The weak-field Einstein hierarchy: linearized Einstein tensor, stress components and the Poisson equation

Let $(\mathrm{\Omega },K,L,C,{\epsilon }_0,\phi ,r)$ be stationary weak-field data with spatial base $U=\pi (\mathrm{\Omega })$ and compact $K\subseteq U$, and suppose the data form an Einstein family with coupling $\kappa >0$ (Stationary weak-field data: the diagonal ansatz, its bounds, stationarity and the Einstein-family condition clauses (a)-(c)). Write $\mathrm{\Phi }=\epsilon c^2\phi :U\to \mathbb{R}$, ${\rho }_{\epsilon }=T_{00}\mathrm{/}{c}^2:U\to \mathbb{R}$, and let $G^{(1)}$ be the first-order Einstein coefficient of Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge. Let $c>0$ and $G_N:=\kappa c^4\mathrm{/}(8\pi )$. There are ${\epsilon }_{\ast }^{\mathrm{\prime }}>0$, depending only on ${\epsilon }_0$, $C$, $L$ and the dimension, and $C_2<\mathrm{\infty }$, depending only on $C$, $L$ and the dimension, such that for every $0<\epsilon \le {\epsilon }_{\ast }^{\mathrm{\prime }}$ the following inequalities hold at every spacetime point $p\in \mathrm{\Omega }$ with $\pi (p)\in K$. Spatial derivatives and $\mathrm{\Delta }$ are taken on $U$ (The Laplacian of a $C^2$ function and of a $C^2$ vector field), and maxima are over the free component indices.

(a) Linearized approximation. With $A_{ab}(\epsilon ):=g_{ab}(\epsilon )-{\eta }_{ab}$,

$\mid G_{ab}(g(\epsilon ))-G_{ab}^{(1)}(A(\epsilon ))\mid \le \frac{C_2{\epsilon }^2}{L^2}\mathrm{.}$

(b) Component hierarchy of the Einstein tensor.

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

(c) Stress hierarchy and independent dust source. The Einstein equation gives

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

For every Einstein family, the definition ${\rho }_{\epsilon }=T_{00}\mathrm{/}{c}^2$ therefore gives $\mathrm{\mid }\mathrm{\Delta }\mathrm{\Phi }-4\pi G_N{\rho }_{\epsilon }\mathrm{\mid }\le \frac{C_2{\epsilon }^2{c}^2}{2L^2}\mathrm{.}$

For the additional conclusion below, let ${\mathrm{\nabla }}^{\epsilon }$ denote the metric-selected connection of $g(\epsilon )$ and assume the Einstein source is an independent admissible nonnegative dust input: for each $\epsilon$, there are smooth fields ${\mathcal{E}}_{\epsilon }:\mathrm{\Omega }\to [0,\mathrm{\infty })$ and $u_{\epsilon }$ on $\mathrm{\Omega }$, with $u_{\epsilon }$ future-directed and $g(\epsilon )(u_{\epsilon },u_{\epsilon })=-1$, such that $T_{ab}={\mathcal{E}}_{\epsilon }(u_{\epsilon }{)}_a(u_{\epsilon }{)}_b$ and ${\mathrm{\nabla }}_a^{\epsilon }({\mathcal{E}}_{\epsilon }{u}_{\epsilon }^a)=0,u_{\epsilon }^a{\mathrm{\nabla }}_a^{\epsilon }{u}_{\epsilon }^b=0.$ Where ${\mathcal{E}}_{\epsilon }>0$, this is the dust model of The dust matter model: rest density, four-velocity, stress-energy and dust system; where it vanishes, the smooth field $u_{\epsilon }$ remains specified and satisfies the geodesic equation by the explicit hypotheses above. The stress conservation identity follows from these equations (and agrees on the positive-density region with Divergence-free dust is exactly the dust system, and divergence-free perfect fluid is exactly the fluid equations). No comoving condition is imposed: the spatial tilt may be nonzero. Define the independent rest-mass density ${\mu }_{\epsilon }:={\mathcal{E}}_{\epsilon }\mathrm{/}{c}^2[\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}],$ where ${\mathcal{E}}_{\epsilon }$ has the rest-energy-density unit $\mathrm{J}{\mathrm{m}}^{-3}$ and ${\mu }_{\epsilon }$ is independent of the coordinate density ${\rho }_{\epsilon }$. It factors through $\pi$: normalization gives ${\mathcal{E}}_{\epsilon }=-g^{ab}{T}_{ab}=-{\kappa }^{-1}{g}^{ab}{G}_{ab}$, whose components factor through the spatial projection by the Einstein-family hypothesis. Thus ${\mu }_{\epsilon }$ also denotes its resulting smooth spatial function on $U$, including on disconnected fibers. On $K$, assume the explicit weak-source and tilted slow-motion bounds $0\le {\mathcal{E}}_{\epsilon }\le \frac{C\epsilon c^4}{G_N{L}^2},s_{\epsilon }\le C\sqrt{\epsilon }c,$ where on the region over $K$ the time orientation is chosen so that ${\mathrm{\partial }}_0$ is future (and $g(u_{\epsilon },{\mathrm{\partial }}_0)<0$), and $n_{\epsilon }:={\mathrm{\partial }}_0\mathrm{/}\sqrt{-g_{00}}$ is the resulting future unit coordinate observer, ${\gamma }_{\epsilon }:=-g(u_{\epsilon },n_{\epsilon })$, $h_{n_{\epsilon }}(u_{\epsilon }):=u_{\epsilon }+g(n_{\epsilon },u_{\epsilon })n_{\epsilon }$, and $s_{\epsilon }$ is the norm of $c{h}_{n_{\epsilon }}(u_{\epsilon })\mathrm{/}{\gamma }_{\epsilon }$ in the observer's rest space (Observers, rest spaces and the spatial projection at an event, Low-velocity expansions of observer energy and momentum with uniform error bounds). Choose ${\epsilon }_{\ast }^{\mathrm{\prime }}\le \min \{{\epsilon }_0,1,(128C{)}^{-1},(4C^2{)}^{-1}\},$ also small enough for the two-jet estimate in the proof. This threshold condition is dimensionless. Then, for a constant $C_d$ depending only on $C$,

$\mathrm{\mid }{T}_{00}(\epsilon )-{\mu }_{\epsilon }{c}^2\mathrm{\mid }\le \frac{C_d{\epsilon }^2{c}^4}{G_N{L}^2},\mathrm{\mid }{\rho }_{\epsilon }-{\mu }_{\epsilon }\mathrm{\mid }\le \frac{C_d{\epsilon }^2{c}^2}{G_N{L}^2},$

and the distinct dust rest-mass density satisfies the Poisson estimate

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

The energy- and mass-density conventions of The dust matter model: rest density, four-velocity, stress-energy and dust system give the same tensor here: $T_{ab}={\mathcal{E}}_{\epsilon }{u}_a{u}_b=c^2{\mu }_{\epsilon }{u}_a{u}_b$, with ${\mathcal{E}}_{\epsilon }=c^2{\mu }_{\epsilon }$.

(d) Scaled dust source. Let $X:=\mathbf{x}\mathrm{/}L$ and $U_L:=\{X\in {\mathbb{R}}^3:LX\in U\}$. Define $\psi (X):=\phi (LX)$ and $R(X):=G_N{L}^2{\mu }_{\epsilon }(LX)\mathrm{/}(\epsilon c^2)$ on $U_L$, with ${\mathrm{\Delta }}_X$ the Laplacian in $X$. Then

$\mathrm{\mid }{\mathrm{\Delta }}_X\psi -4\pi R\mathrm{\mid }\le C_2\epsilon \text{on }K\mathrm{/}L\mathrm{.}$

• Two distinct source densities. Every Einstein family has the component estimate for ${\rho }_{\epsilon }=T_{00}\mathrm{/}{c}^2$, which is a coordinate energy-density reading divided by $c^2$. Matching the Poisson source to the independent dust rest-mass density ${\mu }_{\epsilon }$ requires the separately stated dust, energy-size and tilt hypotheses; the density estimate is proved from those hypotheses and is not an identity.
• Admissibility and existence. The dust continuity and geodesic equations make its stress tensor divergence-free, including across smooth zero-density regions with the velocity field still specified. The theorem assumes the coupled Einstein-dust family; it does not construct one from Newtonian data or assert existence for prescribed initial data.
• Nonlogical comparison with later evolution theory. The density symbol $\rho$ used in Local Einstein-dust evolution: smooth constrained data with prescribed velocity admit a smooth local development, unique up to a data-fixing isometry ↗ is mass-normalized and corresponds to ${\mu }_{\epsilon }={\mathcal{E}}_{\epsilon }\mathrm{/}{c}^2$ here, so its tensor convention $c^2\rho u_a{u}_b$ matches the energy-density convention ${\mathcal{E}}_{\epsilon }{u}_a{u}_b$. This is a notation comparison only; the present theorem assumes its coupled family and its smooth velocity even at zero density. Its proof uses no existence or evolution conclusion from that later theorem.
• Gauge independence. No harmonic-gauge condition is used here: the component estimates are computed directly in the spatially factored chart, and the linearized coefficient is evaluated on the actual perturbation. The flat gauge material of First-order curvature of a flat linearized family, gauge transformations and the quadratic remainder supplies the formula; the uniform two-jet remainder is established here.