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.”
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. That is, in order to compare the observation with the theoretically predicted result, the trajectory of Mercury should be observed and described in terms of coordinates based on the basis vectors.
Similarly, it seems that the extended Euclidean coordinate system is also used in describing galaxies and so forth. Specifically, if a distance to a star is expressed based on a traveling time of light traveling from the star to us, a distance to a star, which is located beyond the massive black hole at the center of our Milky Way, will be infinity. However, the distance to the star is not expressed in such a manner. Rather, it seems that the distance to the star is expressed based on the extended Euclidean coordinate system. Is this my understanding correct?
I am a self-taught person who is not good in English. Thus, any comment on technical contents in this article or any improved edition on English expressions would be very welcome.