Oblate spheroidal coordinates

With parameter $a\ge 0$, oblate spheroidal coordinates on Euclidean three-space areWriting $\Sigma =r^2+a^2{\cos }^2\theta$ and $\Delta =r^2+a^2$, their spatial line element is

Set $x=a\cosh \rho \cos u\cos v$, $y=a\cosh \rho \cos u\sin v$, $z=a\sinh \rho \sin u$, with $a>0$, $\rho \ge 0$ and $|u|\le \pi /2$. Differentiating the position vector gives pairwise orthogonal coordinate tangents and the displayed scale factors of orthogonal coordinates. Thus the positive volume density is $a^3\cosh \rho \cos u({\sinh }^2\rho +{\sin }^{2}u)$. The ordered Jacobian determinant is negative for the order $(\rho ,u,v)$; volume uses its absolute value. Positive constant $\rho$ surfaces are oblate spheroids. For $0<|u|<\pi /2$, constant $u$ gives the half of the one-sheet hyperboloid $(x^2+y^2)/(a^2{\cos }^{2}u)-z^2/(a^2{\sin }^{2}u)=1$ with the sign of $z$ fixed by $u$. At $\rho =0$ the coordinates cover the equatorial disk, while $u=0$ covers the exterior equatorial plane.

Starting from the Minkowski metric in oblate spheroidal coordinates, set $u=t-r$, $R=r$, and choose an azimuth $\phi$ satisfyingThenso the coordinate vector ${\partial }_R$ is null.