singularities - Where does matter go after reaching singularity in uncharged black holes?

I am a layman in physics and just read about black holes on the internet. I read that matter encounters geodesic incompleteness in the singularity in an uncharged black hole.

I heard an analogy of geodesic incompleteness as a straight line on a paper reach a hole on the paper, so it cannot continue. But in this analogy, isn't the straight line possible to continue into 3D (continue down the paper)? So, if matter reaches the singularity, is it possible too (to reach another dimension)?

I also heard that the matter is annihilated when reaching the singularity, does it mean it disappear from this world, and violates conservation of energy?

Answer

Strictly speaking geodesic incompleteness doesn't mean the worldline of the particle ends at the singularity, but rather that we can't predict what happens to it. The trajectory of a freely falling particle is given by an equation called the geodesic equation:

$$ \frac{d^2x^\alpha}{d\tau^2} = -\Gamma^\alpha_{\,\,\mu\nu}\frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau} $$

It's a scary looking equation but you don't need to understand all the details to see what the problem is. What happens at the singularity in a black hole is that some of the parameters $\Gamma^\alpha_{\,\,\mu\nu}$ become infinitely large and we're left with an equation that has infinity on the right hand side. Since we can't do arithmetic with infinity (because it's not a number) we have no way to calculate the trajectory of the particle at the singularity.

Incidentally much the same happens when we try to work backwards in time towards the Big Bang, and that's why it's commonly said that time started at the Big Bang. See my answer to How can something happen when time does not exist? for more on this.

Anyhow, the upshot is that GR cannot tell us what happens to matter falling into a black hole when it hits the singularity. However most of us believe that general relativity ceases to be a good description of the physics when we get close to the singularity and some form of quantum gravity theory will take over. The trouble is that we currently have no theory of quantum gravity.