Properties of the instantaneous Ergo Surface of a Kerr Black Hole

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 $16\pi m^2+4\pi a^2$ (compared to the familiar $16 \pi m^2 - 4 \pi a^2$ for the event horizon). Whereas the total curvature of the instantaneous event horizon is $4\pi$, on the ergo surface it ranges from $4 \pi$(for$a=0$)to0(for$a=m$) due to conical singularities on the axis ($\theta =0,\pi$) of deficit angle $2\pi (1-\sqrt{1-(a\mathrm{/}m{)}^2})$. 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 $0\le a\mathrm{/}m\le 1$ (compared to $0 \leq a/m \leq \sqrt{3}/2$ for the horizon).