Smooth null electrovac wave candidate data

Use length coordinates $(u,v,x,y)$ on ${\mathbb{R}}^4$. Let $a,b:\mathbb{R}\to \mathbb{R}$ be smooth profiles with inverse-length units, and let $h:{\mathbb{R}}^3\to \mathbb{R}$ be a smooth dimensionless function satisfying $h_{xx}+h_{yy}=0$ at every $(u,x,y)$. Define the dimensionless function

$P(u,x,y)=-(a(u{)}^2+b(u{)}^2)(x^2+y^2)+h(u,x,y)\mathrm{.}$

With orientation $du\wedge dv\wedge dx\wedge dy$, define

$g=-2dudv+d{x}^2+d{y}^2+Pd{u}^2,\mathcal{A}=-(a(u)x+b(u)y)du,\mathcal{F}=d\mathcal{A}=a(u)du\wedge dx+b(u)du\wedge dy\mathrm{.}$

The vector field $N={\mathrm{\partial }}_u+\frac{P+1}{2}{\mathrm{\partial }}_v$ has $g(N,N)=-1$ and selects the future direction. The $u,v$ block of $g$ has determinant $-1$, so together with the two positive transverse directions this is a smooth Lorentzian metric of signature $(-,+,+,+)$ and $\sqrt{\mathrm{\mid }\det g\mathrm{\mid }}=1$. The tuple $({\mathbb{R}}^4,g,N,o,\mathcal{F})$ is the null-wave candidate data for the equations of Geometric electrovac equations and flux normalization; no field-equation claim is part of this definition. The metric coefficients are dimensionless, and ${\mathcal{F}}_{ui}$ has inverse-length units.