Naked singularity

When a massive star undergoes a gravitational collapse due to its own immense gravity, the ultimate outcome of this persistent collapse can manifest as either a black hole or a naked singularity. This holds true across a diverse range of physically plausible scenarios allowed by general relativity. The Oppenheimer–Snyder–Datt (OSD) model illustrates the collapse of a spherical cloud composed of homogeneous dust (pressureless matter).{{Cite journal |last1=Oppenheimer |first1=J. R. |last2=Snyder |first2=H. |date=1939-09-01 |title=On Continued Gravitational Contraction |journal=Physical Review |volume=56 |issue=5 |pages=455–459 |doi=10.1103/PhysRev.56.455|doi-access=free |bibcode=1939PhRv...56..455O }}{{Cite journal |last=Datt |first=B. |date=1938-05-01 |title=Über eine Klasse von Lösungen der Gravitationsgleichungen der Relativität |url=https://doi.org/10.1007/BF01374951 |journal=Zeitschrift für Physik |language=de |volume=108 |issue=5 |pages=314–321 |doi=10.1007/BF01374951 |issn=0044-3328|url-access=subscription }} In this scenario, all the matter converges into the spacetime singularity simultaneously in terms of comoving time. Notably, the event horizon emerges before the singularity, effectively covering it. By allowing an inhomogeneous initial density profile, one can demonstrate a significant alteration in the behavior of the horizon. This leads to two distinct potential outcomes arising from the collapse of generic dust: the formation of a black hole, characterized by the horizon preceding the singularity; or the emergence of a naked singularity, where the horizon is delayed. In the case of a naked singularity, this delay enables null geodesics or light rays to escape the central singularity, where density and curvatures diverge, reaching distant observers.{{Cite journal |last1=Waugh |first1=B. |last2=Lake |first2=Kayll |date=1988-08-15 |title=Strengths of shell-focusing singularities in marginally bound collapsing self-similar Tolman spacetimes |url=https://link.aps.org/doi/10.1103/PhysRevD.38.1315 |journal=Physical Review D |volume=38 |issue=4 |pages=1315–1316 |doi=10.1103/PhysRevD.38.1315|pmid=9959270 |url-access=subscription }}{{Cite journal |last1=Waugh |first1=B. |last2=Lake |first2=Kayll |date=1989-09-15 |title=Shell-focusing singularities in spherically symmetric self-similar spacetimes |url=https://link.aps.org/doi/10.1103/PhysRevD.40.2137 |journal=Physical Review D |volume=40 |issue=6 |pages=2137–2139 |doi=10.1103/PhysRevD.40.2137|pmid=10012048 |url-access=subscription }}{{Cite journal |last1=Joshi |first1=P. S. |last2=Dwivedi |first2=I. H. |date=1993-06-15 |title=Naked singularities in spherically symmetric inhomogeneous Tolman-Bondi dust cloud collapse |url=https://link.aps.org/doi/10.1103/PhysRevD.47.5357 |journal=Physical Review D |volume=47 |issue=12 |pages=5357–5369 |doi=10.1103/PhysRevD.47.5357|pmid=10015558 |arxiv=gr-qc/9303037 }} In exploring more realistic scenarios of collapse, one avenue involves incorporating pressures into the model. The consideration of gravitational collapse with non-zero pressures and various models including a realistic equation of state, delineating the specific relationship between the density and pressure within the cloud, has been thoroughly examined and investigated by numerous researchers over the years.Examples include:

• {{Cite journal |last1=Ori |first1=Amos |last2=Piran |first2=Tsvi |date=1987-11-09 |title=Naked singularities in self-similar spherical gravitational collapse |url=http://dx.doi.org/10.1103/physrevlett.59.2137 |journal=Physical Review Letters |volume=59 |issue=19 |pages=2137–2140 |doi=10.1103/physrevlett.59.2137 |pmid=10035434 |issn=0031-9007|url-access=subscription }}
• {{Cite journal |last1=Ori |first1=Amos |last2=Piran |first2=Tsvi |date=1990-08-15 |title=Naked singularities and other features of self-similar general-relativistic gravitational collapse |url=http://dx.doi.org/10.1103/physrevd.42.1068 |journal=Physical Review D |volume=42 |issue=4 |pages=1068–1090 |doi=10.1103/physrevd.42.1068 |pmid=10012941 |issn=0556-2821|url-access=subscription }}
• {{Cite journal |last=Magli |first=Giulio |date=1997-07-01 |title=Gravitational collapse with non-vanishing tangential stresses: a generalization of the Tolman - Bondi model |url=http://dx.doi.org/10.1088/0264-9381/14/7/026 |journal=Classical and Quantum Gravity |volume=14 |issue=7 |pages=1937–1953 |doi=10.1088/0264-9381/14/7/026 |issn=0264-9381|url-access=subscription }}
• {{Cite journal |last=Magli |first=Giulio |date=1998-10-01 |title=Gravitational collapse with non-vanishing tangential stresses: II. A laboratory for cosmic censorship experiments |url=http://dx.doi.org/10.1088/0264-9381/15/10/022 |journal=Classical and Quantum Gravity |volume=15 |issue=10 |pages=3215–3228 |doi=10.1088/0264-9381/15/10/022 |issn=0264-9381|arxiv=gr-qc/9711082 }}
• {{Cite journal |last=Harada |first=Tomohiro |date=1998-10-09 |title=Final fate of the spherically symmetric collapse of a perfect fluid |url=http://dx.doi.org/10.1103/physrevd.58.104015 |journal=Physical Review D |volume=58 |issue=10 |page=104015 |doi=10.1103/physrevd.58.104015 |issn=0556-2821|arxiv=gr-qc/9807038 }}
• {{Cite journal |last1=Harada |first1=Tomohiro |last2=Nakao |first2=Ken-ichi |last3=Iguchi |first3=Hideo |date=1999-07-20 |title=Nakedness and curvature strength of a shell-focusing singularity in spherically symmetric spacetime with vanishing radial pressure |url=http://dx.doi.org/10.1088/0264-9381/16/8/315 |journal=Classical and Quantum Gravity |volume=16 |issue=8 |pages=2785–2796 |doi=10.1088/0264-9381/16/8/315 |issn=0264-9381|arxiv=gr-qc/9904073 }}
• {{Cite journal |last1=Joshi |first1=P. S. |last2=Dwivedi |first2=I. H. |date=January 1999 |title=Initial data and the end state of spherically symmetric gravitational collapse |url=https://dx.doi.org/10.1088/0264-9381/16/1/003 |journal=Classical and Quantum Gravity |language=en |volume=16 |issue=1 |pages=41 |doi=10.1088/0264-9381/16/1/003 |issn=0264-9381|arxiv=gr-qc/9804075 }}
• {{Cite journal |last1=Jhingan |first1=S. |last2=Magli |first2=G. |date=2000-05-09 |title=Black holes versus naked singularities formation in collapsing Einstein clusters |url=http://dx.doi.org/10.1103/physrevd.61.124006 |journal=Physical Review D |volume=61 |issue=12 |page=124006 |doi=10.1103/physrevd.61.124006 |issn=0556-2821|arxiv=gr-qc/9902041 }}
• {{Cite journal |last1=Gonçalves |first1=Sérgio M. C. V. |last2=Jhingan |first2=Sanjay |date=December 2001 |title=Singularities in Gravitational Collapse with Radial Pressure |url=http://dx.doi.org/10.1023/a:1015285531320 |journal=General Relativity and Gravitation |volume=33 |issue=12 |pages=2125–2149 |doi=10.1023/a:1015285531320 |issn=0001-7701|arxiv=gr-qc/0107054 }}
• {{Cite journal |last1=Harada |first1=T. |last2=Iguchi |first2=H. |last3=Nakao |first3=K.-I. |date=2002-03-01 |title=Physical Processes in Naked Singularity Formation |journal=Progress of Theoretical Physics |volume=107 |issue=3 |pages=449–524 |doi=10.1143/ptp.107.449 |issn=0033-068X|doi-access=free |arxiv=gr-qc/0204008 }}
• {{Cite journal |last1=Giambò |first1=Roberto |last2=Giannoni |first2=Fabio |last3=Magli |first3=Giulio |last4=Piccione |first4=Paolo |date=April 2003 |title=New Solutions of Einstein Equations in Spherical Symmetry: The Cosmic Censor to the Court |url=http://dx.doi.org/10.1007/s00220-003-0793-9 |journal=Communications in Mathematical Physics |volume=235 |issue=3 |pages=545–563 |doi=10.1007/s00220-003-0793-9 |issn=0010-3616|arxiv=gr-qc/0204030 }}
• {{Cite journal |last1=Giambò |first1=Roberto |last2=Giannoni |first2=Fabio |last3=Magli |first3=Giulio |last4=Piccione |first4=Paolo |date=June 2004 |title=Naked Singularities Formation in the Gravitational Collapse of Barotropic Spherical Fluids |url=http://dx.doi.org/10.1023/b:gerg.0000022388.11306.e1 |journal=General Relativity and Gravitation |volume=36 |issue=6 |pages=1279–1298 |doi=10.1023/b:gerg.0000022388.11306.e1 |issn=0001-7701|arxiv=gr-qc/0303043 }} They all result in either a black hole or a naked singularity depending on the initial data.

From concepts drawn from rotating black holes, it is shown that a singularity, spinning rapidly, can become a ring-shaped object. This results in two event horizons, as well as an ergosphere, which draw closer together as the spin of the singularity increases. When the outer and inner event horizons merge, they shrink toward the rotating singularity and eventually expose it to the rest of the universe.

A singularity rotating fast enough might be created by the collapse of dust or by a supernova of a fast-spinning star. Studies of pulsars{{Cite web|last=Crew|first=Bec|title=Naked Singularities Can Actually Exist in a Three-Dimensional Universe, Physicists Predict|url=https://www.sciencealert.com/naked-singularities-can-actually-exist-in-a-three-dimensional-universe-physicists-predict|access-date=2020-09-02|website=ScienceAlert|date=23 May 2017 |language=en-gb}} and some computer simulations (Choptuik, 1997) have been performed.{{cite journal|last1=Garfinkle|first1=David|title=Choptuik scaling and the scale invariance of Einstein's equation|journal=Phys. Rev. D|date=1997|volume=56|issue=6|pages=R3169–R3173|doi=10.1103/PhysRevD.56.R3169|arxiv = gr-qc/9612015 |bibcode = 1997PhRvD..56.3169G }} Intriguingly, it is recently reported that some spinning white dwarfs could realistically transmute into rotating naked singularities or black holes with a wide range of near- and sub-solar-mass values by capturing asymmetric dark matter particles.{{cite journal |last1=Chakraborty |first1=C. |last2=Bhattacharyya |first2=S. |date=2024-06-05 |title=Near- and sub-solar-mass naked singularities and black holes from transmutation of white dwarfs|url=https://iopscience.iop.org/article/10.1088/1475-7516/2024/06/007 |journal=Journal of Cosmology and Astroparticle Physics |volume=2024 |issue=6 |pages=007|doi=10.1088/1475-7516/2024/06/007|arxiv=2401.08462 }} Similarly, spinning neutron stars could also be transmuted to slowly spinning near–solar-mass naked singularities by capturing asymmetric dark matter particles, if the accumulated cloud of dark matter particles in the core of a neutron star can be modeled as an anisotropic fluid.{{cite journal|last1=Chakraborty |first1=C. |last2=Bhattacharyya |first2=S. |last3=Joshi |first3=P. S.|date=2024-07-22 |title=Low mass naked singularities from dark core collapse|url=https://iopscience.iop.org/article/10.1088/1475-7516/2024/07/053 |journal=Journal of Cosmology and Astroparticle Physics |volume=2024 |issue=7 |pages=053|doi=10.1088/1475-7516/2024/07/053|arxiv=2405.08758 }} In general, the precession of a gyroscope and the precession of orbits of matter falling into a rotating black hole or a naked singularity can be used to distinguish these exotic objects.{{Cite journal |last1=Chakraborty |first1=C. |last2=Kocherlakota |first2=P. |last3=Joshi |first3=P. S. | date=2017-02-06 |title=Spin precession in a black hole and naked singularity spacetimes |url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.95.044006 |journal=Physical Review D |volume=95 |issue=4 |pages=044006|doi=10.1103/PhysRevD.95.044006|arxiv=1605.00600 }}{{Cite journal |last1=Chakraborty|first1=C. |last2=Kocherlakota |first2=P. |last3=Patil |first3=M. |last4=Bhattacharyya |first4=S. |last5=Joshi |first5=P. S. |last6=Krolak |first6=A.| date=2017-04-12 |title=Distinguishing Kerr naked singularities and black holes using the spin precession of a test gyro in strong gravitational fields|url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.95.084024 |journal=Physical Review D |volume=95 |issue=8 |pages=084024|doi=10.1103/PhysRevD.95.084024|arxiv=1611.08808 }}

Mathematician Demetrios Christodoulou, a winner of the Shaw Prize, has shown that contrary to what had been expected, singularities which are not hidden in a black hole could also occur.{{cite journal|author=D.Christodoulou|title=Examples of naked singularity formation in the gravitational collapse of a scalar field|journal=Ann. Math.|volume=140|pages=607–653|year=1994|doi=10.2307/2118619|issue=3|jstor=2118619}} However, he then showed that such "naked singularities" are unstable.{{cite journal|author=D. Christodoulou|title=The instability of naked singularities in the gravitational collapse of a scalar field|journal=Ann. Math.|volume=149|pages=183–217| year=1999|doi=10.2307/121023|issue=1|jstor=121023|arxiv=math/9901147|s2cid=8930550}}

File:Naked.Singularity,Overextremal.Kerr.Newman,Raytracing.png image of a hypothetical naked singularity in front of a Milky Way background. The parameters of the singularity are M=1, a²+Q²=2M². The singularity is viewed from its equatorial plane at θ=90° (edge on).]]

File:Black.Hole,Extremal.Kerr.Newman,Raytracing.png

Disappearing event horizons exist in the Kerr metric, which is a spinning black hole in a vacuum. Specifically, if the angular momentum is high enough, the event horizons could disappear. Transforming the Kerr metric to Boyer–Lindquist coordinates, it can be shownHobson, et al., General Relativity an Introduction for Physicists, Cambridge University Press 2007, p. 300-305 that the $r$ coordinate (which is not the radius) of the event horizon is

$$r\_\backslash pm\; =\; \backslash mu\; \backslash pm\; (\backslash mu^2-a^2)^\{1/2\},$$

where $\backslash mu\; =\; G\; M\; /\; c^2$, and $a=J/M\; c$. In this case, "event horizons disappear" means that the solutions are complex for $r\_\backslash pm$, or . However, this corresponds to a case where $J$ exceeds $GM^\{2\}/c$ (or in Planck units, {{Nowrap|$J\; >\; M^2$)}}, i.e. the spin exceeds what is normally viewed as the upper limit of its physically possible values.

Disappearing event horizons can also be seen with the Reissner–Nordström geometry of a charged black hole. In this metric, it can be shownHobson, et al., General Relativity an Introduction for Physicists, Cambridge University Press 2007, p. 320-325 that the horizons occur at

$$r\_\backslash pm\; =\; \backslash mu\; \backslash pm\; (\backslash mu^2-q^2)^\{1/2\},$$

where $\backslash mu\; =\; G\; M\; /\; c^2$, and $q^2\; =\; G\; Q^2/(4\; \backslash pi\; \backslash varepsilon\_0\; c^4)$. Of the three possible cases for the relative values of $\backslash mu$ and $q$, the case where causes both $r\_\backslash pm$ to be complex. This means the metric is regular for all positive values of $r$, or in other words, the singularity has no event horizon. However, this corresponds to a case where $Q/\backslash sqrt\{4\; \backslash pi\; \backslash varepsilon\_0\}$ exceeds $M\backslash sqrt\{G\}$ (or in Planck units, {{Nowrap|$Q\; >\; M$)}}, i.e. the charge exceeds what is normally viewed as the upper limit of its physically possible values.

See Kerr–Newman metric for a spinning, charged ring singularity.

A naked singularity could allow scientists to observe an infinitely dense material, which would under normal circumstances be impossible according to the cosmic censorship hypothesis. That is, without an event horizon of any kind, naked singularities could actually emit light.{{cite news|url=https://www.newscientist.com/article/dn12707-is-a-naked-singularity-lurking-in-our-galaxy.html|title=Is a 'naked singularity' lurking in our galaxy?|author=Battersby|first=Stephen|date=1 October 2007|work=New Scientist|access-date=2008-03-06|author-link=Stephen Battersby (science journalist)}}

{{Main|Cosmic censorship hypothesis}}

The cosmic censorship hypothesis says that every gravitational singularity will remain hidden by its event horizon. LIGO events, including GW150914, are consistent with these predictions. Although data anomalies would have resulted in the case of a singularity, the nature of those anomalies remains unknown.{{Cite journal|last=Pretorius|first=Frans|date=2016-05-31|title=Viewpoint: Relativity Gets Thorough Vetting from LIGO|url=https://physics.aps.org/articles/v9/52|journal=Physics|language=en|volume=9|page=52 |doi=10.1103/Physics.9.52|doi-access=free}}

Some research has suggested that if loop quantum gravity is correct, then naked singularities could exist in nature,M. Bojowald, [http://relativity.livingreviews.org/Articles/lrr-2008-4/ Living Rev. Rel. 8, (2005), 11] {{webarchive|url=https://web.archive.org/web/20151221011231/http://relativity.livingreviews.org/Articles/lrr-2008-4/ |date=2015-12-21 }}{{cite journal | last1=Goswami | first1=Rituparno | last2=Joshi | first2=Pankaj S. | title=Spherical gravitational collapse in N dimensions | journal=Physical Review D | volume=76 | issue=8 | date=2007-10-22 | issn=1550-7998 | doi=10.1103/physrevd.76.084026 | page=084026|arxiv=gr-qc/0608136| bibcode=2007PhRvD..76h4026G | s2cid=119441682 }}{{cite journal | last1=Goswami | first1=Rituparno | last2=Joshi | first2=Pankaj S. | last3=Singh | first3=Parampreet | title=Quantum Evaporation of a Naked Singularity | journal=Physical Review Letters | volume=96 | issue=3 | date=2006-01-27 | issn=0031-9007 | doi=10.1103/physrevlett.96.031302 | page=031302| pmid=16486681 |arxiv=gr-qc/0506129| bibcode=2006PhRvL..96c1302G | s2cid=19851285 }} implying that the cosmic censorship hypothesis does not hold. Numerical calculations{{cite journal | last1=Eardley | first1=Douglas M. | last2=Smarr | first2=Larry | title=Time functions in numerical relativity: Marginally bound dust collapse | journal=Physical Review D | publisher=American Physical Society (APS) | volume=19 | issue=8 | date=1979-04-15 | issn=0556-2821 | doi=10.1103/physrevd.19.2239 | pages=2239–2259| bibcode=1979PhRvD..19.2239E }} and some other arguments{{cite journal | last=Królak | first=Andrzej | title=Nature of Singularities in Gravitational Collapse | journal=Progress of Theoretical Physics Supplement | volume=136 | year=1999 | issn=0375-9687 | doi=10.1143/ptps.136.45 | pages=45–56| arxiv=gr-qc/9910108 | bibcode=1999PThPS.136...45K |doi-access=free}} have also hinted at this possibility.

• M. John Harrison's Kefahuchi Tract trilogy of science fiction novels (Light, Nova Swing and Empty Space) centre upon humanity's exploration of a naked singularity.
• "Dark Peril", by James C. Glass (published in Analog March 2005), is a story about space travelers on an exploratory mission. While they investigate a strange cosmological phenomenon, their two small spacecraft begin to shake, and they are unable to leave the area. One crew member realizes that they are trapped in the ergosphere of a black hole or naked singularity. The story describes a cluster of multiple black holes or singularities, and what the crew does to try to survive this seemingly inescapable situation.
• Stephen Baxter's Xeelee Sequence features the Xeelee, who create a massive ring that produces a naked singularity. It is used to travel to another universe.
• In the episode titled "Daybreak", the finale of the 2004 reimagined television series Battlestar Galactica, the Cylon colony orbits a naked singularity.{{Citation needed|date=April 2021}}
• The Sleeping God in Peter Hamilton's The Night's Dawn Trilogy is believed to be a naked singularity.
• In Christopher Nolan's Interstellar the nonexistence of a naked singularity hinders humanity from completing a theory of quantum gravity due to the inaccessibility of experimental data from inside the event horizon.
• In the visual novel Steins;Gate, a naked singularity is used to compress the digitalized memories of the protagonist into a smaller size, to then be sent back in time with an improvised "time leap machine".
• In Vonda McIntyre's 1981 Star Trek novel The Entropy Effect, a naked singularity is found to be a side effect of time travel experimentation, and threatens to destroy the universe if the time travel experiments are not stopped before they started.

• Black hole electron
• Gravitational singularity
• Ring singularity

{{Reflist|30em}}

• {{cite journal | last1=Werner | first1=M. C. | last2=Petters | first2=A. O. | title=Magnification relations for Kerr lensing and testing cosmic censorship | journal=Physical Review D | volume=76 | issue=6 | date=2007-09-24 | issn=1550-7998 | doi=10.1103/physrevd.76.064024 | page=064024|arxiv=0706.0132v2| bibcode=2007PhRvD..76f4024W | s2cid=119647924 }}
• Pankaj S. Joshi, [https://archive.today/20120525044855/http://www.sciam.com/article.cfm?id=naked-singularities "Do Naked Singularities Break the Rules of Physics?"], Scientific American, January 2009.
• Marcus Chown, [https://www.newscientist.com/article/mg20327204.700-fastspinning-black-holes-might-reveal-all.html "Fast-spinning black holes might reveal all"] New Scientist, August 2009.

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