= Newtonian connection from an exponential lapse
{c}
{title2=$\Gamma^i{}_{tt}=\partial_i\phi$}
For smooth spatial $\phi$ and $g(c)=-c^2e^{2\phi/c^2}dt^2+\delta_{ij}dx^idx^j$, the only nonzero <Christoffel symbols> are $\Gamma^t{}_{ti}=\Gamma^t{}_{it}=c^{-2}\partial_i\phi$ and $\Gamma^i{}_{tt}=e^{2\phi/c^2}\partial_i\phi$. 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 $\Gamma^i{}_{tt}=\partial_i\phi$ and gives $d^2x^i/dt^2=-\partial_i\phi$ 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 $(\Delta\phi)dt\otimes dt$: quadratic connection terms vanish and $R^i{}_{tjt}=\partial_j\partial_i\phi$. Thus zero Ricci tensor is the <Laplace equation>, while full flatness would require the entire Hessian to vanish.
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