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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.

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  • $\begingroup$ Please don't post formulas as pictures or plain text, but use MathJax instead. MathJax is easy for people on all devices to read, and can show up clearer on different screen sizes and resolutions.Look at this Math SE meta post for a quick tutorial. $\endgroup$ – user191954 Oct 14 '18 at 12:52
  • $\begingroup$ Thank you for your comment. This was my first question,and i will amend it as soon as possible. $\endgroup$ – SOQEH Oct 14 '18 at 12:55
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You never actually seem to define what you mean by an "extended Euclidean coordinate system," but no, the kind of thing you seem to be describing is not what we actually do if we want to describe the motion of Mercury using general relativity.

GR allows us to use any coordinate system we like, as long as the coordinates behave in a smooth way. I assume you're singling out Mercury because of the historical importance of its anomalous perihelion shift. If you want to describe this effect, the most straightforward thing to do is to use a Schwarzschild spacetime, described in Schwarzschild coordinates. This produces the anomalous part of the perihelion shift, but does not reproduce the perihelion shift that occurs because of the gravity of the other planets (which is larger).

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  • $\begingroup$ Thank you for your answer, which allows me to understand how my unskilled description is interpreted by experts. Obviously, in order to get help from the experts including you, I must explain my difficulties in a way that at least one of the experts can understand. For this, considering your answer, I revised the original question. $\endgroup$ – SOQEH Oct 16 '18 at 1:37

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