Constrained electrovacuum initial data

Let $\mathrm{\Sigma }$ be a nonempty, smooth, oriented three-manifold without boundary. Let $h$ be a smooth Riemannian metric on $\mathrm{\Sigma }$, let $K$ be a smooth symmetric covariant two-tensor, and let $e,b$ be smooth vector fields tangent to $\mathrm{\Sigma }$. For data induced on an embedded spacelike slice $i:\mathrm{\Sigma }\hookrightarrow M$, let $n$ be its future unit normal, let $F$ be the smooth two-form of Electrovacuum tensor Cauchy system, and define spatial covectors $\mathcal{E},\mathcal{B}\in {\mathrm{\Omega }}^1(\mathrm{\Sigma })$ by $\mathcal{E}(X):=F(X,n)$ and $\mathcal{B}(X):=-({\star }_{g}F)(X,n)$ for $X\in T\mathrm{\Sigma }$. Identify these with vector fields using the positive metric $h$: $e:={\mathrm{♯}}_h\mathcal{E}$ and $b:={\mathrm{♯}}_h\mathcal{B}$, so in any coordinates $e^i=h^{ij}{\mathcal{E}}_j$ and $b^i=h^{ij}{\mathcal{B}}_j$. Equivalently, $\mathcal{E}$ is the restriction of the observer electric covector $E\mathrm{/}c$ and $\mathcal{B}$ the restriction of the observer magnetic covector $B$ of Observer electric/magnetic reconstruction and invariants. For an abstract data tuple not assumed to come from $F$, $e,b$ are simply the specified tangent vector fields. This fixes the convention without adding a potential or an existence assumption.

For the observer-covector comparison above, the smooth normal along $\mathrm{\Sigma }$ has local smooth ambient extensions by The future unit normal, induced metric and second fundamental form of a spacelike hypersurface; shrinking and normalizing an extension gives an observer congruence to which Observer electric/magnetic reconstruction and invariants applies. The restrictions $\mathcal{E},\mathcal{B}$ depend only on $n{\mathrm{\mid }}_{\mathrm{\Sigma }}$, so they are independent of the chosen extension.

Write $h^{ij}$ for the inverse matrix to $h_{ij}$ and use $h$ to raise every spatial index. Let $D$ and $R(h)$ denote the Levi-Civita connection and scalar curvature of $h$, set $\tau :=h^{ij}{K}_{ij}$ and $\mathrm{\mid }K{\mathrm{\mid }}_h^2:=h^{ik}{h}^{j\mathrm{\ell }}{K}_{ij}{K}_{k\mathrm{\ell }}$, and choose the induced spatial orientation so ${\mathrm{vol}}_g={\theta }^0\wedge {\mathrm{vol}}_h=-n^{\mathrm{♭}}\wedge {\mathrm{vol}}_h$ in a positively oriented normal coframe with ${\theta }^0(n)=1$. Define the spatial cross product by $h(e{\times }_{h}b,w)={\mathrm{vol}}_h(e,b,w)$ for each tangent vector $w$.

For the fixed parameters $(\kappa ,\mathrm{\Lambda },{\mu }_0)$ of Electrovacuum tensor Cauchy system, the tuple $(\mathrm{\Sigma },h,K,e,b)$ is constrained electrovacuum initial data when

$D_i{e}^i=0,D_i{b}^i=0,$

$R(h)+{\tau }^2-\mathrm{\mid }K{\mathrm{\mid }}_h^2=2\mathrm{\Lambda }+\frac{\kappa }{{\mu }_0}(\mathrm{\mid }e{\mathrm{\mid }}_h^2+\mathrm{\mid }b{\mathrm{\mid }}_h^2),$

$D^j{K}_{ji}-D_i\tau =-\frac{\kappa }{{\mu }_0}(e{\times }_{h}b{)}_i\mathrm{.}$

Here the last display is an equality of covectors. The scalar-curvature form is the Gauss-Codazzi form of the normal projections in Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form; the sign and $K$ convention are those of Spacelike initial data, the constraint equations, developments and maximal developments and Electrovacuum tensor Cauchy system. This definition asserts no existence or uniqueness of a spacetime development.

For the periodic high-regularity class, use the dimensionless coordinate ${\theta }^i$ and length coordinate $X^i=L_0{\theta }^i$ of Periodic high-Sobolev state spaces, for a fixed $L_0>0$. The arrays $h_{ij}$ and $K_{ij}$ are the length-coordinate covariant components with $h_{ij}\in H^{s+1}$ and $K_{ij}\in H^s$. The Sobolev components of $e,b$ are $e_A=h(e,{\mathsf{e}}_A)$ and $b_A=h(b,{\mathsf{e}}_A)$ in the canonical positively oriented $h$-orthonormal frame ${\mathsf{e}}_A$ specified there; thus $e_A,b_A\in H^s$. If ${\mathsf{e}}_A=E_A^i{\mathrm{\partial }}_{X^i}$, their coordinate components are $e^i=E_A^i{e}_A$ and $b^i=E_A^i{b}_A$. The coefficients $E_A^i$ are smooth functions of the positive matrix $h_{ij}$; on the compact range of each such metric, Smooth composition, difference, coefficient product and commutator estimates on bounded H^s data puts them in $H^{s+1}$ and The periodic Sobolev space H^k of the three-torus is a Banach algebra for k at least four makes the coordinate components $H^s$ as well.

For these representatives define the spatial connection coefficients by $\mathrm{\Gamma }(h{)}^k{}_{ij}=\frac{1}{2}{h}^{k\mathrm{\ell }}({\mathrm{\partial }}_{X^i}{h}_{j\mathrm{\ell }}+{\mathrm{\partial }}_{X^j}{h}_{i\mathrm{\ell }}-{\mathrm{\partial }}_{X^{\mathrm{\ell }}}{h}_{ij})$, with ${\mathrm{\partial }}_{X^i}=L_0^{-1}{\mathrm{\partial }}_{{\theta }^i}$. On coordinate vector components put $D_i{e}^j={\mathrm{\partial }}_{X^i}{e}^j+\mathrm{\Gamma }(h{)}^j{}_{ik}{e}^k$ and $D_i{b}^j={\mathrm{\partial }}_{X^i}{b}^j+\mathrm{\Gamma }(h{)}^j{}_{ik}{b}^k$; on the covariant tensor $K$ put $D_i{K}_{jk}={\mathrm{\partial }}_{X^i}{K}_{jk}-\mathrm{\Gamma }(h{)}^{\mathrm{\ell }}{}_{ij}{K}_{\mathrm{\ell }k}-\mathrm{\Gamma }(h{)}^{\mathrm{\ell }}{}_{ik}{K}_{j\mathrm{\ell }}$. Define $R(h)$ by $R^a{}_{bij}={\mathrm{\partial }}_{X^i}\mathrm{\Gamma }(h{)}^a{}_{jb}-{\mathrm{\partial }}_{X^j}\mathrm{\Gamma }(h{)}^a{}_{ib}+\mathrm{\Gamma }(h{)}^a{}_{ic}\mathrm{\Gamma }(h{)}^c{}_{jb}-\mathrm{\Gamma }(h{)}^a{}_{jc}\mathrm{\Gamma }(h{)}^c{}_{ib}$, followed by the Ricci and scalar contractions of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. Sobolev embedding gives $h\in C^{s-1}$ and $K,e,b\in C^{s-2}$; the uniform positive lower eigenvalue makes $h^{-1}$ well defined, and the smooth-composition and algebra suppliers give $\mathrm{\Gamma }(h)\in H^s$, $R(h)\in H^{s-1}$, and $DK,De,Db\in H^{s-1}$. Since $s\ge 5$, these constraint terms have continuous representatives, and their displayed equalities mean pointwise equality of those representatives. In the SI interpretation, $K$ and $D$ carry one inverse-length factor, $R(h)$ and $\mathrm{\Lambda }$ carry inverse-length squared, and the orthonormal components $e_A,b_A$ have units tesla; the factors $\kappa \mathrm{/}{\mu }_0$ give matching units in the constraints. No numerical parameter values are assigned here.