general relativity - How to calculate angular momentum (J) in the Kerr parameter equation?

The Kerr-parameter is given by the following equation: $a = \frac{J}{Mc}$

How to calculate J? Can we just calculate the cartesian product which simplifies to $J = r * M * v$ ? with $r$ = Schwarzschild Radius, $r$ = filament velocity of the BH at the Schwarzschild Radius?

Answer

In asymptotically flat space-times, $J$ is calculated by the Komar integrals

$$J = - \frac{1}{8\pi} \int_{\partial \Sigma} d^2 x \sqrt{\gamma^{(2)}} n_\mu \sigma_\nu \nabla^\mu (\partial_\phi)^\nu$$ Here $\partial \Sigma$ is the two sphere at $i^0$ (spatial infinity) with induced metric $\gamma^{(2)}$. $n$ and $\sigma$ are time-like and space-like unit vectors on $\partial \Sigma$ respectively. $(\partial_\phi)^\mu = \delta^\mu_\phi$ is the Killing vector associated with angular momentum.