Weak-field approximation

The weak-field approximation in general relativity is used to describe the gravitational field very far from the source of gravity. In this approximation, we assume the metric for spacetime ($g$) be written in coordinates as:

$g_{ab}=\eta _{ab}+\epsilon \gamma _{ab}$

where $\eta _{ab}$ are the Minkowski metric components, $\gamma$ is the deviation from the Minkowski metric and $\epsilon$ is taken to be a non-zero real constant. Calculating the Christoffel symbols $\Gamma ^{a}{}_{44}$, we get (upon ignoring terms of order higher than $\epsilon$:

$\Gamma ^{a}{}_{44}=-{\frac {\epsilon }{2}}g^{ad}\gamma _{44,d}$

From this last equation, we find that:

$\Gamma ^{4}{}_{44}=0$ ($i=1,2,3$)

$\Gamma ^{1}{}_{44}=-{\frac {\epsilon }{2}}g$ ($i=1,2,3$)