Does relativity apply also to rotating reference frames?
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?
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.
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.
A rotating reference frame is different. A local experiment can tell you whether you're rotating and how fast you're rotating.