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...).
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!
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....