Causal And Initial Value Boundaries

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model), with the pointwise causal classification of Timelike, null, spacelike and causal vectors of a Lorentz vector space and the equivalence of "future-directed" with "pairing negatively with $N$" supplied by Classification of future and past causal vectors by a timelike field. All curves below take values in $M$; the physical reading of the model is not part of this definition.

(a) The curve class. Let $I\subseteq \mathbb{R}$ be an interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and let $\gamma :I\to M$ be a continuous map, where $I$ has its relative topology. A chart representative on a parameter neighborhood $J\subseteq I$ is $x\circ \gamma {\mathrm{\mid }}_J$, where $(U,x)$ is a smooth chart and $\gamma (J)\subseteq U$ (Smooth manifolds and their smooth charts, $C^r$ and smooth maps between smooth manifolds). The map $\gamma$ is locally absolutely continuous when for each $t\in I$ there are $\delta >0$ and a chart $(U,x)$ such that $\gamma (I\cap (t-\delta ,t+\delta ))\subseteq U$ and every coordinate of $x\circ \gamma$ is absolutely continuous on every compact subinterval of $I\cap (t-\delta ,t+\delta )$ in the sense of Absolute continuity on a compact interval. This includes one-sided neighborhoods at included endpoints. Equivalently, for every compact subinterval $[a,b]\subseteq I$ and every chart $(U,x)$ with $\gamma ([a,b])\subseteq U$, the coordinates of $x\circ \gamma$ are absolutely continuous on $[a,b]$. The equivalence and independence of the chart and smooth atlas are proved in Chart independence, concatenation, reparametrization and transitivity for causal curves ↗ clause (a); continuity is part of both formulations.

(b) Velocity and causal curves. Let $\gamma$ be locally absolutely continuous and let $t$ be an interior point of $I$ at which some — equivalently every — chart representative is differentiable. The velocity ${\gamma }^{\mathrm{\prime }}(t)\in T_{\gamma (t)}M$ is the derivation whose components in a chart $(U,x)$ around $\gamma (t)$ are ${\gamma }^{\mathrm{\prime }}(t)={\sum }_{a=0}^3\frac{d}{dt}(x^a\circ \gamma )(t){\mathrm{\partial }}_a{\mathrm{\mid }}_{\gamma (t)}$ (The velocity derivation of a smooth curve, Derivations at a point and the tangent space); it is independent of the chart by the chain rule, as recorded in Chart independence, concatenation, reparametrization and transitivity for causal curves ↗ clause (b). A future causal curve is a locally absolutely continuous $\gamma$ such that the set of $t\in I$ for which ${\gamma }^{\mathrm{\prime }}(t)$ exists, is nonzero and is future causal is of full measure in $I$ (its complement meets every compact subinterval of $I$ in a set of Lebesgue measure zero). A future timelike curve (also called future chronological) is defined in the same way with "future causal" replaced by "future timelike". Past-directed versions are defined with $g_p(v,N_p)>0$. Since a future causal nonzero vector is nonzero by definition, the velocity is required to be nonzero at almost every parameter value; in particular no nondegenerate subinterval of $I$ can be mapped to a single point: a constant chart representative has derivative zero everywhere in its interior, contradicting the full-measure nonzero-velocity requirement.

(c) Reachability. For events $p,q\in M$ define:

• $p\le q$, read "$q$ is in the future of $p$", when $p=q$, or when there are $a<b$ and a future causal curve $\gamma :[a,b]\to M$ with $\gamma (a)=p$ and $\gamma (b)=q$;
• $p\ll q$, read "$p$ chronologically precedes $q$", when there are $a<b$ and a future timelike curve $\gamma :[a,b]\to M$ with $\gamma (a)=p$ and $\gamma (b)=q$.

The causal future and chronological future of a subset $S\subseteq M$ are $J^{+}(S):={\bigcup }_{p\in S}{J}^{+}(p),I^{+}(S):={\bigcup }_{p\in S}{I}^{+}(p),$ where $J^{+}(p):=\{q\in M:p\le q\}$ and $I^{+}(p):=\{q\in M:p\ll q\}$; past versions are $J^{-}(S):=\{p:\mathrm{\exists }q\in S,q\in J^{+}(p)\}$ and $I^{-}(S):=\{p:\mathrm{\exists }q\in S,p\ll q\}$. By the displayed definition $p\in J^{+}(p)$ for every $p$. Whether $p\in I^{+}(p)$ is a property of the model, not a convention: $p\ll p$ holds exactly when there are $a<b$ and a future timelike curve $\gamma :[a,b]\to M$ with $\gamma (a)=\gamma (b)=p$, that is, exactly when a closed future timelike curve through $p$ exists. No irreflexivity of $\ll$ is asserted, and closed timelike curves are not excluded by the definition.

(d) Inextendibility. A locally absolutely continuous causal curve $\gamma :I\to M$ is future-inextendible when $\gamma$ does not converge to any point of $M$ as its parameter tends to $\sup I$, that is, there is no $x\in M$ with $\gamma (t)\to x$ as $t\to \sup I$; in particular a curve whose parameter range is unbounded above is future-inextendible exactly when it does not converge as $t\to +\mathrm{\infty }$. Past-inextendible is defined dually with $\inf I$, and $\gamma$ is inextendible when it is both. This is the convention of Minguzzi's Definition 2.16; for causal curves parametrized by arclength of a complete auxiliary Riemannian metric it agrees with "the domain cannot be extended", and a causal curve that converges at an endpoint can be extended by a short geodesic segment, as proved in Convergent causal ends are extendible, maximal geodesics have no limit, and Cauchy sets are acausal ↗ clauses (a) and (b). A causal curve passes through $p$ when $p\in \gamma (I)$.

(e) Scope. The definitions are mathematical. They record no postulate about nature, no measurement and no dynamical law; the physical reading of $(M,g,N)$ and of the causal structure as the model's signalling structure is adopted in the framework postulates and is not asserted here.

Assume the Axioms of Countable Choice and Dependent Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with the causal-curve class of Causal curves, chronological and causal reachability, and inextendibility.

(a) Chart independence. Local absolute continuity in the sense of Causal curves, chronological and causal reachability, and inextendibility clause (a) is independent of the chosen chart and of the smooth atlas. Every compact parameter interval admits a finite subdivision on each of whose pieces a chart representative is absolutely continuous. For a continuous curve, the relative neighborhood formulation and the compact chart-interval formulation of that definition are equivalent.

(b) Velocity and smooth composition. At every interior parameter value $t$ at which one chart representative is differentiable, every chart representative is differentiable, the velocity ${\gamma }^{\mathrm{\prime }}(t)$ of Causal curves, chronological and causal reachability, and inextendibility clause (b) is well defined and independent of the chart, and the set of parameter values at which some chart representative is differentiable has full measure in the domain. For every smooth map $F$ of an open subset of $M$ into a finite-dimensional smooth manifold $P$ (in particular $P=\mathbb{R}$) whose domain contains the image of $\gamma$, the composite $F\circ \gamma$ is locally absolutely continuous in charts of $P$ using the same relative neighborhood definition, and $(F\circ \gamma {)}^{\mathrm{\prime }}(t)=d{F}_{\gamma (t)}({\gamma }^{\mathrm{\prime }}(t))$ at almost every parameter value $t$.

(c) Piecewise $C^1$ curves. Every piecewise $C^1$ curve (with the finite admissible partition and one-sided endpoint convention of clause (a) of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold) is locally absolutely continuous; a piecewise $C^1$ curve all of whose tangents are future causal is a future causal curve, and if in addition its tangents are future timelike, it is a future timelike curve.

(d) Concatenation. If ${\gamma }_1:[a,b]\to M$ and ${\gamma }_2:[b,c]\to M$ are future causal curves (respectively future timelike curves) with ${\gamma }_1(b)={\gamma }_2(b)$, then the concatenation $\gamma :[a,c]\to M$, $\gamma (t):={\gamma }_1(t)$ for $t\le b$ and $\gamma (t):={\gamma }_2(t)$ for $t\ge b$, is a future causal (respectively future timelike) curve; the same holds for any finite chain of such curves.

(e) Reparametrization. If $\phi :J\to I$ is a strictly increasing $C^1$ diffeomorphism of intervals with ${\phi }^{\mathrm{\prime }}>0$ and $\gamma :I\to M$ is locally absolutely continuous, then $\gamma \circ \phi$ is locally absolutely continuous and $\gamma \circ \phi$ is future causal (respectively future timelike) exactly when $\gamma$ is. Moreover $\gamma$ is future-inextendible if and only if $\gamma \circ \phi$ is future-inextendible, and this equivalence holds for every strictly increasing continuous surjection $\phi$ of intervals, not only the $C^1$ ones.

(f) Relations. The relation $\le$ of Causal curves, chronological and causal reachability, and inextendibility is reflexive and transitive, and $p\ll q$ implies $p\le q$ for all $p,q$; the relations $J^{\pm }$ and $I^{\pm }$ of that definition are therefore nested, $I^{\pm }(S)\subseteq J^{\pm }(S)$ for every $S\subseteq M$.

• Measurability conventions. The absolute-continuity and almost-everywhere statements are one-dimensional Lebesgue statements on compact parameter intervals, used through [L1]; the only choice principles used are ${\mathrm{A}\mathrm{C}}_{\omega }$ and $\mathrm{D}\mathrm{C}$, assumed in the statement.

• The $C^1$ reparametrization hypothesis. In (e) the $C^1$ hypothesis with ${\phi }^{\mathrm{\prime }}>0$ for the forward implication cannot be relaxed to mere continuity: composing an absolutely continuous curve with a continuous increasing function that is not absolutely continuous need not be absolutely continuous. The inextendibility equivalence is stated separately because it needs only strict monotonicity and surjectivity.

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with the causal curves and the relations $\le$, $\ll$, $I^{\pm }$, $J^{\pm }$ of Causal curves, chronological and causal reachability, and inextendibility. All curves below are curves in that sense.

(a) Chronology and causality. $(M,g,N)$ is chronological when there is no $p\in M$ with $p\ll p$, equivalently when there is no nonconstant future timelike curve $\gamma :[a,b]\to M$ with $\gamma (a)=\gamma (b)$. It is causal when there is no nonconstant future causal curve $\gamma :[a,b]\to M$ with $\gamma (a)=\gamma (b)$; a nonconstant future causal curve with equal endpoints is called a closed causal curve, and a closed future timelike curve is called a closed timelike curve.

(b) Strong causality. $(M,g,N)$ is strongly causal when for every $p\in M$ and every open set $U\subseteq M$ with $p\in U$ there is an open set $V$ with $p\in V\subseteq U$ such that for every future causal curve $\gamma :[a,b]\to M$ and all $a\le s<t\le b$ with $\gamma (s)\in V$ and $\gamma (t)\in V$ one has $\gamma ([s,t])\subseteq V$. Thus no causal curve can leave $V$ and later re-enter it; a set $V$ with this property for all causal curves is called causally convex for causal curves that meet it twice.

(c) Global hyperbolicity. $(M,g,N)$ is globally hyperbolic when it is strongly causal and for all $p,q\in M$ the causal diamond $J^{+}(p)\cap J^{-}(q)$ is compact in the subspace topology (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

(d) Time functions. A time function on $(M,g,N)$ is a function $u:M\to \mathbb{R}$ which is continuous from the manifold topology to the usual topology of $\mathbb{R}$: for every $p\in M$ and every open neighborhood $W$ of $u(p)$ there is an open neighborhood $V$ of $p$ with $u(V)\subseteq W$ (Continuity of a map of topological spaces at a point and globally). It also satisfies $u(\gamma (a))<u(\gamma (b))$ for every future causal curve $\gamma :[a,b]\to M$ with $a<b$ that is not constant on $[a,b]$.

(e) Temporal functions. A temporal function on $(M,g,N)$ is a smooth function $u:M\to \mathbb{R}$ ($C^r$ and smooth maps between smooth manifolds) whose gradient is past-directed timelike: writing $du$ for its differential (The differential of a smooth map) and $\mathrm{\nabla }u$ for the vector field with $g(\mathrm{\nabla }u,X)=du(X)$ for every smooth vector field $X$, constructed from the inverse metric of The inverse metric, musical isomorphisms and index raising, one has $g(\mathrm{\nabla }u,\mathrm{\nabla }u)<0\text{and}du(N)=g(\mathrm{\nabla }u,N)>0.$ The first inequality says that the gradient is nowhere zero and timelike; the second fixes it as past-directed for the chosen time orientation. No level set of a temporal function is asserted here to be a hypersurface or to be spacelike, and no relation between temporal functions and compactness is asserted; those claims are proved separately where they are used.

(f) Scope and conventions. These are mathematical conditions on the Lorentzian model. Different sources define global hyperbolicity by equivalent but differently phrased conditions; this page fixes the formulation of (c) and proves the implications it uses. In particular no equivalence between global hyperbolicity and the existence of a Cauchy hypersurface is asserted by this definition, and the physical reading of strong causality or global hyperbolicity as predictability of a classical model is adopted only in the framework postulates.

Assume the Axioms of Countable Choice and Dependent Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold, let $u$ be a temporal function on it in the sense of Chronology, causality, strong causality, global hyperbolicity and time functions clause (e), and write $\mathrm{\nabla }u$ for its gradient. Then:

(a) $du(v)=g(\mathrm{\nabla }u,v)>0$ for every nonzero future causal vector $v$.

(b) For every future causal curve $\gamma :I\to M$ in the class of Causal curves, chronological and causal reachability, and inextendibility, the composite $u\circ \gamma$ is locally absolutely continuous, its derivative satisfies $(u\circ \gamma {)}^{\mathrm{\prime }}(t)>0$ at almost every $t\in I$, and consequently $u(\gamma (a))<u(\gamma (b))\text{whenever }a<b\text{ and }\gamma \text{ is not constant on }[a,b]\mathrm{.}$

(c) $u$ is a time function on $(M,g,N)$ in the sense of Chronology, causality, strong causality, global hyperbolicity and time functions clause (d); every level set $u^{-1}(c)$, $c\in \mathbb{R}$, is achronal; and $(M,g,N)$ is causal and chronological: it admits no nonconstant future causal curve with equal endpoints.

• Hypotheses. The proof uses only the temporal condition, the choice assumptions carried by the absolutely continuous fundamental theorem of calculus, and the causal-curve calculus; no completeness, compactness or dynamical hypothesis enters.
• Sharpness. The lemma gives no converse: a causal spacetime need not admit a temporal function, and a time function need not be smooth or have timelike gradient; the construction of smooth temporal functions on globally hyperbolic spacetimes is a separate theorem.

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with the causal curves of Causal curves, chronological and causal reachability, and inextendibility, and let $S\subseteq M$.

(a) Spacelike hypersurfaces. A subset $\mathrm{\Sigma }\subseteq M$ is a spacelike embedded hypersurface when it is an embedded three-dimensional submanifold of $M$ (Embedded submanifolds and slice charts) such that the restriction of $g_p$ to $T_p\mathrm{\Sigma }\times T_p\mathrm{\Sigma }$ is positive definite for every $p\in \mathrm{\Sigma }$; equivalently, every nonzero vector tangent to $\mathrm{\Sigma }$ is spacelike (Timelike, null, spacelike and causal vectors of a Lorentz vector space). A spacelike embedded hypersurface is automatically a nondegenerate hypersurface, so its $g$-orthogonal complement in $T_{p}M$ is one-dimensional and nondegenerate, and it carries a unique future-directed unit timelike normal; that normal is constructed where it is used, not here.

(b) Achronality and acausality. $S$ is achronal when no two distinct points $p,q\in S$ satisfy $p\ll q$ or $q\ll p$; $S$ is acausal when no two distinct points $p,q\in S$ satisfy $p\le q$ or $q\le p$. Every acausal set is achronal, since $p\ll q$ implies $p\le q$.

(c) Cauchy sets. $S$ is a Cauchy set when for every inextendible causal curve $\gamma :I\to M$ the parameter set ${\gamma }^{-1}[S]:=\{t\in I:\gamma (t)\in S\}$ has exactly one element. Because a strictly increasing continuous surjection of intervals induces a bijection of parameter sets, this condition is invariant under reparametrization (Chart independence, concatenation, reparametrization and transitivity for causal curves clause (e)); it is stated in terms of parameters rather than of the point set $\gamma (I)\cap S$, so that a closed causal curve counts two visits to the same point as two parameter values. A Cauchy hypersurface is a Cauchy set that is also a spacelike embedded hypersurface in the sense of clause (a).

(d) Domains of dependence. For any $S\subseteq M$ define $D^{+}(S):=\{p\in M:\text{every past-inextendible causal curve through }p\text{ meets }S\},$ and dually $D^{-}(S):=\{p\in M:\text{every future-inextendible causal curve through }p\text{ meets }S\}$, where "meets $S$" means that some parameter value is mapped into $S$, and $D(S):=D^{+}(S)\cup D^{-}(S)$ is the domain of dependence of $S$.

(e) Cauchy horizon. For a closed set $S\subseteq M$ the future Cauchy horizon is $H^{+}(S):=D^{+}(S)\setminus I^{-}(D^{+}(S)),$ the set of points of $D^{+}(S)$ that are not chronologically preceded by any point of $D^{+}(S)$; dually $H^{-}(S):=D^{-}(S)\setminus I^{+}(D^{-}(S))$, and the Cauchy horizon is $H(S):=H^{+}(S)\cup H^{-}(S)$.

(f) Scope. These are mathematical definitions. Neither existence of Cauchy sets nor any relation between the objects defined here is asserted.

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.

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) and let $\mathrm{\Sigma }\subseteq M$ be a spacelike embedded hypersurface in the sense of Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons clause (a), so that $\mathrm{\Sigma }$ is an embedded three-dimensional submanifold (Embedded submanifolds and slice charts) and the restriction of $g_p$ to $T_p\mathrm{\Sigma }$ is positive definite for every $p\in \mathrm{\Sigma }$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space). In the notation of Spacelike initial data, the constraint equations, developments and maximal developments clause (a), whose claims this lemma establishes:

(a) The future unit normal. For every $p\in \mathrm{\Sigma }$ the orthogonal complement $T_p{\mathrm{\Sigma }}^{\perp }$ is one-dimensional, nondegenerate for $g_p$, and negative definite: every nonzero $u\in T_p{\mathrm{\Sigma }}^{\perp }$ is timelike, $g_p(u,u)<0$. Hence there is exactly one $n_p\in T_p{\mathrm{\Sigma }}^{\perp }$ with $g_p(n_p,n_p)=-1$ and $g_p(n_p,N_p)<0$.

(b) Smoothness along the hypersurface. The assignment $p\mapsto n_p$ is smooth along $\mathrm{\Sigma }$: for every $p\in \mathrm{\Sigma }$ there are an open neighborhood $O$ of $p$ in $M$ and a smooth vector field $\tilde{n}$ on $O$ (A smooth vector field is a smooth section of the tangent bundle) with $\tilde{n}{\mathrm{\mid }}_{\mathrm{\Sigma }\cap O}=n{\mathrm{\mid }}_{\mathrm{\Sigma }\cap O}$, and for any two such local extensions the covariant derivative at a point of $\mathrm{\Sigma }$ in a direction tangent to $\mathrm{\Sigma }$ agrees. Thus for $V\in T_p\mathrm{\Sigma }$ the vector ${\mathrm{\nabla }}_{V}n\in T_{p}M$ is well defined as ${\mathrm{\nabla }}_{\tilde{X}}\tilde{n}{\mathrm{\mid }}_p$ for any smooth extensions $\tilde{n}$ of $n$ and $\tilde{X}$ of a tangent field with ${\tilde{X}}_p=V$ (Covariant derivative along a curve, Metric compatibility, torsion freedom and the metric-selected connection class).

(c) The induced metric and second fundamental form. The induced metric $h_p(V,W):=g_p(V,W)$ is a smooth Riemannian metric on $\mathrm{\Sigma }$, and $K_p(V,W):=g_p({\mathrm{\nabla }}_{V}n,W)(V,W\in T_p\mathrm{\Sigma })$ (the second fundamental form) defines a smooth symmetric covariant $2$-tensor field $K$ on $\mathrm{\Sigma }$ (Smooth covariant 2-tensor fields on a smooth manifold) with $K_p(V,W)=-g_p({\mathrm{\nabla }}_{V}W,n_p),$ where ${\mathrm{\nabla }}_{V}W$ is the ambient covariant derivative at $p$ of any smooth extension of a tangent field with value $W$ at $p$, in the direction of any extension with value $V$ at $p$.

(d) The shape operator. The endomorphism $S_p:T_p\mathrm{\Sigma }\to T_p\mathrm{\Sigma }$ defined by $h_p(S_{p}V,W)=K_p(V,W)$ for all $W\in T_p\mathrm{\Sigma }$ is well defined, linear and self-adjoint for the positive definite form $h_p$, and the Weingarten relation ${\mathrm{\nabla }}_{V}n=S_{p}V$ holds for every $V\in T_p\mathrm{\Sigma }$.