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

1. Jun 12, 2009

### redmosquito

I've found a fairly concise review of the Kerr metric at http://www.physics.mcmaster.ca/phys3a03/The Kerr Metric.ppt

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 $r = 0$, which when converted back to Cartesian coordinates is a ring of radius $r_{{\it cartesian}} = a$.

On slide 14 they discuss the outer event horizon at $r = M + \sqrt{M^2 - a^2}$, 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.

Last edited: Jun 12, 2009
2. Jun 16, 2009

### Julian M

Just off the top of my head, oblate spheroid sounds correct (for both the event horizon and the ergosphere)... and ring singularities are certainly also possible

(I'm not aware that all Kerr singularities are rings - I think they can be points... but don't quote me on that)