Internal structure of charged black holes

We investigate the internal structure of charged black holes with a spherically symmetric model in the double-null coordinate system. Hawking radiation is considered using the S-wave approximation of semiclassical backreaction and discharge is simulated by supplying an oppositely charged matter to the black hole. In the stage of formation, the internal structure is determined by the mass and charge of collapsing matter. When the charge-mass ratio is small, a wormhole-like internal structure is observed. The structure becomes analogous to the static limit as the ratio reaches unity. After the formation, mass inflation induces large curvature in the internal structure, which makes the structure insensitive to the late-time perturbations. The internal structure determined from the formation seems to be maintained during a substantial mass reduction. The discharge and neutralization of charged black holes is also investigated for both nonevaporating and evaporating cases. Finally, we discuss the implications of the wormhole-like structure inside of charged black holes.

Figure 1

Penrose diagrams of static charged black holes for (a) $M>Q$, (b) $M=Q$, and (c) $M<Q$.

Figure 2

Charged black holes with mass inflation scenario. The collapsing matter generates mass inflation which induces the spacelike singularity, and the inner-Cauchy horizon becomes a null curvature singularity.

Figure 3

Causal structure of dynamical charged black holes. Evaporation and neutralization are considered.

Figure 4

Evolution of charged black holes [15]. (a) If $E_{+}\gg E_c$, discharge dominates and the black hole approaches $E_{+}=E_c$. (b) If $E_{+}\ll E_c$, discharge is exponentially suppressed. If $Q/M\ll 1$, Hawking radiation dominates and $Q/M$ decreases. (c) If $Q\gg E_c^{-1}$, the black hole will approach and follow the extreme state. (d) If $Q\ll E_c^{-1}$, the black hole will approach and follow $E_{+}=E_c$. (e) When the charge reduces to a few quanta, $Q\sim q_e$ and $T_H\sim M^{-1}\sim m_e$ on the $E_{+}=E_c$ track. Then, the black hole will emit its final quanta of charge via Hawking radiation. (f) The final neutral black hole will lose mass via Hawking radiation.

Figure 5

Contour diagram of radial function $r$ and horizon structure for $\delta =0.15\pi$ and $P=0$. This diagram corresponds to a dynamically formed charged black hole within the static limit.

Figure 6

Contour diagram of radial function $r$ and horizon structure for $\delta =0$. This diagram corresponds to an evaporating neutral black hole.

Figure 7

Contour diagrams of radial function $r$ and horizon structures for $\delta =0.05\pi$. As we include Hawking radiation, both $r_{,u}=0$ and $r_{,v}=0$ types of inner horizons are observed. The resulting structure is similar to a throat of a wormhole.

Figure 8

Contour diagrams of radial function $r$ and horizon structures for $\delta =0.15\pi$. As the charge is increased, the $r_{,u}>0$ region inside of the black hole is decreased. On the other hand, the qualitative behavior of the $r_{,v}=0$-type inner horizon is unchanged.

Figure 9

Plot of $\log |M|$ along $u=10$ surface for black holes with $\delta =0.05\pi$, ${\delta }_1=0.1\pi$, and ${\delta }_1=0.15\pi$. This plot clearly shows the exponential behavior of the mass inflation effect, which becomes stronger as $Q/M$ decreases.

Figure 10

Contour diagram of radial function $r$ and horizon structure for $\delta =0.3\pi$. Among massy horizons, there exists a distinct $r_{,v}=0$-type inner horizon. The $r_{,u}>0$ region becomes negligible.

Figure 11

Contour diagram of radial function $r$ and horizon structure for $\delta =0.5\pi$. The black hole is nearly extreme. Its structure is analogous to the static limit in Fig. 5.

Figure 12

Formation of charged black holes with various $Q/M$. Diagrams in the left column show the horizon structures and the right column show schematic diagrams of the area measured by an ingoing null observer along the dotted line in the corresponding left diagram.

Figure 13

Contour diagrams of radial function $r$ and horizon structures for $A_3=0.001$. As the exotic matter field is supplied from $v=20$, the outer horizon becomes timelike, and both $r_{,u}=0$ and $r_{,v}=0$ types of inner horizons arise. The resulting structure is similar to the one in Fig. 8, where the semiclassical backreaction is used.

Figure 14

Contour diagrams of radial function $r$ and horizon structures for $A_3=0.01$. As a larger amount of negative energy is supplied, the $r_{,u}>0$ region inside of the black hole is increased. The relation between the negative energy and the wormhole-like structure is discussed in Sec. 7. The outer horizon approaches to the inner horizon more quickly.

Figure 15

Contour diagrams of radial function $r$ and horizon structures for $A_3=0.02$. As $Q/M$ reaches unity, the outer horizon and the $r_{,v}=0$-type inner horizon meet together and disappear, which is consistent with the RN metric in Fig. 1c. On the other hand, the $r_{,u}=0$-type inner horizon maintains its structure.

Figure 16

$Q/M$ measured at the outer horizon along the advanced time $v$.

Figure 17

Long-term evaporation of charged black holes. Diagrams in the lower row show schematic diagrams of the area measured by an ingoing null observer along the dotted lines in the upper diagram.

Figure 18

Contour diagram of radial function $r$ and horizon structure for $A_3=0$. The charged black hole in Fig. 5 discharges and becomes a neutral black hole.

Figure 19

Contour diagrams of radial function $r$ and horizon structures for $A_3=0.01$. An exotic matter field is added to the discharging black hole in Fig. 18 to consider Hawking radiation.

Figure 20

Neutralization of evaporating charged black holes. Diagrams in the lower row show schematic diagrams of the area measured by an ingoing null observer along the dotted lines in the upper diagram.

Figure 21

The neutralization of charged black holes can be identified with the separation between the inside and the outside of the wormhole-like structure. (a) If the inside region is regular and does not have a timelike singularity, this process can cause issues related to the information loss problem. (b) In other cases, semiclassical analysis cannot be applied in this region.

Figure 22

$|r_{(1)}-r_{(2)}|/r_{(2)}$ and $4|r_{(2)}-r_{(4)}|/r_{(4)}$ along a few constant $u$ lines, where $r_{(n)}$ is the radial function calculated in $n\times n$ times finer grid than $r_{(1)}$. It converges to the second order with errors $\lesssim 0.1\%$.

Figure 23

Differences in radial function $r$ between two integration schemes along a few constant $u$ lines. $r_{(v)}$ is calculated by integrating $g$ using Eq. (22), and $r_{(u)}$ is calculated by integrating $f$ using Eq. (20). The error is $\lesssim 1\%$.