We are told that at the centre of a black hole lies a singularity; very small and very heavy. We here a lot about the mass but not the size (diameter) of this object,I asume this is not zero So, incredibly tiny though it may be , it would still bel greater than 0. Questions;- What then is the size of a singularity?. Does it , or could it, vary according to the age and mass of the black hole? Could the singularity be continually shrinking, ie getting denser with time.It would only get to 0, of course, after an infinitely long time, ie, never.
The only working theory of gravity that we have is general relativity. According to GR, the size of the singularity is zero. That is basically the definition of the word "singularity." People are working on extending GR to include quantum mechanics, but we don't yet have a theory of quantum gravity. If you ask a physicist to bet a six-pack on it, the most popular guess is that a theory of quantum gravity would give a size for the singularity that was on the order of the Planck length: http://en.wikipedia.org/wiki/Planck_length
My understanding of the definition of a singularity is: a point of infinite density in space. With that definition I have always come to the conclusion that it has no size due to the fact a "point" is quantifiable only with only 1 dimension. It may have width, for example, but no height, no depth. After that the math seems easy, 1 x 0 x 0 = 0.
WannabeNewton made a good point that I was oversimplifying, and SYahoo beat me to answering :-) I think the most general definition of a singularity in GR is that a spacetime has a singularity if it's geodesically incomplete. What that means is basically that you can have free-falling observers for whom the time measured by a ticking clock cannot be extended arbitrarily far into the past (big bang singularity) or future (black hole singularity). Singularities are not points or sets of points in spacetime, so it's not actually obvious to me how you go about defining the dimension of a singularity in general. (And I think for this reason it also doesn't make sense to define a singularity as SYahoo proposes, as a point of infinite density in space -- it's not a point in space.)
In any event, singularities are nonsense. They are areas where the theory breaks down and cannot be taken seriously. I strongly suspect that as you get closer and closer to what General Relativity describes as a singularity, the correct description of what is going on in reality will start to differ dramatically from the General Relativity description. I doubt you have to go all the way to the Planck length for the discrepancies to start.
When using elliptical coordinates, the coordinate radius of the ring singularity is equal to the spin parameter a, see the last two images on this page, the proper radius would be zero at the ring edge. From what I can gather, only the space-like qualities of the Schwarzschild solution dictate a true singularity, in the case of Kerr metric (and Kerr-Newman & Reissner–Nordström metrics) the supposed singularity resides in time-like space so therefore matter might reside at a stable r, I have seen one or two suggestions that the ring may be super dense matter similar to neutron or quark matter, there also might be a weak singularity at the inner horizon (Cauchy horizon) which supposedly marks the boundary of predictability.
Are there any observable signatures of singularities? Something that can testify their existence in nature rather than mathematical artifacts.
Are you asking that: Is it true that a black hole always contains a point of singularity inside its horizon? I think the answer is yes. Hope somebody will explain it further.
No, absolutely not. A black hole, as we understand it, is an event horizon. In General Relativity, there exists a singularity inside this event horizon. But that's just a nonsensical statement mathematically, so it can't actually be true. Instead, there must exist some other theory of gravity which accurately describes the state of matter deep inside a black hole. It may be very dense, but it won't be singular.
The only time I've heard a physicist go into detail on it (not sure if I should be quoting him or if I'm even allowed to on this site) he basically said that at all times in reality, every time math predicts something at infinity, nature finds a tricky way out of it. The example he gave was that water spinning down a hole at the exact center should be spinning at an infinite rate, instead what you see is no water at the center.
Are there any other possible stable states for collapsed stars between a neutron star and a black hole pseudo singularity? I recall reading about quark stars but could there be others? Could some objects that we think are Black Holes be these instead? If so perhaps these could solve the singularity dilemma? http://en.wikipedia.org/wiki/Quark_star
The presumed super-massive black holes at galactic centers are not very dense (thus quark star models are irrelevant) but have event horizons. The indirect evidence for event horizons is compelling, and they may soon be observed directly. See: https://www.physicsforums.com/showthread.php?t=510860 esp. Bcrowell's last post with references.
Pallen thanks for the link. I also found this link: http://en.wikipedia.org/wiki/Exotic_star I was mainly trying to find out if there were other denser but stable forms of matter that still had some finite size ie. avoided need for infinitessimals, singularities or planck sized lengths etc.
Pallen, arent all Black Holes pseudo singularities? Therefore their density goes up as their mass increases, and all of which are the most dense forms of matter possible?
Black holes as a plausible observable phenomenon refer to the event horizon. The galactic black holes show that dense matter explanations to avoid dealing with an event horizon do not work. Best, and steadily increasing evidence, suggests event horizons are a real part of our universe. What happens inside an event horizon is another matter altogether. For one thing, it is unobservable, in principle, under current theory. GR, as a literal, classical theory says there must be some form of singularity inside, but it is most likely a very chaotic one, not one of the neat exact solutions of GR (still within the realm of classical GR - the exact solutions are unstable - any slight deviation magnifies inside the horizon). [I ignore the possibility if naked singularities; they are not currently predicted by plausible models within our universe, so far as I know]. What really happens inside the horizon? I know of no physicist who thinks classical GR will continue to hold. So far as I know, no candidate models make meaningful predictions. Inside an event horizon, a quark star provides no additional insight without a working theory of this regime. Whether a quark star is a possible evolutionary state of some stellar process that, in a few circumstances, avoids an event horizon is an interesting problem in stellar dynamics, but irrelevant as a model of black holes.