Tunneling from the past horizon

We investigate a tunneling and emission process of a thin-shell from a Schwarzschild black hole, where the shell was initially located beyond the Einstein-Rosen bridge and finally appears at the right side of the Penrose diagram. In order to obtain such a solution, we should assume that the areal radius of the black hole horizon increases after the tunneling. Hence, there is a parameter range such that the tunneling rate is exponentially enhanced, rather than suppressed. We may have two interpretations regarding this. First, such a tunneling process from the past horizon is improbable by physical reasons; second, such a tunneling is possible in principle, but in order to obtain a stable Einstein-Rosen bridge, one needs to restrict the parameter spaces. If such a process is allowed, this can be a nonperturbative contribution to Einstein-Rosen bridges as well as eternal black holes.

Figure 1

An example of $V(r)$ (black), ${\beta }_{+}(r)$ (red), and ${\beta }_{-}$ (blue), where $M_{+}=1$, $M_{-}=1.046$, ${\ell }_{+}=\infty$, ${\ell }_{-}=15$, and $\sigma =0.001$.

Figure 2

Dashed curves denote a shell trajectory of a symmetric collapsing solution and thick curves denote a shell trajectory of a symmetric bouncing solution; left is for the inside and right is for the outside the shell.

Figure 3

Tunneling of a shell from inside to outside the past horizon.

Figure 4

Parameter spaces of ${\ell }_{-}$ and $\sigma$ by varying $M_{+}$ and $M_{-}$ (left: $M_{+}=1,M_{-}=5$, middle: $M_{+}=2.5,M_{-}=5$, right: $M_{+}=1,M_{-}=2.5$) that satisfy Eq. (16) (red), Eq. (17) (yellow), and Eq. (18) (blue). The overlapped region is denoted by the black colored region.

Figure 5

$2B$ and $r_2-r_1$ by varying ${\ell }_{-}$ for a given $M_{+}=1$, $M_{-}=2.5$, and $\sigma =0.001$. There exists an exponentially preferred region ($2B<0$), while for the large ${\ell }_{-}$ limit, the probability is exponentially suppressed.

Figure 6

$2B$ and $r_2-r_1$ by varying $\sigma$ for a given $M_{+}=1$, $M_{-}=2.5$, and ${\ell }_{-}=2$.