SINGULARITIES: THE HEART OF BLACK HOLES AND THE BIG BANG
Singularities are fascinating objects and places, yet entirely 'inaccessible' in the sense that you can't actually go there on vacation and send back a postcard, or travel to one on a government grant as a scientific expedition and report back via a peer-reviewed article in a technical scientific journal about the local environment, geography, inhabitants, etc. In a sense singularities are like Heaven in terms of accessibility. You have to rely on intuition or theory or second-hand observations as to what's what and who's who.
Okay, for those readers I've already befuddled, I'd better tell you exactly what a singularity is! You've all heard of the phrase 'Black Hole' and not the one in Calcutta either! I refer to astronomical or cosmic Black Holes. Black Holes are 'black' because they have packed inside them so much stuff, so much mass, and hence so much gravity that not even particles of electromagnetic energy (photons) can escape their gravitational clutches. If photons, and that includes visible light photons, are jailed, can not pass go, can not collect $200, then they might as well as, as far as your perception of them is concerned, not exist. If what you don't see exists, that existence is of no matter (well lots of matter actually). Translated, Black Holes are black because visible light can't get from them to your eyeball! The absence of light is well, blackness.
So, is a Black Hole just a big lump of stuff, albeit stuff you can't see? Well, 'yes' and 'no'. First off, we can 'see' Black Holes indirectly because of their gravitational influence on stellar objects we can see. I mean if you see a star whirling around something you can't see, then the logical interpretation is that the star you can see is in orbit around something you can't see - i.e. a Black Hole. Well 'no', you can't 'see' a Black Hole because light from the Black Hole can't get away from the crush of that Black Hole's gravity.
What's all this got to do with singularities? Well, the stuff composing a Black Hole, all that stuff that clumps together and is the centre of the massive all-encompassing gravity that prohibits the photons to escape the house (Hole) that Jack built is the Black Hole's singularity. An analogy: The extent of the Black Hole is the extent of the Earth's outer atmosphere; the singularity is the solid Earth proper. So think of a nebulous outer edge with a solid core of stuff in the middle. The stuff in the middle generates the intense gravity; the nebulous outer edge marks the boundary between gravity below the threshold of light escaping and light not escaping. That boundary is referred to as the 'event horizon'; the stuff in the middle is the singularity.
Now the idea of a singularity doesn't stop with the idea behind an astronomical Black Hole. No, a singularity is any concentration of stuff or mass that has such a massive amount of gravity as to prevent photons from leaving the gravitational well or prison so created. What's the ultimate Black Hole - the Mother of all Black Holes? Well, if bits of our Universe can clump together to form astronomical Black Holes, then our entire Universe, when clumped together and in a relativity tiny state, would have been the Mother of all Black Holes and hence the Mother of all singularities. When was our Universe in such a state? Well, in the beginning!
Our Universe is expanding. That's verified by direct cosmological observation. Every cluster of galaxies has such astronomically bad 'body odour' that every other cluster of galaxies is moving out of the vicinity quick-smart! Well actually you can't have 'body odour' in space, so that's not the real reason. The real reason is that in the beginning or once upon a time, there was some sort of explosive oomph event that started the expansion process. We call that the 'Big Bang' event. At the time of the Big Bang event, our entire Universe had a close encounter with, well, our entire Universe. Our entire Universe was roughly all in the same space at the same time. Translated, if you run the film of an expanding Universe backwards, you eventually get the entire contents of our Universe on collectively very intimate terms. Such a massive collection of stuff, matter, mass, hence gravity, all of the stuffs, matter or mass that the Universe possesses, well let's just say you'd have the Mother of all singularities - in the Big Bang beginning; or anyway once upon a Big Bang time at least .
Well surely one didn't have this Mother of all singularities just sitting around for aeons then for no apparent reason go 'poof' and thus have an explosive oomph moment which kick-started things off as far as our Universe is concerned. The intense gravity of the Mother of all singularities probably would have muted any oomph to begin with; the birth of our Universe stalled at the onset.
But, let's throw some momentum into the mix. What's the opposite of a Big Bang? It's a Big Crunch! So let's propose that we have this other universe which, the bits and pieces thereof, under all those mutual gravitational attractions, is slowly, ever so slowly, but ever so surely, coming together. And as it comes together, the contracting velocity gets faster, and faster, and faster. Eventually, you have this massive collection of stuff rushing together to meet at a single point in space and in time at a fantastic velocity. There is such momentum present that the contracting Big Crunch universe just can't stop on a dime any more than an automobile going a hundred miles an hour can stop with inside of a foot of having the brakes applied. The Big Crunch at the omega point obviously forms the Mother of all Black Holes and singularities, but the sheer momentum of that contracting universe just tears the fabric of things (space and time) apart, and like a glove turning inside-out, the contraction passes through the omega point, spewing its guts out, becoming an alpha point, which is our Big Bang event and the start of our new expanding Universe.
Okay, so we have two sources that have singularities - singularities at the centre of astronomical Black Holes, and the Mother of all singularities residing inside the Mother of all Black Holes, the one that existed at the Big Bang beginning of the Universe.
We of course can't see a singularity directly (unless you're willing to take a one-way trip down a Black Hole, but even if you survived that and landed safely on the singularity, you couldn't ever broadcast back your findings - that speed of light restriction that by definition a Black Hole imparts regarding sending stuff out). So, we have to rely 100% on what theoretical equations predict a singularity to be. Unfortunately, those equations, when pushed to the sorts of mass and gravitational extremes that a singularity would represent, well you get nonsense answers. Translated, if taken at face value, the equations note that the intense gravity crushes the stuff that itself is responsible for that gravity down to a point of zero dimensions and hence infinite density.
The essential problem behind this nonsense is that gravity is represented by Einstein's Theory of General Relativity which is a classical physics smooth continuum phenomenon. That is, you can have this gravitational value, and that gravitational value, and every possible value in-between. However, tiny objects, which is what a singularity is postulated to be, is in the realm of the quantum, which is a non-continuum phenomena. Think of a staircase. You can be on this step, or the next step, but there is no step in-between the two. That is, you can have this value, or that value, but only certain other values in-between. It's also like money - you can have a five dollar bill, and a ten dollar bill, but not a six-and-a-third dollar bill, or an eight-and-three-quarters dollar bill, or even a seven or a nine dollar bill. Money and staircases are non-continuum quantum-like; money and staircases are not a smooth continuum like gravity is.
So, to adequately come to terms with the really real properties of singularities, you need a theory of quantum gravity. Alas, despite the best efforts of thousands of theoretical physicists over many, many decades, no quantum gravity theory to be had. There's no quantum gravity dice.
So, let's abandon that theoretical track and go back to common sense predictions.
Either Black Hole singularities, or the Big Bang singularity, are infinitely dense and have zero volume, or they do not. If they do not (and the alternative defies common sense and is IMHO ridiculous), then singularities have a finite volume and can grow in size as more stuff is added on. You have an original tiny singularity with extremely high, but not infinite density. You keep piling stuff onto it. For a while, the density keeps on increasing, but since it can't become infinite, there will be a point reached where further increases cease. As more and more stuff continues to be piled on, the only other option is that the size of the singularity must grow. The volume increases, and increases and increases. The upshot is that singularities can reach a size where quantum effects become negligible. Or, in other words, singularities can grow to where they aren't quantum objects any more, and while theories of quantum gravity might be still be useful in explaining their properties, it's probably no longer essential. Singularities have entered the realm of classical physics.
One property of singularities I find interesting is that the stuff that eventually forms the singularity isn't the same sort of stuff that went down the Black Hole's throat in the first place. There's been a phase transition of one kind of stuff to another kind of stuff. You're quite familiar with phase transitions in your day-to-day life. There's nothing mysterious about the concept. The most common example is steam or water vapour condensing to liquid water condensing or freezing to ice; ice melting to liquid water hence boiling or evaporating into steam or water vapour. Apart from your division into solid, liquid and gas, there's also plasma. Now the sort of matter that composes a singularity is probably something else yet again, a phase transition that only extreme gravity can achieve. That such a new state of matter exists is predicted by the following: If you have an ordinary matter star, and if it should happen to collide with an antimatter star, what you get is one hell of a big Ka-Boom; the annihilation of matter/antimatter into pure energy. However, say your matter star implodes into a Black Hole with singularity. And say your antimatter star implodes into a Black Hole with singularity. Now have these two Black Holes collide. No Ka-Boom results, just a larger Black Hole!