This month, April 2009, the Spanish edition of Scientific American(investigación y ciencia)has an article about naked singularities (the English version was dated on February).

The author, Pankaj S. Joshi, seems to be an total expert in the subject (a common issue in Scientific American)and has a recent book- from 2008- about the particular,

Gravitational Collapse and Spacetime Singularities.

I didn´t read the book, but I found the article interesting so I searched in wikipedia and some of the papers linked there. I´ll try to explain some of the aspects now.

In general relativity there are two well known places where singularities appear, black holes and cosmology. The most naive way of thinking/defining singularities is to characterise them as points where the metric (or curvature) of space-time becomes infinite. According to that the event horizon of a Schwarschild black hole would be a singularity. It was realized that that infinity was due to a bad choice of coordinates. In that solution the centrer of the black hole, the R=0 point, also get an infinity value and this can´t be overcome by any other choice of coordinates so it is a genuine singularity.

As far as we like to have coordinate free definitions there is more technical definition, a singularity is defined to be one which contains geodesics which cannot be extended in a smooth manner. The end of such a geodesic is considered to be the singularity. That definition is useful to probe theorems (as did Penrose and Hawkings in the 70´s) about how singularities can´t be avoided in classical general relativity in cosmological sceneries. It also permit to make a diferentiation about coordinate singularities, the ones I talked before, related to black holes, and what are known as conical singularities, related to things such as cosmic strings. A conical singularity occurs when there is a point where the limit of every diffeomorphism invariant quantity is finite. In which case, spacetime is not smooth at the point of the limit itself. Thus, spacetime looks like a cone around this point, where the singularity is located at the tip of the cone. The metric can be finite everywhere if a suitable coordinate system is used.

The S.A article treats about singularities associated to gravitational collapse. There exists what is known like the cosmic censorship conjecture (or hypothesis) , own to Roger Penrose, which states that there are not naked singularities. That is, every singularity must be hidden behind an event horizon and cant be seen from the outside. It has the status of conjecture because it hasn´t be proved in a rigorous way under physically reasonable assumptions. In fact it hasn´t even stated in a mathemathical rigorous way.

The article, obviously, try to answer the conjecture in the negative. Before describing it's arguments I´ll talk about an aspect closely related to the CCC. If one examines the solutions for rotating black holes (Kerr solution) one sees that if is allowed that J, the angular momentum is greater than the mass M of the black hole, that is J/M>1 a naked singularity (a ringed shaped one) appears. A similar thing goes for charged (electric of whatever associated to U(1) gauge symmetry)black hole solutions (Reissner-Nordstom black holes) where a naked singularity appears if the charge, Q, is greater than the mass of the b-h, i.e., Q/M>1. It is commonly assumed that the CCC holds and the case where the equality would arise are called extremal black holes. The possibility of a reverse, negative, sign in the above expressions is not even contemplated in most theoretical considerations.

That´s a reason why I was somewhat surprised when I read the article and saw that in fact when one makes actual calculations, in classical general relativity, of how gravitational collapse behaves when some oversimplifying assumptions about the state of the star are neglected naked singularities actually are shown to be possible.

If the star is perfectly spherically symmetric and of uniform density everything is o.k and the black hole is formed. But relaxing one of the assumptions separately(or both at once) it can be shown that naked singularities actually appear.

Intuitively the reason is, for the case of non uniform density, that it can happen that the rate of accretion into the centre is never fast enough to actually form an even horizon and a central, unique, singularity appears. In the case of non sphericity it is shown that the collapse is neither spheric so the mass is concentrated in two points that become singularities at the end of the collapse avoiding the formation of the event horizon because of the oblong shape of the infalling matter that forbids the concentration of enough mass inside the Schwarschild radius.

Once that this facts are established one can wonder about how realistic and stable are. After all they mean that a large amount of the mass of a big star is concentrated in a point. The precise nature of what a singularity actually is usually is thought to be a question related to quantum gravity. But for regions relatively close (but not too much) to the singularity classical relativity still holds and it is expected that neighbouring matter would be attracted to the singularity. The inexistence of the event horizon means that there is the possibility of going arbitrarily near the singularity and returning to the original point (well, if tide forces don´t kill you and such that). In particular light can go near the singularity and scape to a distant observer so we can see what happens there. But even thought some matter in certain trajectories could scape I think that it is reasonable that most of the matter would be trapped in the singularity. Intuitively one would think that that increases the mass of the singularity and that it sooner or later it will become a black hole. Possibly that is a too naive way of think and that is one of the particularities of the singularity (but I am not sure about it).

Anyway, if singularities re shown to be possible (at least for a certain time) one could try to consider if they are distinguishable from black holes. The answer is in the positive. Even one could try a little bit further and consider the possibility that an existing black hole could break and leave behind the singularity. The most natural case would be kerr black hole which is led to rotate faster than it´s extremal limit. Because astrophysical black holes are usually believed to be Kerr ones one immediately can answer for particular observable signatures of this breaking. ONe arxiv article where one can read the details is this: Magnification relations for Kerr lensing and testing Cosmic Censorship. There are described some mathematical details of calculations made on the pna (post Newtonian approximation)of some optical effects. The author claim that the differences in behaviour among black holes and naked singularities could be observed with the incoming new generation of available technology.

From the viewpoint of an string theorist 4 dimensions are very restrictive, what about the influence of additional dimensions? You can read a paper about the particular: Spherical gravitational collapse in N-dimensions. It is co-authored by Joshi and the answer is mildly positive. I recommend to read the considerations that he makes in the conclusions.

A later thing I am going to discuss is the role of quantum gravity. If naked singularities actually exist they are a window to do observations of quantum gravitational effects (or at least one so expects). But before going there one could answer if quantum gravity considerations modify the classical predictions of formation of naked singularities. I don´t know the "asscendence" of Joshi, tat is, if he is an string theoretic oriented or an LQG oriented researcher. Being an specialist in general relativity one, maybe, would expect him being an LQG researcher, but reading his papers I guess that a better fir would be to consider him a "naked singularity phenomenologist". Anyway, LQG is easier to learn that string theory and is accepted by a plausible quantum gravity by hundreds of people with a tenures/investigation positions in universities so it is reasonable to expect some paper using LQG to investigate the question. And, effectively, there is such paper: Quantum evaporation of a naked singularity.

This papers point in a different direction that the classical results. Using a toy model with an scalar field (in a way similar to loop quantum cosmology calculations) it is shown that near the should be singularity gravity becomes a repulsive force and the naked singularity isn't formed. I guess that one must understand that this calculations make sense in the case where the classical equations point to the formation of the singularity. That is, classically one expect the formation of the naked singularity, but looking at quantum phenomena one sees that actually the singularity is avoided. I must clarify that this calculations are not claimed to be fully quantum by the authors and they still believe that in full quantum sceneries the naked singularity would easily reappear.

Still this last scenery could have relevant observational consequences in the form of powerful gamma ray bursts that result in the evaporation of the should be singularity. The precise signature of that bursts depend in some free factors of the theory, and, in particular, the claim that can be used to estimate a value of LQG, j, the value of the representation of (complex) SU(2) used.

Well, certainly I don´t believe that string theory people would take too seriously this considerations, but claiming possible near future experimental results I believe it well deserves to say something about the subject.

In fact actually there are some results in string theory about singularities, particularly the enhanchon mechanism, but it is related to "educated" singularities inside a black hole who are prudent enough to not show themselves naked. Abut black holes in string theory I hope to write a post soon.

To end this post to leave a link to a self claimed naked singularity who is kind enough have a blog (in Spanish), that links to this: La Singularidad Desnuda.

## Wednesday, April 22, 2009

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