Hoo boy, this is getting weird. People treating me like an expert an' stuff, me and siren gearing up for a slug-out of the Lords of Space-time, and now requests for me to review nodes about physics... oy...

I didn't scream, but that's cos it's 2 am and my sister's ill, and plus the WU's not that bad, iainb.

General relativity suggests that the best explanation for certain inertial phenomena is a 4-dimensional space-time which is curved by the presence of mass. The more mass in an area, the more curved space-time in that area is.

Simply put, a singularity is an infinitesimal point of infinite space-time curvature. It doesn't have infinite mass (if it did it would eat the rest of the universe - the whole curvature of space-time thing would go berko everywhere to an infinite degree) but it does have infinite density, because zero volume. It's not considered part of 'our' space-time beacause all the laws of physics that apply in our space-time break down at a singularity - you may as well think of it a being a gateway to the end of the universe.

Singularities are important in differential geometryand algebraic geometry. Roughly speaking they are points at which a geometric object fails to be smooth.

I'll give a precise definition and then a discussion with some examples. Suppose that we have a subset X of Cn that is defined by the simultaneous vanishing+ of m polynomials f1(x1,...,xn),..., fm(x1,...,xn). Note that in one favourite model of the universe (as a Calabi-Yau manifold) space locally has this form.

Now take a point x on X. Form the Jacobian matrix J which is the mxn matrix with i,j entry dfi / dxj and evaluate at the point x, to obtain J(x). Then the point x is a singular point iff J(x) has rank strictly less than n - the dimension* of X. A point which is not singular is smooth.

OK that's the definition, what does it mean? Consider the special case of a hypersurface, that is we just have one defining equation f=f1. In this case the Jacobian is just a row (df/dx1 ... df/dxn) and X has dimension n-1. Thus for a singularity at x we are asking that the Jacobian matrix should vanish identically at x, that is all the partial derivatives of f have to vanish at that point.

For example, consider a quadratic cone with defining polynomial x2 + y2 + z2. This has a sharp looking point at the vertex of the cone at the origin but looks smooth everywhere else. This intuition fits with the definition because the Jacobian (2x 2y 2z) clearly vanishes at the vertex and nowhere else.

What we are seeing here illustrates a general principle, at most points of X the Jacobian matrix will have the correct rank n - the dimension of X and be smooth, and only at the smaller dimensional set defined by the vanishing of the appropriate minors of the Jacobian can it be singular.

Here's another example, consider the plane algebraic curve defined by y2-x3, (the cuspidal cubic curve). If you sketch this curve then you'll see it is symmetric about the x-axis. In the first quadrant as it moves away from the origin y is growing much faster than x. Thus the curve has a pronounced sharp point at the origin, which is indeed a singularity. At all other points of the curve you can draw a well-defined tangent line but at the origin this tangent line is not well-defined, there is a two-dimensional space in which we could draw tangents. In general, a singularity occurs where the tangent space is bigger than the dimension of the space under consideration.

+ Strictly speaking when I talk about X being defined by the simultaneous vanishing of f1,...,fm what I meam is that when I consider the ideal in the polynomial ring C[x1,...,xn] consisting of all polynomials which vanish on X then this ideal is generated by f1,...,fm.
* The dimension of such a set X given by the simultaneous vanishing of polynomials is defined as the maximum length of a chain of similarly defined subsets.

In GR there are two types of singularity, actual and coordinate. This wasn't really figured out until the 1960's when people really began to get interested in black holes.

A physical singularity is a place in space-time where the density tends to become infinte, generally because the radius of the system goes to 0. We believe that to describe such things properly we will need new physics.

A coordinate singularity is a place that looks like a singularity in one set of coordinates, but that can be transformed away by using another set of coordinates. It is not physical. The event horizon is an example of a coordinate singularity.

Introduction

This write-up will be primarily concerned with the concept of a space-time singularity, for example the one that lies at the heart of a black hole. As noted above, the singularity is a region of space and time where the current accepted physical models; relativity and the quantum mechanical standard model, can make no predictions and offer no insight to what lies within that region.

When an object forms a black hole its gravity overwhelms its matter, crushing it into a smaller and smaller region, and physics as yet provides no mechanism for this collapse to stop. Logically the matter occupies a smaller and smaller region of space-time, all the way down to infinitely small. In 1965 Roger Penrose in fact proved that singularities must occur in gravitational collapses, regardless of the symmetry or other properties of the initial mass.
Right now physics has enough tools to model this collapse right down to the Planck length.When you use a microscope to observe something small, you are limited by the wavelength you are using. A light microscope can only resolve details the size of the wavelength of visible light. Due to the wave particle duality nature of quantum mechanics however, you can use particles such as electrons to resolve smaller details. If you boost these particles to higher energies, their frequency increases, and you can again resolve smaller details. In a way that's what particle accelerators are for; particles of such high energy are used, you can resolve the very fundamental particles that make up everything else. When you get down to regions the size of the Planck length however, the wavelength you need has an mass/energy (E=mc^2) sufficiently large enough to form a black hole! However, if there is (as relativity and the standard model suggest), no limit to the 'smallness' of space, then there is still (in the same way you have an infinite number of integers, and an infinite number of reals between zero and one), the planck volume has still has 'space' for an infinity of things to happen!

The next physical models (such as super-string theory, loop quantum gravity) do not have this problem, as space/time is quantised; there is a fundamental unit to everything, past which or course you can't see. In fact there's hints that the 'theory of everything' must be 'background independent', that is the physics doesn't take place on some abstract mathmatical background, called space/time, rather space/time is a patchwork of discreet entities. As you've defined what the smallest bit of space is, then this must be a singularity in this formulation.

Do Singularities really matter anyway?

As I said physics can offer no answers as to what lies within the (possibly) infinitely large region from the Planck size to the singularity. For a long time physicists hoped the question was irrelevant as it appears anything massive enough to collapse its matter down to infinity will be a black hole and therefore have an event horizon associated with it. Originally this event horizon was a one-way membrane, you can put matter, energy and information past it, but nothing can come back out of it. Any singularity hidden inside the black hole can therefore never affect the rest of the universe, ever, and can therefore be forgotten about. This cosmic censorship hypothesis (suggested by Penrose in 1969) says you can never have a naked singularity; it must always be clothed by an event horizon.

Sleeping dogs have a habit of waking up....

Of course really you can't just let the problem lie there, several important hypothesis that stem from the accepted correct (if incomplete) physics mean you have to seriously think about the consequences of allowing physics to 'make' a singularity.
Firstly cosmology has long sought to explain the origin and evolution of the universe. Observations by Edwin Hubble seemed to suggest all the galaxies are moving away from each other, at a rate proportional to their distance from us. This implied that once they were very close together, in fact tracing backwards infinitely close together...This lead to the formulation of the big bang theory, where the entire universe essentially exploded from an infinitely small region; of course this is a singularity!. Every observation made has so far has confirmed some kind of big bang occurred, refinements such as cosmological inflation don't alter the fundamental fact that the theory must have contain a singularity, a fact proved by Hawking. As you can't see past the Planck length, you can't make predictions what came out of the singularity at the dawn of time, in fact as I said above you could regard the evolution of the universe from the singularity up past the Planck length to have as rich a history as our own universe since the Planck length. As what came before must determine what comes after, cosmology has a real problem with singularities..... Secondly work by Professor Stephen Hawking and other have shown black holes are not in fact completely black, and do in fact radiate at a wavelength proportional to their size. (Please see Hawking radiation for more). A consequence of this is they might radiate away, (over time) all their energy, which could leave a naked singularity behind. What effect a naked singularity would have on the rest of the universe, I don't think anybody knows, I'm pretty sure you can't in fact calculate the effect of this space-time infinity.

Also it's just occurred to me, if you allow sizes smaller than the Planck length to exist, (even if you can't measure them) then when a black hole decays past a certain point, it can emit radiation/particles of sufficient energy to again be black hole, containing a singularity. This would be a self-perpetuating growth, a free lunch of infinite size, something, which I personally do not believe, is possible.

Out with the old in with the new?

So the 'old' physics seems to predict singularities as a logical consequence, but cannot offer any theories of their behaviour; the mathematics simply breaks down. The current hot 'new' physics is superstring theory and its partner m-theory. In these space-time is quantised in that it can only come in 'packets' limited to about the Planck length in size. These strings or branes do away with the concept of infinitely small and in doing solve a lot of problems in physics. In these theories (and there are many, and no-one knows how the one that describes our universe came to be chosen) a black hole would collapse to a string or a brane and no further. The question then arises can the string/brane that is the end product of the gravitational collapse a.k.a. the singularity, contain the necessary energy and information necessary to describe the black hole?

My own two-penneth

I think in this string/brane picture the cosmic censorship can be maintained, I believe that the singularity becomes a topologically complicated knot of 11 dimensional space-time. The emission of particles from the event horizon represents decay of this 'singular' knot, as it decays, it loses energy/mass and the horizon shrinks. At the point where the horizon shrinks to nothing, the 'singularity' finally decays in a flash of Hawking radiation. I think by redefining the singularity in such a way might help some of the problems involved in black hole entropy also, (I humbly refer to my node there...).
Recent work in knot theory has shown that knots may in fact be quantised also. This would mean that some modes of decay for knot/singularity might well be forbidden, which could give rise to 'absorbance lines' in the spectra of black hole radiation. If two cosmic rays of sufficient energy were to collide they could form a black hole in the order of the Planck size, and the above effect could be seen as it decays....

This w/u was done in response to the The content rescue team:nodes Largely of the top of my head, any errors/typos/glaring omissions please msg me!

In the transhumanist definition, a singularity is basically when computers begin designing progressively more intelligent computers, and eventually they end up smarter than humans. Then, as Vinge famously put it, "the human era comes to an end."

Folks like Eliezer Yudkowsky have written lots about this concept, and it seems as though they're right. I haven't thought about it too much, mainly because I get really shaken whenever I do. Maybe that's what fundamentalists go through - I don't know, and I'm not sure I want to.

What Yudkowsky, and many others, are trying for is a friendly singularity - one in which the AI will act in a manner similiar to how humanity would prefer it to act. An unfriendly AI could be worse than any dictator, or use nanotechnology to turn the solar system into a machine to do number-crunching to help with someone's thesis - it's impossible to be sure. The consensus among transhumanists is that a singularity is more or less inevitable, but that we need to be careful how and when it occurs.

Singularity by WinAbility
Alternate file manager

Singularity by default uses a vertical-split double pane view, but it can easily be made to look just like an explorer window, or even a horizontal split. The main toolbar, drivebar and the two sub-toolbars for each pane are completely editable, and you can choose if you want each pane to have it's own titlebar independent of the other. I would prefer it if I was able to add my own 'favorite' links to a user toolbar, like several others allow. Singularity uses the standard Windows Explorer context menus, but I had a little problem using it with PicaView (acdsystems.com), but that wasn't the fault of Singularity. The built-in Find command launches the explorer find, which can be a problem for those of other who use alternate shells. On my system, it does take slightly longer to load than Servant Salamander or 2xExplorer to load or refresh a directory, but for a P200+ it should be negligible. A nice touch: The window icon animates (can be turned off.)

(Shareware, \$39.95 USD)

Official Site
http://www.winability.com/singularity

Also see File Manager.

Sin`gu*lar"i*ty (?), n.; pl. Singularities (#). [L. singularitas: cf. F. singularit'e.]

1.

The quality or state of being singular; some character or quality of a thing by which it is distinguished from all, or from most, others; peculiarity.

Pliny addeth this singularity to that soil, that the second year the very falling down of the seeds yieldeth corn. Sir. W. Raleigh.

I took notice of this little figure for the singularity of the instrument. Addison.

2.

Anything singular, rare, or curious.

Your gallery Have we passed through, not without much content In many singularities. Shak.

3.

Possession of a particular or exclusive privilege, prerogative, or distinction.

No bishop of Rome ever took upon him this name of singularity [universal bishop]. Hooker.

Catholicism . . . must be understood in opposition to the legal singularity of the Jewish nation. Bp. Pearson.

4.

Celibacy.

[Obs.]

Jer. Taylor.