# Naked singularity

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Naked singularity

In general relativity, a naked singularity is a gravitational singularity, without an event horizon. In a black hole, there is a region around the singularity, the event horizon, where the gravitational force of the singularity is strong enough so that light cannot escape. Hence, the singularity cannot be directly observed. A naked singularity, by contrast, is observable from the outside.

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 in the presence of a naked singularity, general relativity cannot make predictions about the future evolution of spacetime.

Some research has suggested that if loop quantum gravity is correct, then naked singularities could exist in nature,[1][2][3] implying that the cosmic censorship hypothesis does not hold. Numerical calculations[4] and some other arguments[5] have also hinted at this possibility.

To this date, no naked singularities (and no event horizons) have been observed.

## Predicted formation

From concepts drawn of 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[citation needed] and some computer simulations (Choptuik, 1997) have been performed.

This is, of course, an example of a mathematical difficulty (divergence to infinity of the density) which reveals a more profound problem in our understanding of the relevant physics involved in the process. A workable theory of quantum gravity should be able to solve problems such as these.

## 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 will disappear. Transforming the Kerr metric to Boyer-Lindquist coordinates, it can be shown[6] 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 μ = GM / c2, and a = J / Mc. In this case, "event horizons disappear" means when the solutions are imaginary for $r_{\pm}$, or μ2 < a2.

Disappearing event horizons can also be seen with the Reissner-Nordström geometry of a charged black hole. In this metric it can be shown[7] that the singularities occur at

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

where μ = GM / c2, and $q = G Q^{2}/(4 \pi \epsilon_{0} c^{4})$. Of the three possible cases for the relative values of μ and q, the case where μ2 < q2 causes both $r_{\pm}$ to be imaginary. This means the metric is regular for all positive values of r, or in other words the singularity has no event horizon.

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.[8]

## Cosmic Censorship Hypothesis

The cosmic censorship hypothesis says that a naked singularity cannot arise in our universe from realistic initial conditions.

## References

1. ^ M. Bojowald, Living Rev. Rel. 8, (2005), 11 (http://relativity.livingreviews.org/Articles/lrr-2008-4/)
2. ^ R. Goswami & P. Joshi, Phys. Rev. D, (2008) (http://arxiv.org/abs/gr-qc/0608136)
3. ^ R. Goswami, P. Joshi, & P. Singh, Phys. Rev. Letters, (2006), 96 (http://arxiv.org/abs/gr-qc/0506129)
4. ^ D. Eardley & L. Smarr, Phys. Rev. D., (1979), 19, (http://prola.aps.org/abstract/PRD/v19/i8/p2239_1)
5. ^ A. Krolak, Prog. Theor. Phys. Supp., (1999) 136, 45, (http://ptp.ipap.jp/link?PTPS/136/45/)
6. ^ Hobson, et. al, General Relativity an Introduction for Physicists, Cambridge University Press 2007, p. 300-305
7. ^ Hobson, et. al, General Relativity an Introduction for Physicists, Cambridge University Press 2007, p. 320-325
8. ^ Stephen Battersby (1 October 2007). "Is a 'naked singularity' lurking in our galaxy?". New Scientist. Retrieved 2008-03-06.

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