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

(Minkowski is a development of the data.) On ${\mathbb{R}}^{1,3}$ with $\eta =-d{t}^2+\delta$ the curvature vanishes, so the Einstein tensor vanishes and $G+\mathrm{\Lambda }\eta =0=\kappa \cdot 0$: the vacuum field equation holds by [L2]. The slice $\{t=0\}$ is a spacelike embedded hypersurface with induced metric $\delta$ and vanishing second fundamental form, since the unit normal ${\mathrm{\partial }}_t$ is parallel; and every inextendible causal curve in ${\mathbb{R}}^{1,3}$ meets $\{t=0\}$ exactly once, by The flat model is globally hyperbolic and its inertial level sets are Cauchy hypersurfaces: its time coordinate is strictly increasing and has range all of $\mathbb{R}$. Hence $({\mathbb{R}}^{1,3},\eta ,{\mathrm{\partial }}_t,j)$ is a development of the data.

(Completeness makes the embedding onto.) The connected initial manifold ${\mathbb{R}}^3$, the vacuum constraints, the existing development of step 1.1 and the restart obligations checked in [L1] satisfy the maximal theorem. Let $Q$ be its connected maximal development, and let $\mathrm{\Phi }:M\to Q$ be its data-fixing embedding of the Minkowski development from step 1.1. Fix $q\in Q$ and a maximal timelike geodesic through it. By Convergent causal ends are extendible, maximal geodesics have no limit, and Cauchy sets are acausal clause (b) this curve is inextendible and meets the initial Cauchy surface at some parameter $a$. At that crossing, $\mathrm{\Phi }$ has invertible differential and fixes the initial embedding. Lift the crossing velocity to $M$. The Minkowski geodesic with that initial datum is the straight line, defined for every affine parameter by A chart with constant metric components has vanishing metric-selected Christoffel symbols and curvature and The geodesic initial-value problem and the exponential map of the metric-selected connection. Its $\mathrm{\Phi }$-image is a geodesic of $Q$ with the same crossing value and velocity; uniqueness makes it agree with the original geodesic on its entire interval, in particular at $q$. Thus every $q$ lies in $\mathrm{\Phi }(M)$, and the embedding is onto, hence a data-fixing isometry. Maximality and unique data-fixing isometry of $Q$ transfer to Minkowski. This proves the full claimed maximality, rather than inferring it solely from existence of an embedding.

($\psi$ is a data-fixing isometry.) The translation $\psi$ is a diffeomorphism of ${\mathbb{R}}^{1,3}$ preserving $\eta$, the time orientation and (vacuously) the matter fields, and $\psi \circ j=j^{\mathrm{\prime }}$ by definition of $j,j^{\mathrm{\prime }}$. Hence $\psi :M\to M^{\mathrm{\prime }}$ is a development embedding of the whole of $M$, and $(M,\psi )$ is a common subdevelopment (Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clauses (b)-(c), Common subdevelopments, maximal common developments and corresponding boundary points clause (a)).

(Maximality.) By Existence and uniqueness of the maximal common development of two developments the maximal common development $(U^{\ast },{\psi }^{\ast })$ contains every common subdevelopment, so $U^{\ast }=M$ and ${\psi }^{\ast }=\psi$ by step 1.1 applied to $(U^{\ast },{\psi }^{\ast })$ in the reverse direction. In particular $U$ has no boundary in $M$, hence no corresponding boundary points, in accordance with A maximal common development has no corresponding boundary points: the spacelike restart produces a strictly larger common development. This is (a).

(The quotient.) By Gluing two developments along their maximal common development: the quotient is a development containing both the quotient $Q:=(M\bigsqcup M^{\mathrm{\prime }})\mathrm{/}\sim$ with the identifications along $\psi$ is a development of the data and the maps $p_M,p_{M^{\mathrm{\prime }}}$ are data-fixing development embeddings; here every point of $M$ is identified with its image in $M^{\mathrm{\prime }}$ and conversely, so the canonical map $M\to Q$, $p\mapsto [p]$, is a bijection and an isometry: its inverse is the map sending the class of $m\in M$ to $m$ and the class of $m^{\mathrm{\prime }}\in M^{\mathrm{\prime }}$ to ${\psi }^{-1}(m^{\mathrm{\prime }})$, which is well defined because $m^{\mathrm{\prime }}\sim {\psi }^{-1}(m^{\mathrm{\prime }})$ and because $\psi$ is a bijection. Both compositions with $\eta$ agree, so $Q$ is isometric to $M$ by a data-fixing isometry; the two development embeddings are the identity and the inverse translation. This is (b).

(The distance formula.) For $r\in J^{-}(q)$ the displacement $q-r=(\mathrm{\Delta }t,\mathrm{\Delta }x)$ is causal, and for $r\mathrm{\ne }q$ the unique maximizing geodesic image is the straight causal segment, whose length is $\sqrt{(\mathrm{\Delta }t{)}^2-\mathrm{\mid }\mathrm{\Delta }x{\mathrm{\mid }}^2}=\sqrt{(t-1{)}^2-\mathrm{\mid }x{\mathrm{\mid }}^2}$ when the displacement is timelike; for nonzero lightlike displacement the value is $0$; for $r=q$ the degenerate constant-geodesic convention also gives zero; this is the Minkowski instance of the attainability and geodesy of maximizers in Time separation in a globally hyperbolic manifold: finiteness, existence of maximizing geodesics and the radial formula on causally convex normal regions clause (a).

(Smoothness and spacality.) On the open set $U:=\{r:{\tau }_q(r)>0\}=I^{-}(q)$ the function $f(t,x):=(t-1{)}^2-\mathrm{\mid }x{\mathrm{\mid }}^2$ is smooth with Lorentzian gradient $(-2(t-1),-2x)$, which is timelike on $U$ (there $(t-1{)}^2>\mathrm{\mid }x{\mathrm{\mid }}^2$); hence its level set $\{f={\tau }_0^2\}$, which is $S$, is a smooth embedded hypersurface with timelike normal and is therefore spacelike (Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, $C^r$ and smooth maps between smooth manifolds).

(At most one meeting along a causal curve.) Since ${\tau }_q$ is continuous and $g(\mathrm{\nabla }{\tau }_q,\cdot )$ has no nonzero causal kernel on $U$, the function $t\mapsto {\tau }_q(\gamma (t))$ is strictly monotone along every causal curve segment contained in $U$; hence such a segment meets the level ${\tau }_0$ in at most one parameter value, and a causal curve segment with ${\tau }_q$-values straddling ${\tau }_0$ meets $S$ at least once by the intermediate value theorem (Time separation in a globally hyperbolic manifold: finiteness, existence of maximizing geodesics and the radial formula on causally convex normal regions clause (b), Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point). If a causal curve met $S$ in two parameter values, the subsegment between them would join two points of $S$ with equal ${\tau }_q$-value, contradicting strict monotonicity; if the subsegment left $U$ it would contain a point $p$ with ${\tau }_q(p)=0$, that is $p\ll ̸q$; but $p\le \gamma (t_2)\ll q$ and push-up gives $p\ll q$, a contradiction (The push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion, Causal curves, chronological and causal reachability, and inextendibility).

(Radial description.) For $r=(1-\sqrt{{\tau }_0^2+\mathrm{\mid }{x}_s{\mathrm{\mid }}^2},x_s)\in S$ the future segment to $q$ is $s\mapsto r+\frac{s}{{\tau }_0}(\sqrt{{\tau }_0^2+\mathrm{\mid }{x}_s{\mathrm{\mid }}^2},-x_s)$, which is a geodesic with timelike velocity of constant norm $1$, hence a future-directed timelike geodesic segment of length ${\tau }_0$ ending at $q$. Conversely every past endpoint of a future-directed timelike geodesic segment of length ${\tau }_0$ ending at $q$ lies on $S$.

(One exponential step produces the rational points with nonzero time coordinate.) By the flat geodesic calculation in [L2], the exponential map is ${\exp }_d(v)=d+v$ for every tangent vector. For $d=(0,q)\in \{0\}\times {\mathbb{Q}}^3=D_0$ and a rational vector $v=(a,w)\in {\mathbb{Q}}^4$ the point $d+v=(a,q+w)$ lies in $D_1$ whenever $v$ is timelike, i.e. $a^2>\mathrm{\mid }w{\mathrm{\mid }}^2$. Given $(a,b)\in {\mathbb{Q}}^4$ with $a\mathrm{\ne }0$, choose $q:=b$ and $w:=0$; then $v=(a,0)$ satisfies $\eta (v,v)=-a^2<0$, so it is timelike, and $(a,b)\in D_1$. Every exponential step from a rational point using a rational vector remains rational, so induction gives $D\subseteq {\mathbb{Q}}^4$. Hence $\{r\in {\mathbb{Q}}^4:r^0\mathrm{\ne }0\}\subseteq D_1$, and $D_0$ supplies the points with $r^0=0$, so ${\mathbb{Q}}^4\subseteq D$.

(Countability and density.) $D$ is countable by the construction (A countable dense subset of the universal development by rational timelike exponential steps from a dense subset of the initial surface, Finite, countably infinite, countable, uncountable) and contains ${\mathbb{Q}}^4$, which is dense in ${\mathbb{R}}^{1,3}$ by the density of $\mathbb{Q}$ in $\mathbb{R}$ coordinatewise (Both $\mathbb{Q}$ and $\mathbb{R}\setminus \mathbb{Q}$ are dense in $\mathbb{R}$, and every nonempty open subset of $\mathbb{R}$ is uncountable); hence $D$ is dense and meets $j(\mathrm{\Sigma })=\{t=0\}$ in $\{0\}\times {\mathbb{Q}}^3$.

(The basis.) The repaired abstract family uses translated rational tangent balls: here ${\exp }_d(B(a,r))=B(d+a,r)$. Since $d,a$ are rational, its centers are rational; conversely every rational center is obtained by taking $a=0$ and that center as $d$. Thus it is exactly the Euclidean rational-center family declared above. For $d\in D$ and rational $r>0$ the set ${\exp }_d(B(0;r))=d+B(0;r)$ is the open Euclidean ball of radius $r$ around $d$ centred at a rational point with rational radius; the family of all such balls is a countable basis of the Euclidean topology of ${\mathbb{R}}^4$ (Basis and subbasis for a topology, and the topology generated by a family of sets, Second countability: an at most countable basis for the topology). In particular $\mathcal{B}$ as defined in the statement is a countable basis of the topology of $M$, so $M$ is second countable, in accordance with An explicit rational-exponential basis for the universal development quotient.