Particle emission rates from a black hole. III. Charged leptons from a nonrotating hole

Abstract

The Hawking emission rates from a nonrotating black hole of small charge are calculated for electrons and muons and their antiparticles. During the stochastic emission of these charged leptons, the charge of the hole fluctuates. Assuming that the only type of charged particle emitted significantly by a hole of mass $M$ is one of these spin-1/2 species with mass $\ensuremath{\mu}$ and charge $e$, the probability distribution for the charge of the hole is computed for $0 \ensuremath{\le} \frac{\mathrm{GM}\ensuremath{\mu}}{\ensuremath{\hbar}c} \ensuremath{\le} 0.4$. The rms value varies from $6.14e$ for $\frac{\mathrm{GM}\ensuremath{\mu}}{\ensuremath{\hbar}c} = 0$ to $2.76e$ for $\frac{\mathrm{GM}\ensuremath{\mu}}{\ensuremath{\hbar}c}=0.4$ and is predicted to be $2.34e$ for $\frac{\mathrm{GM}\ensuremath{\mu}}{\ensuremath{\hbar}c}>>1$. The electrostatic attraction between the emitted particle and its antiparticle, along with the charge fluctuations, causes the average emission rate and power to be lower than for otherwise-similar uncharged particles. This effect of the charge is calculated (ignoring radiative and self-energy corrections, which are of the same order in $e$) to be a few percent, depending upon $\frac{\mathrm{GM}\ensuremath{\mu}}{\ensuremath{\hbar}c}$. The particle rest mass $\ensuremath{\mu}$ also impedes the emission, but by factors which can become much greater for a large enough hole: The average power is reduced to 50% of its value for massless spin-1/2 particles at $\frac{\mathrm{GM}\ensuremath{\mu}}{\ensuremath{\hbar}c} = 0.160$ ($M = 8.33\ifmmode\times\else\texttimes\fi{}{10}^{16}$ g for electrons, 4.03 \ifmmode\times\else\texttimes\fi{} ${10}^{14}$ g for muons) and to 10% at $\frac{\mathrm{GM}\ensuremath{\mu}}{\ensuremath{\hbar}c} = 0.271$ ($M = 1.41\ifmmode\times\else\texttimes\fi{}{10}^{17}$ g for electrons, 6.82 \ifmmode\times\else\texttimes\fi{} ${10}^{14}$ g for muons). It is estimated that muons and heavier particles would contribute about 14% of the power of a nonrotating black hole of $M = 5\ifmmode\times\else\texttimes\fi{}{10}^{14}$ g, helping it to decay away in nearly 16 \ifmmode\times\else\texttimes\fi{} ${10}^{9}$ yr, roughly the present age of the universe.