantum electrodynamics (QED) on a Schwarzschild spacetime. We prove that (except for a small Fermi blocking term) the semiclassical variance $⟨Z^2⟩$ agrees with the variance of a quantum operator $⟨{\widehat{\mathcal{Z}}}^2⟩$, where $\widehat{\mathcal{Z}}$ may be thought of as an “atomic number” that includes the black hole as well as the charge near it (weighted by a factor of $2M/r$). In QED, the fluctuations in $\widehat{\mathcal{Z}}$ do not arise from the black hole itself (whose charge remains fixed), but rather as a collective effect in the Hawking-emitted particles mediated by the long-range electromagnetic interaction. We find the rms charge $⟨Z^2{⟩}^{1/2}$ asymptotes to 3.44 at small PBH masses $M\lesssim 2\times {10}^{16}\mathrm{g}$, declining to 2.42 at $M=5.2\times {10}^{17}\mathrm{g}$.

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• Quantum aspects of black holes
• Quantum fields in curved spacetime
• Primordial black holes