Traversing through a black hole singularity

We show that the Kantowski-Sachs model of a Schwarzschild black hole interior can be slightly generalized in order to accommodate spatial metrics of different orientations, and in this formulation the equations of motion admit a variable redefinition that makes the system regular at the singularity. This system will then traverse the singularity in a deterministic way (information will be conserved through it), and evolve into a time-reversed and orientation-flipped Schwarzschild white hole interior.

Figure 1

The shaded region corresponds to the patch of Schwarzschild spacetime that is covered by the $s$, $\rho$ coordinates as it appears in the Penrose-Carter diagram. The borders of this coordinate patch are represented by the two red dots, the singularity and the horizon.

Figure 2

Examples of solutions of the Kantowski-Sachs system in the $x–y$ plane, with a plot of the potential on the side. The curves asymptotes to the singularity on the left, and the horizon on the right. The gradient on the background represents the values of the volume degree of freedom $v$ ($\mathrm{darker}=\text{larger}$), and the black dots represent the points of maximum volume.

Figure 3

The same solutions of Fig. 2, this time plotted in the $w–z$ plane. The singularity is reached asymptotically as $w\to -\infty$, while the horizon is at $w\to +\infty$.

Figure 4

Shape space with orientation; each hemisphere represents an orientation, and each point on the sphere represents different values of the shape degrees of freedom $(w,u)$. The poles coincide with the value $u=w=0$, while the equator corresponds to the border of the $(w,u)$ plane at infinity. A solution curve is shown on the top plane, together with its projection on the northern hemisphere.

Figure 5

The continuation of the Schwarzschild solution. At the singularity, the shape system remains well defined, and connects two Schwarzschild interiors described by right- and left-handed triads.