Embedding of the Kerr-Newman Black Hole Surface in Euclidean Space

In this paper we discuss the problem of isometric embedding of the surface of a rapidly rotating black hole in a flat space. It is well known that intrinsically defined Riemannian manifolds can be isometrically embedded in a flat space. According to the Cartan-Janet [1, 2] theorem, every analytic Riemannian manifold of dimension n can be locally real analytically isometrically embedded into E with N = n(n+ 1)/2. The so called Fundamental Theorem of Riemannian geometry (Nash, 1956 [3]) states that every smooth Riemannian manifold of dimension n can be globally isometrically embedded in a Euclidean space E with N = (n+ 2)(n+ 3)/2. The problem of isometric embedding of 2D manifolds in E is well studied. It is known that any compact surface embedded isometrically in E has at least one point of positive Gauss curvature. Any 2D compact surface with positive Gauss curvature is always isometrically embeddable in E, and this embedding is unique up to rigid rotations. (For general discussion of these results and for further references, see e.g. [4]). It is possible to construct examples when a smooth geometry on a 2D ball with negative Gauss curvature cannot be isometrically embedded in E (see e.g. [5, 6]). On the other hand, it is easy to construct an example of a global smooth isometric embedding for a surface of the topology S which has both, positive and negative Gauss curvature ball-regions, separated by a closed loop where the Gauss curvature vanishes. An example of such an embedding is shown in Figure 1 [10]. The surface geometry of a charged rotating black hole and its isometric embedding in E was studied long time ago by Smarr [7]. He showed that when the dimensionless rotation parameter α = J/M is sufficiently large, there are two regions near poles of the horizon surface where the Gauss curvature becomes negative. Smarr proved that these regions cannot be isometrically embedded (even locally) in E as a revolution surface, but such local embedding is possible in a 3D Minkowsky space. More recently different aspects of the embedding of a surface of a rotating black hole and its ergosphere in E