Kerr Metric in Cartesian Coordinates

You can find the Kerr metric in pseudo-Cartesian coordinates for instance in "The Kerr Spacetime", by Wiltshire et al, as

begin{eqnarray} ds^2 = &&-dt^2 + dx^2 + dy^2 + dz^2 &+& frac{2mr^3}{r^4 + a^2 z^2} left[ dt + frac{r(x dx + y dy)}{a^2 + r^2} + frac{a(y dx - x dy)}{a^2 + r^2} + frac{z}{r} dz right]^2 end{eqnarray}

with

begin{equation} x^2 + y^2 + z^2 = r^2 + a^2(1 - frac{z^2}{r^2}) end{equation}

and for the angular coordinates,

begin{eqnarray} x &=& (r cos phi + a sin phi) sin theta y &=& (r sin phi - a cos phi) sin theta z &=& r cos theta end{eqnarray}

This gives the appropriate limits, of giving the Schwarzschild metric in Cartesian coordinates for $a to 0$, and the Minkowski metric for $m to 0$.