Extremal Limits and Kerr Spacetime

TL;DR

This paper investigates the discontinuity in the extremal limit of Kerr spacetime, showing that the near-extremal and exactly extremal geometries differ fundamentally, especially regarding the ISCO and horizon properties.

Contribution

It demonstrates the nontrivial difference between the extremal Kerr spacetime and its near-extremal limits, highlighting the order of limits and the absence of trapping horizons in the extremal case.

Findings

Discontinuity in the extremal limit of Kerr geometry's maximal extension.

On the extremal Kerr horizon, the ISCO coincides with the null generator, having zero energy and angular momentum.

No ISCO exists in the near-extremal geometry, indicating a fundamental difference.

Abstract

The fact that one must evaluate the near-extremal and near-horizon limits of Kerr space-time in a specific order, is shown to a lead to discontinuity in the extremal limit, such that this limiting space-time differs nontrivially from the precisely extremal space-time. This is established by first showing a discontinuity in the extremal limit of the maximal analytic extension of the Kerr geometry, given by Carter. Next, we examine the ISCO of the exactly extremal Kerr geometry and show that on the event horizon of the extremal Kerr black hole, it coincides with the principal null geodesic generator of the horizon, having vanishing energy and angular momentum. We find that there is no such ISCO in the near-extremal geometry, thus garnering additional support for our primary contention. We relate this disparity between the two geometries to the lack of a trapping horizon in the extremal situation.