The simplest answer is that the curvature of space-time is a smoothly changing function of distance up to, and through, the event horizon. There is no indication from the curvature (Riemann's Curvature Tensor, or even Ricci's for that matter) that anything serious is happening just inside the event horizon. Einstein's 'equations' work just fine so long as the local curvature of space-time (the strength of the gravitational field) does not become singular. This does not happen just inside an event horizon, but only happens as you approach the 'r=0' singularity itself.

What all of this means is that the mathematical properties of spacetime that matter (its curvature) change smoothly through the event horizon, much like a ride in a sled down a snow-covered hill. Now, to prove that this is in fact the case will probably not be possible because we can never extract information from inside a black hole withough dying, or never being able to return!

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