Newtonian limit of general relativity

For a weak, static field produced by nonrelativistic matter, $T_{00}=\rho$ is the dominant stress-energy component. The field equations and the slow-particle geodesic equation implywhere the Poisson equation is ${\nabla }^2\Phi =4\pi G\rho$.

For smooth spatial $\varphi$ and $g(c)=-c^2{e}^{2\varphi /c^2}d{t}^2+{\delta }_{ij}d{x}^{i}d{x}^j$, the only nonzero Christoffel symbols are ${\Gamma }^t{}_{ti}={\Gamma }^t{}_{it}=c^{-2}{\partial }_i\varphi$ and ${\Gamma }^i{}_{tt}=e^{2\varphi /c^2}{\partial }_i\varphi$. Consequently the Levi-Civita connection converges smoothly on compact sets as $c\to \infty$ although the metric itself diverges in these coordinates. The limiting torsion-free connection has only ${\Gamma }^i{}_{tt}={\partial }_i\varphi$ and gives $d^2{x}^i/d{t}^2=-{\partial }_i\varphi$ when $t$ is an affine parameter. With $R^a{}_{bcd}={\partial }_c{\Gamma }^a{}_{db}-{\partial }_d{\Gamma }^a{}_{cb}+\cdots$, its Ricci tensor is $(\Delta \varphi )dt\otimes dt$: quadratic connection terms vanish and $R^i{}_{tjt}={\partial }_j{\partial }_i\varphi$. Thus zero Ricci tensor is the Laplace equation, while full flatness would require the entire Hessian to vanish.

For a point mass in a positive cosmological constant, the inward Newtonian acceleration $GM/r^2$ balances the outward cosmological acceleration $\Lambda r/3$ at the static radius $r_{\mathrm{s}}=(3GM/\Lambda {)}^{1/3}$.

For a point mass and $r>0$, writeThe vacuum scalar-curvature equation gives $r{f}^{\prime }+3f=0$, so $f=C/r^3$. The remaining vacuum equations fix $C=2GM$ in the corresponding Schwarzschild gauge.