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Vary the geodesic Lagrangian before imposing the normalization of proper time. With $u^{\mu }=d{x}^{\mu }/d\tau$,The Euler-Lagrange equation isequivalently the affinely parametrized geodesic equation. For the static weak metric, its leading spatial connection coefficient isAll terms with two spatial velocities are suppressed by $v^2$. Therefore $d^2{x}^i/d{\tau }^2\simeq -{\partial }_i\Phi (dt/d\tau {)}^2$.

Because $t$ is cyclic, the same Euler-Lagrange equation gives $(1+2\Phi )dt/d\tau =\text{constant}$. On converting to coordinate time, the additional term $(d^{2}t/d{\tau }^2)(d{x}^i/dt)$ is of order $v^2\nabla \Phi$ and can be dropped. Henceto leading order, the Newtonian limit of general relativity. Restoring SI units gives $\Phi ={\Phi }_{\mathrm{SI}}/c^2$, so the acceleration is $-\nabla (c^2\Phi )$. Substituting $L=1$ from proper-time normalization before varying would incorrectly erase the dynamics.

For a slowly moving particle in the weak-field approximation, $dt/d\tau =1+O(\Phi ,v^2)$ and terms containing two spatial velocities may be neglected. The spatial geodesic equation reduces toFor $g_{00}=-(1+2\Phi )$, the Levi-Civita connection isHence the Newtonian limit of general relativity is

In the Newtonian limit of general relativity, time derivatives are negligible and $T_{00}=\rho$ is the only leading component. With signature $(-,+,+,+)$,The $00$ component of the trace-reversed equation becomesIdentifying $h_{00}=-2\Phi$ therefore gives the Poisson equation of Newtonian gravity:

Move the positive cosmological constant to the source side and defineTo first order about Minkowski spacetime, $T^{(\Lambda )}=-\Lambda /(2\pi )$. Its contribution to the trace-reversed $00$ source isRepeating the Newtonian limit of general relativity calculation givesFor a point mass, ${\nabla }^2{r}^2=6$, soThe radial acceleration isThus positive $\Lambda$ supplies an outward acceleration and balances the point-mass attraction at the static radiusThe weak-field approximation requires both $M/r\ll 1$ and $\Lambda r^2\ll 1$, giving parametrically

For a slowly moving particle in a static weak field, the spatial part of the geodesic equation reduces toTo first order in the perturbation,and henceNewtonian acceleration is $-{\delta }^{ij}{\partial }_j\Phi$, so $h_{00}=-2\Phi$. Substitution into part (a), followed by the Poisson equation ${\nabla }^2\Phi =4\pi G\rho$, givesThus the Newtonian limit of general relativity fixes