Explicit stationary weak-field data with a dimensionless quadratic potential: exact Poisson source and the first-order Einstein coefficient

Work on the flat model of The flat model spacetime: the Minkowski metric on R^4 and its inertial chart with inertial chart $x$, let $L>0$ be a length, take spacetime region $\mathrm{\Omega }=\mathbb{R}\times B_L(0)\subseteq {\mathbb{R}}^4$ with spatial base $U=B_L(0)\subseteq {\mathbb{R}}^3$, and let $K={\bar{B}}_{L\mathrm{/}2}(0)\subset U$. Set $c>0$, $G_N>0$, $\kappa :=8\pi G_N\mathrm{/}{c}^4$, $C:=1$, ${\epsilon }_0:=1\mathrm{/}2$, choose the stationary-data remainder interval parameter ${\epsilon }_1^{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}:=1$, set $r\equiv 0$ on that interval, and

$\phi (\mathbf{x}):=\frac{(x^1{)}^2+(x^2{)}^2+(x^3{)}^2}{2L^2}\mathrm{.}$

Then:

1. $(\mathrm{\Omega },K,L,C,{\epsilon }_0,\phi ,r)$ is stationary weak-field data in the sense of Stationary weak-field data: the diagonal ansatz, its bounds, stationarity and the Einstein-family condition clause (a), with $g_{ab}(\epsilon )={\eta }_{ab}+\epsilon h_{ab}$, $h_{00}=-2\phi$, $h_{ij}=-2\phi {\delta }_{ij}$, $h_{0i}=0$; the family $({\delta }_1,h,q)$ with ${\delta }_1:=1\mathrm{/}2$ and $q\equiv 0$ is a flat linearized family and $g(\epsilon )-\eta =\epsilon h$.

2. Defining $T_{ab}(\epsilon ):=G_{ab}(g(\epsilon ))\mathrm{/}\kappa$ and ${\rho }_{\epsilon }:=T_{00}(\epsilon )\mathrm{/}{c}^2$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors) makes the data an Einstein family with coupling $\kappa$ (Stationary weak-field data: the diagonal ansatz, its bounds, stationarity and the Einstein-family condition clause (c)). This smooth symmetric source is defined by the metric; no dust model or independent rest-mass density is asserted.

3. For $\mathrm{\Phi }=\epsilon c^2\phi$, $\mathrm{\Delta }\mathrm{\Phi }=3\epsilon c^2\mathrm{/}{L}^2$ exactly. The constant Newtonian source

   $\bar{\rho }:=\frac{3\epsilon c^2}{4\pi G_N{L}^2}$

   satisfies $\mathrm{\Delta }\mathrm{\Phi }=4\pi G_N\bar{\rho }$ exactly on $U$. The distinct coordinate density read from the Einstein-family tensor satisfies $\mathrm{\mid }{\rho }_{\epsilon }-\bar{\rho }\mathrm{\mid }\le C_2{\epsilon }^2{c}^2\mathrm{/}(8\pi G_N{L}^2)$ on $K$ for $0<\epsilon \le {\epsilon }_{\ast }^{\mathrm{\prime }}$, by the general $00$-component hierarchy; this does not use the hierarchy theorem's independent-dust conclusion.

4. The first-order Einstein coefficient of $A(\epsilon ):=\epsilon h$ is

   $G_{00}^{(1)}(A(\epsilon ))=\frac{6\epsilon }{L^2}=\frac{2\mathrm{\Delta }\mathrm{\Phi }}{c^2},G_{0i}^{(1)}(A(\epsilon ))=G_{ij}^{(1)}(A(\epsilon ))=0,$

   so the leading term of the hierarchy is exactly the first-order coefficient of the family.

• Non-vacuity and scope. The example exhibits a data tuple satisfying the general Einstein-component hypotheses of the hierarchy theorem with $C=1$ and arbitrary $L>0$; it asserts no independent dust realization, and computes the dictionary quantities $\mathrm{\Phi }$, $\bar{\rho }$ and $G^{(1)}$ for it. It is a conditional statement about the model family: it asserts that the quadratic potential solves the Poisson equation with constant density, not that any physical body realizes the density, and it involves no observation.