Vacuum conformal constraint data

Let $\mathrm{\Sigma }$ be a nonempty smooth three-manifold without boundary that is compact: for every family $(U_a{)}_{a\in A}$ of open subsets of $\mathrm{\Sigma }$ with $\mathrm{\Sigma }={\bigcup }_{a\in A}{U}_a$, there is a finite subset $F\subseteq A$ such that $\mathrm{\Sigma }={\bigcup }_{a\in F}{U}_a$. In this item, “vacuum” means that the stress-energy tensor is zero and the cosmological constant is fixed at $\mathrm{\Lambda }=0$. Let $\tilde{h}$ be a smooth Riemannian metric on $\mathrm{\Sigma }$, let $\tilde{\sigma }$ be a smooth symmetric covariant two-tensor field, let $\tau \in \mathbb{R}$ be constant, and let $\varphi \in C^{\mathrm{\infty }}(\mathrm{\Sigma },\mathbb{R})$ satisfy $\varphi (p)>0$ at every $p\in \mathrm{\Sigma }$. Use the positive scalar $\varphi$ as a dimensionless conformal factor.

Write $\tilde{D}$ for the Levi-Civita connection of $\tilde{h}$ and $D$ for that of the metric $h$ defined below. On covariant tensor fields, these symbols also denote their induced product connections, with the dual and tensor-factor rules of the product-connection supplier in Definition references. In a chart, ${\tilde{h}}^{ij}$ denotes the inverse matrix to ${\tilde{h}}_{ij}$. It is smooth because $\det ({\tilde{h}}_{ij})>0$ and the matrix inverse is the adjugate divided by the determinant. For any smooth Riemannian metric $m$ on $\mathrm{\Sigma }$, with Levi-Civita connection $D^m$, define $R^m(X,Y)Z:=D_X^m{D}_Y^{m}Z-D_Y^m{D}_X^{m}Z-D_{[X,Y]}^{m}Z\mathrm{.}$ In a chart, $R[m{]}^a{}_{bij}$ denotes the components of $R^m({\mathrm{\partial }}_i,{\mathrm{\partial }}_j){\mathrm{\partial }}_b$. Define $\mathrm{Ric}(m{)}_{ij}:=R[m{]}^k{}_{ikj}$ and $R(m):=m^{ij}\mathrm{Ric}(m{)}_{ij}$. In particular, $\tilde{R}:=R(\tilde{h})$, with the same curvature sign convention. Define the scalar Laplacian by $\tilde{\mathrm{\Delta }}f:={\tilde{h}}^{ij}({\mathrm{\partial }}_i{\mathrm{\partial }}_{j}f-{\tilde{\mathrm{\Gamma }}}^k{}_{ij}{\mathrm{\partial }}_{k}f),$ where ${\tilde{\mathrm{\Gamma }}}^k{}_{ij}$ are the Christoffel symbols of $\tilde{D}$, given by the Levi-Civita Christoffel formula in Definition references. The curvature sign is the one displayed above. All traces, norms, raising of indices and contractions in this item use the metric indicated by the subscript. In particular, ${\mathrm{tr}}_{\tilde{h}}\tilde{\sigma }:={\tilde{h}}^{ij}{\tilde{\sigma }}_{ij},\mathrm{\mid }\tilde{\sigma }{\mathrm{\mid }}_{\tilde{h}}^2:={\tilde{h}}^{ik}{\tilde{h}}^{j\mathrm{\ell }}{\tilde{\sigma }}_{ij}{\tilde{\sigma }}_{k\mathrm{\ell }},({\mathrm{div}}_{\tilde{h}}\tilde{\sigma }{)}_j:={\tilde{h}}^{ik}{\tilde{D}}_i{\tilde{\sigma }}_{kj}\mathrm{.}$ The tensor $\tilde{\sigma }$ is transverse-traceless (TT) with respect to $\tilde{h}$ when it is symmetric and both displayed trace and divergence vanish. The input tensor in this definition is required to be TT.

Define $h:={\varphi }^4\tilde{h},K:={\varphi }^{-2}\tilde{\sigma }+\frac{\tau }{3}h\mathrm{.}$ Because $\varphi >0$, $h$ is a smooth Riemannian metric; because $\tilde{\sigma }$ and $h$ are symmetric, $K$ is a smooth symmetric tensor. Here $h$ is the candidate induced Riemannian metric and $K$ is the candidate second fundamental form, with the library convention $K(X,Y)=g({\mathrm{\nabla }}_{X}n,Y)$ when the data are induced by a spacelike hypersurface with future unit normal $n$. The pair is vacuum conformal constraint data. It is abstract data on $\mathrm{\Sigma }$; no embedding into a spacetime is part of this definition. For its embedded Lorentzian interpretation assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, as in the supplied smooth tangent-bundle convention; the algebraic tensor definitions themselves use no choice principle. Define the $h$-trace, squared norm and divergence by ${\mathrm{tr}}_{h}K:=h^{ij}{K}_{ij},\mathrm{\mid }K{\mathrm{\mid }}_h^2:=h^{ik}{h}^{j\mathrm{\ell }}{K}_{ij}{K}_{k\mathrm{\ell }},({\mathrm{div}}_{h}K{)}_j:=h^{ik}{D}_i{K}_{kj}\mathrm{.}$ With $R(h)$ computed using $h$, define the Hamiltonian and momentum residuals $\mathcal{H}(h,K):=R(h)+({\mathrm{tr}}_{h}K{)}^2-\mathrm{\mid }K{\mathrm{\mid }}_h^2,{\mathcal{M}}_j(h,K):=h^{ik}{D}_i{K}_{kj}-D_j({\mathrm{tr}}_{h}K)\mathrm{.}$ The data satisfy the vacuum constraint equations when $\mathcal{H}(h,K)=0$ and $\mathcal{M}(h,K)=0$. For an embedded hypersurface, these are the normal and normal-tangential Einstein constraints in the specified $K$ convention.

Tensoriality of curvature and basis-independent contraction make these scalar curvatures independent of the chart. For dimensional bookkeeping in the SI length convention, $[\tau ]={\mathrm{m}}^{-1}$, $[R(h)]=[\tilde{R}]=[\mathrm{\mid }\tilde{\sigma }{\mathrm{\mid }}_{\tilde{h}}^2]=[\tilde{\mathrm{\Delta }}]={\mathrm{m}}^{-2}$, and $[\varphi ]=1$. Thus all terms in the scalar constraint equation have units ${\mathrm{m}}^{-2}$.

Smooth manifolds and their smooth charts

A smooth tensor field

Riemannian metric and riemannian manifold

Levi civita connection

Product connection on tensor and hom bundles

Fundamental theorem of riemannian geometry

Christoffel symbols of an affine connection

Christoffel formula for the levi civita connection

Curvature of an affine connection

Curvature is a type (1,3) tensor

The contraction of a mixed tensor

Contraction is independent of the basis formula

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

The future unit normal, induced metric and second fundamental form of a spacelike hypersurface

Integer powers $a^m$

Laws of integer exponents

Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$

$C^k$ Euclidean maps are closed under componentwise algebra and composition

Identity maps and composites of smooth maps are smooth

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Spacelike initial data, the constraint equations, developments and maximal developments

Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form