The word "singularity" is taken from math, where quite well-behaved functions sometimes go off the deep end. The reciprocal function, for example, which turns a number x into 1/x, delivers perfectly good values as x goes from bigger to smaller numbers.
- 1/10 is 0.1
- 1/2 is 0.5
- 1/0.4 is 2.5
- 1/0.001 is 1,000
… and so on. When you get to zero, though, suddenly "the function has no value." You can't say anything about 1/0, because math does not permit division by zero. Since the operation is forbidden, nothing meaningful can be said about the result. All a mathematician can say is: The function has no value for that number.
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The term "singularity" was picked up by astrophysicists in the mid-20th century. The substance of a star is in a state of equilibrium. Pushing it out is the pressure of radiation surging up from the star's deep interior; pulling it in is gravitation. Even nuclear fires can't go on burning for ever, though. The equilibrium is lost at last and gravitation wins.
What happens then depends on the size of the star. The star will collapse in on itself until … until what? If the star is massive enough, the astounding answer is: until nothing. There is nothing known to physics that will arrest the collapse. The collapsed star will dwindle in size, with a corresponding increase in density, without limit — to a singularity.
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Now, if you stand on the surface of any very large solid body — the Moon, say — and throw a rock up in the air, the body's gravity will slow the rock's speed until it stops and falls back. The faster the rock leaves your hand, the longer this takes to happen. If it leaves your hand fast enough — "escape velocity" — the rock will never fall back.
On the surface of the Moon you'd need to throw that rock vertically upwards at 5,400 miles per hour for it to escape. That's escape velocity from the Moon's surface. Note the following important things:
- The more mass, the stronger the gravity. If the Moon were the same size but more dense, and so having a bigger mass, the escape velocity would be greater.
- The further you are from a body, the weaker its gravitational pull on you. If you could position yourself in midair (except that there is, of course, no air) several miles above the Moon's surface, where gravity is slightly weaker, the escape velocity from that point would be less.
On the first of those points, with the size held constant, escape velocity from the surface increases as the square root of the mass. If the Moon were the same size but a hundred times as dense, escape velocity would be ten times greater: 54,000 mph. If the Moon were sixteen billion times as dense as it is, escape velocity would be 680 million mph.
That's greater than the speed of light, and it's a well-know fact that nothing can exceed the speed of light. If the Moon were that dense, therefore, then nothing whatsoever could escape from its surface, not even light.
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Now go back to the second numbered point above. A few miles above the Moon's surface, where gravity is weaker, there's a point at which light would be able to escape, just.
You can therefore imagine the sixteen-billion-times-denser Moon hidden inside a somewhat larger sphere, outside which things are fairly normal, but inside which it's hard to say what is going on. That sphere is called the "event horizon." Outside event horizon: normal. Inside: weird. This arrangement is formally known as a black hole.
I've over-simplified disgracefully there. At these extremes, relativity and quantum mechanics kick in. Outside the event horizon but close to it, things are far from normal, though still within the bounds of imagination. Inside the event horizon, the weirdness is acute. Time and space, for example, have changed places … And of course, matter can't maintain its familiar structure at sixteen billion times normal density. It collapses, and goes on collapsing, until … nobody really knows what. You can't say. The functions have no values — at least, none that our current understandings can compute. It's a singularity.
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Physics is not math, and there is a certain sleight of hand involved in dragging the notion of "singularity" from one region to the other. A mathematical singularity breaks the rules; but they are (I'm speaking loosely here) our rules. A singularity in physics is a confession that we don't know what the rules are governing this situation. It's passed beyond our current understanding. Since, on historical precedent, we can always hope to expand the scope of our understanding, our ignorance may be a temporary state of affairs. If we can get a good theory of quantum gravitation worked out, we may be able to make confident pronouncements about what's going on at the center of a black hole.
We might even be able to say something intelligible about the grandaddy of all singularities in physics, the Initial Singularity, a.k.a. the Big Bang. Our present understanding is that in the very remote past the universe was much hotter and denser than it is now, that in even earlier times it was even hotter and denser, that … (repeat previous eleven words for ever). If this is the case, there will always be some point at which the heat and density confound our current understanding of physical laws, so that we can't say anything about earlier times.
That may not be right, though. Perhaps that sequence does not go on for ever. Possibly, after some finite number of iterations, it will "turn a corner" and we shall find ourselves in some different kind of universe. Alternatively, if the sequence "… in even earlier times it was even hotter and denser …" really does go on for ever, then the Initial Singularity will for ever be beyond our understanding.
The other candidate for a really intractable singularity in the world is consciousness, the innermost core of self-awareness, the ghost in the machine that puts on such a convincing show of not being part of the physical world. Is it, or isn't it? We don't know, though practically all working neuroscientists seem to think it is. If it isn't, then it is a singularity, which physical enquiries can never unravel.