Stationary weak-field data: the diagonal ansatz, its bounds, stationarity and the Einstein-family condition

Let $(M,g_{\eta },N_{\eta })$ be the flat model of The flat model spacetime: the Minkowski metric on R^4 and its inertial chart with inertial chart $x=(x^0,x^1,x^2,x^3)$ and constant components ${\eta }_{ab}=\mathrm{diag}(-1,1,1,1)$. Let $\mathrm{\Omega }\subseteq M$ be a nonempty open spacetime region, identified with its image in ${\mathbb{R}}^4$, and let $\pi :\mathrm{\Omega }\to {\mathbb{R}}^3$ be the spatial projection $\pi (x^0,\mathbf{x})=\mathbf{x}$. Its image $U:=\pi (\mathrm{\Omega })$ is open: a coordinate ball in $\mathrm{\Omega }$ about any point projects onto a ball about that point's spatial coordinates. Let $K\subseteq U$ be compact. Let $c>0$ and $G_N>0$ be the constants of the framework's conventions and put $\kappa :=8\pi G_N\mathrm{/}{c}^4$. The coordinates have units of metres, $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$, and $\epsilon$ is dimensionless.

(a) Structural data. A tuple $(\mathrm{\Omega },K,L,C,{\epsilon }_0,\phi ,r)$ is stationary weak-field data when

• $L>0$ is a length, $C>0$ is dimensionless, and ${\epsilon }_0>0$ is dimensionless;
• $\phi :U\to \mathbb{R}$ is a smooth dimensionless spatial function and $r=(r_{ab})$ is a family of smooth dimensionless spatial functions $r_{ab}:(-{\epsilon }_1,{\epsilon }_1)\times U\to \mathbb{R}$ for some ${\epsilon }_1>{\epsilon }_0$ ($C^r$ and smooth maps between smooth manifolds);
• for every $0<\epsilon \le {\epsilon }_0$ the component field

$g_{ab}(\epsilon ,p)=\{\begin{array}{ll}-(1+2\epsilon \phi (\pi (p)))+r_{00}(\epsilon ,\pi (p)),& a=b=0,\\ r_{0i}(\epsilon ,\pi (p)),& \{a,b\}=\{0,i\},\\ (1-2\epsilon \phi (\pi (p))){\delta }_{ij}+r_{ij}(\epsilon ,\pi (p)),& a=i,b=j,\end{array}$

is the component field of a smooth Lorentzian metric $g(\epsilon )$ on $\mathrm{\Omega }$ of signature $(-,+,+,+)$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model);

• (spatial factorization and stationarity) all metric components factor through $\pi$ by construction. Thus points of $\mathrm{\Omega }$ with the same spatial coordinates have the same metric components, even when they lie in different connected components of that spatial fiber, and ${\mathrm{\partial }}_0{g}_{ab}=0$ on $\mathrm{\Omega }$;
• (spatial bounds) for every multiindex $\alpha \in {\mathbb{N}}^3$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le 2$ and every $\mathbf{x}\in K$,

$\mathrm{\mid }{\mathrm{\partial }}^{\alpha }\phi (\mathbf{x})\mathrm{\mid }\le \frac{C}{L^{\mathrm{\mid }\alpha \mathrm{\mid }}},\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{r}_{00}(\epsilon ,\mathbf{x})\mathrm{\mid }+\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{r}_{ij}(\epsilon ,\mathbf{x})\mathrm{\mid }\le \frac{C{\epsilon }^2}{L^{\mathrm{\mid }\alpha \mathrm{\mid }}},\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{r}_{0i}(\epsilon ,\mathbf{x})\mathrm{\mid }\le \frac{C{\epsilon }^{3\mathrm{/}2}}{L^{\mathrm{\mid }\alpha \mathrm{\mid }}}\mathrm{.}$

(Derivatives are with respect to $x^1,x^2,x^3$; here $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{r}_{ij}\mathrm{\mid }$ is the maximum over $i,j$ and $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{r}_{0i}\mathrm{\mid }$ is the maximum over $i$.)

(b) Derived data. For stationary weak-field data define the spatial potential

$\mathrm{\Phi }:=\epsilon c^2\phi \text{on }U,$

a smooth spatial function of dimension ${\mathrm{m}}^2{\mathrm{s}}^{-2}$ in the framework's unit conventions; its pullback to $\mathrm{\Omega }$ is stationary. The symmetric component family $g(\epsilon )-\eta$ is the perturbation of the family.

(c) Einstein families. Stationary weak-field data form an Einstein family with coupling $\kappa$ when for every $0<\epsilon \le {\epsilon }_0$ there is a smooth symmetric component field $T_{ab}(\epsilon ,\cdot )$ on $\mathrm{\Omega }$ (Smooth covariant 2-tensor fields on a smooth manifold) satisfying

$G_{ab}(g(\epsilon ))=\kappa T_{ab}(\epsilon )\text{on }\mathrm{\Omega },$

where the connection of $g(\epsilon )$ is the unique torsion-free, metric-compatible connection supplied for any smooth nondegenerate metric by Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clauses (a)-(b), with that supplier's ${\mathrm{A}\mathrm{C}}_{\omega }$ convention. Its Einstein tensor $G$ is defined without assuming a global time orientation as follows. For each $p\in \mathrm{\Omega }$, fix a timelike $v\in T_p\mathrm{\Omega }$ and extend its inertial-chart components constantly. By continuity of $g(\epsilon )$, this smooth field $N$ remains timelike on an open neighborhood $V\ni p$. Apply The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors clauses (a)-(c) to $(V,g(\epsilon ){\mathrm{\mid }}_V,N)$, using the restricted connection, which is the metric-selected connection there by uniqueness. In particular, $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$, $R({\mathrm{\partial }}_c,{\mathrm{\partial }}_d){\mathrm{\partial }}_b=R^a{}_{bcd}{\mathrm{\partial }}_a$, and

${\mathrm{Ric}}_{bd}={\sum }_{a=0}^3{R}^a{}_{bad},S={\sum }_{b,d=0}^3{g}^{bd}{\mathrm{Ric}}_{bd},G_{ab}={\mathrm{Ric}}_{ab}-\frac{1}{2}S{g}_{ab},$

where $(g^{ab})$ is the inverse metric matrix. These local tensor fields agree on every overlap: the restricted connection is the same, and the curvature operator and metric contractions contain no $N$. They therefore define a unique smooth tensor $G$ on $\mathrm{\Omega }$, even if $g(\epsilon )$ is not time-orientable. In that case define the spatial density read from the time-time component

${\rho }_{\epsilon }(\mathbf{x}):=\frac{T_{00}(\epsilon ,x^0,\mathbf{x})}{c^2}\text{on }U,$

where any $x^0$ with $(x^0,\mathbf{x})\in \mathrm{\Omega }$ may be used. This is well-defined: the metric is the pullback of spatial functions, so its coordinate curvature components, and hence $T_{00}={\kappa }^{-1}{G}_{00}$, are constant on each fiber of $\pi$, including when that fiber is disconnected. It has units $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$; it is a coordinate-component density and is not thereby the dust rest-mass density.

(d) Scope. This item fixes a class of metric families and the vocabulary of the weak-field bridge: the spatial factorization, component form, stationarity, spatial derivative bounds, potential, and the equation that upgrades the data to an Einstein family. The stationarity condition gives ${\mathcal{L}}_{{\mathrm{\partial }}_0}g=0$ by the local translation pullback, independently of any time orientation. It asserts neither existence for prescribed data nor that ${\mathrm{\partial }}_0$ is timelike, that the inverse metrics are uniformly bounded, or that ${\rho }_{\epsilon }$ is positive; the latter properties are consequences proved separately where they are used. The family is an explicit input-family hypothesis: neither its existence for prescribed Newtonian initial data nor its construction from other data is claimed. The physical reading of the components, of $G_{ab}=\kappa T_{ab}$, of $\mathrm{\Phi }$ and of ${\rho }_{\epsilon }$ belongs to the framework's postulates and unit conventions, which are separate items.