Kerr metric, singularities in Boyer-Lindquist and Cartesian coordinates

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SUMMARY

The discussion centers on the Kerr metric, specifically its representation in Boyer-Lindquist and Cartesian coordinates. Key points include the central singularity at r = 0, which translates to a ring of radius r_cartesian = a in Cartesian coordinates, and the outer event horizon defined by r = M + √(M² - a²), represented as an oblate spheroid in Cartesian coordinates. The confusion arises from the common practice of depicting the central singularity as a ring while illustrating the outer event horizon as a sphere, which is technically inconsistent. The participants emphasize the need for clarity in these representations to avoid misconceptions.

PREREQUISITES
  • Understanding of the Kerr metric and its significance in black hole physics.
  • Familiarity with Boyer-Lindquist and Cartesian coordinate systems.
  • Knowledge of general relativity and black hole geometry.
  • Basic grasp of singularities and event horizons in astrophysics.
NEXT STEPS
  • Research the mathematical derivation of the Kerr metric and its implications for rotating black holes.
  • Study the differences between Boyer-Lindquist and Cartesian coordinates in the context of general relativity.
  • Explore the properties of singularities in black hole physics, focusing on ring versus point singularities.
  • Examine graphical representations of black hole geometries to understand common misconceptions.
USEFUL FOR

Astrophysicists, students of general relativity, and anyone interested in the geometric properties of black holes and their representations in different coordinate systems.

redmosquito
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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.
 
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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)
 

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