There is very much time after crossing the event. It happens to not correspond to the "t" coordinate that a distant observer would call time, but it still exists.
The mathematics describes 2 singularities with black holes. 1 singularity occurs when describing the black hole using the coordinate system of an observer at constant distance. This is a coordinate singularity and goes away under an appropriate change of coordinates.
Another singularity occurs at what the constant distance observer would describe as the "center" of the black hole. This is not a coordinate singularity. The curvature there goes infinite regardless of your coordinate system.
Whether or not you want to call this singularity the "center" is a matter of opinion. An observer within the event horizon would describe the singularity as a time, not a location, so "center" certainly has some connotations that you may want to avoid. On the other hand. In the coordinate system that I normally see used to describe the interior of black holes, the singularity occurs at the surface r=0; which sounds kind of center-ish.
The mathematics describes 2 singularities with black holes. 1 singularity occurs when describing the black hole using the coordinate system of an observer at constant distance. This is a coordinate singularity and goes away under an appropriate change of coordinates.
Another singularity occurs at what the constant distance observer would describe as the "center" of the black hole. This is not a coordinate singularity. The curvature there goes infinite regardless of your coordinate system.
Whether or not you want to call this singularity the "center" is a matter of opinion. An observer within the event horizon would describe the singularity as a time, not a location, so "center" certainly has some connotations that you may want to avoid. On the other hand. In the coordinate system that I normally see used to describe the interior of black holes, the singularity occurs at the surface r=0; which sounds kind of center-ish.