Consider the Kerr metric given by:

$$ds^2 = -(1-\frac{2GMr}{\rho^2})dt^2 - \frac{2GMar\sin^2{\theta}}{\rho^2}(dtd\phi+d\phi dt)+ \frac{\rho^2}{\Delta}dr^2+\rho^2d\theta^2+ \frac{\sin^2\theta}{\rho^2}[(r^2+a^2)^2-a^2\Delta\sin^2\theta]d\phi^2$$

$\Delta(r) = r^2-2GMr+a^2$

$\rho^2(r,\theta) = r^2+a^2\cos^2\theta$

In calculating the angular momentum using the Komar integral formula, we need to calculate (as given in How to calculate angular momentum (J) in the Kerr parameter equation?): $$ J = - \frac{1}{8\pi} \int_{\partial \Sigma} d^2 x \sqrt{\gamma^{(2)}} n_\mu \sigma_\nu \nabla^\mu (\partial_\phi)^\nu $$ 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 $\Sigma$ and $\partial \Sigma$ respectively. $(\partial_\phi)^\mu = \delta^\mu_\phi$ is the Killing vector associated with angular momentum.

So I have troubles at finding the induced metric $\gamma^{(2)}$ and the normal vectors $n$ and $\sigma$. Is the induced metric given by: $$\gamma_{ij}dx_{i}dx_{j}=\rho^2d\theta^2+ \frac{\sin^2\theta}{\rho^2}[(r^2+a^2)^2-a^2\Delta\sin^2\theta]d\phi^2$$ Or is it just the normal metric on an $S^2$-sphere: $ds^2 = r^2(d\theta^2 + \sin^2\theta d\phi^2)$?

And for the normal vector I have troubles at finding them. So $n$ is normal to $\Sigma$ and $\sigma$ to $\partial \Sigma$ so is it just: $$n_{\mu} = (-(1-\frac{2GMr}{\rho^2})^{\frac{1}{2}},0,0,0)$$ $$\sigma_{\mu} = (0,(\frac{\rho^2}{\Delta})^{\frac{1}{2}},0,0)$$

I have the feeling that I'm missing some terms here or a wrong normalization, but don't know how to fix it.


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