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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.

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Robert Cooper
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In GR you can use any frame that's convenient to solve the Einstein equation and calculate the curvature and/or metric. That's because these tensors are co-ordinate independant. If you choose polar co-ordinates centred on the Sun you get a mildly perturbed Schwarzschild metric, which is indeed nice and simple. Once you have the curvature you can transform it to any co-ordinate frame you want e.g. geocentric co-ordinates, then use it to calculate the evolution of the system.

But I'm not sure this would be any easier than using a Newtonian geostationary frame and all the fictitious forces it creates. I suspect that trying to write down a representation of the curvature or metric tensors in a co-ordinate frame ill suited to the task would be just as messy and difficult.

A simple test might be to take a model Solar System with just one extremely light planet, so you can take the Schwarzschild metric as the solution, then try to express this metric in a co-ordinate frame centred on the planet. I can't help with this I'm afraid - it's far beyond my skills!

John Rennie
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    Right. Well, all the fictitious forces are still there, together with all of their conceivable general relativistic corrections that need us to use the complicated metric tensor field (all components)... Philosophically, the geocentric system is "equally good" in GR because in all coordinate systems, the Solar System and the Earth are given by some curved metric tensor field. Some quasi-inertial and heliocentric fields are still a bit simpler which makes the geocentric system "more true" from a pragmatic viewpoint. – Luboš Motl May 04 '12 at 18:08