ISCO and Principal Null Congruences in Extremal Kerr Spacetime

TL;DR

This paper derives the effective potential in smooth coordinates for extremal Kerr spacetime, revealing that stable circular geodesics exist precisely on the event horizon along principal null congruences, which are degenerate null geodesic structures.

Contribution

It explicitly analyzes the ISCO in extremal Kerr spacetime using universal coordinates and links stable orbits to principal null congruences at the horizon.

Findings

Stable circular geodesics exist on the event horizon.

Principal null congruences are doubly degenerate and nullify Weyl tensor components.

ISCO coincides with principal null geodesic congruences in extremal Kerr.

Abstract

The effective potential in universal like coordinates$(U, V, \theta, \phi)$, which are smooth across the event horizon is derived and investigated the ISCO(Innermost Stable Circular Orbits) explicitly in these coordinates for extremal Kerr spacetime. Extremization of the effective potential for timelike circular orbit shows that the existence of a stable circular geodesics in the extremal spacetime for direct orbit, precisely {\it on} the event horizon in terms of the radial coordinate which coincides with the \emph{principal null geodesic congruences} of the event horizon. These null geodesic congruences mold themselves to the spacetime curvature in such a way that Weyl conformal tensor and its dual are vanished, that is why they are in fact \emph{doubly degenerate principal null congruences}.