The Newtonian comparison model and the weak-field correspondence identification

This item defines the auxiliary framework used to formulate the Newtonian bridge, and the dictionary by which the relativistic weak-field data are read against it. It asserts no approximation; the approximation statement is a separate theorem of this page.

(a) Newtonian comparison model. A Newtonian comparison model is a tuple $(O,{\mathrm{\Phi }}_N,{\rho }_N,G_{\mathrm{c}\mathrm{m}\mathrm{p}})$ consisting of

• a nonempty open set $O\subseteq {\mathbb{R}}^3$, the spatial region, whose points are read as positions in metres;
• a $C^{\mathrm{\infty }}$ function ${\mathrm{\Phi }}_N:O\to \mathbb{R}$, the Newtonian potential, whose values are read in ${\mathrm{m}}^2{\mathrm{s}}^{-2}$;
• a $C^{\mathrm{\infty }}$ function ${\rho }_N:O\to \mathbb{R}$, the Newtonian mass density, whose values are read in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$; and
• a real constant $G_{\mathrm{c}\mathrm{m}\mathrm{p}}>0$ with the dimension ${\mathrm{m}}^3{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$,

such that the Newtonian field equation

$\mathrm{\Delta }{\mathrm{\Phi }}_N=4\pi G_{\mathrm{c}\mathrm{m}\mathrm{p}}{\rho }_N\text{on }O$

holds, where $\mathrm{\Delta }$ is the Laplacian on ${\mathbb{R}}^3$ (The Laplacian of a $C^2$ function and of a $C^2$ vector field). A comparison motion of the model is a curve $q:J\to O$ on a nonempty interval $J\subseteq \mathbb{R}$ with nonempty interior, whose points are read as times in seconds. Here $C^2$ on $J$ means that $q$ is continuous on $J$, is $C^2$ on its open interior, and $q^{\mathrm{\prime }}$, $q^{\mathrm{\prime }\mathrm{\prime }}$ extend continuously to each included finite endpoint. At those endpoints derivatives mean these unique limits from the interior. A comparison motion is $C^2$ in this sense and satisfies the Newtonian equation of motion

$q^{\mathrm{\prime }\mathrm{\prime }}(t)=-\mathrm{\nabla }{\mathrm{\Phi }}_N(q(t))(t\in J),$

where in the standard coordinates of ${\mathbb{R}}^3$ $\mathrm{\nabla }{\mathrm{\Phi }}_N(q):=({\mathrm{\partial }}_1{\mathrm{\Phi }}_N(q),{\mathrm{\partial }}_2{\mathrm{\Phi }}_N(q),{\mathrm{\partial }}_3{\mathrm{\Phi }}_N(q))$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case) and the total derivative is the covector represented by this vector through the Euclidean inner product: $D{\mathrm{\Phi }}_N(q)[v]=⟨\mathrm{\nabla }{\mathrm{\Phi }}_N(q),v⟩$ (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder, A total derivative computes every directional derivative, and its matrix is the Jacobian, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). The model is an auxiliary classical model: its equations are adopted as part of the definition of the comparison framework and are not derived here from the relativistic framework or from anything else.

(b) The correspondence identification. Let $(\mathrm{\Omega },K,L,C,{\epsilon }_0,\phi ,r)$ be stationary weak-field data over the spacetime region $\mathrm{\Omega }$, with spatial base $U=\pi (\mathrm{\Omega })$, parameters $c>0$ and $G_N>0$, an Einstein family, spatial potential $\mathrm{\Phi }=\epsilon c^2\phi$, and coordinate density ${\rho }_{\epsilon }=T_{00}(\epsilon )\mathrm{/}{c}^2$ on $U$ (Stationary weak-field data: the diagonal ansatz, its bounds, stationarity and the Einstein-family condition, SI unit and coordinate conventions of the classical gravitational framework, Stress-energy tensor field of a model). A comparison reading consists of a Newtonian comparison model $(O,{\mathrm{\Phi }}_N,{\rho }_N,G_{\mathrm{c}\mathrm{m}\mathrm{p}})$ and the following adopted dictionary:

1. the Newtonian spatial region is $O=U$, the spatial projection of $\mathrm{\Omega }$, and chart spatial coordinates in metres are read as Newtonian positions;
2. chart time $t=x^0\mathrm{/}c$ in seconds is read as the Newtonian time parameter;
3. the spatial potential $\mathrm{\Phi }:U\to \mathbb{R}$ is identified exactly with ${\mathrm{\Phi }}_N$, both with units ${\mathrm{m}}^2{\mathrm{s}}^{-2}$;
4. the constants $G_N$ and $G_{\mathrm{c}\mathrm{m}\mathrm{p}}$ are identified exactly.

The Newtonian source density ${\rho }_N$ continues to satisfy its exact model equation in clause (a). It is not identified here with either ${\rho }_{\epsilon }=T_{00}\mathrm{/}{c}^2$ or a dust rest-mass density; no density error estimate is part of this definition.

(c) Status. The dictionary of (b) is an adopted correspondence assumption: it fixes which regions, coordinates, potentials and couplings of the two frameworks are to be compared, in which units, and is part of the formulation of any bridge statement built on it. The Newtonian source density ${\rho }_N$, the relativistic coordinate density ${\rho }_{\epsilon }$ and a dust rest-mass density are distinct quantities. Any error-qualified matching between them is a separate theorem under its own hypotheses. This definition neither asserts exact density equality nor agreement of the models, and it does not assert that either framework describes nature.

(d) Scope and exclusions. The comparison model is smooth and source-continuous: point masses and delta densities are excluded, no boundary conditions at spatial infinity are imposed, and no statement is made about the numerical value, sign convention or uncertainty of $G_N$, $G_{\mathrm{c}\mathrm{m}\mathrm{p}}$ or $c$; those belong to the framework's conventions and to empirical work elsewhere. The comparison model has no relativistic content: it carries no metric, no causal structure and no stress-energy tensor, and its identification with the weak-field data is confined to the items listed in (b).