Properties of the instantaneous Ergo Surface of a Kerr Black Hole

TL;DR

This paper analyzes the geometric properties of the instantaneous ergo surface of a Kerr black hole, including its area, curvature, and topological features, revealing differences from the event horizon and conditions for embedding in Euclidean space.

Contribution

It provides a closed-form evaluation of the ergo surface area and examines its curvature, topology, and embeddability, highlighting differences from the event horizon.

Findings

Ergo surface area is approximately $16 \pi m^2 + 4 \pi a^2$ to second order in $a$.

Total curvature of the ergo surface varies from $4 \pi$ to 0 as $a/m$ increases.

The ergo surface remains topologically spherical and is embeddable in Euclidean 3-space for $0 \\leq a/m \\leq 1$.

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 $16 \pi m^2 + 4 \pi a^2$ (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 $a=0$) to 0 (for $a=m$) due to conical singularities on the axis ($\theta=0,\pi$) of deficit angle $2 \pi (1-\sqrt{1-(a/m)^2})$. 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).