Positive harmonic conformastatic electrovac data

Coordinate-units note. Under Oriented Lorentzian fields, observers and coordinate conventions, the coordinates in a length-coordinate chart, including $x^0$, are assigned metres; when $x^0=ct$ is used, $t$ is in seconds and $c>0$ has units $\mathrm{m}{\mathrm{s}}^{-1}$.

Let $D\subseteq {\mathbb{R}}^3$ be open and let $H\in C^{\mathrm{\infty }}(D,(0,\mathrm{\infty }))$ be independent of time. Write $x=(x^1,x^2,x^3)$ for length coordinates, set $x^0=ct$ when a time variable is used, and require $H$ to be dimensionless. The ordinary Euclidean Laplacian is ${\mathrm{\Delta }}_{\delta }H:={\sum }_{i=1}^3{\mathrm{\partial }}_i^{2}H$; call $H$ harmonic when ${\mathrm{\Delta }}_{\delta }H=0$. Take $\mathcal{M}={\mathbb{R}}_{x^0}\times D$, with orientation $d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$ and future direction selected by $N_H=H{\mathrm{\partial }}_{x^0}$.

For a sign $\sigma \in \{-1,+1\}$ define the metric and global potential one-form

$g_H=-H^{-2}(d{x}^0{)}^2+H^2{\sum }_{i=1}^3(d{x}^i{)}^2,{\mathcal{A}}_{\sigma }=\sigma (H^{-1}-1)d{x}^0\mathrm{.}$

The associated two-form is

${\mathcal{F}}_{\sigma }:=d{\mathcal{A}}_{\sigma }=-\sigma {\sum }_{i=1}^3\frac{H_i}{H^2}d{x}^i\wedge d{x}^0,H_i:={\mathrm{\partial }}_{i}H\mathrm{.}$

The positive harmonic conformastatic electrovac data are the tuple $(\mathcal{M},g_H,N_H,o,{\mathcal{F}}_{\sigma })$ for these choices with the additional condition ${\mathrm{\Delta }}_{\delta }H=0$ on $D$. The displayed metric, potential and two-form remain defined for every smooth positive $H$, but only the harmonic choices are given this name. The metric is the chart tensor of Positive-scalar conformastatic chart data and is Lorentzian with the stated future direction by The positive-scalar conformastatic chart tensor is Lorentzian. The term “electrovac data” names a candidate for the equations in Geometric electrovac equations and flux normalization; this definition by itself does not assert that the equations hold. The field components have inverse-length units in length coordinates, so the Ricci and quadratic-field components both have inverse-length-squared units.