 Solution

Choose signature $(-,+,+,+)$, with $e^0$ timelike. The orthonormal coframe in spacetime reconstructs the Lorentzian metric asBecause $A$ and $B$ are nowhere zero on the domain, the coframe is invertible and the metric nondegenerate; either sign of each function is allowed. In retarded and advanced null coordinates $u=t-z$, $v=t+z$ the metric is $-dudv+A(u{)}^{2}d{x}^2+B(u{)}^{2}d{y}^2$, a diagonal example of Rosen coordinates for a plane gravitational wave. Multiplication by $-1$ would give the opposite overall signature convention; all subsequent signs here use the displayed one.