First of all, in physics 'geocentric' or 'heliocentric' is not of a view of how things are, it is simply a choice of coordinate system, which is always based on some one point as center, and a choice of reference frame which takes one thing or actually one frame as stationary and everything else is measured relative to it (re thing vs frame: one generally needs objects to specify a frame, and the frame can be chosen so that one specific thing/object is stationary).

However the real physics is without coordinates and reference frames and General Relativity is formulated in mathematical symbols/concepts that are coordinate-free and reference-frame independent. Of course in order to make a prediction as to what we would find if we measured such and such or saw such and such phenomenon, one needs to know from what frame and coordinate system the measurement or observation will be made - eg if one will be standing on Earth and looking at the sky then the coordinate system and reference frame will be 'geocentric' -  and then we canchoose that frame and coordinate system in the calculations of the prediction of how things would look from that coordinate system/frame, but the resulting statement or prediction is not a physics theory. It is simply a statement of physics as expressed in a particular choice of coordinates/reference frame.

So physics does not support the statement that the universe is centered on the sun or Earth or other body or point in space or event in spacetime – after all, if the sun is the center or Earth is, or any other thing or point, physics would ask what keeps it at the center, and what made it the center, and what makes the sun go around the Earth rather than vice versa, or if it is a point in space then why is that place special - none of this is explicable or sensible within physics, so this is not a physics view at all, instead it is a choice of coordinate system/reference frame. What physics (Einstein) tells us, and general relativity formulates is that it is not wrong to choose any coordinate system centered on any given point, any reference frame anchored to any object. The physics is consistent in any given choice but the true equations of the theory itself are formulated in a way that is independent of these specific choices.

So to answer the question: general relativity is a local theory, where local is a subtle concept. In this context the local frame of the satellite includes a small region of spacetime close to it and not including the Earth. It is in free fall, as are all orbiting objects (the Earth and the satellite are both in free fall), and does not experience inertial accelerations or forces so there is no need for it to explain anything at all, it is to all intents and purposes either stationary or moving with constant speed. That it is in orbit is only known when including the Earth in the picture and then the frame is not local anymore. To really understand this one would need to read a lot more on the subject.

Looked at another way, if one attempted to explain things from a reference frame anchored on the Earth+sattelite one would have to account for the general relativistic effects on the sattelite of all the rest of the universe whirling about the Earth at a high speed, which might just pull it away from the Earth enough to keep it at a certain height. But more correctly, if one tries to includes both the Earth and the satellite in one ‘local reference frame’ since they are not in motion relative to each other and one tries to really take this frame seriously then one is going out of the realm of validity of general relativity and physics and if so then whatever keeps the Earth where it is as opposed to everything else in the universe which is moving around it (eg the sun) also keeps the satellite there.

Avi Rabinowitz researches and lectures on physics, most recently at Ben Gurion University of the Negev.

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