The mention of white holes that I have found (other than on this forum), have been a stretched theory that the big bang resulted from a white hole (which of course leads to, what caused the white hole? ... )

In regards to matter nearing the speed of light as it approaches the singularity in a black hole (using the context singularities exist), I believe this is explained by a similar analogy (for which I do not know if it has been given a name).

Take for example the age old race between the rabbit and turtle. We all know that the rabbit is supposed to win, however not if the turtle is allowed to get a minor head start (some small measurement, i.e. an inch). The reason why the turtle wins in this scenario, is because (mathematically) the rabbit can never surpass the turtle because he/she is always having to reach the halfway point between itself and the turtle before continuing. Infact, if the turtle only is an inch ahead of the rabbit and stops, this concept still applies. The rabbit would have to reach a distance of 0.5 inches before reaching the turtle, but then after reaching 0.5 inches, he/she then has to reach 0.75 inches (another halfway mark) before continuing, and so on and so forth etc. There is a law that ends this seemingly never ending dilema that states there is a reachable finish/ending.

I use the above law (whatever it is, I really don't know the name, only that it exists to describe these situations), to state that matter collides and then builds upon the singularity in black holes. The larger the singularity becomes, the less stable the black hole becomes and eventually dissepates.

-Tchzarmok