Gravitational singularity

{{short description|Condition in which spacetime itself breaks down}}

A '''gravitational singularity''', '''spacetime singularity''', or simply '''singularity''', is a theoretical condition in which [[gravitational field|gravity]] is predicted to be so intense that [[spacetime]] itself would break down catastrophically. As such, a singularity is by definition no longer part of the regular spacetime and cannot be determined by "where" or "when". Gravitational singularities exist at a junction between [[general relativity]] and [[quantum mechanics]]; therefore, the properties of the singularity cannot be described without an established theory of [[quantum gravity]]. Trying to find a complete and precise definition of singularities in the theory of general relativity, the best theory of gravity available, remains a difficult problem.<ref>{{harvnb|Earman|1995|loc=Section 2.2 ''What is a singularity?''|pp=28–31}}</ref><ref name="curiel" /> A singularity in general relativity can be defined by the [[Curvature invariant (general relativity)|scalar invariant]] [[Curvature of Riemannian manifolds|curvature]] becoming [[Infinity|infinite]]<ref>{{cite web|url=http://www.physicsoftheuniverse.com/topics_blackholes_singularities.html|title=Singularities |website=Physics of the Universe }}</ref> or, better, by a [[Geodesics in general relativity|geodesic]] being [[Geodesic manifold#Non-examples|incomplete]].<ref>{{Cite journal |last=Uggla |first=Claes |year=2006 |title=Spacetime Singularities |url=http://www.einstein-online.info/spotlights/singularities |journal=[[Einstein Online]] |publisher=[[Max Planck Institute for Gravitational Physics]] |volume=2 |archive-url=https://web.archive.org/web/20170124030605/http://www.einstein-online.info/spotlights/singularities |archive-date=2017-01-24 |access-date=2015-10-20 |number=1002}}</ref>

General relativity predicts that any object collapsing beyond its [[Schwarzschild radius]] would form a black hole, inside which a singularity will form.<ref name=curiel>{{cite encyclopedia|url=http://plato.stanford.edu/entries/spacetime-singularities/|title=Singularities and Black Holes|last=Curiel|first=Erik|publisher=Metaphysics Research Lab, Stanford University |name-list-style=amp|encyclopedia=Stanford Encyclopedia of Philosophy|date=2021 |access-date=1 October 2021}}</ref> A black hole singularity is, however, covered by an [[event horizon]], so it is never in the [[causal past]] of any outside observer, and at no time can it be objectively said to have formed.<ref>{{cite journal | last1 = Narlikar | first1 = J. V. | author-link1 = Jayant Narlikar | last2 = Padmanabhan | first2 = Th. | author-link2 = Thanu Padmanabhan | date = June 1988 | title = The Schwarzschild solution: Some conceptual difficulties | journal = [[Foundations of Physics|Found Phys]] | volume = 18 | issue = 6 | publisher = [[Springer Nature]] | pages = 659–668 | doi = 10.1007/BF00734568 | bibcode = 1988FoPh...18..659N }}</ref> General relativity also predicts that the initial state of the [[universe]], at the beginning of the [[Big Bang]], was a singularity of infinite density and temperature.<ref>{{harvnb|Wald|1984|p=99}}.</ref>{{Obsolete source|reason=this source is forty years old - a lot has changed since then.|date=October 2024}} However, [[classical field theory|classical]] gravitational theories are not expected to be accurate under these conditions, and a quantum description is likely needed.<ref>{{cite web |last=Hawking |first=Stephen |title=The Beginning of Time |url=http://www.hawking.org.uk/the-beginning-of-time.html |work=Stephen Hawking: The Official Website |publisher=[[Cambridge University]] |access-date=26 December 2012 |archive-date=6 October 2014 |archive-url=https://web.archive.org/web/20141006200729/http://www.hawking.org.uk/the-beginning-of-time.html }}</ref> For example, quantum mechanics does not permit particles to inhabit a space smaller than their [[Compton wavelength]]s.<ref>{{cite book |last=Zebrowski |first=Ernest |url=https://books.google.com/books?id=2twRfiUwkxYC |title=A History of the Circle: Mathematical Reasoning and the Physical Universe |date=2000 |publisher=[[Rutgers University Press]] |isbn=978-0-8135-2898-4 |location=Piscataway New Jersey |page=180}}</ref>

==Interpretation==

Many theories in physics have [[mathematical singularities]] of one kind or another. Equations for these physical theories predict that the ball of mass of some quantity becomes infinite or increases without limit. This is generally a sign for a missing piece in the theory, as in the [[ultraviolet catastrophe]], [[renormalization|re-normalization]], and instability of a hydrogen atom predicted by the [[Larmor formula]].

In classical field theories, including special relativity but not general relativity, one can say that a solution has a singularity at a particular point in spacetime where certain physical properties become ill-defined, with spacetime serving as a background field to locate the singularity. A singularity in general relativity, on the other hand, is more complex because spacetime itself becomes ill-defined, and the singularity is no longer part of the regular spacetime manifold. In general relativity, a singularity cannot be defined by "where" or "when".<ref>{{Cite book |last=Ashtekar |first=Abhay |title=100 years of relativity: space-time structure, Einstein and beyond |date=2005 |publisher=World Scientific |isbn=978-981-256-394-1 |editor-last=Ashtekar |editor-first=Abhay |location=Singapore |chapter=3: "The nature of spacetime singularities" by Alan D. Randall}}</ref>

Some theories, such as the theory of [[loop quantum gravity]], suggest that singularities may not exist.<ref>{{Cite journal |last1=Gambini |first1=Rodolfo |last2=Olmedo |first2=Javier |last3=Pullin |first3=Jorge |date=2014-05-07 |title=Quantum black holes in loop quantum gravity |journal=Classical and Quantum Gravity |volume=31 |issue=9 |article-number=095009 |arxiv=1310.5996 |bibcode=2014CQGra..31i5009G |doi=10.1088/0264-9381/31/9/095009 |issn=0264-9381 |s2cid=119247455}}</ref> This is also true for such classical unified field theories as the Einstein–Maxwell–Dirac equations. The idea can be stated in the form that, due to [[quantum gravity]] effects, there is a minimum distance beyond which the force of gravity no longer continues to increase as the distance between the masses becomes shorter, or alternatively that interpenetrating particle waves mask gravitational effects that would be felt at a distance.

==In black holes==

Mathematical models of [[black holes]] based on general relativities have singularities at their centers—points where the curvature of spacetime becomes infinite, and [[geodesic]]s terminate within a finite [[proper time]]. However, it is unknown whether these singularities truly exist in real black holes.<ref name="cd22">{{Cite book |title=The Foundations of Spacetime Physics |last1=Crowther |first1=Karen |date=2022-09-29 |isbn=978-1-003-21901-9 |location=New York |url=https://philarchive.org/archive/CROFAT-5 |last2=De Haro |first2=Sebastian |editor-last=Vassallo |editor-first=Antonio |chapter=Four Attitudes Towards Singularities in the Search for a Theory of Quantum Gravity |doi=10.4324/9781003219019 |arxiv=2208.05946 }}</ref> Some physicists believe that singularities do not exist, and that their existence, which would make spacetime [[causality (physics)|unpredictable]], signals a breakdown of general relativity and a need for a more complete understanding of [[quantum gravity]].<ref>{{Cite journal |last1=Alesci |first1=Emanuele |last2=Bahrami |first2=Sina |last3=Pranzetti |first3=Daniele |title=Quantum gravity predictions for black hole interior geometry |journal=Physics Letters B |date=2019 |volume=797 |article-number=134908 |doi=10.1016/j.physletb.2019.134908 |arxiv=1904.12412 |bibcode=2019PhLB..79734908A }}</ref><ref>{{Cite journal |last1=Koshelev |first1=Alexey S. |last2=Tokareva |first2=Anna |title=Nonperturbative quantum gravity denounces singular black holes |journal=Physical Review D |date=2025 |volume=111 |issue=8 |article-number=086026 |doi=10.1103/PhysRevD.111.086026 |bibcode=2025PhRvD.111h6026K |doi-access=free }}</ref><ref>{{Cite journal |last1=Olmedo |first1=Javier |last2=Saini |first2=Sahil |last3=Singh |first3=Parampreet |title=From black holes to white holes: A quantum gravitational, symmetric bounce |journal=Classical and Quantum Gravity |date=2017 |volume=34 |issue=22 |doi=10.1088/1361-6382/aa8da8 |arxiv=1707.07333 |bibcode=2017CQGra..34v5011O }}</ref> Others believe that such singularities could be resolved within the current framework of physics, without having to introduce quantum gravity.<ref name="cd22" /> There are also physicists, including Kip Thorne<ref name="scienceofinterstellar" /> and [[Charles Misner]],<ref>{{Cite journal |last1=Misner |first1=Charles W. |title=Absolute Zero of Time |journal=Physical Review |date=1969 |volume=186 |issue=5 |pages=1328–1333 |doi=10.1103/PhysRev.186.1328 |bibcode=1969PhRv..186.1328M }}</ref> who believe that not all singularities can be resolved, and that some likely still exist in the real universe despite the effects of quantum gravity.<ref name="cd22" /><ref name="dlc07">{{Cite journal |last1=Doran |first1=Rosa |last2=Lobo |first2=Francisco S. N. |last3=Crawford |first3=Paulo |title=Interior of a Schwarzschild Black Hole Revisited |journal=Foundations of Physics |date=2008 |volume=38 |issue=2 |pages=160–187 |doi=10.1007/s10701-007-9197-6 |arxiv=gr-qc/0609042 |bibcode=2008FoPh...38..160D }}</ref> Finally, still others believe that singularities do not exist, and that their existence in general relativity does not matter, since general relativity is already believed to be an incomplete theory.<ref name="cd22" />

According to general relativity, every black hole has a singularity inside.<ref name="Carroll-2004">{{cite book |last1=Carroll |first1=Sean M. |author-link=Sean M. Carroll |title=Spacetime and Geometry: An Introduction to General Relativity |title-link=Spacetime and Geometry |date= |publisher=Addison-Wesley |year=2003 |isbn=978-0-8053-8732-2}}, the lecture notes on which the book was based are available for free from Sean Carroll's [https://www.preposterousuniverse.com/spacetimeandgeometry/ website] {{Webarchive|url=https://web.archive.org/web/20170323013522/http://www.preposterousuniverse.com/spacetimeandgeometry/|date=23 March 2017}}</ref>{{rp|p=205}}<ref name="hp70">{{Cite journal |last1=Hawking |first1=S. W. |last2=Penrose |first2=R. |title=The singularities of gravitational collapse and cosmology |journal=Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences |date=1970 |volume=314 |issue=1519 |pages=529–548 |doi=10.1098/rspa.1970.0021 |bibcode=1970RSPSA.314..529H }}</ref> For a non-rotating black hole, this region takes the shape of a single point; for a rotating black hole it is smeared out to form a [[ring singularity]] that lies in the plane of rotation.<ref name="Carroll-2004"/>{{rp|p=264}} In both cases, the singular region has zero volume. All of the mass of the black hole ends up in the singularity.<ref name=Carroll-2004/>{{rp|p=252}} Since the singularity has nonzero mass in an infinitely small space, it can be thought of as having infinite [[mass density|density]].<ref>{{cite news |title=Sizes of Black Holes? How Big is a Black Hole? |url=https://www.skyandtelescope.com/astronomy-resources/how-big-is-a-black-hole/ |access-date=9 October 2018 |work=[[Sky & Telescope]] |date=22 July 2014 |archive-date=3 April 2019 |archive-url=https://web.archive.org/web/20190403035741/https://www.skyandtelescope.com/astronomy-resources/how-big-is-a-black-hole/ |url-status=live }}</ref>

[[File:BKL dynamics - video.ogg|thumbtime=0|thumb|alt=A sphere being continuously warped into different ellipsoids, with the direction of the warping constantly changing.|Chaotic oscillations of spacetime experienced by an object approaching a gravitational singularity]]

Observers falling into a Schwarzschild black hole (i.e., non-rotating and not charged) cannot avoid being carried into the singularity once they cross the event horizon.<ref>{{cite journal |last1=Lewis |first1=G. F. |last2=Kwan |first2=J. |title=No Way Back: Maximizing Survival Time Below the Schwarzschild Event Horizon |url=https://www.cambridge.org/core/journals/publications-of-the-astronomical-society-of-australia/article/no-way-back-maximizing-survival-time-below-the-schwarzschild-event-horizon/2A1CCF5CB13E7BEFA6441B3038C635A3 |journal=Publications of the Astronomical Society of Australia |volume=24 |issue=2 |pages=46–52 |date=2007 |doi=10.1071/AS07012 |arxiv=0705.1029 |bibcode=2007PASA...24...46L |s2cid=17261076}}</ref><ref>{{Cite journal |last1=Toporensky |first1=Alexei |last2=Popov |first2=Sergei |title=How to Delay Death and Look Further into the Future if You Fall into a Black Hole |journal=Resonance |date=2023 |url=https://link.springer.com/article/10.1007/s12045-023-1602-8 |volume=28 |issue=5 |pages=737–749 |doi=10.1007/s12045-023-1602-8|url-access=subscription }}</ref> As they fall further into the black hole, they will be torn apart by the growing [[tidal force]]s in a process sometimes referred to as [[spaghettification]] or the ''noodle effect''. Eventually, they will reach the singularity and be crushed into an infinitely small point.<ref name=JCWheeler-2007>{{cite book |last=Wheeler |first=J. Craig | author-link = J. Craig Wheeler |title=Cosmic Catastrophes |edition=2nd |publisher=Cambridge University Press |date=2007 |isbn=978-0-521-85714-7}}</ref>{{rp|p=182}}

Although in theory, the interior of a Schwarzschild black hole curves inwards towards a sharp point at the singularity, this model is only true when the spacetime inside the black hole had not been perturbed. Any perturbations, such as those caused by matter or radiation falling in, would cause space to [[BKL singularity|oscillate chaotically]] near the singularity. Any matter falling in would experience intense tidal forces rapidly changing in direction, all while being compressed into an increasingly small volume.<ref>{{Cite book |last1=Belinskii |first1=V.A. |last2=Lifshitz |first2=E.M. |last3=Khalatnikov |first3=I.M. |last4=Agyei |first4=A.K. |title=Perspectives in Theoretical Physics |chapter=The oscillatory mode of approach to a singularity in homogeneous cosmological models with rotating axes |date=1992 |pages=677–689 |doi=10.1016/B978-0-08-036364-6.50048-X |isbn=978-0-08-036364-6 }}</ref><ref name="scienceofinterstellar">{{Cite book |title=The Science of Interstellar |last=Thorne |first=Kip |date=7 November 2014 |publisher=W. W. Norton & Company |isbn=978-0-393-35137-8}}</ref><ref>{{Cite web |title=Of singularities and breadmaking |url=https://www.einstein-online.info/en/spotlight/singularities_bkl/ |last=Garfinkle |first=David |url-status=live |archive-url=https://web.archive.org/web/20250709022836/https://www.einstein-online.info/en/spotlight/singularities_bkl/ |archive-date=9 July 2025 |access-date=14 October 2025 |website=Einstein Online |year=2007}}</ref>

In the case of a charged or rotating black hole, it is possible to avoid the singularity. Extending these solutions as far as possible reveals the hypothetical possibility of exiting the black hole into a different spacetime with the black hole acting as a [[wormhole]].<ref name="Carroll-2004"/>{{rp|p=257}} The possibility of travelling to another universe is, however, only theoretical, since any perturbation would destroy this possibility.<ref>{{cite journal |title=Black holes: the inside story |first1=S. |last1=Droz |first2=W. |last2=Israel |first3=S. M. |last3=Morsink |journal=Physics World |volume=9 |issue=1 |pages=34–37 |date=1996|bibcode=1996PhyW....9...34D |doi=10.1088/2058-7058/9/1/26}}</ref> It also appears to be possible to follow [[closed timelike curve]]s (returning to one's own past) around the Kerr singularity, which leads to problems with [[causality (physics)|causality]] like the [[grandfather paradox]].<ref name=Carroll-2004/>{{rp|p=266}}<ref name="thorne93">{{Cite conference |title=Closed Timelike Curves |conference=General relativity and gravitation |last=Thorne |first=Kip S. |url=https://www.its.caltech.edu/~kip/scripts/ClosedTimelikeCurves-II121.pdf |author-link=Kip Thorne |year=1993}}</ref> However, processes inside the black hole, such as quantum gravity effects or mass inflation, might prevent closed timelike curves from arising.<ref name="thorne93" />

To solve technical issues with general relativity, some models of gravity do not include black hole singularities. These theoretical black holes without singularities are called ''regular'', or ''nonsingular'', black holes.<ref>{{Cite journal |last1=Lan |first1=Chen |last2=Yang |first2=Hao |last3=Guo |first3=Yang |last4=Miao |first4=Yan-Gang |title=Regular Black Holes: A Short Topic Review |journal=International Journal of Theoretical Physics |date=2023 |volume=62 |issue=9 |article-number=202 |doi=10.1007/s10773-023-05454-1 |arxiv=2303.11696 |bibcode=2023IJTP...62..202L }}</ref><ref>{{Cite journal |last1=Olmo |first1=Gonzalo |last2=Rubiera-Garcia |first2=Diego |title=Nonsingular Black Holes in ƒ (R) Theories |journal=Universe |date=2015 |volume=1 |issue=2 |pages=173–185 |doi=10.3390/universe1020173 |arxiv=1509.02430 |bibcode=2015Univ....1..173O |doi-access=free }}</ref> For example, the [[Fuzzball (string theory)|fuzzball]] model, based on [[String theory|string theory]], states that black holes are actually made up of [[quantum state|quantum microstate]]s and need not have a singularity or an event horizon.<ref>{{cite journal |last=Mathur |first=Samir D. |title=The fuzzball proposal for black holes: an elementary review |journal=Fortschritte der Physik |volume=53 |issue=7–8 |page=793 |date=2005 |doi=10.1002/prop.200410203 |arxiv=hep-th/0502050 |bibcode=2005ForPh..53..793M |s2cid=15083147}}</ref><ref>{{Cite journal |last1=Avery |first1=Steven G. |last2=Chowdhury |first2=Borun D. |last3=Puhm |first3=Andrea |title=Unitarity and fuzzball complementarity: "Alice fuzzes but may not even know it!" |journal=Journal of High Energy Physics |date=2013 |issue=9 |article-number=12 |doi=10.1007/JHEP09(2013)012 |arxiv=1210.6996 |bibcode=2013JHEP...09..012A }}</ref> The theory of [[loop quantum gravity]] proposes that the curvature and density at the center of a black hole is large, but not infinite.<ref>{{Cite journal |last1=Bojowald |first1=Martin |title=Black-Hole Models in Loop Quantum Gravity |journal=Universe |date=2020 |volume=6 |issue=8 |page=125 |doi=10.3390/universe6080125 |arxiv=2009.13565 |bibcode=2020Univ....6..125B |doi-access=free }}</ref>

==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 the central singularity of 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 in general relativity|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 |article-number=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 |article-number=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 |article-number=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 [[Kerr metric]], which is a spinning black hole in a vacuum, if the [[angular momentum]] (<math>J</math>) is high enough. Transforming the Kerr metric to [[Boyer–Lindquist coordinates]], it can be shown<ref>{{harvnb|Hobson|Efstathiou|Lasenby|2013|pp=300-305}}.</ref> 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 <math>\mu = G M / c^{2}</math>, and <math>a=J/M c</math>. In this case, "event horizons disappear" means when the solutions are complex for <math>r_{\pm}</math>, or <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 [[Reissner–Nordström metric|Reissner–Nordström]] geometry of a charged black hole if the charge (<math>Q</math>) is high enough. In this metric, it can be shown<ref>{{harvnb|Hobson|Efstathiou|Lasenby|2013|pp=320-325}}.</ref> that the singularities occur at <math>r_{\pm}= \mu \pm \left(\mu^{2} - q^{2}\right)^{1/2}</math>, where <math>\mu = G M / c^{2}</math>, and <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 <math>\mu</math> and <math>q</math>, the case where <math>\mu^{2} < q^{2}</math> causes both <math>r_{\pm}</math> to be complex. This means the metric is regular for all positive values of <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]].

==See also== {{div col|colwidth=30em}} * 0-dimensional singularity: [[magnetic monopole]] * 1-dimensional singularity: [[cosmic string]] * 2-dimensional singularity: [[Domain wall (string theory)|domain wall]] * [[Fuzzball (string theory)]] * [[Penrose–Hawking singularity theorems]] * [[White hole]] * [[BKL singularity]] * [[Initial singularity]] * [[Shock singularity]] * [[Mass inflation]] {{div col end}}

== References == {{Reflist|30em}}

== Bibliography == {{refbegin}} * {{cite book |last1=Earman |first1=John |title=Bangs, crunches, whimpers, and shrieks: Singularities and acausalities in relativistic spacetimes |date=1995 |publisher=Oxford University Press |isbn=0-19-509591-X}} * {{cite book |last1=Joshi |first1=Pankaj S |title=Gravitational collapse and spacetime singularities |date=2007 |publisher=Cambridge University Press |location=New York |isbn=978-1-107-40536-3}} * {{cite book | first1 = Charles W. | last1 = Misner | author-link = Charles W. Misner | first2 = Kip | last2 = Thorne | author-link2 = Kip Thorne | first3 = John Archibald | last3 = Wheeler | author-link3 = John Archibald Wheeler | title = Gravitation | publisher = [[W. H. Freeman]] | date = 1973 | isbn = 0-7167-0344-0 | title-link = Gravitation (book) }} §31.2 The nonsingularity of the gravitational radius, and following sections; §34 Global Techniques, Horizons, and Singularity Theorems * {{cite book |last1=Wald |first1=Robert M. | author-link = Robert Wald | title = General Relativity | publisher = [[University of Chicago Press]] | date = 1984 | isbn = 0-226-87033-2 | title-link = General Relativity (book) }} *{{Cite journal |last1=Hawking |first1=S. W. |author-link=Stephen Hawking |last2=Penrose |first2=R. |author-link2=Roger Penrose |date=1970 |title=The singularities of gravitational collapse and cosmology |journal=Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences |language=en |volume=314 |issue=1519 |pages=529–548 |bibcode=1970RSPSA.314..529H |doi=10.1098/rspa.1970.0021 |issn=0080-4630 |doi-access=free}} * {{Cite journal |last1=Shapiro |first1=Stuart L. |last2=Teukolsky |first2=Saul A. |author-link2=Saul Teukolsky |date=1991 |title=Formation of naked singularities: The violation of cosmic censorship |journal=Physical Review Letters |language=en |volume=66 |issue=8 |pages=994–997 |bibcode=1991PhRvL..66..994S |doi=10.1103/PhysRevLett.66.994 |issn=0031-9007 |pmid=10043968 |s2cid=7830407|url=https://resolver.caltech.edu/CaltechAUTHORS:20180629-153305570 }} * {{Cite journal |last=Penrose |first=Roger |url=http://www.ias.ac.in/jarch/jaa/17/213-231.pdf |title=Chandrasekhar, black holes, and singularities |journal=Journal of Astrophysics and Astronomy |date=December 1996 |volume=17 |issue=3–4 |pages=213–231 |language=en |doi=10.1007/BF02702305 |bibcode=1996JApA...17..213P |issn=0250-6335}} * {{Cite journal |last=Penrose |first=Roger |url=http://www.ias.ac.in/jarch/jaa/20/233-248.pdf |title=The question of cosmic censorship |journal=Journal of Astrophysics and Astronomy |date=December 1999 |volume=20 |issue=3–4 |pages=233–248 |language=en |doi=10.1007/BF02702355 |bibcode=1999JApA...20..233P |issn=0250-6335}} * {{Cite journal |last=Singh |first=T. P. |url=http://www.ias.ac.in/jarch/jaa/20/221-232.pdf |title=Gravitational collapse, black holes and naked singularities |journal=Journal of Astrophysics and Astronomy |date=December 1999 |volume=20 |issue=3–4 |pages=221–232 |language=en |doi=10.1007/BF02702354 |arxiv=gr-qc/9805066 |bibcode=1999JApA...20..221S |issn=0250-6335}} * {{Cite book |last1=Hobson |first1=Michael P. |title=General relativity: an introduction for physicists |last2=Efstathiou |first2=George |last3=Lasenby |first3=Anthony |date=2013 |publisher=Cambridge University Press |isbn=978-0-521-82951-9 |edition=1. publ., 6. print |location=Cambridge}} {{refend}}

==Further reading== * {{Cite journal |last=Senovilla |first=José M. M. |date=2022-05-02 |title=A critical appraisal of the singularity theorems |url=https://royalsocietypublishing.org/doi/10.1098/rsta.2021.0174 |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |language=en |volume=380 |issue=2222 |article-number=20210174 |doi=10.1098/rsta.2021.0174 |pmid=35282689 |issn=1364-503X|arxiv=2108.07296 |bibcode=2022RSPTA.38010174S }} * ''[[The Elegant Universe]]'' by [[Brian Greene]]. This book provides a [[layman]]'s introduction to string theory, although some of the views expressed have already become outdated. His use of common terms and his providing of examples throughout the text help the layperson understand the basics of string theory.

[[Category:Gravitational singularities]] [[Category:General relativity]] [[Category:Lorentzian manifolds]] [[Category:Physical paradoxes]] [[Category:Physical phenomena]] [[Category:Concepts in astronomy]]