Stable space-like singularity formation for axi-symmetric and polarized near-Schwarzschild black hole interiors

Understand

We show a stability result for the Schwarzschild singularity (inside the black hole region) for the Einstein vacuum equations.

• The result is proven in the class of polarized axial symmetry, under perturbations of the Schwarzschild data induced on a hypersurface $\{r=\e\}$, $\e<<2M$.
• Our result is only partly a stability result, in that we show that while a (space-like) singularity persists under perturbations as above, the behaviour of the metric approaching the singularity is much more involved than for the Schwarzschild solution.
• Indeed, we find that the solution displays asymptocially-velocity-term-dominated dynamics and approaches a different Kasner solution at each point of the singularity.