"Rotating universes"
Hi, Always curious,
Just thought I'd recommend some more reading:
You can find a new arXiv eprint with fabulous illustrations of closed timelike curves CTCs in the Goedel lambdadust solution as
http://www.arxiv.org/abs/gr-qc/0611093. The classic description is in Hawking and Ellis, The Large Scale Structure of Space-Time, and if you search the arXiv you should find a dozen or so eprints discussing various aspects of this fascinating solution.
I also recommend Cuifolini and Wheeler, Gravitation and Inertia, for more about Mach's principle, frame dragging, gravitomagnetism, and rotating cosmological models, even though I don't feel this book comes up the high standards of exposition set in the classic textbook by Misner, Thorne, and Wheeler, Gravitation, Freeman, 1973.
JesseM said:
A rotating universe in the context of general relativity isn't rotating in quite the sense we usually think of it--there's no center of rotation
That's not true in general; the so-called LRS (locally rotationally symmetric models) and the well-known Van Stockum dust are examples of cosmological models in which the matter (source of the gravitational field) is rotating about an axis with a definite location "in space". However, the Goedel lambdadust is homogeneous (not isotropic), so it does have the property you mentioned.
The issue of how to define "rotation" in curved spacetimes gets quite tricky and much ink has been spilled--- unfortunately, some contributions seem to involve "independently committing" the same errors which have been made (and corrected) in the past, so discussion can easily become contentious.
As always, the local versus global distinction is critical. There are distinct notions of "rotation" which are local in the sense of "jet space", principally vorticity (MTW, Hawking and Ellis, the book by Eric Poisson, A Relativist's Toolkit, are all good sources for the kinematical quantities known as the vorticity tensor, expansion tensor, and acceleration vector), and neccessarily more sophisticated notions which are global.
One thing to watch out for (for those who know what these buzzwords mean): in many "rotating" dust models, the "obvious" coframe read off the metric is often already inertial and even comoving with the dust particles, but the frame may be spinning. With luck, as in the Goedel lambdadust, you can "despin" the frame by applying, as you run along the world line of each dust particle, just the right rotation as a function of proper time about one of the spatial frame vectors. Here, note that in curved spacetimes, nonspinning inertial frames correspond to Lorentz frames in flat spacetime. Spinning but inertial frames correspond to, well, you probably get the idea.
Similar remarks hold for circularly polarized plane waves. Speaking of which, the Osvath-Shuecking circularly polarized gravitational plane wave solution is often touted as a "rotating" cosmological solution (although it is an exact vacuum solution, indeed a Petrov type N vacuum, not a fluid solution, so no matter is anywhere in sight!).
Chris Hillman
