The two well-known predictions about black holes today are two. First, from inside a black hole you cannot escape because the way out has gone back in time. Second, you can see the whole future of the universe because the time in the universe you left is now like space. Too bad you can't tell anyone.
The idea that we live inside a black hole is not as crazy as it seems. Black holes deform space and time until their roles are reversed. For anyone who falls into a black hole, the radial dimension, towards the singularity, becomes time and the temporal dimension becomes like space. The basic idea of the theory that the Big Bang is a black hole (Black Hole Big Bang Theory, or BHBBT) is that the matter of a mother universe collapses into a black hole. The singularity of this black hole is in a single point in space relative to anyone in the parent universe. But, due to the reversal of time and space for anyone within the child universe, that point in space r = 0 becomes their starting point in time, t = 0. So what was a singularity in space is now a singularity in time, just like the Big Bang.
This means that any matter that falls from the "mother" universe into a black hole will disappear from that universe and emerge at a completely confused starting point of the "daughter" universe.
How much matter from the "mother" universe would have ended up in the "daughter" universe? And how much can it finish? All too? Can one universe end up inside another? The reason an entire universe can be contained within another has to do with the strange way in which time and space can be warped, stretched, compressed and twisted. What appeared to be a dead center at the center of a black hole may instead be a step in the birth of a new universe.
You can have many universes interconnected in this way, where mothers give birth to daughters who give birth to other daughters, and so on to infinity. So far from being "old" only 13 billion years, the entire interconnected cosmos can be infinitely old.
The standard Big Bang model
The standard model of the universe's Big Bang (SBB) is that the universe, including time, space and matter, it was born in a single point about 13 billion years ago. From the standpoint of general relativity, Einstein's theory of gravity, space itself has been compressed at that point. As time went by, space began to expand taking matter with it as it went. This process would continue today and we know it because when we observe distant galaxies, they are moving away from us. The further away a galaxy is, the faster it moves away from us. This is consistent with a theory of the universe in which space is expanding. The greater the space between two points, the faster they can move apart. The standard example of this is a set of dots on a balloon. Blow up the balloon and all the dots move away.
Where is the center of the universe?
The center of the universe, where the Big Bang happened, is not for us in space, but at a point in time, t = 0, in the Big Bang. The balloon analogy is useful here because the center of the balloon is obviously not on the balloon. Space, therefore, is like the surface of the balloon with an additional dimension, so it is 3 dimensions instead of 2. The past is like the inside of the balloon.
Black holes, however, have their centers at a point in space, r = 0 in coordinates centered on the singularity of the black hole. Therefore, they are fundamentally different from the Big Bang singularity.
So how can we be inside a black hole? Time becomes space
One of the strange characteristics of general relativity is its ability to bend space and time to the point where time and space can swap roles. Intense concentrations of matter can warp space and time so that it changes the meaning of space and time for different observers. For an observer outside a black hole, called a distant observer, the singularity is at a point in space. For the observer within the event horizon, however, the singularity is at some point in the time, some time in the future.
The theory that the Big Bang is a black hole, BHBBT, suggests that at least for some types of singularities, once it reaches matter, it enters a new universe where time at singularity is the starting point of that universe.
Imagine you are an ant crawling on a table. As you crawl, go down a slope. The slope gets steeper and steeper until it becomes completely vertical. Suddenly, it ends at a point. This is typical of how black holes are depicted. But now, instead of ending up at that point, beyond this point the space expands again. Passing through that single point you re-emerge in an expanding cone: a new universe perpendicular to the one you left behind.
A black hole as the origin of the universe
BHBBT is a compelling theory that can be rigorously formulated within the limits of Einstein's theory of general relativity. It does not require new physics. It also explains why the Big Bang happened. And on a philosophical and anthropic principle level, he reassuringly explains that we were not born "from nothing", but were generated by something else, and perhaps we are in a certain sense infinite.
References:
Smolin, Lee. "Did the universe evolve ?." Classical and Quantum Gravity 9.1 (1992): 173.
Stuckey, WM "The observable universe inside a black hole." American Journal of Physics 62.9 (1994): 788-795.
Easson, Damien A., and Robert H. Brandenberger. "Universe generation from black hole interiors." Journal of High Energy Physics 2001.06 (2001): 024.
