Attempt to answer some questions...
wilgory said:
I understand that motion is relative
That is a caricature of what relativity theory says, one which unfortunately can be particularly misleading in the context of cosmology.
It would be better to say that relativistic kinematics (str) and Galilean kinematics agree that
inertial motion is relative.
wilgory said:
Is there such a thing as universal motion? In other words, is the space-time continuum rotating or moving in any way as a whole entity?
Well, there are a couple of phenomena you might be asking about.
First, roughly speaking, our solar system is in a sense moving linearly with respect to the CMB, and this can be detected and has been. For those who are familiar with the FRW models, which are some of the simplest models employed in relativistic cosmology, as an exercise you can write down a frame field which is moving linearly wrt a frame comoving with the dust particles; the latter in the FRW idealization in some sense provides a kind of notion of absolute rest wrt the CMB.
Second, there are many different things one might mean by "rotating matter" in relativistic cosmology, and unfortunately it is not easy to briefly explain the distinctions. One possibility: the FRW models are homogeneous and isotropic (that's why they are so simple), but one can consider more elaborate models which are neither homogeneous nor isotropic. In particular, one can consider models in which the matter in the universe is a sense rotating about a preferred axis (which you can think of as a definite place in the universe).
wilgory said:
Can this be determined using the cosmic background radiation?
Such models can in fact be obtained as perturbations of the FRW models, which are good approximations to the universe we observe, so these models can be plausible candidates to try to match to observations. When that is done, one can come up with limits to the rate at which the universe could be "rotating" in the sense intended. See Wheeler and Cuifolini,
Gravitation and Inertia.
wilgory said:
Would this motion have an effect on space-time?
According to gtr (and many closely related theories), rotating matter (in most of the various senses I mentioned) produces characteristic gravitational effects.
wilgory said:
Well, frame-dragging, for example. One can write down models in which observers would in principle find that their local inertial gyrostabilized frame is slowly spinning with respect to the distant stars. The buzzword you are probably looking for is "Mach principles" (note that many Mach principles have been suggested). See
http://www.arxiv.org/abs/gr-qc/9607009