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 (
) be written in coordinates as:
where
are the Minkowski metric components,
is the deviation from the Minkowski metric and
is taken to be a non-zero real constant.
We can obtain a relation between the Newtonian gravitational potential
and the deviation term above. Calculating the Christoffel symbols
, we get (upon ignoring terms of order higher than
):
From this last equation, we find that:
(
)
The geodesic equation becomes
where
is the Newtonian gravitational potential and
is the speed of light. Thus: