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

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\mathrm{\nabla }$ and Einstein tensor $G$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors), let $\mathrm{\Sigma }\subseteq M$ be a spacelike embedded hypersurface (Embedded submanifolds and slice charts, Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons clause (a)), and fix real constants $\kappa$ and $\mathrm{\Lambda }$ (the framework coupling constants, purely parameters for this definition).

(a) The geometric data. For $p\in \mathrm{\Sigma }$ the orthogonal complement $T_p{\mathrm{\Sigma }}^{\perp }$ is one-dimensional and nondegenerate (eta-orthogonality and orthogonal complements in a Lorentz vector space, Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)); there is therefore a unique $n_p\in T_p{\mathrm{\Sigma }}^{\perp }$ with $g_p(n_p,n_p)=-1$ and $g_p(n_p,N_p)<0$ and it depends smoothly on $p$, the future unit normal field of $\mathrm{\Sigma }$; this and the properties of $h$ and $K$ below are proved, not assumed, in The future unit normal, induced metric and second fundamental form of a spacelike hypersurface ↗. Define on $\mathrm{\Sigma }$ the induced metric $h$ and the second fundamental form $K$ by $h_p(V,W):=g_p(V,W),K_p(V,W):=g_p({\mathrm{\nabla }}_{V}n,W)(V,W\in T_p\mathrm{\Sigma }),$ where ${\mathrm{\nabla }}_{V}n$ is the covariant derivative at $p$ of the ambient field $n$ in the tangential direction $V$; $h$ is a smooth Riemannian metric on $\mathrm{\Sigma }$ and $K$ a smooth symmetric covariant $2$-tensor field on $\mathrm{\Sigma }$ (Smooth covariant 2-tensor fields on a smooth manifold).

(b) Matter data and constraints. Let $T$ be a smooth symmetric covariant $2$-tensor field on $M$ (the stress-energy field of a matter branch). Its matter data on $\mathrm{\Sigma }$ are the smooth fields $\rho :=T(n,n),j(V):=-T(n,V)(V\in T\mathrm{\Sigma })\mathrm{.}$ The tuple $(\mathrm{\Sigma },h,K,\rho ,j)$ satisfies the constraint equations for $(\kappa ,\mathrm{\Lambda })$ when $G(n,n)-\mathrm{\Lambda }=\kappa \rho ,G(n,V)=\kappa T(n,V)\text{for every }V\in T_p\mathrm{\Sigma },p\in \mathrm{\Sigma }\mathrm{.}$ The first is the normal projection of the equation $G+\mathrm{\Lambda }g=\kappa T$ evaluated on $(n,n)$, namely $G(n,n)+\mathrm{\Lambda }g(n,n)=\kappa T(n,n)$, with $g(n,n)=-1$; the second is the normal-tangential projection, the term in $\mathrm{\Lambda }$ dropping because $g(n,V)=0$. That a solution of the field equation satisfies these two projections is proved in The normal and normal-tangential projections of the field equation are the constraint equations ↗. The purely tangential projection $G(V,W)+\mathrm{\Lambda }h(V,W)=\kappa T(V,W)$ is not part of the constraints and is not required here.

(c) Developments. A development of the data $(\mathrm{\Sigma },h,K,\rho ,j)$ is a time-oriented smooth Lorentzian four-manifold $(M^{\mathrm{\prime }},g^{\mathrm{\prime }},N^{\mathrm{\prime }})$ together with a smooth embedding $\phi :\mathrm{\Sigma }\to M^{\mathrm{\prime }}$ whose image is a spacelike embedded hypersurface with induced metric ${\phi }^{\ast }{g}^{\mathrm{\prime }}=h$ and second fundamental form $K$ with respect to its future unit normal $n^{\mathrm{\prime }}$. Write $T^{\mathrm{\prime }}$ for the smooth symmetric covariant $2$-tensor field included as the development's stress-energy field. Require the tensor equation $G^{\mathrm{\prime }}+\mathrm{\Lambda }{g}^{\mathrm{\prime }}=\kappa T^{\mathrm{\prime }}$ at every point of $M^{\mathrm{\prime }}$, where $G^{\mathrm{\prime }}$ is the Einstein tensor of $g^{\mathrm{\prime }}$ with its metric-selected connection, defined as above. The constants $\kappa$ and $\mathrm{\Lambda }$ are the fixed parameters of the initial data. The matter-data matching condition is the pair of normal projections $\rho (p)=T_{\phi (p)}^{\mathrm{\prime }}(n_{\phi (p)}^{\mathrm{\prime }},n_{\phi (p)}^{\mathrm{\prime }}),j_p(V)=-T_{\phi (p)}^{\mathrm{\prime }}(n_{\phi (p)}^{\mathrm{\prime }},d{\phi }_p(V))(p\in \mathrm{\Sigma },V\in T_p\mathrm{\Sigma })\mathrm{.}$ Equivalently, the scalar field $T^{\mathrm{\prime }}(n^{\mathrm{\prime }},n^{\mathrm{\prime }})$ along $\phi (\mathrm{\Sigma })$ pulls back to $\rho$, and the covector field $-T^{\mathrm{\prime }}(n^{\mathrm{\prime }},\cdot )$ restricted to $T\phi (\mathrm{\Sigma })$ pulls back to $j$. This tuple prescribes only these stress-energy projections; any additional matter-field data are not included in $(\mathrm{\Sigma },h,K,\rho ,j)$. The development is a globally hyperbolic development when $\phi (\mathrm{\Sigma })$ is a Cauchy hypersurface of $(M^{\mathrm{\prime }},g^{\mathrm{\prime }},N^{\mathrm{\prime }})$. The implication from this Cauchy property to global hyperbolicity of $(M^{\mathrm{\prime }},g^{\mathrm{\prime }},N^{\mathrm{\prime }})$ is a separate theorem, deferred to the later causal theory; it is not established by this definition (Chronology, causality, strong causality, global hyperbolicity and time functions, Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons clause (c)).

(d) Maximality. A globally hyperbolic development $(M_{\ast },g_{\ast },N_{\ast };{\phi }_{\ast })$ is maximal for the data when for every globally hyperbolic development $(M^{\mathrm{\prime }},g^{\mathrm{\prime }},N^{\mathrm{\prime }};{\phi }^{\mathrm{\prime }})$ there is a smooth isometric embedding $\psi :M^{\mathrm{\prime }}\to M_{\ast }$ with $\psi \circ {\phi }^{\mathrm{\prime }}={\phi }_{\ast }$, $\psi$ preserving the time orientations and carrying the selected stress-energy tensor $T^{\mathrm{\prime }}$ to $T_{\ast }$, that is, ${\psi }^{\ast }{T}_{\ast }=T^{\mathrm{\prime }}$. This is a universal embedding property: every globally hyperbolic development embeds into the selected maximal one with its data and stress-energy tensor preserved. It does not assert that arbitrary globally hyperbolic developments are mutually isometric.

(e) Scope. The definition asserts neither existence nor uniqueness of developments, and it does not assert that $T^{\mathrm{\prime }}$ is generated by matter fields solving any equation: those claims require the local and global existence theory of the Einstein-matter system and are not made on this page. The constraint equations are displayed only as equations satisfied or not by given data. The equivalent formulation of the constraints in terms of the induced scalar curvature of $h$ and the divergence of $K$, and the existence theory for solutions of the constraints, rest on the Gauss-Codazzi relations and on PDE theory and are deferred to the later pages of this track; no such equivalence is asserted here.