Global geometry of two-dimensional charged black holes

The semiclassical geometry of charged black holes is studied in the context of a two-dimensional dilaton gravity model where effects due to pair-creation of charged particles can be included in a systematic way. The classical mass-inflation instability of the Cauchy horizon is amplified and we find that gravitational collapse of charged matter results in a spacelike singularity that precludes any extension of the spacetime geometry. At the classical level, a static solution describing an eternal black hole has timelike singularities and multiple asymptotic regions. The corresponding semiclassical solution, on the other hand, has a spacelike singularity and a Penrose diagram like that of an electrically neutral black hole. Extremal black holes are destabilized by pair-creation of charged particles. There is a maximally charged solution for a given black hole mass but the corresponding geometry is not extremal. Our numerical data exhibits critical behavior at the threshold for black hole formation.

Figure 1

The Penrose diagram of a classical $1+1$-dimensional charged black hole is the same as for a Reissner-Nordström black hole in $3+1$ dimensions. The thick lines represent the timelike singularities and the dashed lines are the horizons. The structure repeats itself in the vertical direction.

Figure 2

The Penrose diagram for a semiclassical charged black hole in $1+1$ dimensions is the same as that of a $3+1$-dimensional Schwarzschild black hole.

Figure 3

(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) In the extremal limit $\xi$ has a double zero at $y=0$ and the two horizons coincide.

Figure 4

(a) $\psi$, $\xi$, and $Z$ plotted as a function of $y$ for a semiclassical black hole. The inner horizon of the classical solution is replaced by a singularity where $\psi$, $\xi$, and $Z$ all approach zero. (b) A corresponding plot for a classical black hole solution, repeated from Fig. 3 for comparison.

Figure 5

Two cases in which the sign change of Eq. (52) occurs. (a) The field $Z$ has a local minimum at some $y$ in the interval $-y_0<y<0$, or (b) the field $Z$ goes through zero at some $y$ in the same interval.

Figure 6

(a) The asymptotic energy density $\epsilon$ as a function of the electric field at the horizon $f(0)$. (b) The asymptotic energy density $\epsilon$ as a function of $\psi (0)$ for four values of $f(0)/\sqrt{8}=0.2$, 0.4, 0.6, 0.8.

Figure 7

Initial data for dynamical collapse.

Figure 8

The numerical evolution scheme is a staggered grid leapfrog.

Figure 9

Density plots of the charge distribution $Z(u,v)$ for the collapse of massless (a) and massive (b) fermions. The density plots of (c) dilaton $\psi (u,v)$ and (d) scalar curvature $R(u,v)$ are for collapse of massless fermions. The corresponding plots for collapse of massive fermions are not substantially different, and are not provided here. The singularity is shown by a thick black line, while the thin line indicates the apparent horizon. The white stripe just below the singularity in (d) is a region of negative curvature.

Figure 10

Spacetime curvature along a constant $v$ profile that runs into the black hole singularity.

Figure 11

Scaling behavior of the area of the apparent horizon.

Figure 12

The absolute value of $Z_{\mathrm{AH}}$, the bosonized matter field at the apparent horizon, in the scaling regime. The sign of $Z_{\mathrm{AH}}$ changes between adjacent humps in the graph.

Figure 13

The asymptotic energy density ${\epsilon }_H$ of Hawking radiation as a function of the electric field at the horizon $f(0)$.

Figure 14

Typical solutions for the area function $\psi$ inside the horizon for $e^2\gg \kappa$. The different curves correspond to different values of the electric field at the horizon $f(0)/\sqrt{8}=0.0$, 0.1, …, 0.9. In all cases the area function decreases to its critical value inside the black hole.

Figure 15

The area function $\psi$ inside the horizon for $e^2\ll \kappa$ for different values of the electric field at the horizon $f(0)/\sqrt{8}=0.0$, 0.1, …, 0.9. For nonvanishing $f(0)$ the area function bounces before it reaches the critical value ${\psi }_{\mathrm{cr}}=\kappa /4$.

Figure 16

Density plot of $\psi (u,v)$ for a collapse of charged matter with the backreaction due to both the Schwinger effect and Hawking radiation taken into account.