What I've got so far is the metric tensor for flat-space in bispherical coordinates with a stationary focal distance term "a".
## (1, 2, 3, 4) = (σ, τ, ϕ, t) ##
## g = \begin{pmatrix}
L^2 & 0 & 0 & 0\\
0 & L^2 & 0 & 0\\
0 & 0 & L^2 sin^2 σ & 0\\
0 & 0 & 0 & c^2
\end{pmatrix} ##
Where
## L =...