On the Wikipedia Article on “Geodesics in general relativity”, it says the following: “Thus, for example, the path of a planet orbiting a star is the projection of a geodesic of the curved 4-D spacetime geometry around the star onto 3-D space.”
I want to know how the geodesic of curved 4-D spacetime geometry is projected onto 3-D space.
First, let’s suppose that the planet is, for example, the Mercury far from us. In this case, I think that, for the above “projection onto 3-D space”, it is necessary to extend a Euclidean coordinate system, which is constructed by us living on the Earth, to a region near the Mercury in a simple and continuous manner. (Here, "the extension in the simple and continuous manner" means that three orthogonal basis vectors, defined by an observer on Earth, are used to span the entire universe beyond the earth.)
For example, as far as I know, in the cases of the anomalous precession of Mercury or the deflection of light caused by the sun, theoretically predicted results are initially calculated from the geodesic equation using the Schwarzschild metric, but they are finally written in terms of coordinates in the laboratory frame through transformation.
In this sense, in order to compare the observation with the theoretically predicted result (i.e., to realize the projection in the Wikipedia article), the trajectory of Mercury should be observed and described in terms of Euclidean coordinates constructed based on the basis vectors.
Is this my understanding correct?
Any comment would be very welcome.