The controlled weak-field slow-motion correspondence with the Newtonian comparison model

Adopt the postulates Adopted spacetime model and classical domain of the gravitational framework, Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions, Adopted Einstein field equation in the direct-equation presentation in the direct-equation presentation and Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework. Let stationary weak-field data $(\mathrm{\Omega },K,L,C,{\epsilon }_0,\phi ,r)$ over spacetime region $\mathrm{\Omega }$ have spatial base $U=\pi (\mathrm{\Omega })$, form an Einstein family with $\kappa =8\pi G_N\mathrm{/}{c}^4$, and satisfy the independent admissible nonnegative slow-dust hypotheses in clause (c) of The weak-field Einstein hierarchy: linearized Einstein tensor, stress components and the Poisson equation. Write ${\mu }_{\epsilon }$ for that dust rest-mass density. Let a comparison reading with a Newtonian comparison model $(U,{\mathrm{\Phi }}_N,{\rho }_N,G_{\mathrm{c}\mathrm{m}\mathrm{p}})$ be adopted under The Newtonian comparison model and the weak-field correspondence identification, so that ${\mathrm{\Phi }}_N=\mathrm{\Phi }=\epsilon c^2\phi$ and $G_{\mathrm{c}\mathrm{m}\mathrm{p}}=G_N$; the Newtonian model satisfies its exact Poisson equation $\mathrm{\Delta }{\mathrm{\Phi }}_N=4\pi G_{\mathrm{c}\mathrm{m}\mathrm{p}}{\rho }_N$ by definition.

Let ${\epsilon }_{\ast }^{\mathrm{\prime }}$, $C_2$ and $C_3$ be the thresholds and constants of The weak-field Einstein hierarchy: linearized Einstein tensor, stress components and the Poisson equation, Uniform Christoffel-symbol bounds for stationary weak-field data and Coordinate-time acceleration of weak-field geodesics: the Newtonian leading term with uniform error. Let $S_{\ast }>0$, $0<\epsilon \le {\epsilon }_{\ast }^{\mathrm{\prime }}$, and put $T_{\epsilon }=L{S}_{\ast }\mathrm{/}(\sqrt{\epsilon }c)$ and $U_L=\{X\in {\mathbb{R}}^3:LX\in U\}$, $\psi (X)=\phi (LX)$ on $U_L$. Assume there are an ideal freely falling massive test body worldline ${\gamma }_{\epsilon }$ and its spatial coordinate curve $x_{\epsilon }:[0,T_{\epsilon }]\to U$ such that ${\gamma }_{\epsilon }(t)=(ct,x_{\epsilon }(t))\in \mathrm{\Omega }$, $x_{\epsilon }(t)\in K$, and $\mathrm{\mid }{\dot{x}}_{\epsilon }(t)\mathrm{\mid }\le C\sqrt{\epsilon }c$. Suppose also that hypotheses (H2)-(H3) of Scaled trajectory limit: weak-field geodesics converge to the Newtonian system, with rate O(epsilon) for identical initial data hold on a compact convex $K^{\mathrm{\prime }}\subseteq U_L$ containing both scaled trajectory ranges, with $K_2={\sup }_{X\in K^{\mathrm{\prime }}}\mathrm{\parallel }{D}^2\psi (X){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}<\mathrm{\infty }$, using the Euclidean vector norm and induced Hessian operator norm fixed in Scaled trajectory limit: weak-field geodesics converge to the Newtonian system, with rate O(epsilon) for identical initial data.

Then the following are conditional consequences of the adopted postulates, the independent coupled Einstein-dust input, the comparison dictionary and the stated mathematical hypotheses. No family or dust solution is asserted to exist for prescribed Newtonian data.

(a) Field and source correspondence. At every $\mathbf{x}\in K$,

$\mid \kappa T_{00}(\epsilon )-\frac{2\mathrm{\Delta }\mathrm{\Phi }}{c^2}\mid \le \frac{C_2{\epsilon }^2}{L^2},\mathrm{\mid }{G}_{ij}(g(\epsilon ))\mathrm{\mid }\le \frac{C_2{\epsilon }^2}{L^2},\mathrm{\mid }{G}_{0i}(g(\epsilon ))\mathrm{\mid }\le \frac{C_2{\epsilon }^{3\mathrm{/}2}}{L^2},$

and the distinct source densities satisfy

$\mathrm{\mid }\mathrm{\Delta }\mathrm{\Phi }-4\pi G_N{\mu }_{\epsilon }\mathrm{\mid }\le \frac{C_2{\epsilon }^2{c}^2}{L^2},\mathrm{\mid }{\rho }_N-{\mu }_{\epsilon }\mathrm{\mid }\le \frac{C_2{\epsilon }^2{c}^2}{4\pi G_N{L}^2},\mathrm{\mid }{\rho }_{\epsilon }-{\mu }_{\epsilon }\mathrm{\mid }\le \frac{C_d{\epsilon }^2{c}^2}{G_N{L}^2},$

where ${\rho }_{\epsilon }=T_{00}\mathrm{/}{c}^2$ is the relativistic coordinate density and ${\mu }_{\epsilon }$ is the independently specified dust rest-mass density. The comparison model's Poisson equation remains exact; the approximation is the density matching.

(b) Free-fall kinematics. The assigned worldline ${\gamma }_{\epsilon }$ is a future timelike geodesic of $g(\epsilon )$ with $g(U_{\epsilon },U_{\epsilon })=-c^2$, proper time an affine parameter, and initial data determining the worldline. On the stated coordinate-time interval,

$\mid {\ddot{x}}_{\epsilon }^i(t)+{\mathrm{\partial }}_i\mathrm{\Phi }(x_{\epsilon }(t))\mid \le \frac{C_3{c}^2{\epsilon }^2}{L}(t\in [0,T_{\epsilon }])\mathrm{.}$

(c) Trajectory correspondence. With $X_{\epsilon }(S)=x_{\epsilon }(LS\mathrm{/}(\sqrt{\epsilon }c))\mathrm{/}L$, $W_{\epsilon }(S)={\dot{x}}_{\epsilon }(LS\mathrm{/}(\sqrt{\epsilon }c))\mathrm{/}(\sqrt{\epsilon }c)$, and $(X_{\ast },W_{\ast })$ the comparison motion of (H2), let $D(S)=\mathrm{\parallel }{X}_{\epsilon }(S)-X_{\ast }(S){\mathrm{\parallel }}_2+\mathrm{\parallel }{W}_{\epsilon }(S)-W_{\ast }(S){\mathrm{\parallel }}_2$ using the same Euclidean norm convention. Then

$D(S)\le (D(0)+C_3\epsilon S_{\ast })e^{(1+K_2)S_{\ast }}(0\le S\le S_{\ast })\mathrm{.}$

Thus convergent scaled initial data give uniform convergence on the fixed scaled interval, and identical scaled data give a uniform $O(\epsilon )$ error. The corresponding coordinate times $t=LS\mathrm{/}(\sqrt{\epsilon }c)$ grow like ${\epsilon }^{-1\mathrm{/}2}$.

(d) Status and qualifications. This is a conditional comparison of an Einstein family containing an independently specified conserved tilted dust source with an auxiliary Newtonian model. It does not derive the Newtonian model, does not establish that either model describes nature, and uses no observations, apparatus, calibration or measured constant. The input family and dust fields are hypotheses; the theorem proves neither their existence nor uniqueness for prescribed Newtonian data. Any empirical value of $G_N$ and comparison with data would require separate uncertainty-qualified empirical premises.

• What is proved. The Newtonian model satisfies its exact Poisson equation by definition; under the weak-field Einstein-dust hypotheses its exact source density matches the independent dust rest-mass density within the stated uniform error. The coordinate density $T_{00}\mathrm{/}{c}^2$ also matches that rest-mass density within its own error. Ideal free-fall worldlines converge to the comparison motions with a uniform $O(\epsilon )$ error on a fixed scaled interval. These are conditional correspondences, not identities of the frameworks.
• Hypotheses are not redundant. Spatial factorization, stationarity, the metric and matter bounds, dust conservation, the slow dust tilt, the slow test-body motion, and existence of the comparison motion on the interval are assumptions. The dust stress estimate is derived from its independent rest-energy density and tilt; the theorem does not assume the tautological identity $T_{00}=c^2{\rho }_{\epsilon }$ as matter input.
• Empirical status. The conclusion is about models, not nature. Any comparison with reported Newtonian or relativistic data would require the measured value of $G_N$, the apparatus and calibration model, and a statistical account, none of which is part of this deduction.