Semiclassical geometry of charged black holes

At the classical level, two-dimensional dilaton gravity coupled to an abelian gauge field has charged black hole solutions, which have much in common with four-dimensional Reissner-Nordström black holes, including multiple asymptotic regions, timelike curvature singularities, and Cauchy horizons. The black hole spacetime is, however, significantly modified by quantum effects, which can be systematically studied in this two-dimensional context. In particular, the back-reaction on the geometry due to pair-creation of charged fermions destabilizes the inner horizon and replaces it with a spacelike curvature singularity. The semiclassical geometry has the same global topology as an electrically neutral black hole.

Figure 1

(a) Density plot of the area function $\psi$ of a black hole formed by the gravitational collapse of charged matter. The curves indicate the apparent horizon (thin) and the curvature singularity (thick). (b) Density plot of the bosonized matter field $Z$. Screening due to pair-creation prevents the electric charge from penetrating into the strong-coupling region.

Figure 2

(a) $\psi$ and $\xi$ plotted as a function of $y$ for a classical black hole solution. The two horizons are at the zeroes of $\xi$ and the curvature singularity is where $\psi$ goes to zero. (b) Semiclassical black hole. The inner horizon is replaced by a singularity where $\psi$, $\xi$, and $Z$ all approach zero.