One thing I’ve always found a bit of a curiosity is how a rapidly rotating star might collapse to a ring singularity relative to the speed of light and what the final parameters of the ring singularity might be (i.e. reduced circumference considering r=0 at the ring edge). Due to the conservation of angular momentum, I have seen suggestions that the reduced circumference would be based on [itex]J=vmr[/itex] (see ref. 1 below). Based on [itex]c[/itex] being the maximum velocity, this is rearranged to give the quantity [itex]a[/itex] where [itex]a=J/mc[/itex] but this puts the ring singularity between the outer and inner event horizon, implying that the RS would hold a ‘stable’ orbit in space-like spacetime which would be like us hovering ‘constantly’ at 1.00 pm. This didn’t seem correct but the only other option is that the tangential velocity of the collapsing star would exceed [itex]c[/itex] locally (the proper tangential velocity would exceed c due to frame-dragging) in order for it to collapse within the Cauchy horizon, again, this had it’s own problems (i.e. local superluminal velocity).(adsbygoogle = window.adsbygoogle || []).push({});

I had a look on the web and found the following question/answer- http://answers.yahoo.com/question/i...HDCpA4jzKIX;_ylv=3?qid=20081218053035AAT9JKE" which I thought gave a fairly decent answer. Basically, as the star collapses, the Lorentz parameter has an effect on the linear momentum part of the angular momentum equation-

Newtonian equations for angular momentum-

[tex]J=rp[/tex]

Where [itex]J[/itex] is angular momentum, [itex]r[/itex] is radius and [itex]p[/itex] is momentum where

[tex]p=mv[/tex]

Where [itex]m[/itex] is mass and [itex]v[/itex] is velocity (in the case of a rotating object, tangential velocity)

Combining both equations produces-

[tex]J=vmr[/tex]

In special relativity, linear momentum becomes

[tex]p=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}[/tex]

Which means the coordinate radius of a ring singularity might be expressed as

[tex]r=\frac{J\sqrt{1-\frac{v^2}{c^2}}}{mv}[/tex]

Based on 3 sol mass black hole with a spin parameter of a/M=0.95 and a potential tangential velocity of 0.99c for the ring singularity, provides an approximation for the reduced circumference of the RS of r~600 mm which puts it well within the Cauchy horizon (r=0 considered to be at the outer edge of the ring singularity in some models of a rotating black hole and any space inside the ring would be considered negative) while a=J/mc still plays a part in the Kerr metric (having parallels with the gravitational radius). While it's apparent that what goes on inside a black hole is not fully understood, this seems a better proposal than simply r=J/mv.

I'd be interested to hear other peoples opinions regarding this.

(1)The Kerr Black Hole by Max Camenzind & A. Müller

http://www.lsw.uni-heidelberg.de/users/mcamenzi/GR_07.pdf [Broken]

p 206 fig. 7.4 p 209 fig. 7.8

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# Collapse to a ring singularity

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