Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality

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.