Editing Naked singularity (section)

==Metrics==
[[File:Naked.Singularity,Overextremal.Kerr.Newman,Raytracing.png|thumb|[[Ray tracing (physics)|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).]]
[[File:Black.Hole,Extremal.Kerr.Newman,Raytracing.png|thumb|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<ref>Hobson, et al., ''General Relativity an Introduction for Physicists'', Cambridge University Press 2007, p. 300-305</ref> that the <math>r</math> coordinate (which is not the radius) of the event horizon is
<math display="block">r_\pm = \mu \pm (\mu^2-a^2)^{1/2},</math>
where <math>\mu = G M / c^2</math>, and <math>a=J/M c</math>.  In this case, "event horizons disappear" means that the solutions are complex for <math>r_\pm</math>, or <math>\mu^2 < a^2</math>. However, this corresponds to a case where <math>J</math> exceeds <math>GM^{2}/c</math> (or in [[Planck units]], {{Nowrap|<math>J > M^2</math>)}}, 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 metric|Reissner–Nordström]] geometry of a charged black hole.  In this metric, it can be shown<ref>Hobson, et al., ''General Relativity an Introduction for Physicists'', Cambridge University Press 2007, p. 320-325</ref> that the horizons occur at
<math display="block">r_\pm = \mu \pm (\mu^2-q^2)^{1/2},</math>
where <math>\mu = G M / c^2</math>, and <math>q^2 = G Q^2/(4 \pi \varepsilon_0 c^4)</math>.  Of the three possible cases for the relative values of <math>\mu</math> and <math>q</math>, the case where <math>\mu^2 < q^2</math> causes both <math>r_\pm </math> to be complex.  This means the metric is regular for all positive values of <math>r</math>, or in other words, the singularity has no event horizon. However, this corresponds to a case where <math>Q/\sqrt{4 \pi \varepsilon_0}</math> exceeds <math>M\sqrt{G}</math> (or in Planck units, {{Nowrap|<math>Q > M</math>)}}, 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.