Interior of black holes and information recovery

We analyze time evolution of a spherically symmetric collapsing matter from a point of view that black holes evaporate by nature. We first consider a spherical thin shell that falls in the metric of an evaporating Schwarzschild black hole of which the radius $a(t)$ decreases in time. The important point is that the shell can never reach $a(t)$ but it approaches $a(t)-a(t)\frac{\mathrm{d}a(t)}{\mathrm{d}t}$. This situation holds at any radius because the motion of a shell in a spherically symmetric system is not affected by the outside. In this way, we find that the collapsing matter evaporates without forming a horizon. Nevertheless, a Hawking-like radiation is created in the metric, and the object looks the same as a conventional black hole from the outside. We then discuss how the information of the matter is recovered. We also consider a black hole that is adiabatically grown in the heat bath and obtain the interior metric. We show that it is the self-consistent solution of $G_{\mu \nu }=8\pi G⟨T_{\mu \nu }⟩$ and that the four-dimensional Weyl anomaly induces the radiation and a strong angular pressure. Finally, we analyze the internal structures of the charged and the slowly rotating black holes.

Figure 1

A test particle in the time-dependent Schwarzschild metric. $r(t)$ cannot catch up with $a(t)$ as long as $\frac{\mathrm{d}a}{\mathrm{d}t}(t)<0$.

Figure 2

Time evolution of the core with $a^{\prime }$ and the shell with $\epsilon (t=0)\sim \frac{\hslash }{a}$ in the view of the local time $t$.

Figure 3

The interior picture of an evaporating black hole.

Figure 4

The Penrose diagram of the evaporating black hole in the vacuum.

Figure 5

A closer look at the outermost region.

Figure 6

The black hole that is in equilibrium with the heat bath. The matter in the outermost region has been replaced with radiation from the heat bath.

Figure 7

The Penrose diagram for the stationary black hole in Fig. 6.

Figure 8

$\frac{\sigma }{l_p^2}$ quanta with energy ${\epsilon }_1\sim \frac{\hslash }{a}$ approaching the core.

Figure 9

Scattering process between ingoing matter and outgoing radiation in the outermost region.

Figure 10

The black hole that is formed adiabatically in the heat bath.

Figure 11

The meaning of $f(r)$.

Figure 12

The evaporation of the adiabatically formed black hole and the entropy.

Figure 13

A picture of the Reissner-Nordstrom black hole consistent with the condition for thermodynamical integrability. Electric charge exists only in the outermost region with the width $\sim \frac{1}{r_{+}}$.

Figure 14

The numerical result of $a(t)[r(t)-a(t)]$ for $a(0)=100$, $r(0)=110$, $\sigma =k=1$. It approaches $2k=2$.

Figure 15

An evaporating null shell. In this coordinate $(u,r)$, an outgoing null ray is depicted by a line with $u=\mathrm{const}$.

Figure 16

A continuous distribution modeled by many shells.

Figure 17

A continuously distributed matter that collapses and evaporates and a trajectory of a field from the flat spacetime to the matter. Here, an outgoing null line is depicted as $u=\mathrm{const}$.

Figure 18

Examples of two-shell collapsing process.