Kerr metric, singularities in Boyer-Lindquist and Cartesian coordinates

[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.

 

[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)

 

[Entropia]

Yes, this is correct. The Kerr metric is a solution to Einstein's field equations that describes the spacetime around a rotating, electrically neutral black hole. It is the most common case of black hole geometry and is often used as a model for studying the properties of black holes.

The Kerr metric is usually expressed in Boyer-Lindquist coordinates, which are adapted to the symmetry of the rotating black hole. These coordinates are useful for understanding the geometry of the black hole, but they can be difficult to work with mathematically. That's why it's often helpful to convert the Kerr metric into Cartesian coordinates, which are much easier to work with.

Slide 6 of the PowerPoint presentation shows the formulas for converting between Boyer-Lindquist and Cartesian coordinates. This allows us to see the relationship between the two coordinate systems and how they describe the same spacetime.

Slide 12 discusses the central singularity of the Kerr metric, which is located at r = 0 in Boyer-Lindquist coordinates. When we convert this to Cartesian coordinates, we see that the singularity becomes a ring with a radius of r_{{\it cartesian}} = a. This is because the singularity is actually a ring-shaped region around the black hole, rather than a single point.

On slide 14, the outer event horizon of the Kerr metric is discussed. In Boyer-Lindquist coordinates, this horizon is located at r = M + \sqrt{M^2 - a^2}. However, when we convert this to Cartesian coordinates, we see that the outer event horizon becomes an oblate spheroid. This is because the event horizon is the boundary beyond which nothing, including light, can escape the black hole's gravitational pull. In Cartesian coordinates, this boundary becomes a three-dimensional shape rather than a single point.

It is technically incorrect to draw a diagram of the Kerr spacetime where the central singularity is a ring in Cartesian coordinates, but the outer event horizon is a sphere in Boyer-Lindquist coordinates. This is because the two coordinate systems are describing the same spacetime, and the shape of the singularity and event horizon should be consistent in both systems. It may be helpful to draw separate diagrams for each coordinate system to better understand the geometry of the Kerr spacetime.