The conceptual key here is that time dilation is not something that happens to the infalling matter. Gravitational time dilation, like special-relativistic time dilation, is not a physical process but a difference between observers. When we say that there is infinite time dilation at the event horizon we don't mean that something dramatic happens there. Instead we mean that something dramatic *appears* to happen according to an observer infinitely far away. An observer in a spacesuit who falls through the event horizon doesn't experience anything special there, sees her own wristwatch continue to run normally, and does not take infinite time on her own clock to get to the horizon and pass on through. Once she passes through the horizon, she only takes a finite amount of clock time to reach the singularity and be annihilated. (In fact, this ending of observers' world-lines after a finite amount of their own clock time, called geodesic incompleteness, is a common way of *defining* the concept of a singularity.)

When we say that a distant observer never sees matter hit the event horizon, the word "sees" implies receiving an optical signal. It's then obvious as a matter of definition that the observer never "sees" this happen, because the definition of a horizon is that it's the boundary of a region from which we can never see a signal.

People who are bothered by these issues often acknowledge the external *unobservability* of matter passing through the horizon, and then want to pass from this to questions like, "Does that mean the black hold never really forms?" This presupposes that a distant observer has a uniquely defined notion of simultaneity that applies to a region of space stretching from their own position to the interior of the black hole, so that they can say what's going on inside the black hole "now." But the notion of simultaneity in GR is even more limited than its counterpart in SR. Not only is simultaneity in GR observer-dependent, as in SR, but it is also local rather than global.

> Is there a connected solution of 3+1 dimensional general relativity with one space-like slice not have a singularity, and another space-like slice having one.

This is a sophisticated formulation, but I don't think it succeeds in getting around the fundamental limitations of GR's notion of "now." Here's a Penrose diagram for the Schwarzschild spacetime. The Schwarzschild spacetime represents an eternal black hole, not one that formed by gravitational collapse, but even so, I think this example is adequate to show why this way of getting at "now" doesn't help.

![Penrose diagram of Schwarzschild spacetime][1]

On this type of diagram, light cones look just like they would on a normal spacetime diagram of Minkowski space, but distance scales are highly distorted. The diamond represents the entire spacetime outside the horizon, with the distortion fitting this entire infinite region into that finite area on the page. The upper left edge of the diamond is the black hole's event horizon, which is a lightlike surface. The triangle is the spacetime inside the event horizon. The dashed line is the singularity, which is spacelike.

E is an event on the world-line of an observer. The red spacelike slice is one possible "now" for this observer. According to this slice, no test particle has ever fallen in and reached the singularity; every such test particle has a world-line that intersects the red slice, and therefore it's still on its way in. For example, the green line could be the world-line of such a particle. It intersects the red slice, so the particle is still infalling according to the red definition of "now."

The blue spacelike slice is another possible "now" for the same observer at the same time. According to this definition of "now," the green particle has already hit the singularity.

If this was SR, then we could decide whether red or green was the correct notion of simultaneity for the observer, based on the observer's state of motion. But in GR, this only works locally (which is why I made the red and blue slices coincide near E). There is no well-defined way of deciding whether red or blue is the correct way of extending this notion of simultaneity globally.

So the literal answer to the quoted part of the question is yes, but I think it should be clear that this doesn't establish whether infalling matter has already hit the singularity at some "now" for a distant observer.

Although we refer to a Schwarzschild spacetime as a description of an eternal black hole, i.e., one that has always existed and always will, we can see in this diagram that for the red definition of "now," the singularity doesn't even exist yet.

  [1]: https://i.sstatic.net/feHHJ.jpg