Gr15 Exact Checked Sbierski Smooth Mghd Proof With Complete Local Prerequisites And Quotient Second Countability

Data. Fix $\kappa >0$, $\mathrm{\Lambda }\in \mathbb{R}$, a smooth three-manifold $\mathrm{\Sigma }$ without boundary, a smooth Riemannian metric $h$ and smooth symmetric covariant $2$-tensor $K$. The normal convention is $K(V,W)=g({\mathrm{\nabla }}_{V}n,W)$ for the future unit normal. Specify one fixed covariant matter model, consisting of a finite list of smooth tensor fields $\psi$, fixed equations $\mathcal{E}(g,\psi )=0$ and a fixed stress expression $\mathcal{T}(g,\psi )$, with pullback by local diffeomorphisms commuting with the equations and stress. Fix its complete smooth Cauchy data $d$: these determine all independent matter initial variables, not only the normal stress projections. Initial tensors with normal directions are specified on the abstract bundle $\mathcal{V}=\mathbb{R}\nu \oplus T\mathrm{\Sigma }$ with metric $g_0(a\nu +X,b\nu +Y)=-ab+h(X,Y)$. In a development, identify $a\nu +X$ with $an+dj(X)$.

An admissible fixed coupled model is one satisfying the complete initialization, hyperbolic reduction, recovery and comparison hypotheses of Local geometric development for a fixed covariant Einstein-matter model with complete constrained data, or a specified independent local existence and uniqueness reduction such as the dust branch below. The same fixed model, complete data and admissibility conditions are used in every comparison and at every restart. Admissibility includes the applicability of its supplied complete-data local existence and geometric uniqueness theorem to the data realized on each spacelike restart surface of the compared solutions. This must be checked for the actual model and surface; it is not inferred from stress conservation. This class includes the following four concrete branches; they do not exhaust it.

1. Vacuum: no matter fields, $\mathcal{T}=0$, and empty matter data.
2. Scalar field: fix one smooth potential $V$ and one normalization $K_{\varphi }>0$. The field is $\phi$, its equation is ${\mathrm{\square }}_g\phi -V^{\mathrm{\prime }}(\phi )=0$, its stress is the expression in The scalar-field matter model of the classical gravitational framework clause (c), and its complete data are $({\phi }_0,{\pi }_0)$ with ${\pi }_0=n(\phi ){\mathrm{\mid }}_{\mathrm{\Sigma }}$. The fixed potential has no sign restriction (Local Einstein-scalar evolution: smooth constrained data admit a local development, unique up to a data-fixing isometry).
3. Strict barotropic fluid: fix a smooth EOS $P:J\to \mathbb{R}$ on an open interval with $ϵ+P(ϵ)>0$ and $0<P^{\mathrm{\prime }}(ϵ)<1$. The fields are $ϵ\in J$ and future unit velocity $u$, the stress is $(ϵ+P(ϵ))u^{\mathrm{♭}}\otimes u^{\mathrm{♭}}+P(ϵ)g$, and the equations are the continuity and Euler equations of The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy clause (c), with this fixed EOS. Complete data are $({ϵ}_0,v_0)$, where ${ϵ}_0:\mathrm{\Sigma }\to J$ and $h(v_0,v_0)<1$, equivalently the bundle velocity $U_0=(1-h(v_0,v_0){)}^{-1\mathrm{/}2}(\nu +v_0)$. No small-tilt restriction is imposed (Local evolution for a strict barotropic perfect fluid: smooth constrained data admit a development, unique up to a data-fixing isometry).
4. Geodesic nonnegative dust: the fields are an energy-coefficient density $\epsilon \ge 0$ and a smooth future unit timelike field $u$, including on $\{\epsilon =0\}$. Set $\mathcal{T}=\epsilon u^{\mathrm{♭}}\otimes u^{\mathrm{♭}}$ and require both ${\mathrm{\nabla }}_{u}u=0$ and $u(\epsilon )+\epsilon {\mathrm{\nabla }}_a{u}^a=0$ everywhere. Complete data are $({\epsilon }_0,U_0)$, with ${\epsilon }_0\ge 0$ and prescribed smooth future unit bundle velocity $U_0$, equivalently arbitrary smooth subluminal tilt. On positive-density regions this is The dust matter model: rest density, four-velocity, stress-energy and dust system with its density parameter equal to $\epsilon$. Through empty regions the geodesic velocity is separately prescribed and matched; conservation of stress alone is not substituted for this full system. For a fixed $c>0$, the mass-normalized theorem Local Einstein-dust evolution: smooth constrained data with prescribed velocity admit a smooth local development, unique up to a data-fixing isometry applies with ${\rho }_0={\epsilon }_0\mathrm{/}{c}^2$ and reconstructs $\epsilon =c^2\rho$.

Require the constraints for the initial stress $T_0$ computed from this complete data on $\mathcal{V}$: $S_h+({\mathrm{tr}}_{h}K{)}^2-\mathrm{\mid }K{\mathrm{\mid }}_h^2=2\mathrm{\Lambda }+2\kappa T_0(\nu ,\nu ),{\mathrm{div}}_{h}K-d{\mathrm{tr}}_{h}K=\kappa T_0(\nu ,\cdot )\mathrm{.}$ These use the convention of Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form. Equality of data includes equality of the model, constants, potential or EOS, and every component of $d$.

The local Cauchy-development and uniqueness suppliers for vacuum, scalar and strict fluid are the fixed-model theorem Local geometric development for a fixed covariant Einstein-matter model with complete constrained data and the two concrete branch theorems above. Its harmonic route requires the same fixed extraction/reconstruction rules and smooth symmetric-hyperbolic coefficient family for every compared physical solution, a positive time matrix on an admissible open state set, complete compatible initialization from $(h,K,d)$, propagation of auxiliary compatibility with recovery of the full matter equations on the reduced solution, and stress conservation on that reduced solution before Einstein recovery. These are the three explicit hypotheses of that theorem; conservation alone does not supply a Cauchy evolution law. Dust uses its independent common-flow/parallel-frame initialization, hyperbolic reduction, subsidiary recovery and uniqueness of Local Einstein-dust evolution: smooth constrained data with prescribed velocity admit a smooth local development, unique up to a data-fixing isometry, rather than being asserted to satisfy the harmonic metric-first-derivative matter contract. Countable Choice and Dependent Choice are carried wherever these local theorems are invoked. Every additional fixed coupled model must supply the same complete-data local existence, admissibility, recovery and uniqueness obligations before it is used. This definition makes no local or global existence assertion.

(a) Development. A development of this data is a tuple $(M,g,N,j,\psi )$ consisting of:

1. a time-oriented smooth Lorentzian four-manifold $(M,g,N)$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model);
2. a smooth embedding $j:\mathrm{\Sigma }\to M$ whose image $j(\mathrm{\Sigma })$ is a spacelike embedded hypersurface (Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons clause (a));
3. the smooth tensor fields $\psi$ of the fixed model over $M$;
4. the equations on $M$: $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa {\mathcal{T}}_{ab}(g,\psi )$ with the Einstein tensor of $g$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors) and $\mathcal{E}(g,\psi )=0$;
5. the initial condition on $j(\mathrm{\Sigma })$: $j(\mathrm{\Sigma })$ is a Cauchy set of $(M,g,N)$ in the sense of Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons clause (c), that is, every inextendible causal curve of Causal curves, chronological and causal reachability, and inextendibility meets $j(\mathrm{\Sigma })$ in exactly one parameter value;
6. the induced data: $j^{\ast }(g)=h$, the second fundamental form of $j$ with respect to the future unit normal equals $K$ in the convention $K(V,W)=g({\mathrm{\nabla }}_{V}n,W)$ of The future unit normal, induced metric and second fundamental form of a spacelike hypersurface, and the initial data of $\psi$ on $j(\mathrm{\Sigma })$ transported by $j$ equal $d$.

No completeness of $(M,g)$, no inextendibility of $M$, and no claim that the equations hold beyond $M$ are part of the definition. Whether a development is globally hyperbolic is not part of the definition either; by clause 5 it follows from The existence of a Cauchy set implies strong causality and compact causal diamonds.

(b) Data-fixing isometry. A data-fixing isometry between developments $(M,g,N,j,\psi )$ and $(M^{\mathrm{\prime }},g^{\mathrm{\prime }},N^{\mathrm{\prime }},j^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }})$ of the same data is a smooth diffeomorphism $\phi :M\to M^{\mathrm{\prime }}$ such that

1. ${\phi }^{\ast }(g^{\mathrm{\prime }})=g$;
2. $g_{\phi (p)}^{\mathrm{\prime }}(d{\phi }_p{N}_p,N_{\phi (p)}^{\mathrm{\prime }})<0$ for every $p\in M$: the isometry carries the future cone of $N$ onto that of $N^{\mathrm{\prime }}$;
3. $\phi \circ j=j^{\mathrm{\prime }}$;
4. ${\phi }^{\ast }{\psi }^{\mathrm{\prime }}=\psi$, with the natural pullback action on every listed tensor type (and the inverse differential on contravariant slots). Equality includes every listed field, including the prescribed dust velocity at zero density.

(c) Development embedding. A development embedding of $(M,g,N,j,\psi )$ into $(M^{\mathrm{\prime }},g^{\mathrm{\prime }},N^{\mathrm{\prime }},j^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }})$ is a smooth map $\phi :M\to M^{\mathrm{\prime }}$ that is a diffeomorphism onto an open subset of $M^{\mathrm{\prime }}$ and satisfies clauses (b)1-4. Development embeddings are isometric immersions by clause (b)1.

(d) Maximality. A development $(M,g,N,j,\psi )$ of the given data is maximal when for every development $(M^{\mathrm{\prime }},g^{\mathrm{\prime }},N^{\mathrm{\prime }},j^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }})$ of the same data there exists a development embedding $\phi :M^{\mathrm{\prime }}\to M$. A maximal development is also called a maximal globally hyperbolic development of the data.

(e) Scope. This is a mathematical definition of the solution class used by the Cauchy problem. The physical reading of $(M,g,N)$ and $\psi$ and the status of $(h,K,d)$ as initial data of a model of nature are supplied by the framework postulates of the physics library, not here; and no existence, uniqueness or maximality assertion is made by this definition.

Two developments of the same data are compared only through maps that fix the initial manifold pointwise (clause (b)3), so a data-fixing isometry is determined by its action away from $j(\mathrm{\Sigma })$ only after its action on $j(\mathrm{\Sigma })$ is prescribed. The direction of a development embedding is from the smaller development to the larger one: clause (d) embeds every development into a maximal one.

The field $N$ represents the future cone and is not an additional dynamical field. Replacing it by any smooth timelike field in the same future cone does not change which maps are data-fixing. An isometry preserving that cone carries the future unit normal of the initial surface to the future unit normal of its image; no equality between the arbitrary representing fields is needed.

Let $(M,g,N,j,\psi )$ and $(M^{\mathrm{\prime }},g^{\mathrm{\prime }},N^{\mathrm{\prime }},j^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }})$ be two developments of the same data in the sense of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality.

(a) Common subdevelopment. A common subdevelopment of $M$ and $M^{\mathrm{\prime }}$ is a pair $(U,\phi )$ in which

1. $U\subseteq M$ is an open subset with $j(\mathrm{\Sigma })\subseteq U$ such that the restriction $(U,g{\mathrm{\mid }}_U,N{\mathrm{\mid }}_U,j,\psi {\mathrm{\mid }}_U)$ is a development of the same data (with the induced field equations on the open submanifold $U$); and
2. $\phi :(U,g{\mathrm{\mid }}_U,N{\mathrm{\mid }}_U,j,\psi {\mathrm{\mid }}_U)\to (M^{\mathrm{\prime }},g^{\mathrm{\prime }},N^{\mathrm{\prime }},j^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }})$ is a development embedding in the sense of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (c).

Here "the induced field equations on $U$" means the equations of clause (a)4 of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality restricted to $U$; the induced Lorentzian manifold is $(U,g{\mathrm{\mid }}_U)$ with the time orientation $N{\mathrm{\mid }}_U$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

(b) Maximal common development. A common subdevelopment $(U^{\ast },{\phi }^{\ast })$ is a maximal common development (MCD) of $M$ and $M^{\mathrm{\prime }}$ when for every common subdevelopment $(U,\phi )$ one has $U\subseteq U^{\ast }$ and ${\phi }^{\ast }{\mathrm{\mid }}_U=\phi$. Existence is not asserted by this definition; it is proved in Existence and uniqueness of the maximal common development of two developments.

(c) Corresponding boundary points. Let $(U,\phi )$ be a common subdevelopment. Its corresponding boundary points are pairs $(p,p^{\mathrm{\prime }})\in {\mathrm{\partial }}^{M}U\times {\mathrm{\partial }}^{M^{\mathrm{\prime }}}\phi (U)$, where the boundaries are taken in $M$ and in $M^{\mathrm{\prime }}$ respectively, such that for every open neighbourhood $V$ of $p$ in $M$ and every open neighbourhood $V^{\mathrm{\prime }}$ of $p^{\mathrm{\prime }}$ in $M^{\mathrm{\prime }}$ there is $x\in V\cap U$ with $\phi (x)\in V^{\mathrm{\prime }}$.

(d) Side of the initial surface. Write ${\mathrm{\Sigma }}_M=j(\mathrm{\Sigma })$ and ${\mathrm{\Sigma }}_{M^{\mathrm{\prime }}}=j^{\mathrm{\prime }}(\mathrm{\Sigma })$. For a corresponding pair $(p,p^{\mathrm{\prime }})$, openness of $U\supseteq {\mathrm{\Sigma }}_M$ and of $\phi (U)\supseteq {\mathrm{\Sigma }}_{M^{\mathrm{\prime }}}$ implies $p\mathrm{\notin }{\mathrm{\Sigma }}_M$ and $p^{\mathrm{\prime }}\mathrm{\notin }{\mathrm{\Sigma }}_{M^{\mathrm{\prime }}}$. The Cauchy-set trichotomy of Cauchy sets are closed, split the manifold into past, set and future, and are locally Lipschitz graphs clause (a) puts each boundary point in exactly one chronological side. These sides are open because $I^{\pm }({\mathrm{\Sigma }}_M)={\bigcup }_{s\in {\mathrm{\Sigma }}_M}{I}^{\pm }(s)$, and each point future and past is open by Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets clause (a); the same argument applies in $M^{\mathrm{\prime }}$. For $x\in U$, its ambient and intrinsic sides agree: the trichotomy in $U$ gives an intrinsic timelike curve between $x$ and ${\mathrm{\Sigma }}_M$ in one direction, and that curve is also an ambient curve; the disjoint ambient trichotomy excludes the other direction. The development embedding $\phi$ preserves these curves and the initial surface, so corresponding interior points have matching ambient sides in $M$ and $M^{\mathrm{\prime }}$. Choose neighbourhoods of $p,p^{\mathrm{\prime }}$ contained in their respective sides. Condition (c) supplies an interior witness $x$ in both neighbourhoods under $\phi$; hence the two sides must match. Every witness for neighbourhoods contained in these sides then has this same sign. Thus the pair lies either to the future of both initial surfaces or to their past, with no cofinal witnesses of the opposite sign. The future case is used below; the past case is its time reverse.

(e) Scope. These are mathematical relations between two fixed developments and their boundary points. Neither the existence of boundary points, nor the existence of a common subdevelopment with nonempty boundary, nor uniqueness of a corresponding point, is asserted here.

Clause (a) realises the common development as a subset of $M$, breaking the symmetry between $M$ and $M^{\mathrm{\prime }}$; this is the convention of Sbierski's second variant, chosen because the construction below patches maps on subsets of a fixed ambient development. Clause (c) is the definition of a corresponding boundary pair; the equivalent characterisation by timelike curves is proved, not assumed, in Structure of corresponding boundary points: achronality of the future boundary, equality of pasts, and uniqueness.

Assume 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) for the supplied causal-calculus and Cauchy-side results used below.

Let $(M,g,N,j,\psi )$ and $(U,g{\mathrm{\mid }}_U,N{\mathrm{\mid }}_U,j,\psi {\mathrm{\mid }}_U)$ be as in Common subdevelopments, maximal common developments and corresponding boundary points clause (a): $M$ is a development of the given data, $U\subseteq M$ is open with $j(\mathrm{\Sigma })\subseteq U$, and $U$ with the restricted fields is again a development of the same data. Write $\mathrm{\Sigma }:=j(\mathrm{\Sigma })$, and compute $I^{\pm }$, $J^{\pm }$ and $\le$ in $M$ (Causal curves, chronological and causal reachability, and inextendibility). Then:

(a) Past convexity within the future of $\mathrm{\Sigma }$. For every future causal curve $\nu :[a,b]\to M$ with $\nu (a)\in \mathrm{\Sigma }$ and $\nu (b)\in U\cap I^{+}(\mathrm{\Sigma })$ one has $\nu ([a,b])\subseteq U$.

(b) Future convexity within the past of $\mathrm{\Sigma }$. For every future causal curve $\nu :[a,b]\to M$ with $\nu (a)\in U\cap I^{-}(\mathrm{\Sigma })$ and $\nu (b)\in \mathrm{\Sigma }$ one has $\nu ([a,b])\subseteq U$.

(c) Segments with endpoints in $U$. For every future causal curve $\nu :[a,b]\to M$ with $\nu (a),\nu (b)\in U$ one has $\nu ([a,b])\subseteq U$.

Clause (a) is the exact convexity used in Rigidity of isometric immersions of a connected Lorentzian manifold, and agreement of data-fixing embeddings on overlaps and Existence and uniqueness of the maximal common development of two developments: it says that a subdevelopment cannot be entered and left along a causal segment from the common initial surface. The proof uses maximal geodesics of the open submanifold $U$ only as a device that makes the portion of $\nu$ in $U$ inextendible there; no geodesic completeness and no maximality of $U$ among developments is assumed.

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model), let $V\subseteq U\subseteq M$ be open subsets with the induced metric and time orientation, and let $\gamma :I\to U$ be a future causal curve (Causal curves, chronological and causal reachability, and inextendibility) that is inextendible in the Lorentzian manifold $U$: $\gamma$ does not converge to any point of $U$ as its parameter tends to either end of $I$ (clause (d) of Causal curves, chronological and causal reachability, and inextendibility, read with ambient set $U$). Let $J$ be a connected component of ${\gamma }^{-1}[V]:=\{t\in I:\gamma (t)\in V\}\mathrm{.}$ Then $\gamma {\mathrm{\mid }}_J$, regarded as a future causal curve in the Lorentzian manifold $V$ with the restricted fields, is inextendible in $V$.

The ambient inextendibility assumption is used when a component reaches an end of the original domain. At a finite interior endpoint of a component, continuity supplies an ambient value outside the open subset, so Hausdorff uniqueness rules out a limit inside that subset. These are distinct cases; no continuation through a point outside the subset is claimed to stay within it.

Throughout, for a smooth map $\psi :N\to N^{\mathrm{\prime }}$ between smooth Lorentzian four-manifolds write ${\psi }^{\ast }(g^{\mathrm{\prime }})=g$ when $g_p(v,w)=g_{\psi (p)}^{\mathrm{\prime }}(d{\psi }_{p}v,d{\psi }_{p}w)$ for all $p\in N$ and $v,w\in T_{p}N$; such a map is an isometric immersion (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, $C^r$ and smooth maps between smooth manifolds). Let $\mathrm{\nabla },{\mathrm{\nabla }}^{\mathrm{\prime }}$ be the metric-selected connections (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) and let geodesics and exponential maps be those of The geodesic initial-value problem and the exponential map of the metric-selected connection.

(a) Rigidity. Let $N$ be connected and let ${\psi }_1,{\psi }_2:N\to N^{\mathrm{\prime }}$ be isometric immersions with ${\psi }_1(p)={\psi }_2(p)$ and $d{\psi }_1(p)=d{\psi }_2(p)$ for some $p\in N$. Then ${\psi }_1={\psi }_2$, that is, ${\psi }_1(q)={\psi }_2(q)$ and $d{\psi }_1(q)=d{\psi }_2(q)$ for every $q\in N$.

(b) Data-fixing embeddings have equal differentials on the initial surface. Let $(M,g,N,j,\psi )$, $(M^{\mathrm{\prime }},g^{\mathrm{\prime }},N^{\mathrm{\prime }},j^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }})$ and $(U,g{\mathrm{\mid }}_U,N{\mathrm{\mid }}_U,j,\psi {\mathrm{\mid }}_U)$ be as in Common subdevelopments, maximal common developments and corresponding boundary points clause (a), and let ${\psi }_1,{\psi }_2:U\to M^{\mathrm{\prime }}$ be development embeddings in the sense of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (c). Then $d{\psi }_1=d{\psi }_2$ at every point of $\mathrm{\Sigma }:=j(\mathrm{\Sigma })\subseteq U$.

(c) Agreement on overlaps. Assume 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) for the supplied causal-convexity and Cauchy-side arguments in this clause. Let $(M,g,N,j,\psi )$ and $(M^{\mathrm{\prime }},g^{\mathrm{\prime }},N^{\mathrm{\prime }},j^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }})$ be developments of the same data, and let $(U_1,{\psi }_1)$ and $(U_2,{\psi }_2)$ be common subdevelopments in the sense of Common subdevelopments, maximal common developments and corresponding boundary points clause (a). Then ${\psi }_1={\psi }_2$ on $U_1\cap U_2$.