Naked singularity

In general relativity, a naked singularity is a hypothetical gravitational singularity without an event horizon.

When there exists at least one causal geodesic that, in the future, extends to an observer either at infinity or to an observer comoving with the collapsing cloud, and in the past terminates at the gravitational singularity, then that singularity is referred to as a naked singularity.^[1] In a black hole, the singularity is completely enclosed by a boundary known as the event horizon, inside which the curvature of spacetime caused by the singularity is so strong that light cannot escape. Hence, objects inside the event horizon—including the singularity itself—cannot be observed directly. In contrast, a naked singularity would be observable.

The theoretical existence of naked singularities is important because their existence would mean that it would be possible to observe the collapse of an object to infinite density. It would also cause foundational problems for general relativity, because general relativity cannot make predictions about the evolution of spacetime near a singularity. In generic black holes, this is not a problem, as an outside viewer cannot observe the spacetime within the event horizon.

Naked singularities have not been observed in nature. Astronomical observations of black holes indicate that their rate of rotation falls below the threshold to produce a naked singularity (spin parameter 1). GRS 1915+105 comes closest to the limit, with a spin parameter of 0.82-1.00.^[2] It is hinted that GRO J1655−40 could be a naked singularity.^[3]

According to the cosmic censorship hypothesis, gravitational singularities may not be observable. If loop quantum gravity is correct, naked singularities may be possible in nature.

Predicted formation

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 within the framework of the general theory of relativity. The Oppenheimer–Snyder–Datt (OSD) model illustrates the collapse of a spherical cloud composed of homogeneous dust (pressureless matter).^[4]^[5] 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. Considering variations in the initial density (considering inhomogeneous 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, and 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.^[6]^[7]^[8] 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.^[9] 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^[10] and some computer simulations (Choptuik, 1997) have been performed.^[11] Intriguingly, it is recently reported that some spinning white dwarfs can realistically transmute into rotating naked singularities and black holes with a wide range of near- and sub-solar-mass values by capturing asymmetric dark matter particles.^[12] Similarly, the spinning neutron stars could also be transmuted to the slowly-spinning near-solar mass naked singularities by capturing the 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.^[13] 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.^[14] ^[15]

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 also occur.^[16] However, he then showed that such "naked singularities" are unstable.^[17]

Metrics

Ray traced 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).

Comparison with an extremal black hole with M=1, a²+Q²=1M²

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 shown^[18] that the $r$ coordinate (which is not the radius) of the event horizon is

r_{\pm }=\mu \pm (\mu ^{2}-a^{2})^{1/2},

where $\mu =GM/c^{2}$ , and $a=J/Mc$ . In this case, "event horizons disappear" means when the solutions are complex for $r_{\pm }$ , or $\mu ^{2}<a^{2}$ . However, this corresponds to a case where $J$ exceeds $GM^{2}/c$ (or in Planck units, $J>M^{2}$ ), 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 shown^[19] that the horizons occur at

r_{\pm }=\mu \pm (\mu ^{2}-q^{2})^{1/2},

where $\mu =GM/c^{2}$ , and $q^{2}=GQ^{2}/(4\pi \varepsilon _{0}c^{4})$ . Of the three possible cases for the relative values of $\mu$ and $q$ , the case where $\mu ^{2}<q^{2}$ causes both $r_{\pm }$ 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 $Q/{\sqrt {4\pi \varepsilon _{0}}}$ exceeds $M{\sqrt {G}}$ (or in Planck units, $Q>M$ ), 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.

Effects

A naked singularity could allow scientists to observe an infinitely dense material, which would under normal circumstances be impossible by the cosmic censorship hypothesis. Without an event horizon of any kind, some speculate that naked singularities could actually emit light.^[20]

Cosmic censorship hypothesis

The cosmic censorship hypothesis says that a gravitational singularity would remain hidden by the 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.^[21]

Some research has suggested that if loop quantum gravity is correct, then naked singularities could exist in nature,^[22]^[23]^[24] implying that the cosmic censorship hypothesis does not hold. Numerical calculations^[25] and some other arguments^[26] have also hinted at this possibility.

In fiction

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 space crafts 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]}
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.
In the eighth part of the “JoJo's Bizarre Adventure” manga series, “JoJolion”, the main character is able to create small naked singularities. It is not officially stated as such, but the ability “Go Beyond” creates invisible, fast-spinning vortexes out of an infinitely small ‘string’, drawing similarities to the Cosmic Censorship Hypothesis and Loop quantum gravity.

References

^ Joshi, Pankaj S. (1996). Global aspects in gravitation and cosmology. International series of monographs on physics (1. paperback (with corr.) ed.). Oxford: Clarendon Press. ISBN 978-0-19-850079-7.
^ Jeanna Bryne (20 November 2006). "Pushing the Limit: Black Hole Spins at Phenomenal Rate". space.com. Retrieved 2017-11-25.
^ Chakraborty, C.; Bhattacharyya, S. (2018-08-28). "Does the gravitomagnetic monopole exist? A clue from a black hole x-ray binary". Physical Review D. 98 (4): 043021. arXiv:1712.01156. doi:10.1103/PhysRevD.98.043021.
^ Oppenheimer, J. R.; Snyder, H. (1939-09-01). "On Continued Gravitational Contraction". Physical Review. 56 (5): 455–459. doi:10.1103/PhysRev.56.455.
^ Datt, B. (1938-05-01). "Über eine Klasse von Lösungen der Gravitationsgleichungen der Relativität". Zeitschrift für Physik (in German). 108 (5): 314–321. doi:10.1007/BF01374951. ISSN 0044-3328.
^ Waugh, B.; Lake, Kayll (1988-08-15). "Strengths of shell-focusing singularities in marginally bound collapsing self-similar Tolman spacetimes". Physical Review D. 38 (4): 1315–1316. doi:10.1103/PhysRevD.38.1315.
^ Waugh, B.; Lake, Kayll (1989-09-15). "Shell-focusing singularities in spherically symmetric self-similar spacetimes". Physical Review D. 40 (6): 2137–2139. doi:10.1103/PhysRevD.40.2137.
^ Joshi, P. S.; Dwivedi, I. H. (1993-06-15). "Naked singularities in spherically symmetric inhomogeneous Tolman-Bondi dust cloud collapse". Physical Review D. 47 (12): 5357–5369. arXiv:gr-qc/9303037. doi:10.1103/PhysRevD.47.5357.
^
Examples include:
- Ori, Amos; Piran, Tsvi (1987-11-09). "Naked singularities in self-similar spherical gravitational collapse". Physical Review Letters. 59 (19): 2137–2140. doi:10.1103/physrevlett.59.2137. ISSN 0031-9007.
- Ori, Amos; Piran, Tsvi (1990-08-15). "Naked singularities and other features of self-similar general-relativistic gravitational collapse". Physical Review D. 42 (4): 1068–1090. doi:10.1103/physrevd.42.1068. ISSN 0556-2821.
- Magli, Giulio (1997-07-01). "Gravitational collapse with non-vanishing tangential stresses: a generalization of the Tolman - Bondi model". Classical and Quantum Gravity. 14 (7): 1937–1953. doi:10.1088/0264-9381/14/7/026. ISSN 0264-9381.
- Magli, Giulio (1998-10-01). "Gravitational collapse with non-vanishing tangential stresses: II. A laboratory for cosmic censorship experiments". Classical and Quantum Gravity. 15 (10): 3215–3228. arXiv:gr-qc/9711082. doi:10.1088/0264-9381/15/10/022. ISSN 0264-9381.
- Harada, Tomohiro (1998-10-09). "Final fate of the spherically symmetric collapse of a perfect fluid". Physical Review D. 58 (10). arXiv:gr-qc/9807038. doi:10.1103/physrevd.58.104015. ISSN 0556-2821.
- Harada, Tomohiro; Nakao, Ken-ichi; Iguchi, Hideo (1999-07-20). "Nakedness and curvature strength of a shell-focusing singularity in spherically symmetric spacetime with vanishing radial pressure". Classical and Quantum Gravity. 16 (8): 2785–2796. arXiv:gr-qc/9904073. doi:10.1088/0264-9381/16/8/315. ISSN 0264-9381.
- Joshi, P. S.; Dwivedi, I. H. (January 1999). "Initial data and the end state of spherically symmetric gravitational collapse". Classical and Quantum Gravity. 16 (1): 41. arXiv:gr-qc/9804075. doi:10.1088/0264-9381/16/1/003. ISSN 0264-9381.
- Jhingan, S.; Magli, G. (2000-05-09). "Black holes versus naked singularities formation in collapsing Einstein clusters". Physical Review D. 61 (12). arXiv:gr-qc/9902041. doi:10.1103/physrevd.61.124006. ISSN 0556-2821.
- Gonçalves, Sérgio M. C. V.; Jhingan, Sanjay (December 2001). "Singularities in Gravitational Collapse with Radial Pressure". General Relativity and Gravitation. 33 (12): 2125–2149. arXiv:gr-qc/0107054. doi:10.1023/a:1015285531320. ISSN 0001-7701.
- Harada, T.; Iguchi, H.; Nakao, K.-I. (2002-03-01). "Physical Processes in Naked Singularity Formation". Progress of Theoretical Physics. 107 (3): 449–524. arXiv:gr-qc/0204008. doi:10.1143/ptp.107.449. ISSN 0033-068X.
- Giambò, Roberto; Giannoni, Fabio; Magli, Giulio; Piccione, Paolo (April 2003). "New Solutions of Einstein Equations in Spherical Symmetry: The Cosmic Censor to the Court". Communications in Mathematical Physics. 235 (3): 545–563. arXiv:gr-qc/0204030. doi:10.1007/s00220-003-0793-9. ISSN 0010-3616.
- Giambò, Roberto; Giannoni, Fabio; Magli, Giulio; Piccione, Paolo (June 2004). "Naked Singularities Formation in the Gravitational Collapse of Barotropic Spherical Fluids". General Relativity and Gravitation. 36 (6): 1279–1298. arXiv:gr-qc/0303043. doi:10.1023/b:gerg.0000022388.11306.e1. ISSN 0001-7701.
^ Crew, Bec (23 May 2017). "Naked Singularities Can Actually Exist in a Three-Dimensional Universe, Physicists Predict". ScienceAlert. Retrieved 2020-09-02.
^ Garfinkle, David (1997). "Choptuik scaling and the scale invariance of Einstein's equation". Phys. Rev. D. 56 (6): R3169–R3173. arXiv:gr-qc/9612015. Bibcode:1997PhRvD..56.3169G. doi:10.1103/PhysRevD.56.R3169.
^ Chakraborty, C.; Bhattacharyya, S. (2024-06-05). "Near- and sub-solar-mass naked singularities and black holes from transmutation of white dwarfs". Journal of Cosmology and Astroparticle Physics. 2024 (06): 007. arXiv:2401.08462. doi:10.1088/1475-7516/2024/06/007.
^ Chakraborty, C.; Bhattacharyya, S.; Joshi, P. S. (2024-07-22). "Low mass naked singularities from dark core collapse". Journal of Cosmology and Astroparticle Physics. 2024 (07): 053. arXiv:2405.08758. doi:10.1088/1475-7516/2024/07/053.
^ Chakraborty, C.; Kocherlakota, P.; Joshi, P. S. (2017-02-06). "Spin precession in a black hole and naked singularity spacetimes". Physical Review D. 95 (4): 044006. arXiv:1605.00600. doi:10.1103/PhysRevD.95.044006.
^ Chakraborty, C.; Kocherlakota, P.; Patil, M.; Bhattacharyya, S.; Joshi, P. S.; Krolak, A. (2017-04-12). "Distinguishing Kerr naked singularities and black holes using the spin precession of a test gyro in strong gravitational fields". Physical Review D. 95 (8): 084024. arXiv:1611.08808. doi:10.1103/PhysRevD.95.084024.
^ D.Christodoulou (1994). "Examples of naked singularity formation in the gravitational collapse of a scalar field". Ann. Math. 140 (3): 607–653. doi:10.2307/2118619. JSTOR 2118619.
^ D. Christodoulou (1999). "The instability of naked singularities in the gravitational collapse of a scalar field". Ann. Math. 149 (1): 183–217. arXiv:math/9901147. doi:10.2307/121023. JSTOR 121023. S2CID 8930550.
^ Hobson, et al., General Relativity an Introduction for Physicists, Cambridge University Press 2007, p. 300-305
^ Hobson, et al., General Relativity an Introduction for Physicists, Cambridge University Press 2007, p. 320-325
^ Battersby, Stephen (1 October 2007). "Is a 'naked singularity' lurking in our galaxy?". New Scientist. Retrieved 2008-03-06.
^ Pretorius, Frans (2016-05-31). "Viewpoint: Relativity Gets Thorough Vetting from LIGO". Physics. 9: 52. doi:10.1103/Physics.9.52.
^ M. Bojowald, Living Rev. Rel. 8, (2005), 11 Archived 2015-12-21 at the Wayback Machine
^ Goswami, Rituparno; Joshi, Pankaj S. (2007-10-22). "Spherical gravitational collapse in N dimensions". Physical Review D. 76 (8): 084026. arXiv:gr-qc/0608136. Bibcode:2007PhRvD..76h4026G. doi:10.1103/physrevd.76.084026. ISSN 1550-7998. S2CID 119441682.
^ Goswami, Rituparno; Joshi, Pankaj S.; Singh, Parampreet (2006-01-27). "Quantum Evaporation of a Naked Singularity". Physical Review Letters. 96 (3): 031302. arXiv:gr-qc/0506129. Bibcode:2006PhRvL..96c1302G. doi:10.1103/physrevlett.96.031302. ISSN 0031-9007. PMID 16486681. S2CID 19851285.
^ Eardley, Douglas M.; Smarr, Larry (1979-04-15). "Time functions in numerical relativity: Marginally bound dust collapse". Physical Review D. 19 (8). American Physical Society (APS): 2239–2259. Bibcode:1979PhRvD..19.2239E. doi:10.1103/physrevd.19.2239. ISSN 0556-2821.
^ Królak, Andrzej (1999). "Nature of Singularities in Gravitational Collapse". Progress of Theoretical Physics Supplement. 136: 45–56. arXiv:gr-qc/9910108. Bibcode:1999PThPS.136...45K. doi:10.1143/ptps.136.45. ISSN 0375-9687.