Conversion from Spherical to Cartesian:
coordinates...
$x = r sin \theta cos \phi$
$y = r sin \theta sin \phi$
$z = r cos \theta$
inverse...
$r = \sqrt{x^2 + y^2 + z^2}$
$\theta = atan2(\sqrt{x^2 + y^2}, z)$
$\phi = atan2(y, x)$
identities...
$cos\phi = \frac{x}{\sqrt{x^2+y^2}}$
$sin\phi = \frac{y}{\sqrt{x^2+y^2}}$
$cos\theta = \frac{z}{r}$
$sin\theta = \frac{\sqrt{x^2+y^2}}{r}$
differentials...
$dr = \frac{x dx + y dy + z dz}{r}$
$d\theta = \frac{xz dx + yz dy - (x^2 + y^2) dz}{r^2 \sqrt{x^2 + y^2}}$
$d\phi = \frac{-y dx + x dy}{x^2 + y^2}$
...squared
$dr^2 = \frac{x^2 dx^2 + y^2 dy^2 + z^2 dz^2 + 2 xy dx dy + 2 xz dx dz + 2 yz dy dz}{r^2}$
$d\theta^2 = \frac{x^2 z^2 dx^2 + y^2 z^2 dy^2 + (x^2 + y^2)^2 dz^2 + xyz^2 dx dy - xz(x^2 + y^2) dx dz - yz (x^2 + y^2) dy dz}{r^4 (x^2 + y^2)}$
$d\phi^2 = \frac{y^2 dx^2 + x^2 dy^2 - 2xy dx dy}{(x^2 + y^2)^2}$
Kerr Spherical:
$ds^2 =$
$-(1 - \frac{R r}{\Sigma}) dt^2$
$- \frac{2 R a r sin^2 \theta}{\Sigma} dt d\phi$
$+ \frac{\Sigma}{\Delta} dr^2 $
$+ \Sigma d\theta^2 $
$+ (r^2 + a^2 + \frac{R a^2 r sin^2 \theta}{\Sigma}) sin^2 \theta d\phi^2$
for
$\Sigma = r^2 + a^2 cos^2 \phi = r^2 + \frac{a^2 x^2}{x^2 + y^2} = \frac{r^2 x^2 + r^2 y^2 + a^2 x^2}{x^2 + y^2} = \frac{(r^2 + a^2) x^2 + r^2 y^2}{x^2 + y^2}$
$\Delta = r^2 - R r + a^2$
$R = 2 G M = $ Schwarzschild radius
$a = \frac{J}{M} =$ angular momentum per unit mass
Substitute:
$ds^2 =$
$-(1 - \frac{R r}{\Sigma}) dt^2$
$- \frac{2 R a}{r \Sigma} dt (-y dx + x dy)$
$+ \frac{\Sigma}{r^2 \Delta} (x^2 dx^2 + y^2 dy^2 + z^2 dz^2 + 2 xy dx dy + 2 xz dx dz + 2 yz dy dz)$
$+ \frac{\Sigma}{r^4 (x^2 + y^2)} (x^2 z^2 dx^2 + y^2 z^2 dy^2 + (x^2 + y^2)^2 dz^2 + xyz^2 dx dy - xz(x^2 + y^2) dx dz - yz (x^2 + y^2) dy dz)$
$+ (\frac{r^2 + a^2}{r^2 (x^2 + y^2)} + R \frac{a^2}{r^3 \Sigma}) (y^2 dx^2 + x^2 dy^2 - 2xy dx dy)$
Distribute...
$ds^2 =$
$-(1 - \frac{R r}{\Sigma}) dt^2$
$+ \frac{2 y R a}{r \Sigma} dt dx$
$- \frac{2 x R a}{r \Sigma} dt dy$
$+ \frac{x^2 \Sigma}{r^2 \Delta} dx^2$
$+ \frac{y^2 \Sigma}{r^2 \Delta} dy^2$
$+ \frac{z^2 \Sigma}{r^2 \Delta} dz^2$
$+ \frac{2xy \Sigma}{r^2 \Delta} dx dy$
$+ \frac{2xz \Sigma}{r^2 \Delta} dx dz$
$+ \frac{2yz \Sigma}{r^2 \Delta} dy dz$
$+ \frac{x^2 z^2 \Sigma}{r^4 (x^2 + y^2)} dx^2$
$+ \frac{y^2 z^2 \Sigma}{r^4 (x^2 + y^2)} dy^2$
$+ \frac{(x^2 + y^2) \Sigma}{r^4} dz^2$
$+ \frac{xyz^2 \Sigma}{r^4 (x^2 + y^2)} dx dy$
$- \frac{xz \Sigma}{r^4} dx dz$
$- \frac{yz \Sigma}{r^4} dy dz$
$+ y^2 (\frac{r^2 + a^2}{r^2 (x^2 + y^2)} + \frac{R a^2}{r^3 \Sigma}) dx^2$
$+ x^2 (\frac{r^2 + a^2}{r^2 (x^2 + y^2)} + \frac{R a^2}{r^3 \Sigma}) dy^2$
$- 2xy (\frac{r^2 + a^2}{r^2 (x^2 + y^2)} + \frac{R a^2}{r^3 \Sigma}) dx dy$
Group...
$ds^2 =$
$-(1 - \frac{R r}{\Sigma}) dt^2$
$+ \frac{2 y R a}{r \Sigma} dt dx$
$- \frac{2 x R a}{r \Sigma} dt dy$
$+ \frac{1}{r^2} (\frac{x^2 \Sigma}{\Delta} + \frac{x^2 z^2 \Sigma}{r^2 (x^2 + y^2)} + y^2 (\frac{r^2 + a^2}{x^2 + y^2} + \frac{R a^2}{r \Sigma})) dx^2$
$+ \frac{1}{r^2} (\frac{y^2 \Sigma}{\Delta} + \frac{y^2 z^2 \Sigma}{r^2 (x^2 + y^2)} + x^2 (\frac{r^2 + a^2}{x^2 + y^2} + \frac{R a^2}{r \Sigma})) dy^2$
$+ \frac{1}{r^2} (\frac{z^2 \Sigma}{\Delta} + \frac{(x^2 + y^2) \Sigma}{r^2}) dz^2$
$+ 2 \frac{1}{r^2} (\frac{xy \Sigma}{\Delta} + \frac{xyz^2 \Sigma}{2 r^2 (x^2 + y^2)} - xy (\frac{r^2 + a^2}{x^2 + y^2} + \frac{R a^2}{r \Sigma})) dx dy$
$+ 2 \frac{xz \Sigma}{r^2} (\frac{1}{\Delta} - \frac{1}{2 r^2}) dx dz$
$+ 2 \frac{yz \Sigma}{r^2} (\frac{1}{\Delta} - \frac{1}{2 r^2}) dy dz$
Because rotations are in xy, they will have extra crap in their terms.
So will diagonals due to $dr^2$ influence.
The cleanest terms should be $dx dz$ and $dy dz$ -- whatever is in them, proportional to $xz$ or $yz$ should show up in all other terms proportional to $x^i x^j$
$ds^2 =$
$-(1 - \frac{R r}{\Sigma}) dt^2$
$+ \frac{2 y R a}{r \Sigma} dt dx$
$- \frac{2 x R a}{r \Sigma} dt dy$
$+ (\frac{x^2 \Sigma}{r^2} (\frac{1}{\Delta} - \frac{1}{2 r^2}) - \frac{x^2 \Sigma}{2 r^4} + \frac{x^2 \Sigma}{r^2 (x^2 + y^2)} + \frac{y^2}{r^2} (\frac{r^2 + a^2}{x^2 + y^2} + \frac{R a^2}{r \Sigma})) dx^2$
$+ (\frac{y^2 \Sigma}{r^2} (\frac{1}{\Delta} - \frac{1}{2 r^2}) - \frac{y^2 \Sigma}{2 r^4} + \frac{y^2 \Sigma}{r^2 (x^2 + y^2)} + \frac{x^2}{r^2} (\frac{r^2 + a^2}{x^2 + y^2} + \frac{R a^2}{r \Sigma})) dy^2$
$+ (\frac{z^2 \Sigma}{r^2} (\frac{1}{\Delta} - \frac{1}{2 r^2}) - \frac{z^2 \Sigma}{2 r^4} + \frac{\Sigma}{r^2}) dz^2$
$+ 2 (\frac{xy \Sigma}{r^2} (\frac{1}{\Delta} - \frac{1}{2 r^2}) + \frac{xy \Sigma}{2 r^2 (x^2+y^2)} - \frac{xy}{r^2} (\frac{r^2 + a^2}{x^2 + y^2} + \frac{R a^2}{r \Sigma})) dx dy$
$+ 2 \frac{xz \Sigma}{r^2} (\frac{1}{\Delta} - \frac{1}{2 r^2}) dx dz$
$+ 2 \frac{yz \Sigma}{r^2} (\frac{1}{\Delta} - \frac{1}{2 r^2}) dy dz$
Now I'm taking the $dx dz$ and $dy dz$ coefficients out of the spatial metric...
$\gamma_{ij} = x_i x_j (\frac{\Sigma}{r^2} (\frac{1}{\Delta} - \frac{1}{2r^2}) + \delta_{ij} (-\frac{\Sigma}{2r^4} + \frac{\Sigma}{r^2} - 1))$
$ = \frac{x_i x_j}{2r^4(x^2 + y^2)} (\frac{(a^2 x^2 + r^2 (x^2 + y^2))(r^2 - a^2 + R r)}{r^2 + a^2 - R r} + \delta_{ij} (2 a^2 x^2 r^2 - a^2 x^2 - r^2 x^2 - r^2 y^2))$
$ds^2 =$
$(\eta_{tt} + \frac{R r}{\Sigma}) dt^2$
$+ 2 (\eta_{tx} + \frac{y R a}{r \Sigma}) dt dx$
$+ 2 (\eta_{ty} - \frac{x R a}{r \Sigma}) dt dy$
$+ (\eta_{xx} + \gamma_{xx} + \frac{x^2 \Sigma}{r^2 (x^2 + y^2)} + \frac{a^2}{r^2} - \frac{x^2}{x^2 + y^2} - \frac{2 a^2 x^2}{r^2 (x^2 + y^2)} + \frac{y^2 R a^2}{r^3 \Sigma}) dx^2$
$+ (\eta_{yy} + \gamma_{yy} + \frac{y^2 \Sigma}{r^2 (x^2 + y^2)} - \frac{y^2}{x^2 + y^2} + \frac{x^2 R a^2}{r^3 \Sigma}) dy^2$
$+ 2 (\eta_{xy} + \gamma_{xy} + \frac{xy \Sigma}{2 r^2 (x^2+y^2)} - \frac{xy}{r^2} (\frac{r^2 + a^2}{x^2 + y^2} + \frac{R a^2}{r \Sigma})) dx dy$
$+ 2 (\eta_{xz} + \gamma_{xz}) dx dz$
$+ 2 (\eta_{yz} + \gamma_{yz}) dy dz$
$+ (\eta_{zz} + \gamma_{zz}) dz^2$
$g_{\mu\nu} = \left[
\matrix{
\eta_{tt} + \frac{R r}{\Sigma} &
\eta_{tx} + \frac{y R a}{r \Sigma} &
\eta_{ty} - \frac{x R a}{r \Sigma} &
0
\\
\eta_{tx} + \frac{y R a}{r \Sigma} &
\eta_{xx} + \gamma_{xx} + \frac{x^2 \Sigma}{r^2 (x^2 + y^2)} + \frac{a^2}{r^2} - \frac{x^2}{x^2 + y^2} - \frac{2 a^2 x^2}{r^2 (x^2 + y^2)} + \frac{y^2 R a^2}{r^3 \Sigma} &
\eta_{xy} + \gamma_{xy} + \frac{xy \Sigma}{2 r^2 (x^2+y^2)} - \frac{xy}{r^2} (\frac{r^2 + a^2}{x^2 + y^2} + \frac{R a^2}{r \Sigma}) &
\eta_{xz} + \gamma_{xz}
\\
\eta_{ty} - \frac{x R a}{r \Sigma} &
\eta_{xy} + \gamma_{xy} + \frac{xy \Sigma}{2 r^2 (x^2+y^2)} - \frac{xy}{r^2} (\frac{r^2 + a^2}{x^2 + y^2} + \frac{R a^2}{r \Sigma}) &
\eta_{yy} + \gamma_{yy} + \frac{y^2 \Sigma}{r^2 (x^2 + y^2)} - \frac{y^2}{x^2 + y^2} + \frac{x^2 R a^2}{r^3 \Sigma} &
\eta_{yz} + \gamma_{yz} &
\\
0 &
\eta_{xz} + \gamma_{xz} &
\eta_{yz} + \gamma_{yz} &
\eta_{zz} + \gamma_{zz}
}
\right]$
$g_{\mu\nu} = \eta_{\mu\nu} + \gamma_{\mu\nu} + \omega_{\mu\nu}$
For ...
$\eta_{\mu\nu} = \left[\matrix{-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1}\right]$
$\gamma_{\mu\nu} = \left[\matrix{
0 & 0 \\
0 &
x_i x_j (\frac{\Sigma}{r^2} (\frac{1}{\Delta} - \frac{1}{2r^2}) + \delta_{ij} (-\frac{\Sigma}{2r^4} + \frac{\Sigma}{r^2} - 1))
}\right]$
$\omega_{\mu\nu} = \left[\matrix{
\frac{R r}{\Sigma} &
\frac{y R a}{r \Sigma} &
-\frac{x R a}{r \Sigma} &
0
\\
\frac{y R a}{r \Sigma} &
\frac{x^2 \Sigma}{r^2 (x^2 + y^2)} + \frac{a^2}{r^2} - \frac{x^2}{x^2 + y^2} - \frac{2 a^2 x^2}{r^2 (x^2 + y^2)} + \frac{y^2 R a^2}{r^3 \Sigma} &
\frac{xy \Sigma}{2 r^2 (x^2+y^2)} - \frac{xy}{r^2} (\frac{r^2 + a^2}{x^2 + y^2} + \frac{R a^2}{r \Sigma}) &
0
\\
-\frac{x R a}{r \Sigma} &
\frac{xy \Sigma}{2 r^2 (x^2+y^2)} - \frac{xy}{r^2} (\frac{r^2 + a^2}{x^2 + y^2} + \frac{R a^2}{r \Sigma}) &
\frac{y^2 \Sigma}{r^2 (x^2 + y^2)} - \frac{y^2}{x^2 + y^2} + \frac{x^2 R a^2}{r^3 \Sigma} &
0
\\
0 & 0 & 0 & 0
}\right]$
...
Spoilers:
http://arxiv.org/pdf/0706.0622.pdf gives:
$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 + \frac{2mr^3}{r^4 + a^2 z^2} \left[ dt + \frac{r(x dx + y dy)}{a^2 + r^2} + \frac{a(y dx - x dy}{a^2 + r^2} + \frac{z}{r} dz \right]^2$
...for $x^2 + y^2 + z^2 = r^2 + a^2 \left[ 1 - \frac{z^2}{r^2} \right]$