Boyer–Lindquist coordinates

Remarks

Example: for $M = Q = 0$, $\Delta = r^2 + a^2$, the cross term vanishes, and $$g = -dt^2 + \frac{\Sigma}{r^2 + a^2}\, dr^2 + \Sigma\, d\theta^2 + (r^2 + a^2)\sin^2\theta\, d\phi^2.$$ This is flat Minkowski space written with $x = \sqrt{r^2 + a^2}\sin\theta\cos\phi$, $y = \sqrt{r^2 + a^2}\sin\theta\sin\phi$, $z = r\cos\theta$, the oblate spheroidal coordinates. For $Q = 0$ one checks $\frac{\Delta - a^2\sin^2\theta}{\Sigma} = 1 - \frac{2Mr}{\Sigma}$ and $r^2 + a^2 - \Delta = 2Mr$, which gives the usual form of the Kerr metric.

Non-example: the Kerr–Schild coordinates of the Kerr spacetime are not Boyer–Lindquist coordinates. In them the metric is flat Minkowski space plus a multiple of $\ell \otimes \ell$ for a null covector $\ell$, and it has several off-diagonal terms besides $dt\, d\phi$.

The coordinates fail where $\Delta = 0$, which are the horizons $r_\pm = M \pm \sqrt{M^2 - a^2 - Q^2}$, and on the axis $\sin\theta = 0$. The fields $\partial_t$ and $\partial_\phi$ are Killing vector fields, because no coefficient depends on $t$ or $\phi$.