Naked singularity explained

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

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(\mu2-a2)1/2,

where

\mu=GM/c2

, and

a=J/Mc

. In this case, "event horizons disappear" means when the solutions are complex for

r\pm

, or

\mu2<a2

. However, this corresponds to a case where

J

exceeds

GM2/c

(or in Planck units,, 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(\mu2-q2)1/2,

where

\mu=GM/c2

, and

q2=GQ2/(4\pi\varepsilon0c4)

. Of the three possible cases for the relative values of

\mu

and

q

, the case where

\mu2<q2

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\varepsilon0}

exceeds

M\sqrt{G}

(or in Planck units,, 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

See also

Further reading

Notes and References

  1. Book: Joshi, Pankaj S. . Global aspects in gravitation and cosmology . 1996 . Clarendon Press . 978-0-19-850079-7 . 1. paperback (with corr.) . International series of monographs on physics . Oxford.
  2. Web site: Pushing the Limit: Black Hole Spins at Phenomenal Rate. space.com. 2017-11-25. Jeanna Bryne. 20 November 2006 .
  3. 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. 10.1103/PhysRevD.98.043021. 1712.01156 .
  4. Oppenheimer . J. R. . Snyder . H. . 1939-09-01 . On Continued Gravitational Contraction . Physical Review . 56 . 5 . 455–459 . 10.1103/PhysRev.56.455. free .
  5. Datt . B. . 1938-05-01 . Über eine Klasse von Lösungen der Gravitationsgleichungen der Relativität . Zeitschrift für Physik . de . 108 . 5 . 314–321 . 10.1007/BF01374951 . 0044-3328.
  6. 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 . 10.1103/PhysRevD.38.1315.
  7. Waugh . B. . Lake . Kayll . 1989-09-15 . Shell-focusing singularities in spherically symmetric self-similar spacetimes . Physical Review D . 40 . 6 . 2137–2139 . 10.1103/PhysRevD.40.2137.
  8. 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 . 10.1103/PhysRevD.47.5357. gr-qc/9303037 .
  9. Examples include:
  10. Web site: Crew. Bec. Naked Singularities Can Actually Exist in a Three-Dimensional Universe, Physicists Predict. 2020-09-02. ScienceAlert. 23 May 2017 . en-gb.
  11. Garfinkle. David. Choptuik scaling and the scale invariance of Einstein's equation. Phys. Rev. D. 1997. 56. 6. R3169–R3173. 10.1103/PhysRevD.56.R3169. gr-qc/9612015 . 1997PhRvD..56.3169G .
  12. 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. 10.1088/1475-7516/2024/06/007. 2401.08462 .
  13. 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. 10.1088/1475-7516/2024/07/053. 2405.08758 .
  14. 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. 10.1103/PhysRevD.95.044006. 1605.00600 .
  15. 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. 10.1103/PhysRevD.95.084024. 1611.08808 .
  16. D.Christodoulou. Examples of naked singularity formation in the gravitational collapse of a scalar field. Ann. Math.. 140. 607–653. 1994. 10.2307/2118619. 3. 2118619.
  17. D. Christodoulou. The instability of naked singularities in the gravitational collapse of a scalar field. Ann. Math.. 149. 183–217. 1999. 10.2307/121023. 1. 121023. math/9901147. 8930550.
  18. Hobson, et al., General Relativity an Introduction for Physicists, Cambridge University Press 2007, p. 300-305
  19. Hobson, et al., General Relativity an Introduction for Physicists, Cambridge University Press 2007, p. 320-325
  20. News: Is a 'naked singularity' lurking in our galaxy?. Battersby. Stephen. 1 October 2007. New Scientist. 2008-03-06. Stephen Battersby (science journalist).
  21. Pretorius. Frans. 2016-05-31. Viewpoint: Relativity Gets Thorough Vetting from LIGO. Physics. en. 9. 52 . 10.1103/Physics.9.52. free.
  22. M. Bojowald, Living Rev. Rel. 8, (2005), 11
  23. Goswami . Rituparno . Joshi . Pankaj S. . Spherical gravitational collapse in N dimensions . Physical Review D . 76 . 8 . 2007-10-22 . 1550-7998 . 10.1103/physrevd.76.084026 . 084026. gr-qc/0608136. 2007PhRvD..76h4026G . 119441682 .
  24. Goswami . Rituparno . Joshi . Pankaj S. . Singh . Parampreet . Quantum Evaporation of a Naked Singularity . Physical Review Letters . 96 . 3 . 2006-01-27 . 0031-9007 . 10.1103/physrevlett.96.031302 . 031302. 16486681 . gr-qc/0506129. 2006PhRvL..96c1302G . 19851285 .
  25. Eardley . Douglas M. . Smarr . Larry . Time functions in numerical relativity: Marginally bound dust collapse . Physical Review D . American Physical Society (APS) . 19 . 8 . 1979-04-15 . 0556-2821 . 10.1103/physrevd.19.2239 . 2239–2259. 1979PhRvD..19.2239E .
  26. Królak . Andrzej . Nature of Singularities in Gravitational Collapse . Progress of Theoretical Physics Supplement . 136 . 1999 . 0375-9687 . 10.1143/ptps.136.45 . 45–56. gr-qc/9910108 . 1999PThPS.136...45K . free.

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