End of a black hole’s evaporation

At the end of the Hawking evaporation the horizon of a black hole enters a physical region where quantum gravity cannot be neglected. The physics of this region has not been much explored. We characterize its physics and introduce a technique to study it.

Figure 1

The three regions of a black hole spacetime where quantum gravity becomes relevant. In the dark grey region quantum gravity cannot be neglected and the diagram itself becomes unreliable. The future of the locations $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ encounter different quantum gravity phenomena depending, respectively, on the presence of the collapsing matter ($\mathbf{C}$), the horizon ($\mathbf{B}$), or neither ($\mathbf{A}$).

Figure 2

Carter-Penrose causal diagram of the black to white transition.

Figure 3

The bounded curvature scalar (7).

Figure 4

The past portion of the boundary surface.

Figure 5

Carter-Penrose diagram of the $\mathbf{B}$ region with the surface $\mathrm{\Sigma }$ and its components highlighted.

Figure 6

The triangulation of ${\mathrm{\Sigma }}^t$. The brown tetrahedron $t_{-}$ is inscribed into the larger violet tetrahedron $t_{+}$. The blue segments connect vertices of the two tetrahedra radially.

Figure 7

All images represent the triangulation of ${\mathrm{\Sigma }}^t$, but with different tetrahedra highlighted. In (a) no tetrahedron is highlighted (the tetrahedron $t_{-}$ is in brown to remind that it is hollowed inside); in (b) the four $T_{-a}^t$ are highlighted; in (c) three $T_{+a}^t$ out of four are highlighted; in (d) three $T_{ab}^t$ out of six are highlighted; in (e) the remaining three $T_{ab}^t$ are highlighted; in (f) two $T_{-a}^t$, two $T_{+b}^t$ and two $T_{cd}^t$ are highlighted.

Figure 8

Two-dimensional graph ${\mathrm{\Gamma }}_{\mathrm{\Sigma }}$ of the spin-network of $\mathrm{\Sigma }$. The circles are nodes (dual to tetrahedra) and the segments are links (dual to the triangles).

Figure 9

The left figure portrays a two-dimensional representation of the dual of the triangulation of ${\mathrm{\Sigma }}^t$ and the right figure portrays a three-dimensional representation of the dual ${\mathrm{\Gamma }}_{\mathrm{\Sigma }}$ of the full triangulation of $\mathrm{\Sigma }$, with labels omitted; one can easily label the right figure reading the different labels from the left figure and using $t=p$ for the bottom and $t=f$ for the top.

Figure 10

Definition of $k_{+}$.