Approximate Geodesic Motion
True geodesic motion is an idealization where one assumes the existence of test particles. Although in many cases real matter and energy can be approximated as test particles, situations arise where their appreciable mass (or equivalent thereof) can affect the background gravitational field in which they reside.
This creates problems when performing an exact theoretical description of a gravitational system (for example, in accurately describing the motion of two stars in a binary star system). This leads one to consider the problem of determining to what extent any situation approximates true geodesic motion. In qualitative terms, the problem is solved: the smaller the gravitational field produced by an object compared to the gravitational field it lives in (for example, the Earth's field is tiny in comparison with the Sun's), the closer this object's motion will be geodesic.
As Einstein's field equations determine the geometry of spacetime, it should be possible to determine the geodesics of the spacetime as well. For the case of dust, the problem can be solved by using the Bianchi identities. Many attempts have been made to do the same for other matter distributions.
Read more about this topic: Geodesic (general Relativity)
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