Conformal electrovacuum seed data

Let $\mathrm{\Sigma }$ be a nonempty oriented smooth three-manifold without boundary. Let $\bar{h}$ be a smooth Riemannian metric on $\mathrm{\Sigma }$, let $\psi \in C^{\mathrm{\infty }}(\mathrm{\Sigma },(0,\mathrm{\infty }))$, let $\bar{E},\bar{B}$ be smooth contravariant vector fields, let $\bar{A}$ be a smooth symmetric covariant two-tensor with ${\bar{h}}^{ij}{\bar{A}}_{ij}=0$, let $\tau \in C^{\mathrm{\infty }}(\mathrm{\Sigma },\mathbb{R})$, and let $\mathrm{\Lambda }\in \mathbb{R}$ be constant. Let $c,G,{\mu }_0>0$ and $\kappa =8\pi G\mathrm{/}{c}^4$ be the parameters of the inherited SI-to-geometric dictionary, and set $\alpha =\sqrt{\kappa \mathrm{/}(2{\mu }_0)}>0$. Define

$h:={\psi }^4\bar{h},K:=\frac{\tau }{3}h+{\psi }^{-2}\bar{A},E:={\psi }^{-6}\bar{E},B:={\psi }^{-6}\bar{B}\mathrm{.}$

The fields $E$ and $B$ are geometric-normalized spatial vectors: in length coordinates their components have units ${\mathrm{m}}^{-1}$. Their SI-convention counterparts are $e:={\alpha }^{-1}E$ and $b:={\alpha }^{-1}B$. In length coordinates, metric components and $\psi$ are dimensionless, $\tau$ and the covariant components of $\bar{A}$ and $K$ have units ${\mathrm{m}}^{-1}$, and $\mathrm{\Lambda }$ has units ${\mathrm{m}}^{-2}$. The tuple $(\mathrm{\Sigma },\bar{h},\psi ,\bar{A},\tau ,\bar{E},\bar{B},\mathrm{\Lambda })$ with these induced fields is called conformal electrovacuum seed data. This definition imposes no divergence, Hamiltonian, or momentum equation and asserts no existence, uniqueness, or spacetime development.

Constrained electrovacuum initial data

SI and geometrized Einstein–Maxwell parameters