Is the Wikipedia depiction of the ergosphere of a Kerr black hole a Cassini oval?

Question

Glancing at https://en.wikipedia.org/wiki/Rotating_black_hole (current revision) I thought that the frontier of the ergosphere appearing in the picture at the beginning of the considered article looks very much like a Cassini oval. Is it actually one? If yes, how can we prove it?

Answer 1

No, the geometry of that region is not exactly a Cassini oval. The following figure shows the geometry of the Kerr space-time with the proper nomenclature of the relevant surfaces:

Note that there are two horizons two surfaces of infinite redshift. For a Kerr black hole of mass $M$ and spin $a$, the horizon surfaces are given by $$r_h^\pm=M\pm\sqrt{M^2-a^2}$$ and the surfaces of infinite redshift are given by $$r_e^\pm=M\pm\sqrt{M^2-a^2\cos^2\theta}$$ where the plus and minus sign refers to the outer and inner surfaces respectively and $\theta$ is the zenith angle. The above figure is a side-view of the geometry plotted in the $yz-$plane which can be rotated about the $z-$axis to get the complete 3-dimensional picture. One can plot two equations (for a fixed value of the spin parameter $-M\leq a\leq M$) as a function of $\theta$ to obtain the above figure.

EDIT:

For a more technical explanation, see this answer to the same question cross-posted on MathOverflow.