Embedding of the Kerr-Newman black hole surface in Euclidean space

We obtain a global embedding of the surface of a rapidly rotating Kerr-Newman black hole in an Euclidean 4-dimensional space.

Figure 1

This picture shows the “croissant” surface. A solid line separates two regions with opposite signs of the Gauss curvature. Each of these regions has the topology of a 2D ball. The Gauss curvature is negative in the upper ball-region.

Figure 2

The surface shown at this picture is formed by straight lines passing through a point $r=0$. Its Gauss curvature vanishes. The surface has a cone singularity at $r=0$ with a negative angle deficit.

Figure 3

${\Phi }_0$ as a function of the parameter $a$.

Figure 4

Plot for $R$ as a function of $\mu$ for $\beta =0.7$.

Figure 5

Plot for $R$ as a function of $r$ (curve 1) and $\rho$ (curve 2) for $\beta =0.7$.

Figure 6

By gluing these two figures along their edges one obtains a 2D surface without angle deficits and isometric to the surface of a rotating black hole ($\beta =0.7$.)