A short introduction on Angular momentum of Kerr Blackhole

TL;DR

This paper explores the properties of Kerr black holes in general relativity, deriving equations for equatorial motion, effective potential profiles, and analyzing the maximum radius of the innermost stable circular orbit at extremal spin.

Contribution

It provides a detailed derivation of the radial equation, effective potential, and the behavior of angular momentum and energy profiles for Kerr black holes, including the extremal case.

Findings

Effective potential profiles for different rotation parameters

Expression for angular momentum per unit mass squared

Maximum radius of ISCO at extremal spin

Abstract

General relativity (GR) predicts the existence of black hole (BH). The rotating BH called as a Kerr Black hole and GR implies that there is an upper limit on the angular momentum per mass squared of black holes $\leq 1$, above which the event horizon of the Kerr BH is not exist. We find the radial equation for equatorial motion for Kerr BH in terms of the effective potential. We have shown the effective potential profile for different rotation parameter ($a$). We find the solution of the radial equation of the Kerr metric and found the expression of the angular momentum per unit mass squared, $\tilde a = \frac{a}{M}$. We showed the profile of $\tilde a$ as a function of $\frac{r}{M}$. The solution also leads the energy per unit rest mass ($e$) and we showed its behavior as a function of $\frac{r}{M}$. We enumerated the maximum values of radius of innermost stable circular orbit ($r_{ISCO}$) for $\tilde a=1$.