A Model of the Collapse and Evaporation of Charged Black Holes

Abstract: In this paper, a natural generalization of KMY model is proposed for the evaporation of charged black holes. Within the proposed model, the back reaction of Hawking radiation is considered. More specifically, we consider the equation $G_{\mu\nu} = 8\pi \langle T_{\mu\nu}\rangle$, in which the matter content $\langle T_{\mu\nu}\rangle$ is assumed spherically symmetric. With this equation of motion, the asymptotic behavior of the model is analyzed. Two kinds of matter contents are taken into consideration in this paper. In the first case (the thin-shell model), the infalling matter is simulated by a null-like charged sphere collapsing into its center. In the second case, we consider a continuous distribution of spherical symmetric infalling null-like charged matter. It is simulated by taking the continuous limit of many co-centric spheres collapsing into the center. We find that in the thin-shell case, an event horizon forms and the shell passes through the horizon before becoming extremal, provided that it is not initially near-extremal. In the case of continuous matter distribution, we consider explicitly an extremal center covered by neutral infalling matter and find that the event horizon also forms. The black hole itself will become near-extremal eventually, leaving possibly a non-electromagnetic energy residue less than the order of $\ell_{p}^{4}/e_{0}^{3}$. The details of the behavior of these models are explicitly worked out in this paper.