I've found a fairly concise review of the Kerr metric at http://www.physics.mcmaster.ca/phys3a03/The Kerr Metric.ppt(adsbygoogle = window.adsbygoogle || []).push({});

The Kerr Metric for Rotating, Electrically Neutral Black Holes: The Most Common Case of Black Hole Geometry.Ben Criger and Chad Daley.

On slide 6 they give the usual formulas for converting between Boyer-Lindquist and Cartesian coordinates.

On slide 12 they discuss the central singularity at [itex]r = 0[/itex], which when converted back to Cartesian coordinates is a ring of radius [itex]r_{{\it cartesian}} = a[/itex].

On slide 14 they discuss the outer event horizon at [itex]r = M + \sqrt{M^2 - a^2}[/itex], which when converted back to Cartesian coordinates is an oblate spheroid.

Is this correct? If so, wouldn't it be technically incorrect to draw a diagram of the Kerr spacetime where the central singularity is a ring (the Cartesian representation) but the outer event horizon is a sphere (the Boyer-Lindquist representation)? I see this a lot, and have always been confused by it.

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# Kerr metric, singularities in Boyer-Lindquist and Cartesian coordinates

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