In classical mechanics, if I want to view the Earth as the fixed center of the solar system, I must accelerate my reference frame to keep it centered on the Earth. That accelerated reference frame causes all sorts of messy fictitious forces that push the stars and planets around(Why do we say that the earth moves around the sun?). But in GR, since both the Sun and the Earth are in free-fall, I would think that the Earth's reference frame would be just as "natural" as the Sun's reference frame. In GR, would a geocentric model of the solar system would be "messy"*? If so, why?
*Let's avoid rotating reference frames by pretending that the Earth doesn't rotate sidreally.