If the Earth stood still and the rest of the universe rotated around it instead, would its equator still bulge? According to general relativity and Gravity Probe B, the answer is YES. It doesn’t matter if you are spinning or if the universe is revolving around you. Both situations are equivalent.
Hmmmmmmmm. That's reference to Mach's principle.
I'm no general relativity specialist and whatever I learn about general relativity is in form of textbooks with examples and stuff to solve, so I haven't ever came across the statement that rotation was 'relative' to something in GR before. Sure, people tend to think rotation is relative (not noticing that Earth is spinning), but rotation is demonstrably not relative in same sense in which motion is relative; you can sit in locked room without windows yet still deduce rotation of Earth 'relatively to far away stars' - even though you can't deduce motion of Earth relatively to far away stars, or tell apart uniform downward gravity from acceleration (which you could measure relatively to stars, if you wish). You can go beyond acceleration & rotation and consider Jerk (third derivative), then Jounce, those you can also feel in the locked room without looking at far away stars. Is there some dragging effect as well for higher derivatives, so that analogous statement like 'it does not matter if entire universe's being shaked or subway train you're riding hits a bump' would be true? [I really don't think so]. Does anyone know more about that?
Also, in special relativity (which I do know rather well, unlike general relativity), infinite or just very big universe *cannot* simply rotate at any rate of revolution, chiefly because far away parts would have to move at faster than speed of light to appear to be rotating around you. So, under special relativity it is no wonder what so ever that lack of rotation relatively to far away stars coincides with lack of rotation as determined by gyroscope. Special relativity is quite fine with Mach's principle.