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.

We can obtain a relation between the Newtonian gravitational potential $\Phi$ and the deviation term above. Calculating the Christoffel symbols $\Gamma ^{a}{}_{44}$, we get (upon ignoring terms of order higher than $\epsilon$):

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

From this last equation, we find that:

$\Gamma _{44}^{4}=0$

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

The geodesic equation becomes

${\frac {d^{2}x^{i}}{dt^{2}}}=-\Gamma _{44}^{i}c^{2}={\frac {\epsilon c^{2}}{2}}\gamma _{44,i}=-\nabla \Phi$

where $\Phi$ is the Newtonian gravitational potential and $c$ is the speed of light. Thus:

$\Phi ={\frac {\epsilon c^{2}}{2}}\gamma _{44}$