Kerr Black Hole

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The spacetime metric is, in Boyer-Lindquist coordinates,

ds^2 = -\frac{\Lambda^2}{\rho^2} (dt - a\, \sin^2\theta \, d\phi)^2 + \frac{\sin^2\theta}{\rho^2} [ (r^2 + a^2) d\phi - a\, dt]^2 + \frac{\rho^2}{\Lambda^2} dr^2 + \rho^2 d\theta^2

where

\begin{array}{rcl} \Lambda^2 & \equiv & r^2 - 2Mr + a^2 \\ \rho^2 & \equiv & r^2 + a^2\cos^2\theta \\ a & \equiv & J/M \end{array}

When split into 3+1 (space + time), the extrinsic curvature is given by

\begin{array}{rcl} K_{r\phi} &=& \frac{(r \partial_r \Omega - \Omega) M a \sin^2 \theta}{\Lambda \sqrt{\rho^6 \Omega}}\\ K_{\theta \phi} &=& \frac{(\partial_{\theta} \Omega) M a r \sin^2 \theta}{\Lambda \sqrt{\rho^6 \Omega}} \end{array}

where \Omega \equiv (r^2 + a^2) \, \rho^2 + 2 M a^2 r \sin^2 {\theta}.

In horizon-penetrating (Kerr-Schild) coordinates, the four-metric is:

ds^2 = (\eta_{\mu \nu} + 2 H \ell_\mu \ell_\nu) dx^\mu dx^\nu,

where H is a scalar function, and \ell_\mu a null (in Minkowski space) vector field. We can write these for an arbitrarily directed spin vector:

\begin{array}{rcl} H &=& \frac{M r^3}{r^4 + (\vec{a}\cdot\vec{x})^2}\\ \ell_\mu &=& \left( 1, \frac{r^2 \vec{x} - r \vec{a} \times \vec{x} + (\vec{a}\cdot\vec{x})\vec{a}}{r (r^2 + a^2)} \right) \end{array}
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