https://en.wikipedia.org/wiki/Rotating_black_holehttps://en.wikipedia.org/wiki/Naked_singularity#EffectsNaked singularity

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For the novel, see A Naked Singularity.

In general relativity, a naked singularity is a gravitational singularity without an event horizon. In a black hole, the singularity is completely enclosed by a boundary known as the event horizon, inside which the gravitational force of the singularity is so strong that light cannot escape. Hence, objects inside the event horizon—including the singularity itself—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 general relativity cannot make predictions about the future evolution of space-time near a singularity. In generic black holes, this is not a problem, as an outside viewer cannot observe the space-time within the event horizon.

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.

The naked singularity hypothesis is disfavored by data from the first observation of gravitational waves.[6]

Contents [hide]

1 Predicted formation

2 Metrics

3 Effects

4 Cosmic censorship hypothesis

5 In fiction

6 See also

7 References

8 Further reading

Predicted formation[edit]

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[citation needed] and some computer simulations (Choptuik, 1997) have been performed.[7]

This is 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. [speculation?]

Shaw Prize winning mathematician Demetrios Christodoulou has shown that contrary to what had been expected, singularities which are not hidden in a black hole also occur.[8] However, he then showed that such "naked singularities" are unstable.[9]

Metrics[edit]

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[10] that the {\displaystyle r} r coordinate (which is not the radius) of the event horizon is

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

where {\displaystyle \mu =GM/c^{2}} \mu =GM/c^{2}, and {\displaystyle a=J/Mc} a=J/Mc. In this case, "event horizons disappear" means when the solutions are complex for {\displaystyle r_{\pm }} r_{\pm }, or {\displaystyle \mu ^{2}<a^{2}} \mu ^{2}<a^{2}.

Disappearing event horizons can also be seen with the Reissner–Nordström geometry of a charged black hole. In this metric, it can be shown[11] that the singularities occur at

https://en.wikipedia.org/wiki/Kerr_metr ... _wormholesSymmetries[edit]

The group of isometries of the Kerr metric is the subgroup of the ten-dimensional Poincaré group which takes the two-dimensional locus of the singularity to itself. It retains the time translations (one dimension) and rotations around its axis of rotation (one dimension). Thus it has two dimensions. Like the Poincaré group, it has four connected components: the component of the identity; the component which reverses time and longitude; the component which reflects through the equatorial plane; and the component that does both.

In physics, symmetries are typically associated with conserved constants of motion, in accordance with Noether's theorem. As shown above, the geodesic equations have four conserved quantities: one of which comes from the definition of a geodesic, and two of which arise from the time translation and rotation symmetry of the Kerr geometry. The fourth conserved quantity does not arise from a symmetry in the standard sense and is commonly referred to as a hidden symmetry.

Overextreme Kerr solutions[edit]

The location of the event horizon is determined by the larger root of {\displaystyle \Delta =0} \Delta =0. When {\displaystyle {r_{s}/2}<\alpha } {r_{s}/2}<\alpha (i.e. {\displaystyle GM^{2}<Jc} GM^{2}<Jc), there are no (real valued) solutions to this equation, and there is no event horizon. With no event horizons to hide it from the rest of the universe, the black hole ceases to be a black hole and will instead be a naked singularity.[11]

Kerr black holes as wormholes[edit]

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Although the Kerr solution appears to be singular at the roots of Δ = 0, these are actually coordinate singularities, and, with an appropriate choice of new coordinates, the Kerr solution can be smoothly extended through the values of {\displaystyle r} r corresponding to these roots. The larger of these roots determines the location of the event horizon, and the smaller determines the location of a Cauchy horizon. A (future-directed, time-like) curve can start in the exterior and pass through the event horizon. Once having passed through the event horizon, the {\displaystyle r} r coordinate now behaves like a time coordinate, so it must decrease until the curve passes through the Cauchy horizon.

The region beyond the Cauchy horizon has several surprising features. The {\displaystyle r} r coordinate again behaves like a spatial coordinate and can vary freely. The interior region has a reflection symmetry, so that a (future-directed time-like) curve may continue along a symmetric path, which continues through a second Cauchy horizon, through a second event horizon, and out into a new exterior region which is isometric to the original exterior region of the Kerr solution. The curve could then escape to infinity in the new region or enter the future event horizon of the new exterior region and repeat the process. This second exterior is sometimes thought of as another universe. On the other hand, in the Kerr solution, the singularity is a ring, and the curve may pass through the center of this ring. The region beyond permits closed time-like curves. Since the trajectory of observers and particles in general relativity are described by time-like curves, it is possible for observers in this region to return to their past.

While it is expected that the exterior region of the Kerr solution is stable, and that all rotating black holes will eventually approach a Kerr metric, the interior region of the solution appears to be unstable, much like a pencil balanced on its point.[12] This is related to the idea of cosmic censorship.