Scaled trajectory limit: weak-field geodesics converge to the Newtonian system, with rate O(epsilon) for identical initial data

Let $(\mathrm{\Omega },K,L,C,{\epsilon }_0,\phi ,r)$ be stationary weak-field data with spatial base $U=\pi (\mathrm{\Omega })$, compact $K\subseteq U$, spatial potential $\mathrm{\Phi }=\epsilon c^2\phi$ and $c>0$ (Stationary weak-field data: the diagonal ansatz, its bounds, stationarity and the Einstein-family condition), let ${\epsilon }_{\ast }$ and $C_2$ be the threshold and constant of Uniform Christoffel-symbol bounds for stationary weak-field data, let $S_{\ast }>0$ and $0<\epsilon \le {\epsilon }_{\ast }$, and put $T_{\epsilon }:=L{S}_{\ast }\mathrm{/}(\sqrt{\epsilon }c)$, $U_L:=\{X\in {\mathbb{R}}^3:LX\in U\}$, and $\psi (X):=\phi (LX)$ for $X\in U_L$.

Equip each ${\mathbb{R}}^3$ here with the Euclidean inner product $⟨X,Y⟩={\sum }_{i=1}^3{X}^i{Y}^i$ and norm $\mathrm{\parallel }X{\mathrm{\parallel }}_2=\sqrt{⟨X,X⟩}$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). In the standard coordinates, $\mathrm{\nabla }\psi =({\mathrm{\partial }}_{X^1}\psi ,{\mathrm{\partial }}_{X^2}\psi ,{\mathrm{\partial }}_{X^3}\psi )$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case); write $D^2\psi :=D(\mathrm{\nabla }\psi )$ for its derivative and $\mathrm{\parallel }{D}^2\psi (X){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}:={\sup }_{\mathrm{\parallel }v{\mathrm{\parallel }}_2=1}\mathrm{\parallel }{D}^2\psi (X)v{\mathrm{\parallel }}_2$.

Suppose:

• (H1) there is an affinely parametrized geodesic of $g(\epsilon )$ admitting a $C^2$ reparametrization ${\gamma }_{\epsilon }$ by $t=x^0\mathrm{/}c$ on $[0,T_{\epsilon }]$, with $dt\mathrm{/}ds\mathrm{\ne }0$ in its affine parameter $s$, whose length-coordinate components are ${\gamma }_{\epsilon }(t)=(ct,x_{\epsilon }(t))\in \mathrm{\Omega }$, with spatial trajectory $x_{\epsilon }(t)\in K$ and $\mathrm{\mid }{\dot{x}}_{\epsilon }(t)\mathrm{\mid }\le C\sqrt{\epsilon }c$ for every such $t$;
• (H2) setting $t(S):=LS\mathrm{/}(\sqrt{\epsilon }c)$, $X_{\epsilon }(S):=x_{\epsilon }(t(S))\mathrm{/}L$ and $W_{\epsilon }(S):={\dot{x}}_{\epsilon }(t(S))\mathrm{/}(\sqrt{\epsilon }c)$ for $S\in [0,S_{\ast }]$, and letting $K^{\mathrm{\prime }}\subseteq U_L$ be a compact convex set containing every $X_{\epsilon }(S)$, there is a $C^1$ map $(X_{\ast },W_{\ast }):[0,S_{\ast }]\to {\mathbb{R}}^3\times {\mathbb{R}}^3$ with $X_{\ast }([0,S_{\ast }])\subseteq K^{\mathrm{\prime }}$ and

$X_{\ast }^{\mathrm{\prime }}(S)=W_{\ast }(S),W_{\ast }^{\mathrm{\prime }}(S)=-\mathrm{\nabla }\psi (X_{\ast }(S))(0\le S\le S_{\ast });$

• (H3) $\mathrm{\parallel }{D}^2\psi (X){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le K_2$ for every $X\in K^{\mathrm{\prime }}$, where $K_2\ge 0$.

Then for every $S\in [0,S_{\ast }]$, with $D(S):=\mathrm{\parallel }{X}_{\epsilon }(S)-X_{\ast }(S){\mathrm{\parallel }}_2+\mathrm{\parallel }{W}_{\epsilon }(S)-W_{\ast }(S){\mathrm{\parallel }}_2$,

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

where $C_3$ depends only on $C$, $L$ and the dimension. Consequently the following hold.

(i) Convergence. If $D(0)\to 0$ as $\epsilon \downarrow 0$ along the family, then ${\sup }_{0\le S\le S_{\ast }}D(S)\to 0$.

(ii) Rate for identical data. If $D(0)=0$, then ${\sup }_{0\le S\le S_{\ast }}D(S)\le C_3\epsilon S_{\ast }{e}^{(1+K_2)S_{\ast }}=O(\epsilon )$.

(iii) Long coordinate times. The coordinate-time horizon is $T_{\epsilon }=L{S}_{\ast }\mathrm{/}(\sqrt{\epsilon }c)$, so the fixed scaled interval $[0,S_{\ast }]$ corresponds to coordinate times growing like ${\epsilon }^{-1\mathrm{/}2}$; the limit statement is therefore not a large-$t$ statement at fixed $\epsilon$.

• What the limit is. The statement is a two-sequence comparison: a family of geodesics of the metrics $g(\epsilon )$ with data scaled as in (H1)-(H2), and a solution of the Newtonian comparison system. It is not a statement that general relativity reduces to Newtonian mechanics, nor a statement about any particular physical system.
• The exponential factor. The bound grows like $e^{(1+K_2)S_{\ast }}$: on fixed scaled intervals it is uniform in $\epsilon$, but it degrades as the scaled interval or the field gradient grows. No global-in-time limit is asserted.