Not in any normal way; the geocentricists tried to build mechanical models but the increasing number of epicycles was a big clue that the system was in fact heliocentric.
I well understand the history of this topic, after all, I have two years of Caltech physics :-) Nevertheless, as Einstein demonstrated, things look very different depending on the frame of reference.
There's no such thing as a "normal" frame of reference.
> the geocentricists tried to build mechanical models but the increasing number of epicycles was a big clue that the system was in fact heliocentric.
Not exactly, it was a big clue that the planets did not move in perfect circles. The mechanical models did not provide any evidence of heliocentrism. It was Galileo's observations of Venus that torpedoed the geocentric theory.
Galileo didn't really do a good job of proving the heliocentric view though. His advocacy of it was based more on aesthetics than observations. There was a great look at this in an article called "The Great Ptolemaic Smackdown and Down 'n Dirty Mud-Wrassle" in the Jan/Feb 2013 issue of Analog. The author has an expanded version on his blog here [1].
NIST Technical Note 1385 "GPS Receivers and Relativity" by Ashby and Weiss discusses how to solve the GPS positioning equations in a relativistically correct way. It turns out that since the frame of reference (Earth-Centered/Earth-Fixed) is rotating, it is non-inertial and you have to apply some corrections to do the job right.
I mean the principle of relativity (not the theory of relativity), i.e. the fact that the laws of physics are exactly the same in any (inertial) frame of reference and you cannot tell at which speed you're moving except relative to something.
A rotating reference frame is different. A local experiment can tell you whether you're rotating and how fast you're rotating.
Is which object orbits which actually relative to a reference frame though? Does not seem like it should be.
The Sun has a certain mass, the Earth has a certain mass, the center of the point of orbit of both (barycenter? That seems to be the right term) is inside of the Sun no matter what frame you hang out in. What am I missing?
You can set up whatever frame of reference works best for a situation. We do it all the time. It's very convenient for us earthers to use a geocentric framing of the universe for our daily life, where the skies revolve around the Earth. We do it every day.
Such as the word "sunrise" is very geocentric. We don't even have a word for the heliocentric term for the same thing. We also use geocentric phrases like "jets chasing the Sun" and "sundials track the movement of the Sun", etc.
You can use whatever frame of reference you want, sure. Does there exist one in which the Sun orbits the Earth though? I don't think so. Those examples sound either metaphorical or about angular position/velocity, not orbit.
> Does there exist one in which the Sun orbits the Earth though?
Yes. It's what I've been talking about in this thread. You could always, of course, demand that the local TV station change the word "sunrise" to "when the Earth rotates until the Sun shines on the TV station", but I suspect that won't be successful.
