My understanding was that relativistic physics can be expressed in any inertial coordinate system, but not arbitrary systems. That is, no experiment can determine if we are "still" or "moving" at a constant velocity; but we can determine if we are accelerating, or moving in a circle (which by definition involves constant acceleration perpendicular to the current velocity).
Thus, we can clearly state that the Earth is orbiting, and can't view it as relativisticly stationary.
But, to my shock, I recently came across this text http://books.google.com/books?id=lWEmNBaHCJMC&pg=PA211&dq=einstein+infeld+physics+ptolemy+copernicus&hl=en&ei=dWZ_TubbKqn20gH8hNjSDw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q&f=false (which has Einstein as a coauthor) which, on page 212, seems to say that although special relativity requires an inertial coordinate system, general relativity does not! And that therefore we can state that the Earth is stationary and the Sun orbits it! I would reject this as pseudoscientific bunk, if not for the authors of the book.