Editing Gravitational singularity (section)

==Types==
There are multiple types of singularities, each with different physical features that have characteristics relevant to the theories from which they originally emerged, such as the different shapes of the singularities, ''conical and curved''. They have also been hypothesized to occur without event horizons, structures that delineate one spacetime section from another in which events cannot affect past the horizon; these are called ''naked.''

===Conical===

A conical singularity occurs when there is a point where the limit of some [[Diffeomorphism invariance|diffeomorphism invariant]] quantity does not exist or is infinite, in which case spacetime is not smooth at the point of the limit itself. Thus, spacetime looks like a [[Cone (geometry)|cone]] around this point, where the singularity is located at the tip of the cone. The metric can be finite everywhere the [[coordinate system]] is used.

An example of such a conical singularity is a [[cosmic string]] and a [[Schwarzschild metric|Schwarzschild black hole]].<ref>{{cite journal |last1=Copeland |first1=Edmund J. |last2=Myers |first2=Robert C. |last3=Polchinski |first3=Joseph |year=2004 |title=Cosmic F- and D-strings |journal=Journal of High Energy Physics |volume=2004 |issue=6 |page=13 |arxiv=hep-th/0312067 |bibcode=2004JHEP...06..013C |doi=10.1088/1126-6708/2004/06/013 |s2cid=140465}}</ref>

===Curvature===
[[File:Black hole details.svg|thumb|upright=0.7|A simple illustration of a non-spinning [[black hole]] and its singularity ]]
Solutions to the equations of [[general relativity]] or another theory of [[gravity]] (such as [[supergravity]]) often result in encountering points where the [[Metric (general relativity)|metric]] blows up to infinity. However, many of these points are completely [[Smooth function|regular]], and the infinities are merely a result of [[Coordinate singularity|using an inappropriate coordinate system at this point]]. To test whether there is a singularity at a certain point, one must check whether at this point [[Diffeomorphism invariance|diffeomorphism invariant]] quantities (i.e. [[scalar (physics)|scalar]]s) become infinite. Such quantities are the same in every coordinate system, so these infinities will not "go away" by a change of coordinates.

An example is the [[Schwarzschild metric|Schwarzschild]] solution that describes a non-rotating, [[Electric charge|uncharged]] black hole. In coordinate systems convenient for working in regions far away from the black hole, a part of the metric becomes infinite at the [[event horizon]]. However, spacetime at the event horizon is [[Smooth function|regular]]. The regularity becomes evident when changing to another coordinate system (such as the [[Kruskal coordinates]]), where the metric is perfectly [[Smooth function|smooth]]. On the other hand, in the center of the black hole, where the metric becomes infinite as well, the solutions suggest a singularity exists. The existence of the singularity can be verified by noting that the [[Kretschmann scalar]], being the square of the [[Riemann tensor]] i.e. <math>R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}</math>, which is diffeomorphism invariant, is infinite.

While in a non-rotating black hole the singularity occurs at a single point in the model coordinates, called a "point singularity", in a rotating black hole, also known as a [[Kerr black hole]], the singularity occurs on a ring (a circular line), known as a "[[ring singularity]]". Such a singularity may also theoretically become a [[wormhole]].<ref>If a rotating singularity is given a uniform electrical charge, a repellent force results, causing a [[ring singularity]] to form. The effect may be a stable [[wormhole]], a non-point-like puncture in spacetime that may be connected to a second ring singularity on the other end. Although such wormholes are often suggested as routes for faster-than-light travel, such suggestions ignore the problem of escaping the black hole at the other end, or even of surviving the immense [[tidal force]]s in the tightly curved interior of the wormhole.</ref>

More generally, a spacetime is considered singular if it is [[Geodesic (general relativity)#Geodesic incompleteness and singularities|geodesically incomplete]], meaning that there are freely-falling particles whose motion cannot be determined beyond a finite time, being after the point of reaching the singularity. For example, any observer inside the [[event horizon]] of a non-rotating black hole would fall into its center within a finite period of time. The classical version of the [[Big Bang]] [[physical cosmology|cosmological]] model of the [[universe]] contains a causal singularity at the start of [[time]] (''t''=0), where all time-like geodesics have no extensions into the past. Extrapolating backward to this hypothetical time 0 results in a universe with all spatial dimensions of size zero, infinite density, infinite temperature, and infinite spacetime curvature.

===Naked singularity===
{{Main Article|Naked singularity}}
Until the early 1990s, it was widely believed that general relativity hides every singularity behind an [[event horizon]], making naked singularities impossible. This is referred to as the [[cosmic censorship hypothesis]]. However, in 1991, physicists Stuart Shapiro and [[Saul Teukolsky]] performed computer simulations of a rotating plane of dust that indicated that general relativity might allow for "naked" singularities. What these objects would actually look like in such a model is unknown. Nor is it known whether singularities would still arise if the simplifying assumptions used to make the simulation were removed. However, it is hypothesized that light entering a singularity would similarly have its geodesics terminated, thus making the [[naked singularity]] look like a black hole.<ref>{{Cite journal |last=Bojowald |first=Martin |year=2008 |title=Loop Quantum Cosmology |journal=Living Reviews in Relativity |language=en |volume=11 |issue=1 |pages=4 |bibcode=2008LRR....11....4B |doi=10.12942/lrr-2008-4 |issn=2367-3613 |pmc=5255532 |pmid=28163606 |doi-access=free}}</ref><ref>{{Cite journal |last1=Goswami |first1=Rituparno |last2=Joshi |first2=Pankaj S. |year=2008 |title=Spherical gravitational collapse in N dimensions |journal=Physical Review D |language=en |volume=76 |issue=8 |pages=084026 |arxiv=gr-qc/0608136 |bibcode=2007PhRvD..76h4026G |doi=10.1103/PhysRevD.76.084026 |issn=1550-7998 |s2cid=119441682}}</ref><ref>{{Cite journal |last1=Goswami |first1=Rituparno |last2=Joshi |first2=Pankaj S. |last3=Singh |first3=Parampreet |date=2006-01-27 |title=Quantum Evaporation of a Naked Singularity |journal=Physical Review Letters |language=en |volume=96 |issue=3 |pages=031302 |arxiv=gr-qc/0506129 |bibcode=2006PhRvL..96c1302G |doi=10.1103/PhysRevLett.96.031302 |issn=0031-9007 |pmid=16486681 |s2cid=19851285}}</ref>

Disappearing event horizons exist in the&nbsp;[[Kerr metric]], which is a spinning black hole in a vacuum, if the&nbsp;[[angular momentum]]&nbsp;(<math>J</math>) is high enough. Transforming the Kerr metric to&nbsp;[[Boyer–Lindquist coordinates]], it can be shown<ref>{{harvnb|Hobson|Efstathiou|Lasenby|2013|pp=300-305}}.</ref>&nbsp;that the coordinate (which is not the radius) of the event horizon is, <math>r_{\pm} = \mu \pm \left(\mu^{2} - a^{2}\right)^{1/2}</math>, where&nbsp;<math>\mu = G M / c^{2}</math>, and&nbsp;<math>a=J/M c</math>. In this case, "event horizons disappear" means when the solutions are complex for&nbsp;<math>r_{\pm}</math>, or&nbsp;<math>\mu^{2} < a^{2}</math>. However, this corresponds to a case where <math>J</math> exceeds <math>GM^{2}/c</math> (or in [[Planck units]], {{Nowrap|<math>J > M^{2}</math>)}}; i.e. the spin exceeds what is normally viewed as the upper limit of its physically possible values.

Similarly, disappearing event horizons can also be seen with the&nbsp;[[Reissner–Nordström metric|Reissner–Nordström]]&nbsp;geometry of a charged black hole if the charge&nbsp;(<math>Q</math>) is high enough. In this metric, it can be shown<ref>{{harvnb|Hobson|Efstathiou|Lasenby|2013|pp=320-325}}.</ref>&nbsp;that the singularities occur at <math>r_{\pm}= \mu \pm \left(\mu^{2} - q^{2}\right)^{1/2}</math>, where&nbsp;<math>\mu = G M / c^{2}</math>, and&nbsp;<math>q^2 = G Q^2/\left(4 \pi \epsilon_0 c^4\right)</math>. Of the three possible cases for the relative values of&nbsp;<math>\mu</math> and&nbsp;<math>q</math>, the case where&nbsp;<math>\mu^{2} < q^{2}</math>&nbsp;causes both&nbsp;<math>r_{\pm}</math> to be complex. This means the metric is regular for all positive values of&nbsp;<math>r</math>, or in other words, the singularity has no event horizon. However, this corresponds to a case where <math>Q/\sqrt{4 \pi \epsilon_0}</math> exceeds <math>M\sqrt{G}</math> (or in Planck units, {{Nowrap|<math>Q > M</math>)}}; i.e. the charge exceeds what is normally viewed as the upper limit of its physically possible values. Also, actual astrophysical black holes are not expected to possess any appreciable charge.

A black hole possessing the lowest <math>M</math> value consistent with its <math>J</math> and <math>Q</math> values and the limits noted above; i.e., one just at the point of losing its event horizon, is termed [[extremal black hole|extremal]].