= Solution
Use spatial indices $i,j=1,2,3$, set $\phi_i=\partial_i\phi$, and assume $\phi$ is smooth. The <inverse metric> has $g^{tt}=-c^{-2}e^{-2\phi/c^2}$ and $g^{ij}=\delta^{ij}$. The <Christoffel symbols> of the <Levi-Civita connection> follow from $\Gamma^\alpha{}_{\beta\gamma}=g^{\alpha\delta}(\partial_\beta g_{\gamma\delta}+\partial_\gamma g_{\beta\delta}-\partial_\delta g_{\beta\gamma})/2$:
$$
\boxed{\Gamma^t{}_{ti}=\Gamma^t{}_{it}=\frac{\phi_i}{c^2},\qquad \Gamma^i{}_{tt}=e^{2\phi/c^2}\phi_i.}
$$
All other components vanish. Although $g_{tt}=-c^2-2\phi+O(c^{-2})$ diverges, its <affine connection> has a smooth limit on compact subsets:
$$
\boxed{\Gamma^{(\infty)i}{}_{tt}=\phi_i,\qquad \text{all other limiting components}=0.}
$$
This is the <Newtonian connection from an exponential lapse>. It is <torsion-free>, and its <geodesic equation>, using $t$ as an <affine parameter> when $\dot t\ne0$, is $d^2x^i/dt^2=-\partial_i\phi$. Thus $\phi$ has the role of a <Newtonian gravitational potential>. There is no finite limiting nondegenerate <Lorentzian metric> in these fixed coordinates; $-g(c)/c^2$ tends to $dt^2$, a rank-one tensor. The limiting connection preserves $dt$ and the contravariant spatial tensor with components $h^{ij}=\delta^{ij}$, $h^{t\mu}=0$, which describe the degenerate temporal/spatial structures of this limit.
Back to article page
