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Originally Posted by George
Ok, but your parenthetic(the Sun's gravity) is foundational to how the various coordinate systems will look.
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I would say it is foundational to the shape of the vase-- but the coordinate systems can still look very different. For example, issues like which is orbiting which are entirely coordinate dependent issues, and GR tells you how to make any coordinate system function. They may result in awkward or unphysical seeming boundary conditions, but who can say what the "right" boundary conditions are? That's pretty much the road that leads to Mach, which is where the GR journey all started.
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Adjusting the mass, I assume, will change either the coordinate system itself or the coordinate results of motion; cause and effect, in effect.
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What makes a coordinate system useful for doing physics is having a mapping from the coordinates to distances (a metric). Different looking coordinates can have different looking metrics yet still be the same (i.e., they apply to the same vase). The "coordinate invariant" properties will be the only ones that "nature itself knows", so that doesn't include which object is orbiting which. However, given the way we read extra things into the situation, it seems the more "objective" perspective to say that the Earth orbits the Sun, as it shows that the Earth does not need to be stationary, and we do not need to invoke "coordinate forces" (like centrifugal force) to understand what is happening. So on that basis some claim that this is the "truth" of the matter.