Properties of the instantaneous Ergo Surface of a Kerr Black Hole

Abstract

This paper explores properties of the instantaneous ergo surface of a Kerr black hole. The surface area is evaluated in closed form. In terms of the mass ($m$) and angular velocity ($a$), to second order in $a$, the area of the ergo surface is given by $16 πm^2 + 4 πa^2$ (compared to the familiar $16 πm^2 - 4 πa^2$ for the event horizon). Whereas the total curvature of the instantaneous event horizon is $4 π$, on the ergo surface it ranges from $4 π$ (for $a=0$) to 0 (for $a=m$) due to conical singularities on the axis ($θ=0,π$) of deficit angle $2 π(1-\sqrt{1-(a/m)^2})$. A careful application of the Gauss-Bonnet theorem shows that the ergo surface remains topologically spherical. Isometric embeddings of the ergo surface in Euclidean 3-space are defined for $0 \leq a/m \leq 1$ (compared to $0 \leq a/m \leq \sqrt{3}/2$ for the horizon).